adaptive bayesian wavelet shrinkage abstract when fitting wavelet based models shrinkage of the empirical wavelet coefficients is an effective tool for denoising the data this article outlines a bayesian approach to shrinkage obtained by placing priors on the wavelet coefficients the prior for each coefficient consists of a mixture of two normal distributions with different standard deviations the simple and intuitive form of prior allows us to propose automatic choices of prior parameters these parameters are chosen adaptively according to the resolution level of the coefficients typically shrinking high resolution frequency coefficients more heavily assuming a good estimate of the background noise level we obtain closed form expressions for the posterior means and variances of the unknown wavelet coefficients the latter may be used to assess uncertainty in the reconstruction several examples are used to illustrate the method and comparisons are made with other shrinkage methods

the drinfeld realization of the elliptic quantum group bq λ a2 2 we construct a realization of the l operator satisfying the rll relation of the face type elliptic quantum group bq a2 2 the construction is based on the elliptic analog of the drinfeld currents of uq a2 2 which forms the elliptic algebra uq p a2 2 we give a realization of the elliptic currents e z f z and k z as a tensor product of the drinfeld currents of uq a2 2 and a heisenberg algebra in the level one representation we also give a free field realization of the elliptic currents applying these results we derive a free field realization of the uq p a2 2 analog of the bq a2 2 intertwining operators the resultant operators coincide with those of the vertex operators in the dilute al model which is known to be a rsos restriction of the a2 2 face model

a q deformation of wakimoto modules primary fields and screening operators theq vertex operators of frenkel and reshetikhin are studied by means of aq deformation of the wakimoto module for the quantum affine algebrau q widehat mathfrak s mathfrak l 2 at an arbitrary levelk 0 2 a fock module version of theq deformed primary field of spinj is introduced as well as the screening operators which anti commute with the action ofu q widehat mathfrak s mathfrak l 2 up to a total difference of a field a proof of the intertwining property is given for theq vertex operators corresponding to the primary fields of spinj 1 2z 0 a sample calculation of the correlation function is also given

fast division using accurate quotient approximations to reduce the number of iterations a class of iterative integer division algorithms is presented based on look up table and taylor series approximations to the reciprocal the algorithm iterates by using the reciprocal to find an approximate quotient and then subtracting the quotient multiplied by the divisor from the dividend to find a remaining dividend fast implementations can produce an average of either 14 or 27 b per iteration depending on whether the basic or advanced version of this method is implemented detailed analyses are presented to support the claimed accuracy per iteration speed estimates using state of the art ecl components show that this method is faster than the newton raphson technique and can produce 53 b quotients of 53 b numbers in about 25 ns using the basic method and 21 ns using the advanced method in addition these methods naturally produce an exact remainder which is very useful for implementing precise rounding specifications

an r matrix for d 3 4 this letter gives the calculation of a trigonometric r matrix for the twisted affine quantum group of type d 3 4

a new sure approach to image denoising interscale orthonormal wavelet thresholding this paper introduces a new approach to orthonormal wavelet image denoising instead of postulating a statistical model for the wavelet coefficients we directly parametrize the denoising process as a sum of elementary nonlinear processes with unknown weights we then minimize an estimate of the mean square error between the clean image and the denoised one the key point is that we have at our disposal a very accurate statistically unbiased mse estimate stein s unbiased risk estimate that depends on the noisy image alone not on the clean one like the mse this estimate is quadratic in the unknown weights and its minimization amounts to solving a linear system of equations the existence of this a priori estimate makes it unnecessary to devise a specific statistical model for the wavelet coefficients instead and contrary to the custom in the literature these coefficients are not considered random any more we describe an interscale orthonormal wavelet thresholding algorithm based on this new approach and show its near optimal performance both regarding quality and cpu requirement by comparing it with the results of three state of the art nonredundant denoising algorithms on a large set of test images an interesting fallout of this study is the development of a new group delay based parent child prediction in a wavelet dyadic tree

