automorphisms of the fine grading of sl n c associated with the generalized pauli matrices we consider the grading of sl n c by the group n of generalized pauli matrices the grading decomposes the lie algebra into n2 1 one dimensional subspaces in the article we demonstrate that the normalizer of grading decomposition of sl n c in n is the group sl 2 zn where zn is the cyclic group of order n as an example we consider sl 3 c graded by 3 and all contractions preserving that grading we show that the set of 48 quadratic equations for grading parameters splits into just two orbits of the normalizer of the grading in 3

weyl groups of fine gradings on matrix algebras octonions and the albert algebra given a grading a g gag on a nonassociative algebra a by an abelian group g we have two subgroups of aut a the automorphisms that stabilize each component ag as a subspace and the automorphisms that permute the components by the weyl group of we mean the quotient of the latter subgroup by the former in the case of a cartan decomposition of a semisimple complex lie algebra this is the automorphism group of the root system i e the so called extended weyl group a grading is called fine if it cannot be refined we compute the weyl groups of all fine gradings on matrix algebras octonions and the albert algebra over an algebraically closed field of characteristic different from 2 in the case of the albert algebra

grading refinements in the contractions of lie algebras and their invariants all mutually inequivalent toroidal gradings of the lie algebra are constructed each of them provides several different graded contractions to other eight dimensional lie algebras these can be either continuous or discrete the continuous ones are related to generalized wigner inonu contractions the behaviour of casimir operators under the graded contractions is studied and the casimir and generalized casimir operators of the contracted lie algebras are presented

on lie gradings ii abstract to obtain fine gradings of gl n c it is necessary to study maximal abelian subgroups mad subgroups of diagonizable automorphisms in a ut gl n c we described two finite sets h n and k n such that each mad subgroup consisting only from inner automorphisms is conjugated to some element of h n and each mad subgroup containing at least one outer automorphism is conjugated to some element of k n using these results concerning k n we then easily find all mad subgroups in a ut o n c for n 8 and in a ut sp 2n c

extrapolation algorithms for discrete signals with application in spectral estimation in this paper we present some new algorithms useful for extrapolation and spectral estimation of band limited sequences in one and two dimensions first we show that many of the existing extrapolation algorithms for noiseless observations are unified under the criterion of minimum norm least squares mnls extrapolation for example the iterative algorithms proposed in 2 and 8 10 are shown to be special cases of a one step gradient algorithm which has linear convergence convergence and other numerical properties are improved by going to a conjugate gradient algorithm for noisy observations these algorithms could be extended by considering a mean square extrapolation criterion which gives rise to a mean square extrapolation filter and also to a recursive extrapolation filter examples and application of these methods are given extension of these algorithms is made for problems where the signal is known to be periodic a new set of functions called the periodic discrete prolate spheroidal sequences p dpss analogous to dpss 21 22 are introduced and their properties are studied finally several of these algorithms are generalized to two dimensions and the relevant equations are given

the weyl group of the fine grading of sl n c associated with tensor product of generalized pauli matrices we consider the fine grading of sl n c induced by tensor product of generalized pauli matrices in the paper regard pgl n c as the inner automorphism group of sl n c based on the classification of maximal abelian subgroups of pgl n c consisting of diagonalizable automorphisms by havlicek et al we prove that any such subgroup k of pgl n c is a symplectic abelian group and its weyl group which describes the symmetry of the fine grading induced by the action of k is just the isometry group of the symplectic abelian group k for a finite symplectic abelian group it is also proved that its isometry group is always generated by the transvections contained in it

direct projection decoding algorithm for sigma delta modulated signals usually sigma delta modulators spl sigma spl delta m are modeled by replacing the quantizer with an additive white noise source based on this linear model linear decoder structures can be designed however the spl sigma spl delta m is a nonlinear system and corresponding nonlinear decoder structures may be able to achieve a better signal to quantization noise ratio sqnr performance in this paper a new nonlinear decoding algorithm is presented it is based on the projection onto complex sets pocs algorithm developed by hein and zakhor our algorithm reduces the decoding problem to the solution of a single quadratic programming problem based on the state equations of the modulator and the condition that the modulator input signal is bandlimited the bandlimitation constraint is directly applied to the state equations thus the resulting quadratic programming problem needs to be solved only once

