a modified split radix fft with fewer arithmetic operations recent results by van buskirk have broken the record set by yavne in 1968 for the lowest exact count of real additions and multiplications to compute a power of two discrete fourier transform dft here we present a simple recursive modification of the split radix algorithm that computes the dft with asymptotically about 6 fewer operations than yavne matching the count achieved by van buskirk s program generation framework we also discuss the application of our algorithm to real data and real symmetric discrete cosine transforms where we are again able to achieve lower arithmetic counts than previously published algorithms

the decomposition of lie powers let g be a group f a field of prime characteristic p and v a finite dimensional fg module let l v denote the free lie algebra on v regarded as an fg submodule of the free associative algebra or tensor algebra t v for each positive integer r let l r v and t r v be the r th homogeneous components of l v and t v respectively here l r v is called the r th lie power of v our main result is that there are submodules b 1 b 2 of l v such that for all r b r is a direct summand of t r v and whenever m geqslant 0 and k is not divisible by p the module l p mk v is the direct sum of l p m b k l p m 1 b pk l 1 b p mk thus every lie power is a direct sum of lie powers of p power degree the approach builds on an analysis of t r v as a bimodule for g and the solomon descent algebra

on computing the split radix fft this paper presents an efficient fortran program that computes the duhamel hollmann split radix fft an indexing scheme is used that gives a three loop structure for the split radix fft that is very similar to the conventional cooley tukey fft both a decimation in frequency and a decimation in time program are presented an arithmetic analysis is made to compare the operation count of the cooley tukey fft fo several different radixes to that of the split radix fft the split radix fft seems to require the least total arithmetic of any power of two dft algorithm

investigation of continuous time quantum walk via modules of bose mesner and terwilliger algebras the continuous time quantum walk on the underlying graphs of association schemes has been studied via the algebraic combinatorics structures of association schemes namely semi simple modules of their bose mesner and terwilliger algebras it is shown that the terwilliger algebra stratifies the graph into a d 1 disjoint union of strata which is different from the stratification based on distance except for distance regular graphs in underlying graphs of association schemes the probability amplitudes and average probabilities are given in terms of dual eigenvalues of association schemes such that the amplitudes of observing the continuous time quantum walk on all sites belonging to a given stratum are the same therefore there are at most d 1 different observing probabilities the importance of association scheme in continuous time quantum walk is shown by some worked out examples such as arbitrary finite group association schemes followed by symmetric sn dihedral d2m and cyclic groups at the end it is shown that the highest irreducible representations of terwilliger algebras pave the way to use the spectral distributions method of jafarizadeh and salimi 2005 preprint quant ph 0510174 in studying quantum walk on some rather important graphs called distance regular graphs

on schur algebras and related algebras vi some remarks on rational and classical schur algebras abstract in a recent paper dipper and doty 4 introduced certain finite dimensional algebras associated with the natural module of the general linear group and its dual which they call rational schur algebras we give a proof via tilting modules that these algebras are in fact generalised schur algebras using the same technique we show that certain finite dimensional algebras associated with classical groups introduced by doty 20 are quasi hereditary algebras a generalised schur algebras may be viewed as a quotient of the algebra of distributions of a reductive group by a certain ideal we give generators for this ideal

recursive calculation of effective resistances in distance regular networks based on bose mesner algebra and christoffel darboux identity recently jafarizadeh et al j phys a math theor 40 4949 2007 have given a method for calculation of effective resistance two point resistance on distance regular networks where the calculation was based on stratification introduced by jafarizadeh and salimi j phys a 39 1 2006 and stieltjes transform of the spectral distribution stieltjes function associated with the network also jafarizadeh et al j phys a math theor 40 4949 2007 have shown that effective resistances between a node and all nodes belonging to the same stratum with respect to r i belonging to the ith stratum with respect to are the same in this work an algorithm for recursive calculation of the effective resistances in an arbitrary distance regular resistor network is provided where the derivation of the algorithm is based on the bose mesner algebra stratification of the network spectral techniques and christoffel darboux identity it is shown that the effective resistance on a dista

composition factors of tensor products of truncated symmetric powers let g be the general linear group of degree n over an algebraically closed field k of characteristic p 0 we study the m fold tensor product s e m of the truncated symmetric algebra s e of the symmetric algebra s e of the natural module e for g we are particularly interested in the set of partitions occurring as the highest weight of a composition factor of s e m we explain how the determination of these composition factors is related to the determination of the set of composition factors of the m fold tensor product s e m of the symmetric algebra we give a complete description of the composition factors of s e m in terms of distinguished partitions our main interest is in the classical case but since the quantised version is essentially no more difficult we express our results in the general context throughout