empirical bayes selection of wavelet thresholds this paper explores a class of empirical bayes methods for level dependent threshold selection in wavelet shrinkage the prior considered for each wavelet coefficient is a mixture of an atom of probability at zero and a heavy tailed density the mixing weight or sparsity parameter for each level of the transform is chosen by marginal maximum likelihood if estimation is carried out using the posterior median this is a random thresholding procedure the estimation can also be carried out using other thresholding rules with the same threshold details of the calculations needed for implementing the procedure are included in practice the estimates are quick to compute and there is software available simulations on the standard model functions show excellent performance and applications to data drawn from various fields of application are used to explore the practical performance of the approach by using a general result on the risk of the corresponding marginal maximum likelihood approach for a single sequence overall bounds on the risk of the method are found subject to membership of the unknown function in one of a wide range of besov classes covering also the case of f of bounded variation the rates obtained are optimal for any value of the parameter p in 0 simultaneously for a wide range of loss functions each dominating the lq norm of the th derivative with 0 and 0 q 2 attention is paid to the distinction between sampling the unknown function within white noise and sampling at discrete points and between placing constraints on the function itself and on the discrete wavelet transform of its sequence of values at the observation points results for all relevant combinations of these scenarios are obtained in some cases a key feature of the theory is a particular boundary corrected wavelet basis details of which are discussed overall the approach described seems so far unique in combining the properties of fast computation good theoretical properties and good performance in simulations and in practice a key feature appears to be that the estimate of sparsity adapts to three different zones of estimation first where the signal is not sparse enough for thresholding to be of benefit second where an appropriately chosen threshold results in substantially improved estimation and third where the signal is so sparse that the zero estimate gives the optimum accuracy rate

empirical bayes approach to improve wavelet thresholding for image noise reduction wavelet threshold algorithms replace small magnitude wavelet coefficients with zero and keep or shrink the other coefficients this is basically a local procedure because wavelet coefficients characterize the local regularity of a function although a wavelet transform has decorrelating properties structures in images like edges are never decorrelated completely and these structures appear in the wavelet coefficients a classification based on a local criterion like coefficient magnitude is not the perfect method to distinguish important uncorrupted coefficients from coefficients dominated by noise we therefore introduce a geometrical prior model for configurations of important wavelet coefficients and combine this with local characterization of a classical threshold procedure into a bayesian framework the local characterization is incorporated into the conditional model whereas the prior model describes only configurations not coefficient values more precisely local characterization favors co

vertex operators of quantum affine lie algebras u q d n 1 we give an explicit formula for the vertex operators related to the level 1 representations of the quantum affine lie algebrasuq dn 1 in terms of bosons as an application we derive an integral formula for the correlation functions of the vertex models withuq dn 1 symmetry

vertex operators for twisted quantum affine algebras we construct explicitly the q vertex operators intertwining operators for the level one modules v i of the classical quantum affine algebras of twisted types using interacting bosons where i 0 1 for a 2 n 1 i 0 for d 3 i 0 n for d 2 1 and i n for a 2 n a perfect crystal graph for d 3 4 is constructed as a by product

on the fly rounding computing arithmetic in implementations of operations based on digit recurrence algorithms such as division left to right multiplication and square root the result is obtained in digit serial form from most significant digit to least significant to reduce the complexity of the result digit selection and allow the use of redundant addition the result digit has values from a signed digit set as a consequence the result has to be converted to conventional representation which can be done on the fly as the digits are produced without the use of a carry propagate adder the authors describe three ways to modify this conversion process so that the result is rounded the resulting operation is fast because no carry propagate addition is needed the schemes described apply also to online arithmetic operations

staggered polarization of vertex models with u_q widehat sl n symmetry in this paper we give an explicit formula for level 1 vertex operators related to u q widehat sl n as operators on the fock spaces we derive also their commutation relations as an application we calculate with the vector representation of u q widehat sl n thereby extending the recent work on the staggered polarization of the xxz model

commutation relations of vertex operators for uq sl m n we consider commutation relations and invertibility relations of vertex operators for the quantum affine superalgebra uq sl m n by using bosonization we show that vertex operators give a representation of the graded zamolodchikov faddeev algebra by direct computation invertibility relations of type ii vertex operators for n m are very similar to those of type i for m n we consider commutation relations and invertibility relations of vertex operators for the quantum affine superalgebra uq sl m n by using bosonization we show that vertex operators give a representation of the graded zamolodchikov faddeev algebra by direct computation invertibility relations of type ii vertex operators for n m are very similar to those of type i for m n