density of yang lee zeros for the ising ferromagnet the densities of yang lee zeros for the ising ferromagnet on the l x l square lattice are evaluated from the exact grand partition functions l 3 approximately 16 the properties of the density of yang lee zeros are discussed as a function of temperature t and system size l the three different classes of phase transitions for the ising ferromagnet first order phase transition second order phase transition and yang lee edge singularity are clearly distinguished by estimating the magnetic scaling exponent yh from the densities of zeros for finite size systems the divergence of the density of zeros at yang lee edge in high temperatures yang lee edge singularity which has been detected only by the series expansion until now for the square lattice ising ferromagnet is obtained from the finite size data the identification of the orders of phase transitions in small systems is also discussed using the density of yang lee zeros

yang lee circle theorem for an ideal pseudospin 1 2 bose gas in an arbitrary external potential and in an external magnetic field abstract yang lee circle theorem was extended to an ideal homogeneous pseudospin 1 2 bose gas in an external magnetic field x z wang phys rev e 63 2001 046103 in this paper the theorem is extended to the case with an arbitrary external potential the densities of zeros for several cases are determined it is found that the critical exponents of the yang lee edge singularities are the mean field values 1 2 independent of external potentials and spatial dimensions

the pauli matrices in n dimensions and finest gradings of simple lie algebras of type a n 1 properties of the lie algebra gl n c are described for a basis which is a generalization of the 2 2 pauli matrices the 3 3 case is described in detail the remarkable properties of that basis are the grading of the lie algebra it offers each grading subspace is one dimensional and the matrix group it generates it is a finite group with the center of sl n c as its commutator group

on the use of singular value decomposition and decimation in discrete time band limited signal extrapolation the problem of extrapolating a band limited signal in discrete time is viewed as one of solving an underdetermined system of linear equations choosing the minimum norm least squares mnls solution is one criterion for singling out an extrapolation from all the possible solutions to the linear system use of the moore penrose inverse yields the mnls solution and singular value decomposition svd provides a means for implementing the moore penrose inverse an expression for the mean square error incurred in solving a linear system via svd is derived this can be used to estimate the number of singular values needed to form the inverse the error expression also indicates that decimation can be applied in the extrapolation problem to reduce the high computational cost of svd without degrading the extrapolation the results developed for the one dimensional case are extended to higher dimensions examples of the svd approach to extrapolation are given along with examples using other extrapolation techniques for comparison the svd approach compares favorably with known mnls extrapolation methods

graded contractions of lie algebras and central extensions we use the concept of grading of lie algebras to investigate the appearance of central charges during the contraction process as for the usual graded contractions one finds simultaneously the central extensions of classes of algebras rather than specific lie algebras to illustrate the method we consider in detail two physical applications the kinematical algebras of spacetime and the u n bosons limits of some lie algebras

subjoinings of affine kac moody algebras subjoinings among affine kac moody algebras is a relation generalising the algebra subalgebra relation it is analogous to subjoinings among semisimple finite lie algebras a list of maximal subjoinings of equal rank affine kac moody algebras is presented it is conjectured that there are no other maximal subjoinings

fine grading of sl p2 c generated by tensor product of generalized pauli matrices and its symmetries study of the normalizer of the mad group corresponding to a fine grading offers the most important tool for describing symmetries in the system of nonlinear equations connected with contraction of a lie algebra one fine grading that is always present in any lie algebra sl n c is the pauli grading the mad group corresponding to it is generated by generalized pauli matrices for such mad group we already know its normalizer its quotient group is isomorphic to the lie group sl 2 zn z2 in this paper we deal with a more complicated situation namely that the fine grading of sl p2 c is given by a tensor product of the pauli matrices of the same order p p being a prime we describe the normalizer of the corresponding mad group and we show that its quotient group is isomorphic to sp 4 fp z2 where fp is the finite field with p elements

