a polynomial time approximation algorithm for the permanent of a matrix with nonnegative entries we present a polynomial time randomized algorithm for estimating the permanent of an arbitrary n n matrix with nonnegative entries this algorithm technically a fully polynomial randomized approximation scheme computes an approximation that is with high probability within arbitrarily small specified relative error of the true value of the permanent

proof of wahl s conjecture on surjectivity of the gaussian map for flag varieties introduction let x be a smooth projective variety over c with ample line bundles t and 2 on x we can form their external tensor product to get a line bundle t fxi 2 on x x x let 5d be the ideal sheaf of the diagonal d c x x x wahl defines a map which he calls as the gaussian map i21 2 h x x x d 0 1 2 h x lc 4 0 l c 0 2 which is induced from the canonical projection id d d by identifying the cxxxkkd y od module d k supported in d with the sheaf of 1 forms ql on d x wahl has extensively studied this map in w1 w2 w3 and w4 and he made the following conjecture in w41

borel de siebenthal pairs global weyl modules and stanley reisner rings we develop the theory of integrable representations for an arbitrary standard maximal parabolic subalgebra of an affine lie algebra we see that such subalgebras can be thought of as arising in a natural way from a borel de siebenthal pair of semisimple lie algebras we see that although there are similarities with the representation theory of the standard maximal parabolic subalgebra there are also very interesting and non trivial differences including the fact that there are examples of non trivial global weyl modules which are irreducible and finite dimensional we also give a presentation of the endomorphism ring of the global weyl module although these are no longer polynomial algebras we see that for certain parabolics these algebras are stanley reisner rings which are both koszul and cohen macaulay

normality and smoothness of simple linear group compactifications given a semisimple algebraic group g we characterize the normality and the smoothness of its simple linear compactifications namely those equivariant g times g compactifications possessing a unique closed orbit which arise in a projective space of the shape mathbb p mathrm end v where v is a finite dimensional rational g module both the characterizations are purely combinatorial and are expressed in terms of the highest weights of v in particular we show that mathrm sp 2r with r geqslant 1 is the unique non adjoint simple group which admits a simple smooth compactification

multidimensional permanents of polystochastic matrices abstract a d dimensional matrix is called 1 polystochastic if it is non negative and the sum over each line equals 1 such a matrix that has a single 1 in each line and zeros elsewhere is called a 1 permutation matrix a diagonal of a d dimensional matrix of order n is a choice of n elements no two in the same hyperplane the permanent of a d dimensional matrix is the sum over the diagonals of the product of the elements within the diagonal for a given order n and dimension d the set of 1 polystochastic matrices forms a convex polytope that includes the 1 permutation matrices within its set of vertices for even n and odd d we give a construction for a class of 1 permutation matrices with zero permanent consequently we show that the set of 1 polystochastic matrices with zero permanent contains at least n n 3 2 1 2 o 1 1 permutation matrices and contains a polytope of dimension at least c n 3 2 for fixed c d and even n we also provide counterexamples to a conjecture by taranenko 11 about the location of local extrema of the permanent for odd d we give a construction of 1 permutation matrices that decompose into a convex linear sum of positive diagonals these combine with a theorem of taranenko 11 to provide counterexamples to a conjecture by dow and gibson 4 generalising van der waerden s conjecture to higher dimensions

vector adaptive vector quantization with application to speech coding a vector adaptive vector quantization vavq scheme which may be viewed as a generalization of gain adaptive vector quantization is described the proposed scheme adjusts each component of the encoding signal vector according to a statistical estimate of the signal characteristics the vavq scheme can cope with large input dynamic ranges it can be used in either the time domain or the transform domain and exhibits approximately 4 db improvement in segmental signal to noise ratio snr over the fixed vq

the geometric measure of multipartite entanglement and the singular values of a hypermatrix it is shown that the geometric measure of entanglement of a pure multipartite state satisfies a polynomial equation generalizing the singular value equation of the matrix of coefficients of a bipartite state the equation is solved for a class of three qubit states