scaled radix 2 8 algorithm for efficient computation of length n 2 m dfts this paper presents a scaled radix 2 8 fast fourier transform fft sr28fft algorithm for computing length n 2m discrete fourier transforms dfts scaled by complex number rotating factors the idea of the sr28fft algorithm is from the modified split radix fft msrfft algorithm and its purpose is to furnish other algorithms with high efficiency but without shortcomings of the msrfft algorithm a novel radix 2 4 fft nr24fft algorithm and a novel radix 2 8 fft nr28fft algorithm are proposed these two algorithms use sr28fft to calculate their sub dfts of odd indexed terms several aspects of the two algorithms such as computational complexity computation accuracy and coefficient evaluations or accesses to the lookup table all are improved the bit reverse method can be used for their order permutation and no extra memory is required to store their extra coefficients by the two novel algorithms which contribute significantly to the performance of the fft algorithms the sr28fft algorithm can also be applied to other algorithms whose decomposition contains sub dfts of powers of two the appendix presents an algorithm named sr28fft 2 for further reducing the number of arithmetic operations and nr24fft and nr28fft algorithms based on sr28fft 2 requires fewer real operations than that required by the msrfft algorithm

entanglement entropy in the spinless free fermion model and its application to the graph isomorphism problem in this paper the entanglement entropy is investigated in the ground state of a spinless free fermion hamiltonian where its hopping matrix is given by the adjacency matrix of the graph the bipartite entanglement entropy is calculated from the eigenvalues of the correlation matrix the entanglement entropy and mutual information are calculated for some scalable graphs including a complete graph and three sets of strongly regular graphs srgs in addition the volume law scaling has been shown for these graphs then we use the entanglement entropy as a tool for studying the graph isomorphism problem in some non isomorphic pairs of graphs in order to distinguish them from each other two graphs are isomorphic when they are related to each other by relabeling the graph vertices we study the graph isomorphism in non isomorphic srgs distance regular graphs and pseudo distance regular graphs most pairs of non isomorphic srgs can be distinguished from each other by using fermionic entanglement entropy

some remarks on tilting modules double centralisers and lie modules abstract in a recent paper 14 s doty and j m douglass consider the degree r component l r e of the free lie algebra l e on an n dimensional vector space e over a suitable field k of characteristic not dividing r the authors discuss amongst other things the question of when this space regarded as a module for the general linear group of degree n equivalently the schur algebra s n r has the double centraliser property some general classes are described but the question in general is left open here we show that at most one block component of l r e can fail to have the double centraliser property and give some examples of the failure of the property for that block and therefore for the lie module

an accurate error analysis model for fast fourier transform an error propagation model is proposed for the in place decimation in time version of the radix 2 fft algorithm with the model an accurate error expression and error variance for the computation of fft are derived this correspondence deals with fixed point and block floating point arithmetic simulation results agree closely with the theoretical predicted ones we find that some roundoff errors at different stages correlate with each other the density of correlations is closely associated with the round off approach used in butterfly calculations

calculation of reduced density matrices from correlation functions it is shown that for solvable fermionic and bosonic lattice systems the reduced density matrices can be determined from the properties of the correlation functions this provides the simplest way to these quantities which are used in the density matrix renormalization group method

quantum spin chain toeplitz determinants and the fisher hartwig conjecture we consider the one dimensional quantum spin chain which is called the xx model xx0 model or isotropic xy model in a transverse magnetic field we are mainly interested in the entropy of a block of lneighboring spins at zero temperature and of an infinite system we represent the entropy in terms of a toeplitz determinant and calculate the asymptotic analytically we derive the first two terms of the asymptotic decomposition interestingly these two terms of decomposition clearly show a length scale related to the field h

double centralizer properties dominant dimension and tilting modules double centralizer properties play a central role in many parts of algebraic lie theory soergel s double centralizer theorem relates the principal block of the bernstein gelfand gelfand category o of a semisimple complex lie algebra with the coinvariant algebra i e the cohomology algebra of the corresponding flag manifold schur weyl duality relates the representation theories of general linear and symmetric groups in defining characteristic or via the quantized version in nondefining characteristic in this paper we exhibit algebraic structures behind these double centralizer properties we show that the finite dimensional algebras relevant in this context have dominant dimension at least two with respect to some projective injective or tilting modules general arguments which combine methods from ring theory qf 3 rings and dominant dimension with tools from representation theory approximations tilting modules then imply the validity of these double centralizer properties as well as new ones in contrast to the traditional proofs e g by the fundamental theorems of invariant theory no computations are necessary