the construction of solvable lie algebras from equidimensional nilpotent algebras abstract the study of gradings of solvable lie algebras l of finite dimensionover a field f of zero characteristic led the authors to the discovery of equidimensional nilpotent algebras l uniquely determined by l up to f isomorphy conversely for any finite dimensional lie algebra l over f an algorithm is developed which yields in parametric form all solvable lie algebras l determining l as the corresponding nilpotent algebra the exposition is independent of lie grading theory it organizes in a novel way the classification of solvable lie algebras of given dimension around the same task for nilpotent algebras every isomorphy class of solvable f algebras is obtained in this way

solvable lie algebras of dimension 4 over perfect fields abstract the solvable lie algebras of dimension not greater than four over a perfect field of reference are described in terms of their nilpotent frame counted over finite fields of q elements and classified over the real and complex number fields the number dq n of nonisomorphic solvable lie algebras of dimension n over f q roughly speaking grows as a polynomial in n with qn 2 as highest term but there are smaller additional terms depending on the residue class of q modulo n 1 if n 4

non linear reconstruction of delta sigma modulated signals randomized surrogate constraint decoding algorithm delta sigma a d conversion is a popular technique used to achieve high resolution data conversion for low to moderate bandwidth applications it is commonly accepted that the digital circuitry of the a d converters should perform a linear time invariant lti filtering on the output of the modulator in order to reconstruct the input signal however it has been shown that when higher order a d converters are operated at higher oversampling ratios osrs non linear reconstruction algorithms extract more information about the input signal than lti methods still use of non linear algorithms has been limited due to their complexity and stability issues two practical non linear methods presented in the literature are fast projection onto convex sets pocs and direct projection dp in this work we show that the non linear reconstruction process for modulated sequences can be treated as a linear feasibility problem lfp we then show that if the reconstruction process is defined as an lfp its implementation is less complex than in the case of the dp method we further describe how the lfp can be well conditioned and demonstrate the reconstruction process by using the surrogate constraint algorithm with random constraint selection the described algorithm considers a small number of constraints at a time which makes it suitable for serial implementation

a fast block based nonlinear decoding algorithm for spl sigma spl delta modulators previous work has established that the digital output of a spl sigma spl delta modulator as an a d converter contains more information about the analog input than is extracted with conventional linear filtering under reasonable mathematical assumptions optimal nonlinear decoding of the digital output can achieve significantly larger signal to noise ratios than linear filtering however the hitherto proposed decoding algorithms only demonstrate conceptual feasibility and are impractical from a computational point of view we present a new block based decoding algorithm that like previous work employs projections onto convex sets the algorithm owes its speed to a change of projection norm an accelerated convergence scheme and a decimation like subsampling it is on the order of 10 sup 4 10 sup 5 times faster than one previously published algorithm for typical parameter values and about 2 10 times slower than linear decoding the new algorithm is applicable to all currently popular spl sigma spl delta architectures

new gradings of sl 3 c representations two new gradings of the lie algebra sl 3 c are the refined sl 2 c and o 3 gradings the grading in each case utilizes the sl 2 c or o 3 weight together with a new additive modular label a complete set of sl 3 c representation basis states labeled by each to the two sets of additive quantum numbers is found the newlabeling operator in both cases fails to commute with the cubic sl 3 c casimir operator and hence mixes states of the contragredient representations p q and q p

deterministic analysis of oversampled a d conversion and decoding improvement based on consistent estimates discusses the deterministic analysis of oversampled a d conversion adc the properties derivable from such an analysis and the consequences on reconstruction using nonlinear decoding given a band limited input x producing a quantized version c the authors consider the set of all input signals that are band limited and produce c they call any element of this set a consistent estimate of x regardless of the type of encoder simple predictive or noise shaping they show that this set is convex and as a consequence any nonconsistent estimate can be improved they also show that the classical linear decoding estimates are not necessarily consistent numerical tests performed on simple adc single loop and multiloop spl sigma spl delta modulation show that consistent estimates yield a mean square error mse that decreases asymptotically with the oversampling ratio faster than the linear decoding mse by approximately 3 db octave this implies an asymptotic mse of the order of spl oscr r sup spl minus 2n 2 instead of spl oscr r sup spl minus 2n 1 in linear decoding where r is the oversampling ratio and n the order of the modulator methods of improvement of nonconsistent estimates based on the deterministic knowledge of the quantized signal are proposed for simple adc predictive adc single loop and multiloop spl sigma spl delta modulation