the geometric measure of entanglement for a symmetric pure state with non negative amplitudes in this paper for a class of symmetric multiparty pure states we consider a conjecture related to the geometric measure of entanglement for a symmetric pure state the closest product state in terms of the fidelity can be chosen as a symmetric product state we show that this conjecture is true for symmetric pure states whose amplitudes are all non negative in a computational basis the more general conjecture is still open

tropical determinant on transportation polytopes abstract let d k l m n be the set of all the integer points in the transportation polytope of k n l n matrices with row sums lm and column sums km in this paper we find the sharp lower bound on the tropical determinant over the set d k l m n this integer piecewise linear programming problem in arbitrary dimension turns out to be equivalent to an integer non linear in fact quadratic optimization problem in dimension two we also compute the sharp upper bound on a modification of the tropical determinant where the maximum over all the transversals in a matrix is replaced with the minimum

the scaling mean and a law of large permanents abstract in this paper we study two types of means of the entries of a nonnegative matrix the permanental mean which is defined using permanents and the scaling mean which is defined in terms of an optimization problem we explore relations between these two means making use of important results by egorychev and falikman the van der waerden conjecture friedland sinkhorn and others we also define a scaling mean for functions in a much more general context our main result is a law of large permanents a pointwise ergodic theorem for permanental means of dynamically defined matrices that expresses the limit as a functional scaling mean the concepts introduced in this paper are general enough so to include as particular cases certain classical types of means as for example symmetric means and muirhead means as a corollary we reobtain a formula of halasz and szekely for the limit of the symmetric means of a stationary random process

frobenius splitting and cohomology vanishing for schubert varieties let k be an algebraically closed field of characteristic p 0 and x be a projective variety over k we then have the absolute frobenius f x x and an injection ox f 6x given by f fp f e o9x if this makes ox a direct summand in f ox as an x module we call x a frobenius split variety for such a variety the vanishing theorem for ample line bundles follows trivially from serre s vanishing theorem for tensoring ox f ox by an ample line bundle l and noting that l f x f f l f lp projection formula we get that the map in cohomology h x l h x f lp h x lp is an injection iterating this we see that h x l injects into h x lp for every p but for large v the latter is zero thus frobenius split varieties have quite pleasant properties it also turns out that using duality for the frobenius morphism or equivalently the cartier operator one can give a manageable criterion for frobenius splitting the point here is the local nature of duality the relevance of the compatibility of local and global duality was suggested to us by grothendieck s proof of h x y 0 for a noncomplete variety x of dimension n 4 theorem 6 9 and by kempf s paper 10 by this criterion and the bott samelson demazure desingularisation of schubert varieties it follows very easily that schubert varieties in characteristic p are frobenius split the following vanishing theorem is then an immediate consequence let g be a reductive group over the field k of arbitrary characteristic zero or positive let q be a parabolic subgroup and x c g q a schubert variety let l be an ample line bundle on g q then hi x l 0 for i 0 and the restriction map h0 g q l h0 x l is surjective if char k 0 this is a consequence of the compatible frobenius splitting of x in g q and the char k 0 case is handled by semicontinuity the above result for special schubert varieties has been proved by seshadrimusili lakshmibai 12 13 and by kempf 9 by using characteristic free methods for x g q andersen 1 and haboush 5 have given simple proofs using characteristic p methods

concentric shell partition vector quantization with application to image coding a method for designing a vector quantizer by first partitioning the residual image vector space into concentric shells and then searching for the smallest possible codebook to represent the residual image vector space while adhering to the visual perceptive qualities such as edges and textures in the image representation is described the coding method involves the decomposition of the image vector into relatively independent components each with a small dynamic range to permit the design of a small fixed codebook for the vector quantizer optimal partitioning amounts to optimally selecting the set of shell radii which is achieved using a modified form of simulated annealing the results obtained in the coding of images show that the concentric shell partition vector quantizer compares favourably with the gain shape vector quantizer in terms of subjective image quality at the same bit rate while achieving higher snr and lower computational complexity

the ring of sections of a complete symmetric variety we study the ring of sections a x of a complete symmetric variety x that is of the wonderful completion of g h where g is an adjoint semisimple group and h is the fixed subgroup for an involutorial automorphism of g we find generators for pic x we generalize the prv conjecture to complete symmetric varieties and construct a standard monomial theory for a x that is compatible with g orbit closures in x this gives a degeneration result and the rational singularityness for a x 2003 elsevier science usa all rights reserved