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 fixed point accuracy analysis of fft algorithms in this paper we investigate the effect of fixed point arithmetics with limited precision for different fast fourier transform fft algorithms a matrix representation of error propagation model is proposed to analyze the rounding effect an analytic expression of overall quantization loss due to the arithmetic quantization errors is derived to compare the performance with decimation in time dit and decimation in frequency dif configurations from the simulation results the radix 2 dit fft algorithm has better accuracy in term of signal to quantization noise ratio sqnr based on the results a simple criterion of wordlength optimization is proposed to yield comparable accuracy with fewer bit budget

accumulation of coefficient roundoff error in fast fourier transforms implemented with logarithmic number system when a fast fourier transform fft is implemented on a digital computer or with special purpose hardware quantization errors will arise due to finite word lengths in the digital system this correspondence presents an analysis of error accumulation due to coefficient rounding in the fft implemented with a logarithmic number system the theoretical result of the coefficient roundoff error analysis is verified by computer simulations the results indicate that the fft implemented with the logarithmic number system provides better signal to noise performance than that implemented with a fixed point or floating point number system

a note on an accurate error analysis model for fast fourier transform the expressions derived by ma see ibid vol 45 p 1641 1997 for error variance and average power expectation using the model that has been proposed are not accurate in this paper all the possible correlations have not been considered the expressions have been modified to make them exact closed form expressions for the correlation between errors of different stages have been derived the expressions have been verified by comparing the predicted values using these expressions with the simulation values given by ma

brauer algebras symplectic schur algebras and schur weyl duality in this paper we prove the schur weyl duality between the symplectic group and the brauer algebra over an arbitrary infinite field k we show that the natural homomorphism from the brauer algebra b n 2m to the endomorphism algebra of the tensor space k 2m n as a module over the symplectic similitude group gsp 2m k or equivalently as a module over the symplectic group sp 2m k is always surjective another surjectivity that of the natural homomorphism from the group algebra for gsp 2m k to the endomorphism algebra of k 2m n as a module over b n 2m is derived as an easy consequence of s oehms s results s oehms j algebra 1 244 2001 19 44

on the representations of the full matrix semigroup on homogeneous polynomials this paper is the successor of 11 in the closing theorem there we determined the composition factors of w for a regular i 1 6 11 here we are going to describe the structure of wa in the singular case the second theme of this paper is the criterion for distributivity of the m modules wcd since the necessary conditions for that established in 1 8 51 limit d to d p p 1 we just have to sort out the latter case this will be done in the third section of this paper for the convenience of the reader we recall some notation terms and definitions used in 11

optimal transfer of a d level quantum state over pseudo distance regular networks in the previous work jafarizadeh and sufiani 2008 phys rev a 77 022315 by using some techniques such as stratification and spectral distribution associated with the graphs perfect state transfer pst of a qubit spin 1 2 particle over distance regular spin networks was discussed in this paper optimal transfer of an arbitrary d level quantum state qudit over antipodes of more general networks called pseudo distance regular networks is investigated in other words by using the same spectral analysis techniques and algebraic structures of pseudo distance regular graphs we give an explicit analytical formula for suitable coupling constants in the specific hamiltonians so that the state of a particular qudit initially encoded on one site will optimally evolve into the opposite site without any dynamical control i e we show how to analytically derive the parameters of the system so that optimal state transfer can be achieved also for the specific form of hamiltonians that we consider necessary conditions in order for pst to be achieved are given finally for these hamiltonians pst and optimal imperfect st over some important examples of pseudo distance regular networks are discussed

fixed point analysis and parameter selections of msr cordic with applications to fft designs mixed scaling rotation msr coordinate rotation digital computer cordic is an attractive approach to synthesizing complex rotators this paper presents the fixed point error analysis and parameter selections of msr cordic with applications to the fast fourier transform fft first the fixed point mean squared error of the msr cordic is analyzed by considering both the angle approximation error and signal round off error incurred in the finite precision arithmetic the signal to quantization noise ratio sqnr of the output of the fft synthesized using msr cordic is thereafter estimated based on these analyses two different parameter selection algorithms of msr cordic are proposed for general and dedicated msr cordic structures the proposed algorithms minimize the number of adders and word length when the sqnr of the fft output is constrained design examples show that the fft designed by the proposed method exhibits a lower hardware complexity than existing methods