varieties of nilpotent lie algebras of dimension less than six the set of all nilpotent ide algebra multiplications on a fixed vector space of dimension n over a field k forms an affine variety n isomorphism classes of these lie algebras are orbits in i under the change of basis action of gl n k by examining the orbit structure of n for small n we hope to gain a new perspective on the classification of nilpotent lie algebras in this paper we determine completely the orbit structure of the variety n when n 6 5 and k c we also show that the homomorphic image of a nilpotent lie algebra is in the closure of the orbit of that lie algebra taking the direct product with an abelian lie algebra to make the dimension right some of the orbit structure of n for n 6 and other geometric properties of n were described by vergne in vl v2 v3 recently carles has used vergne s results to study the variety of all lie algebras cl our motivation for studying the classification of nilpotent lie algebras is to gain information about the classification of nilpotent groups the correspondence between nilpotent lie algebras and nilpotent groups is described in go

degenerations of filippov algebras we consider the variety of filippov n lie algebra structures on an n 1 dimensional vector space the group gln k acts on it and we study the orbit closures with respect to the zariski topology this leads to the definition of filippov algebra degenerations we present some fundamental results on such degenerations including trace invariants and necessary degeneration criteria finally we classify all orbit closures in the variety of complex n 1 dimensional filippov n ary algebras we consider the variety of filippov n lie algebra structures on an n 1 dimensional vector space the group gln k acts on it and we study the orbit closures with respect to the zariski topology this leads to the definition of filippov algebra degenerations we present some fundamental results on such degenerations including trace invariants and necessary degeneration criteria finally we classify all orbit closures in the variety of complex n 1 dimensional filippov n ary algebras

analysis and further results on adaptive entropy coded quantization buffer underflow and overflow problems associated with entropy coding are completely eliminated by effectively imposing reflecting walls at the buffer endpoints synchronous operation of the aecq adaptive entropy coded quantizer encoder and decoder is examined in detail and it is shown that synchronous operation is easily achieved without side information a method is developed to explicitly solve for the buffer state probability distribution and the resulting average distortion when memoryless buffer state feedback is used as well as when the source is stationary and memoryless this method is then used as a tool in the design of low distortion aecq systems with particular attention given to developing source scale invariant distortion performance it is shown that the introduction of reflecting buffer walls in a properly designed aecq system results in a very small rate distortion performance penalty and that the resulting aecq system can be an extremely simple and effective solution to the stationary memoryless source coding problem for a wide range of source types operation with nonstationary sources is also examined

entropy constrained vector quantization an iterative descent algorithm based on a lagrangian formulation for designing vector quantizers having minimum distortion subject to an entropy constraint is discussed these entropy constrained vector quantizers ecvqs can be used in tandem with variable rate noiseless coding systems to provide locally optimal variable rate block source coding with respect to a fidelity criterion experiments on sampled speech and on synthetic sources with memory indicate that for waveform coding at low rates about 1 bit sample under the squared error distortion measure about 1 6 db improvement in the signal to noise ratio can be expected over the best scalar and lattice quantizers when block entropy coded with block length 4 even greater gains are made over other forms of entropy coded vector quantizers for pattern recognition it is shown that the ecvq algorithm is a generalization of the k means and related algorithms for estimating cluster means in that the ecvq algorithm estimates the prior cluster probabilities as well experiments on multivariate gaussian distributions show that for clustering problems involving classes with widely different priors the ecvq outperforms the k means algorithm in both likelihood and probability of error

adaptive entropy coded subband coding of images the authors describe a design approach called 2 d entropy constrained subband coding ecsbc based upon recently developed 2 d entropy constrained vector quantization ecvq schemes the output indexes of the embedded quantizers are further compressed by use of noiseless entropy coding schemes such as huffman or arithmetic codes resulting in variable rate outputs depending upon the specific configurations of the ecvq and the ecpvq over the subbands many different types of sbc schemes can be derived within the generic 2 d ecsbc framework among these the authors concentrate on three representative types of 2 d ecsbc schemes and provide relative performance evaluations they also describe an adaptive buffer instrumented version of 2 d ecsbc called 2 d ecsbc aec for use with fixed rate channels which completely eliminates buffer overflow underflow problems this adaptive scheme achieves performance quite close to the corresponding ideal 2 d ecsbc system