parameterized complexity of determinant and permanent abstract every square matrix a a u v c n n can be represented as a digraph having n vertices in the digraph a block or 2 connected component is a maximally connected subdigraph that has no cut vertex the determinant and the permanent of a matrix can be calculated in terms of the determinant and the permanent of some specific induced subdigraphs of the blocks in the digraph interestingly these induced subdigraphs are vertex disjoint and they partition the digraph such partitions of the digraph are called the b partitions in this paper first we develop an algorithm to find the b partitions next we analyze the parameterized complexity of matrix determinant and permanent where the parameters are the sizes of blocks and the number of cut vertices of the digraph we give a class of combinations of cut vertices and block sizes for which the parametrized complexities beat the state of art complexities of the determinant and the permanent

a note on affine toda couplings the author proves the clebsch gordan property for the couplings of an untwisted affine toda field theory utilizing dorey s 1991 conjecture

embeddings of semisimple complex lie groups and cohomological components of modules let g g be an embedding of semisimple complex lie groups b b a pair of nested borel subgroups and g b g b the associated embedding of flag manifolds let o be an equivariant invertible sheaf on g b and o o be its restriction to g bg b consider the g equivariant pullback h g b o h g b o the borel weil bott theorem and schur s lemma imply that is either surjective or zero if is nonzero the image of the dual map is a g irreducible component in a g irreducible module called a cohomological component we establish a necessary and sufficient condition for nonvanishing of also we prove a theorem on the structure of the set of pairs of dominant weights with v v cohomological here v v and v denote the respective highest weight modules simplified specializations are formulated for regular and diagonal embeddings in particular we give an alternative proof of a recent theorem of dimitrov and roth beyond the regular and diagonal cases we study equivariantly embedded rational curves and we also show that the generators of the algebra of ad invariant polynomials on a semisimple lie algebra can be obtained as cohomological components our methods rely on kostant s theory of lie algebra cohomology

equidimensionality of convolution morphisms and applications to saturation problems fix a split connected reductive group g over a field k and a positive integer r for any r tuple of dominant coweights i of g we consider the restriction m of the r fold convolution morphism of mirkovic vilonen to the twisted product of affine schubert varieties corresponding to we show that if all the coweights i are minuscule then the fibers of m are equidimensional varieties with dimension the largest allowed by the semi smallness of m we derive various consequences the equivalence of the non vanishing of hecke and representation ring structure constants and a saturation property for these structure constants when the coweights i are sums of minuscule coweights this complements the saturation results of knutson tao and kapovich leeb millson we give a new proof of the p r v conjecture in the sums of minuscules setting finally we generalize and reprove a result of spaltenstein pertaining to equidimensionality of certain partial springer resolutions of the nilpotent cone for gln

bounds on multipartite entangled orthogonal state discrimination using local operations and classical communication we show that entanglement guarantees difficulty in the discrimination of orthogonal multipartite states locally the number of pure states that can be discriminated by local operations and classical communication is bounded by the total dimension over the average entanglement a similar general condition is also shown for pure and mixed states these results offer a rare operational interpretation for three abstractly defined distancelike measures of multipartite entanglement

projective normality of the wonderful compactification of semisimple adjoint groups we prove that if g is a semisimple adjoint group over an algebraically closed field of arbitrary characteristic x is the wonderful compactification of g and widehat g is a simply connected covering of g then for any widehat g times widehat g linearised very ample line bundle l over x the cone over x given by l is normal

on finite state vector quantization for noisy channels finite state vector quantization fsvq over a noisy channel is studied a major drawback of a finite state decoder is its inability to track the encoder in the presence of channel noise in order to overcome this problem we propose a nontracking decoder which directly estimates the code vectors used by a finite state encoder the design of channel matched finite state vector quantizers for noisy channels using an iterative scheme resembling the generalized lloyd algorithm is also investigated simulation results based on encoding a gauss markov source over a memoryless gaussian channel show that the proposed decoder exhibits graceful degradation of performance with increasing channel noise as compared with a finite state decoder also the channel matched finite state vector quantizers are shown to outperform channel optimized vector quantizers having the same vector dimension and rate however the nontracking decoder used in the channel matched finite state quantizer has a higher computational complexity compared with a channel optimized vector quantizer decoder thus if they are allowed to have the same overall complexity encoding and decoding the channel optimized vector quantizer can use a longer encoding delay and achieve similar or better performance finally an example of using the channel matched finite state quantizer as a backward adaptive quantizer for nonstationary signals is also presented