graded contractions of the lie algebra e 2 1 the graded contractions of pseudo euclidean lie algebra are studied the non equivalent gradings of of type and a are extended to the entire lie algebra using the action of on the abelian ideal the translations the graded contractions embed into a large family of six dimensional lie algebras the family includes solvable nilpotent and nonsolvable lie algebras both decomposable and indecomposable ones the distinction between graded and inonu wigner contractions is analysed the physically most interesting lie algebras obtained by the contractions are the inhomogeneous galilei and pseudo galilei

symmetries of finite heisenberg groups for multipartite systems a composite quantum system comprising a finite number k of subsystems which are described with position and momentum variables in i 1 k is considered its hilbert space is given by a k fold tensor product of hilbert spaces of dimensions n1 nk the symmetry group of the respective finite heisenberg group is given by the quotient group of certain normalizer this paper extends our previous investigation of bipartite quantum systems to arbitrary multipartite systems of the above type it provides detailed description of the normalizers and the corresponding symmetry groups the new class of symmetry groups represents a very specific generalization of symplectic groups over modular rings as an application a new proof of existence of the maximal set of mutually unbiased bases in hilbert spaces of prime power dimensions is provided

equivalence of some quadratic programming algorithms we formulate a general algorithm for the solution of a convex but not strictly convex quadratic programming problem conditions are given under which the iterates of the algorithm are uniquely determined the quadratic programming algorithms of fletcher gill and murray best and ritter and van de panne and whinston dantzig are shown to be special cases and consequently are equivalent in the sense that they construct identical sequences of points the various methods are shown to differ only in the manner in which they solve the linear equations expressing the kuhn tucker system for the associated equality constrained subproblems equivalence results have been established by goldfarb and djang for the positive definite hessian case our analysis extends these results to the positive semi definite case

image restoration by the method of convex projections part 1ߞtheory a projection operator onto a closed convex set in hilbert space is one of the few examples of a nonlinear map that can be defined in simple abstract terms moreover it minimizes distance and is nonexpansive and therefore shares two of the more important properties of ordinary linear orthogonal projections onto closed linear manifolds in this paper we exploit the properties of these operators to develop several iterative algorithms for image restoration from partial data which permit any number of nonlinear constraints of a certain type to be subsumed automatically their common conceptual basis is as follows every known property of an original image f is envisaged as restricting it to lie in a well defined closed convex set thus m such properties place f in the intersection e0 ei of the corresponding closed convex sets e1 e2 em given only the projection operators pi onto the individual ei s i 1 m we restore f by recursive means clearly in this approach the realization of the pi s in a hilbert space setting is one of the major synthesis problems section i describes the geometrical significance of the three main theorems in considerable detail and most of the underlying ideas are illustrated with the aid of simple diagrams section ii presents rules for the numerical implementation of 11 specific projection operators which are found to occur frequently in many signal processing applications and the appendix contains proofs of all the major results

weyl groups of some fine gradings on e6 abstract the weyl groups of the fine gradings on the exceptional lie algebra e 6 whose universal grading group is infinite are computed

graded contractions of the pauli graded sl 3 c the lie algebra sl 3 c is considered in the basis of generalized pauli matrices corresponding grading is the pauli grading here it is one of the four gradings of the algebra which cannot be further refined the set s of 48 contraction equations for 24 contraction parameters is solved our main tools are the symmetry group of the pauli grading of sl 3 c which is essentially the finite group sl 2 z3 and the induced symmetry of the system s a list of all equivalence classes of solutions of the contraction equations is provided among the solutions 175 equivalence classes are non parametric and 13 solutions depend on one or two continuous parameters providing a continuum of equivalence classes and subsequently continuum of non isomorphic lie algebras solutions of the contraction equations of pauli graded sl 3 c are identified here as specific solvable lie algebras of dimensions up to 8 earlier algorithms for identification of lie algebras given by their structure constants had to be made more efficient in order to distinguish non isomorphic lie algebras encountered here resulting lie algebras are summarized in tabular form there are 88 indecomposable solvable lie algebras of dimension 8 77 of them being nilpotent there are 11 infinite sets of parametric lie algebra which still deserve further study