improved decimal floating point logarithmic converter based on selection by rounding this paper presents the algorithm and architecture of the decimal floating point dfp logarithmic converter based on the digit recurrence algorithm with selection by rounding the proposed approach can compute faithful dfp logarithm results for any one of the three dfp formats specified in the ieee 754 2008 standard in order to optimize the latency for the proposed design we mainly integrate the following novel features 1 using the redundant carry save representation of the data path 2 reducing the number of iterations by determining the number of initial iteration and 3 retiming and balancing the delay of the proposed architecture the proposed architecture is synthesized with stm 90 nm standard cell library and the results show that the critical path delay and the number of clock cycles of the proposed decimal64 logarithmic converter are 1 55 ns 34 4 fo4 and 19 respectively and the total hardware complexity is 43 572 nand2 gates the delay estimation results of the proposed architecture show that its latency is close to that of the binary radix 16 logarithmic converter and that it has a significant decrease on latency compared with a recently published high performance cordic implementation

generalized incomplete gamma functions with applications abstract in this paper we introduce new functions as generalizations of the incomplete gamma functions the functions are found to be useful in heat conduction probability theory and in the study of fourier and laplace transforms some important properties of the functions are studied we have investigated the asymptotic behavior laplace transforms special cases decomposition formula integral representations convolutions recurrence relations and differentiation formula of these functions applications of these functions in evaluation of certain inverse laplace transforms to the definite integrals and to the infinite series of exponential functions are shown

very high radix division with prescaling and selection by rounding a division algorithm in which the quotient digit selection is performed by rounding the shifted residual in carry save form is presented to allow the use of this simple function the divisor and dividend is prescaled to a range close to one the implementation presented results in a fast iteration because of the use of carry save forms and suitable recodings the execution time is calculated and several convenient values of the radix are selected comparison with other dividers for radices 2 sup 9 to 2 sup 18 is performed using the same assumptions

multiplicative division employing independent factors we introduce a division procedure where a normalized quotient approximation is derived as a product of four independent closed form factors specific derivations of the method for ieee 754 standard binary single and double precision division are developed in detail theoretical error bounds are derived to prove correctness for obtaining properly rounded single precision quotients by rounding the product of these four factors the proofs include exhaustive verification for the single precision extremal cases for double precision quotient approximations the error bounds verify suitable accuracy so that further determination of the remainder sign is sufficient to obtain precisely rounded double precision quotients selected experimental results are provided to illustrate the double precision level quotient approximations a prescaling process is introduced to reduce the table size requirements for these double precision approximations to less than 5 kbytes

very high radix circular cordic vectoring and unified rotation vectoring a very high radix algorithm and implementation for circular cordic is presented we first present in depth the algorithm for the vectoring mode in which the selection of the digits is performed by rounding of the control variable to assure convergence with this kind of selection the operands are prescaled however in the cordic algorithm the coordinate x varies during the execution so several scalings might be needed we show that two scalings are sufficient moreover the compensation of the variable scale factor including the cordic scale factor and the prescaling factors is done by computing the logarithm of the scale factor and performing the compensation by an exponential then we combine in a unified unit the proposed vectoring algorithm and the very high radix rotation algorithm which was previously proposed by the authors we compare with low radix implementations in terms of latency and hardware complexity estimations of the delay for 32 bit precision show a speedup of about two with respect to the radix 4 case with redundant addition this speedup is obtained at the cost of an increase in the hardware complexity which is moderate for the pipelined implementation we also compare at the algorithmic level with other very high radix proposals demonstrating the advantages of our algorithms

a low latency pipelined 2d and 3d cordic processors the unfolded and pipelined cordic is a high performance hardware element that produces a wide variety of one and two argument functions with high throughput the reduction in delay power and area cost are of significant interest regarding this module due to its high demand for resources the linear approximation to rotation has been proposed to achieve such reductions however the schemes for rotation multiplication and vectoring division complicate the implementation in a single unit in this work we improve the linear approximation scheme leading to a unified implementation for rotation and vectoring where fully parallel tree multipliers are used instead of the second half of cordic iterations we also combine the linear approximation to rotation with the scale factor compensation so that the compensation is concurrently performed with the rotation process we then extend the method to 3d cordic such an extension is not straightforward due to the lack of existing analytical expressions for the convergence of the algorithm a comparison using a rough area time model and synthesis results shows that our proposals may achieve significant reductions in delay with no increase in area in actual implementations