brunn minkowski inequalities for contingency tables and integer flows we establish approximate log concavity for a wide family of combinatorially defined integer valued functions examples include the number of non negative integer matrices contingency tables with prescribed row and column sums margins as a function of the margins and the number of integer feasible flows in a network as a function of the excesses at the vertices as a corollary we obtain approximate log concavity for the kostant partition function of type a we also present an indirect evidence that at least some of the considered functions might be genuinely log concave

efficient codebooks for vector quantization image compression with an adaptive tree search algorithm this paper discusses some algorithms to be used for the generation of an efficient and robust codebook for vector quantization vq some of the algorithms reduce the required codebook size by 4 or even 8 b to achieve the same level of performance as some of the popular techniques this helps in greatly reducing the complexity of codebook generation and encoding we also present a new adaptive tree search algorithm which improves the performance of any product vq structure our results show an improvement of nearly 3 db over the fixed rate search algorithm at a bit rate of 0 75 b pixel

toward an understanding of entanglement for generalized n qubit w states we solve stationarity equations of the geometric measure of entanglement for multi qubit w type states in this way we compute analytically the maximal overlap of one parameter n qubit and two parameter four qubit w type states and their nearest product states possible extensions to arbitrary w type states and geometrical interpretations of these results are discussed in detail

geometric source coding and vector quantization a geometric formulation is presented for source coding and vector quantizer design motivated by the asymptotic equipartition principle the authors consider two broad classes of source codes and vector quantizers elliptical codes and quantizers based on the gaussian density function and pyramid codes and quantizers based on the laplacian density function elliptical and weighted pyramid vector quantizers are developed by selecting codewords as points in a lattice that lie on or near a specified ellipse or pyramid the combination of geometric structure and lattice basis allows simple encoding and decoding algorithms

majorization doubly stochastic matrices and comparison of eigenvalues abstract 1 basic properties of majorization 2 isotone maps and algebraic operations 3 double sub and superstochasticity 4 doubly stochastic matrices 5 doubly stochastic matrices with minimum permanent 6 comparison of eigenvalues 7 doubly stochastic maps

random unistochastic matrices an ensemble of random unistochastic orthostochastic matrices is defined by taking squared moduli of elements of random unitary orthogonal matrices distributed according to the haar measure on u n or o n an ensemble of symmetric unistochastic matrices is obtained with use of unitary symmetric matrices pertaining to the circular orthogonal ensemble we study the distribution of complex eigenvalues of bistochastic unistochastic and orthostochastic matrices in the complex plane we compute averages entropy traces over the ensembles of unistochastic matrices and present inequalities concerning the entropies of products of bistochastic matrices

permanents of d dimensional matrices let a ai1i2 id be an n1 n2 nd matrix over a commutative ring the permanent of a is defined by per a n1i 1ai 2 i 3 i d i where the summation ranges over all one to one functions k from 1 2 n1 to 1 2 nk k 2 3 d in this paper it is shown that a number of properties of permanents of 2 dimensional matrices extend to higher dimensional matrices in particular permanents of nonnegative d dimensional matrices with constant hyperplane sums are investigated the paper concludes by introducing s permanents which generalize the definition above that the permanent becomes the 1 permanent and showing that an s permanent can always be converted into a 1 permanent

monotones and invariants for multi particle quantum states we introduce new entanglement monotones which generalize to the case of many parties those which give rise to the majorization based partial ordering of bipartite states entanglement we give some examples of restrictions they impose on deterministic and probabilistic conversion between multipartite states via local actions and classical communication these include restrictions which do not follow from any bipartite considerations we derive supermultiplicativity relations between each state s monotones and the monotones for collective processing when the parties share several states we also investigate polynomial invariants under local unitary transformations and show that a large class of these are invariant under collective unitary processing and also multiplicative putting restrictions for example on the exact conversion of multiple copies of one state to multiple copies of another