re examination of yang lee zeros of the anisotropic ising models on square triangular and honeycomb lattices we have determined the zeros of the partition functions of the anisotropic ising models on square triangular and honeycomb lattices in the absence of a magnetic field with arbitrary combinations of interactions it is found that the zeros are generally distributed over areas however the zeros near the positive real axis are distributed on the unit circle in a complex plane with the zero density g which leads to logarithmic singularity of the free energy our results generalize fisher s results for isotropic cases

the symmetries of the fine gradings of sl nk c associated with direct product of pauli groups a grading of a lie algebra is called fine if it could not be further refined for a fine grading of a simple lie algebra we define its weyl group to describe the symmetry of this grading it is already known that the weyl group of the fine grading of sl n c induced by the action of the group n of the generalized pauli matrices of rank n is sl 2 zn where zn is the cyclic group of order n in this paper we consider the fine grading of sl nk c induced by the action of the group of k fold tensor product of the generalized pauli matrices of rank n we prove that its weyl group is sp 2k zn and is generated by transvections therefore this generalizes the previous result

image restoration by the method of convex projections part 2 applications and numerical results the image restoration theory discussed in a previous paper by youla and webb 1 is applied to a simulated image and the results compared with the well known method known as the gerchberg papoulis algorithm the results show that the method of image restoration by projection onto convex sets by providing a convenient technique for utilizing a priori information performs significantly better than the gerchberg papoulis method

discrete and continuous graded contractions of representations of lie algebras simultaneous grading of lie algebras and their representation spaces is used to develop a new theory of grading preserving contractions of representations of all lie algebras admitting the chosen grading the theory is completely different from the traditional ways of contracting representations the graded contractions fall naturally into two classes discrete and continuous ones related respectively to 2 cocycles and coboundaries of the grading group

reconstruction of oversampled band limited signals from spl sigma spl delta encoded binary sequences considers the application of spl sigma spl delta modulators to analog to digital conversion the authors have previously shown that for constant input signals optimal nonlinear decoding can achieve large gains in signal to noise ratio snr over linear decoding the present paper shows a similar result for band limited input signals the new nonlinear decoding algorithm is based on projections onto convex sets pocs and alternates between a time domain operation and a band limitation to find a signal invariant under both the time domain operation results in a quadratic programming problem the band limitation can be based on singular value decomposition of a certain matrix the authors show simulation results for the snr performance of a pocs based decoder and a linear decoder for the single loop double loop and two stage spl sigma spl delta modulators and for a specific fourth order interpolative modulator depending on the modulator and the oversampling ratio improvements in snr of up to 10 20 db can be achieved

discrete and continuous graded contractions of lie algebras and superalgebras grading preserving contractions of lie algebras and superalgebras of any type over the complex number field are defined and studied such contractions fall naturally into two classes the wigner inonu like continuous contractions and new discrete contractions a general method is described for any abelian grading semigroup and any lie algebra or superalgebra admitting such a grading all contractions preserving z2 z3 and z2 z2 gradings are found examples of these gradings and contractions for the simple lie algebra a2 affine kac moody algebra a 1 and the simple superalgebra osp 2 1 are shown

the four sets of additive quantum numbers of su 3 the four maximal sets of additive quantum numbers and related fine gradings of the lie algebra sl 3 c are described in detail the quantum numbers are determined by a grading of the lie algebra fine gradings providing maximal sets of them two sets of additive quantum numbers are equivalent precisely if the corresponding gradings are equivalent under a transformation from the automorphism group of the lie algebra