new svoboda tung division the paper presents a general theory for developing new svoboda tung or simply nst division algorithms not suffering the drawbacks of the classical svoboda tung or simply st method nst avoids the drawbacks of st by proper recoding of the two most significant digits of the residual before selecting the most significant digit of this recoded residual as the quotient digit nst relies on the divisor being in the range 1 1 spl delta where spl delta is a positive fraction depending upon 1 the radix 2 the signed digit set used to represent the residual and 3 the recoding conditions of the two most significant digits of the residual if the operands belong to the ieee std range 1 2 they have to be conveniently prescaled in that case nst produces the correct quotient but the final residual is scaled by the same factor as the operands therefore nst is not useful in applications where the unsealed residual is necessary an analysis of nst shows that previously published algorithms can be derived from the general theory proposed in the paper moreover nst reveals a spectrum of new possibilities for the design of alternative division units for a given radix b the number of different algorithms of this kind is b sup 2 4

higher level representations of the quantum affine algebrauq ŝl 2 abstract we introduce the notion of quantum z algebras for the quantum affine algebra after developing the quantum z algebra we use it to study the quantum affine algebrauq sl 2 and construct bases for each of its higher level modules

two parameter twisted quantum affine algebras we establish drinfeld realization for the two parameter twisted quantum affine algebras using a new method the hopf algebra structure for drinfeld generators is given for both untwisted and twisted two parameter quantum affine algebras which include the quantum affine algebras as special cases

twisted quantum affine algebras we give a highest weight classification of the finite dimensional irreducible representations of twisted quantum affine algebras as in the untwisted case such representations are in one to one correspondence with n tuples of monic polynomials in one variable but whereas in the untwisted case n is the rank of the underlying finite dimensional complex simple lie algebra in the twisted case n is the rank of the subalgebra of fixed by the diagram automorphism the way in which such an n tuple determines a representation is also more complicated than in the untwisted case

the estimation of laplace random vectors in additive white gaussian noise this paper develops and compares the maximum a posteriori map and minimum mean square error mmse estimators for spherically contoured multivariate laplace random vectors in additive white gaussian noise the mmse estimator is expressed in closed form using the generalized incomplete gamma function we also find a computationally efficient yet accurate approximation for the mmse estimator in addition this paper develops an expression for the mse for any estimator of spherically contoured multivariate laplace random vectors in additive white gaussian noise awgn the development of which again depends on the generalized incomplete gamma function the estimators are motivated and tested on the problem of wavelet based image denoising

on the approximation of a generalized incomplete gamma function arising in heat conduction problems abstract several asymptotic expansions are given for the generalized incomplete gamma function x b x t 1 e t b t dt x 0 re b 0 b 0 as x tends to infinity where the real parameter is allowed to take negative values in the case b i 0 we obtain the expansions of the decomposition functions c x and s x closed form expressions are also given for 3 2 5 2

vertex operators of admissible modules ofuq c 1 n using our recent bosonic realization ofuq sp2n we construct explicitly the vertex operators for the level 1 2 modules ofuq sp2n using bosonic fields our method contains a detailed analysis of all theq intertwining relations

digit recurrence dividers with reduced logical depth in this paper we propose a class of division algorithms with the aim of reducing the delay of the selection of the quotient digit by introducing more concurrency and flexibility in its computation from the proposed class of algorithms we select one that moves part of the selection function out of the critical path with a corresponding reduction in the critical path compared with existing alternatives we present the algorithm and describe the architectures for radix 4 and for radix 16 for radix 16 we use the scheme of overlapping two radix 4 stages in both cases radix 4 and radix 16 we show that our algorithms allow the design of units with well balanced critical paths with consequent decreases of the cycle times moreover in the radix 16 case we include some additional speculation techniques to estimate the speedup we used a rough timing model based on logical effort for both radices we estimate a speedup of about 25 percent with respect to previous implementations in the radix 4 case this is achieved by using roughly the same area while in the radix 16 case the area is increased by about 30 percent we verified our estimations by performing a synthesis of the radix 4 units

free field approach to the dilute al models we construct a free field realization of vertex operators of the dilute al models along with the felder complex for l 3 we also study an e8 structure in terms of the deformed virasoro currents