quantized flag manifolds and irreducible representations we study irreducible representations of a certain quantization of the algebra of polynomial functions on a generalized flag manifold regarded as a real manifold all irreducible representations are classified for a subclass of flag manifolds containing in particular the irreducible compact hermitian symmetric spaces for this subclass it is shown that the irreducible representations are parametrized by the symplectic leaves of the underlying poisson bracket we also discuss the relation between the quantized flag manifolds studied in this paper and the quantum flag manifolds studied by soibel man lakshimibai reshetikhin juraco s ovicek and korogodsky

approximating the permanent a randomised approximation scheme for the permanent of a 0 1s presented the task of estimating a permanent is reduced to that of almost uniformly generating perfect matchings in a graph the latter is accomplished by simulating a markov chain whose states are the matchings in the graph for a wide class of 0 1 matrices the approximation scheme is fully polynomial i e runs in time polynomial in the size of the matrix and a parameter that controls the accuracy of the output this class includes all dense matrices those that contain sufficiently many 1 s and almost all sparse matrices in some reasonable probabilistic model for 0 1 matrices of given density for the approach sketched above to be computationally efficient the markov chain must be rapidly mixing informally it must converge in a short time to its stationary distribution a major portion of the paper is devoted to demonstrating that the matchings chain is rapidly mixing apparently the first such result for a markov chain with genuinely c

matrix permanent and quantum entanglement of permutation invariant states we point out that a geometric measure of quantum entanglement is related to the matrix permanent when restricted to permutation invariant states this connection allows us to interpret the permanent as an angle between vectors by employing a recently introduced permanent inequality by carlen et al methods appl anal 13 1 2006 we can prove explicit formulas of the geometric measure for permutation invariant basis states in a simple way

kronecker products minuscule representations and polynomial identities the special role of the highest long and highest short roots in the derivation of extremum properties of the weights of irreducible representations of semisimple lie algebras is pointed out these properties are used to give an intrinsic and unifying reformulation as well as a new proof of klimyk s theorem on kronecker products ukrain mat z 18 19 1966 this new form of klimyk s theorem reveals the special position of the minuscule representations in kronecker products as an immediate consequence the explicit formula for the kronecker product of an arbitrary representation with a minuscule representation is obtained explicit expressions for the weights of the minuscule representations are given new derivations of theorems of dynkin trudy mosk obsch 1 39 1952 and feingold proc am math soc 70 109 1978 on kronecker products as well as the necessary condition for a kronecker product to decompose in two irreducible components are obtained the proofs are based on a theorem of parth

vector predictive coding of speech at 16 kbits s vector quantization in its simplest form may be regarded as a generalization of pcm independent quantization of each sample of a waveform to what might be called vector pcm where a block of consecutive samples a vector is simultaneously quantized as one unit in theory a performance arbitrarily close to the ultimate rate distortion limit is achievable with waveform vector quantization if the dimension of the vector k is large enough the main obstacle in effectively using vector quantization is complexity a vector quantizer of dimension k operating at a rate of r bits sample requires a number of computations on the order of k2 kr and a memory of the same order however a low dimensional vector quantizer dimensions 4 8 achieves a remarkable improvement over scalar quantization pcm consequently using the vector quantizer as a building block and imbedding it with other waveform data compression techniques may lead to the development of a new and powerful class of waveform coding systems this paper proposes and analyzes a waveform coding system adaptive vector predictive coding avpc in which a low dimensionality vector quantizer is used in an adaptive predictive coding scheme in the encoding process a locally generated prediction of the current input vector is subtracted from the current vector and the resulting error vector is coded by a vector quantizer each frame consisting of many vectors is classified into one of m statistical types this classification determines which one of m fixed predictors and of m vector quantizers will be used for encoding the current frame