simple radix 4 division with operands scaling a radix 4 division algorithm with operands scaling is proposed the algorithm uses a recurrence with redundant addition carry save or signed digit and combines simple scaling with a quotient selection function that depends only on the estimate of the partial remainder and is independent of the divisor the scheme results in a significant speedup with respect to both the radix 2 and radix 4 without scaling

very high radix square root with prescaling and rounding and a combined division square root unit an algorithm for square root with prescaling and selection by rounding is developed and combined with a similar scheme for division since division is usually more frequent than square root the main concern of the combined implementation is to maintain the low execution time of division while accepting a somewhat larger execution time for square root the algorithm is presented in detail including the mathematical development of bounds for the first square root digit and for the scaling factor the proposed implementation is described evaluated and compared with other combined div sqrt units the comparisons show that the proposed scheme potentially produces a significant speed up for division whereas for square root the speed up is small

uq sl 2 1 vertex operators screen currents and correlation functions at an arbitrary level bosonized q vertex operators related to the four dimensional evaluation modules of the quantum affine superalgebra uq sl 2 1 are constructed for arbitrary level k where 0 1 is a complex parameter appearing in the four dimensional evaluation representations they are intertwiners among the level highest weight fock wakimoto modules screen currents which commute with the action of uq sl 2 1 up to total differences are presented integral formulas for n point functions of type i and type ii q vertex operators are proposed

vertex representations of quantum affine algebras we construct vertex representations of quantum affine algebras of ade type which were first introduced in greater generality by drinfeld and jimbo the limiting special case of our construction is the untwisted vertex representation of affine lie algebras of frenkel kac and segal our representation is given by means of a new type of vertex operator corresponding to the simple roots and satisfying the defining relations in the case of the quantum affine algebra of type a we introduce vertex operators corresponding to all the roots and determine their commutation relations this provides an analogue of a chevalley basis of the affine lie algebra unk n in the basic representation

boosting very high radix division with prescaling and selection by rounding an extension of the very high radix division with prescaling and selection by rounding is presented this extension consists of increasing the effective radix of the implementation by obtaining a few additional bits of the quotient per iteration without increasing the complexity of the unit to obtain the prescaling factor or the delay of an iteration as a consequence for some values of the effective radix it permits an implementation with a smaller area and the same execution time of the original scheme details of the algorithm and the implementation are presented estimations of the execution time and area are given for 54 bit and 114 bit quotients and compared with those of other division units

twisted vertex representations of quantum affine algebras recent interests in quantum groups are stimulated by their marvelous relations with quantum yang baxter equations conformal field theory invariants of links and knots and q hypergeometric series besides understanding the reason of the appearance of quantum groups in both mathematics and theoretical physics there is a natural problem of finding q deformations or quantum analogues of known structures quantum groups were first defined by drinfeld 2 and jimbo 9 also see 4 as a q deformation of the universal enveloping algebras of the kacmoody algebras in the work of trigonometric solutions of yang baxter equations in the same spirit it was shown in 13 14 that there exists a 1 1 correspondence between the integrable highest weight representations of symmetrizable kac moody algebras and those of the corresponding quantum groups where both spaces have the same dimension in the case of generic q i e q is not a root of unity moreover one can be very explicit in the case of quantum gl n to write down the irreducible highest weight representations quantum affine algebras are the quantum groups associated to affine lie algebras following drinfeld s realization 3 of q analog of loop algebras the vertex representation of untwisted simply laced quantum affine algebras was constructed in frenkel jing 6 which is a q deformation of frenkel kac 7 and segal 15 construction in the theory of affine lie algebras subsequently the same was done for the quantum affine algebra of type b in 1 in the present work we construct vertex representations of quantum affine algebras twisted by an automorphism of the dynkin diagram which generalizes certain important cases in the ordinary twisted vertex operator calculus 5

digit serial methods with applications to division and square root we present a generic digit serial method dsm to compute the digits of a real number v bounds on these digits and on the errors in the associated estimates of v formed from these digits are derived to illustrate our results we derive such bounds for a parameterized family of high radix algorithms for division and square root these bounds enable a dsm designer to determine for example whether a given choice of parameters allows rapid formation and rounding of its approximation to v