on the number of interference alignment solutions for the k user mimo channel with constant coefficients in this paper we study the number of different interference alignment ia solutions in a k user multiple input multiple output mimo interference channel when the alignment is performed via beamforming and no symbol extensions are allowed we focus on the case where the number of ia equations matches the number of variables in this situation the number of ia solutions is finite and constant for any channel realization out of a zero measure set and as we prove in this paper it is given by an integral formula that can be numerically approximated using monte carlo integration methods more precisely the number of alignment solutions is the scaled average of the determinant of a certain hermitian matrix related to the geometry of the problem interestingly while the value of this determinant at an arbitrary point can be used to check the feasibility of the ia problem its average properly scaled gives the number of solutions for single beam systems the asymptotic growth rate of the number of solutions is analyzed and some connections with classical combinatorial problems are presented nonetheless our results can be applied to arbitrary interference mimo networks with any number of users antennas and streams per user

gain adaptive vector quantization with application to speech coding the generalization of gain adaptation to vector quantization vq is explored in this paper and a comprehensive examination of alternative techniques is presented we introduce a class of adaptive vector quantizers that can dynamically adjust the gain or amplitude scale of code vectors according to the input signal level the encoder uses a gain estimator to determine a suitable normalization of each input vector prior to vq encoding the normalized vectors have reduced dynamic range and can then be more efficiently coded at the receiver the vq decoder output is multiplied by the estimated gain both forward and backward adaptation are considered and several different gain estimators are compared and evaluated gain adaptive vq can be used alone for vector pcm coding i e direct waveform vq or as a building block in other vector coding schemes the design algorithm for generating the appropriate gain normalized vq codebook is introduced when applied to speech coding gain adaptive vq achieves significant performance improvement over fixed vq with a negligible increase in complexity

unique decomposition of tensor products of irreducible representations of simple algebraic groups we show that a tensor product of irreducible finite dimensional representations of a simple lie algebra over a field of characteristic zero determines the individual constituents uniquely this is analogous to the uniqueness of prime factorisation of natural numbers

a note on the determinant and permanent problem abstract in valiant s theory of arithmetic complexity the following question occupies a central position given an integer n what is the minimal m such that the permanent of an n n matrix is the projection of the determinant of an m m matrix more generally for affine linear transformations we ask the similar question what is the minimal m such that the permanent of an n n matrix is the determinant of an m m matrix via affine linear transformation this paper gives the lower bound m 2 n for affine linear transformations the result is an improvement of earlier results by von zur gathen babai and seress it also generalizes a classical theorem of marcus and minc

the solution of van der waerden s problem for permanents in this paper a solution to van der waerden s problem is presented it is hoped that the solution of this problem will stimulate new and vigorous interest in the theory of permanents and related topics permanents are ideal objects of study having both practical applications and abstract mathematical problems problems involving permanents tend to be easily stated but only solvable using deep methods the theory of permanents as an independent branch of mathematics which has developed parallel to the theory of determinants is far from exhausted many difficult questions must still be resolved in particular the problem of calculation and estimation of permanents for specific classes of matrices is of prime importance the solution of van der waerden s problem makes use of the penetrating force of the method of mixed volumes from combinatorial analysis even hilbert noted that there were connections to be uncovered between the theory of mixed volumes and other areas of mathematics the method of mixed volumes is ideally suited to solving extremal problems and problems of uniqueness and obtaining deep new inequalities it is reasonable to assume that in the future the method of mixed volumes will stand with that of generating functions as one of the basic analytic tools of computational combinatorial analysis

product vectors in the ranges of multi partite states with positive partial transposes and permanents of matrices in this paper we consider a system of homogeneous algebraic equations in complex variables and their conjugates which arise naturally from the range criterion for separability of ppt states we examine systematically these equations to get sufficient conditions for the existence of nontrivial solutions this gives us possible upper bounds of ranks of ppt entangled edge states and their partial transposes we will focus on the multi partite cases which are much more delicate than the bi partite cases we use the notion of permanents of matrices as well as techniques from algebraic geometry through the discussion

the cos λ and sin λ transforms as intertwining operators between generalized principal series representations of sl n 1 k abstract in this article we connect topics from convex and integral geometry with well known topics in representation theory of semisimple lie groups by showing that the cos and sin transforms on the grassmann manifolds gr p k su n 1 k s u p k u n 1 p k are standard intertwining operators between certain generalized principal series representations induced from a maximal parabolic subgroup p p of sl n 1 k the index p indicates the dependence of the parabolic on p the general results of knapp and stein 1971 1980 20 21 and vogan and wallach 1990 44 then show that both transforms have meromorphic extension to c and are invertible for generic c furthermore known methods from representation theory combined with a selberg type integral allow us to determine the k spectrum of those operators

strongly multiplicity free modules for lie algebras and quantum groups abstract let u be either the universal enveloping algebra of a complex semisimple lie algebra g or its drinfel d jimbo quantisation over the field c z of rational functions in the indeterminate z we define the notion of strongly multiplicity free smf for a finite dimensional u module v and prove that for such modules the endomorphism algebras end u v r are generic in the sense that in the classical unquantised case they are quotients of kohno s infinitesimal braid algebra t r while in the quantum case they are quotients of the group ring c z b r of the r string braid group b r in the classical case the generators are generalisations of the quadratic casimir operator c of u while in the quantum case they arise from r matrices which may be thought of as square roots of a quantum analogue of c in a completion of u r this unifies many known results and brings some new cases into their context these include the irreducible 7 dimensional module in type g 2 and arbitrary irreducibles for sl 2 the work leads naturally to questions concerning non semisimple deformations of the relevant endomorphism algebras which arise when the ground rings are varied

conjugation properties of tensor product multiplicities it was recently proven that the total multiplicity in the decomposition into irreducibles of the tensor product of two irreducible representations of a simple lie algebra is invariant under conjugation of one of them at a given level this also applies to the fusion multiplicities of affine algebras here we show that in the case of su 3 the lists of multiplicities in the tensor products and are identical up to permutations this latter property does not hold in general for other lie algebras we conjecture that the same property should hold for the fusion product of the affine algebra of su 3 at finite levels but this is not investigated in the present paper

fractional rate multitree speech coding the authors present both forward and backward adaptive speech coders that operate at 9 6 12 and 16 kb s using integer and fractional rate trees weighted squared error distortion measures the m l tree search algorithm and incremental path map symbol release they introduce the concept of multitree source codes and illustrate how the multitree structure allows scalar quantizer based codes and scalar adaptation rules to be used for fractional rate tree coding with a frequency weighted distortion measure the forward and backward adaptive multitree coders produce near toll quality speech at 16 kb s while the backward adaptive 9 6 kb s multitree coder substantially outperforms adaptive predictive coding and has an encoding delay of less than 2 ms performance results are present in terms of unweighted and weighted signal to noise ratio and segmental signal to noise ratio sound spectrograms and subjective listening tests

product code vector quantizers for waveform and voice coding memory and computation requirements imply fundamental limitations on the quality that can be achieved in vector quantization systems used for speech waveform coding and linear predictive voice coding lpc one approach to reducing storage and computation requirements is to organize the set of reproduction vectors as the cartesian product of a vector codebook describing the shape of each reproduction vector and a scalar codebook describing the gain or energy such shape gain vector quantizers can be applied both to waveform coding using a quadratic error distortion measure and to voice coding using an itakura saito distortion measure in each case the minimum distortion reproduction vector can be found by first selecting a shape code word and then based on that choice selecting a gain codeword several algorithms are presented for the design of shape gain vector quantizers based on a traning sequence of data or a probabilistic model the algorithms are used to design shape gain vector quantizers for both the waveform coding and voice coding application the quantizers are simulated and their performance is compared to that of previously reported vector quantization systems

tropical determinant of integer doubly stochastic matrices abstract let d m n be the set of all the integer points in the m dilate of the birkhoff polytope of doubly stochastic n n matrices in this paper we find the sharp upper bound on the tropical determinant over the set d m n we define a version of the tropical determinant where the maximum over all the transversals in a matrix is replaced with the minimum and then find the sharp lower bound on thus defined tropical determinant over d m n

boolean matrices with prescribed row column sums and stable homogeneous polynomials we prove a new efficiently computable lower bound on the coefficients of stable homogeneous polynomials and present its algorithmic and combinatorial applications our main application is the first poly time deterministic algorithm which approximates the partition functions associated with boolean matrices with prescribed row and column sums within simply exponential multiplicative factor this new algorithm is a particular instance of new polynomial time deterministic algorithms related to the multiple partial differentiation of polynomials given by evaluation oracles