adaptive entropy coded pruned tree structured predictive vector quantization of images simpler versions of a previously introduced adaptive entropy coded predictive vector quantization pvq scheme where the embedded entropy constrained vector quantizer ecvq is replaced by a pruned tree structured vq ptsvq are described the resulting encoding scheme is shown to result in drastically reduced complexity at only a small cost in performance coding results for selected real world images are given

the classification of algebras of level one abstract in the present paper we obtain a list of algebras up to isomorphism such that the closure of any complex finite dimensional algebra contains one of the algebras of the given list

axisymmetric viscoplastic deformation by the boundary element method abstract this paper presents a boundary element formulation and numerical implementation of the problem of small axisymmetric deformation of viscoplastic bodies while the extension from planar to axisymmetric problems can be carried out fairly simply for the finite element method fem this is far from true for the boundary element method bem the primary reason for this fact is that the axisymmetric kernels in the integral equations of the bem contain elliptic functions which cannot be integrated analytically even over boundary elements and internal cells of simple shape thus special methods have to be developed for the efficient and accurate numerical integration of these singular and sensitive kernels over discrete elements the accurate determination of stress rates by differentiation of the displacement rates presents another formidable challenge a successful numerical implementation of the boundary element method with elementwise called the mixed approach or pointwise called the pure bem or bem approach determination of stress rates has been carried out a computer program has been developed for the solution of general axisymmetric viscoplasticity problems comparisons of numerical results from the bem and fem for several illustrative problems are presented and discussed in the paper it is possible to get direct solutions for the simpler class of problems for cylinders of uniform cross section and these solutions are also compared with the bem and fem results for such cases

on vector transformation it has been shown that a vector transform is a good tool for digital filtering and image coding this paper addresses some of the issues related to vector transforms first a fundamental difference between vector transforms and conventional transforms is examined second two approaches to constructing vector transforms are discussed finally an application of vector transformation to an efficient implementation of 2 d nonseparable digital filters is proposed

a three dimensional bem solution for plasticity using regression interpolation within the plastic field this paper presents an improved solution of three dimensional plasticity problems using the boundary element method bem the bem formulation for plasticity requires volume as well as boundary discretizations an initial stress formulation is used to satisfy the material non linearity conventionally the plastic field in the volume element or cell is interpolated based on the value of plastic stress at the nodes of the cell in this paper the distribution of the plastic field in the cell is based on a number of points interior to the cell the plastic field is described using regression interpolation polynomials through these interior points the constitutive relation is satisfied at each interior point the number of points can be varied in each cell thus allowing for adaptive volume cells the plastic stresses are computed at the interior points only therefore the need for surface stress computation which uses numerical derivatives at the surface is completely eliminated three dimensional applications are used to compare the present regression interpolation procedure with the conventional method for elasto plasticity problems in all variations of the applications studied regression interpolation based on interior points provided superior results to those determined via the conventional nodal interpolation method

on the degenerations of solvable leibniz algebras abstract the present paper is devoted to the description of rigid solvable leibniz algebras in particular we prove that solvable leibniz algebras under some conditions on the nilradical are rigid and we describe four dimensional solvable leibniz algebras with three dimensional rigid nilradical we show that the grunewald o halloran s conjecture any n dimensional nilpotent lie algebra is a degeneration of some algebra of the same dimension holds for leibniz algebras of dimensions less than four the algebra of level one which is omitted in the 1991 gorbatsevich s paper is indicated

advanced boundary element analysis of two and three dimensional problems of elasto plasticity an advanced formulation of the boundary element method has been developed for inelastic analysis based on an initial stress approach the iterative solution algorithm makes use of an accelerated initial stress approach in which the past history of initial stresses are used to obtain an initial estimate for the current increment in the present analysis the geometry and functions are represented by higher order quadratic shape functions to model complex geometries and rapid functional variations accurately the methods of numerical integration of the kernels particularly the singular type are substantially improved by devising suitable automatic sub segmentation routines that incorporate the recent developments in mapping procedures the formulations have been implemented for two dimensional plane stress plane strain and three dimensional elasto plasticity problems

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

a new bem formulation for tow and three dimensional elastoplasticity using particular integras new two and three dimensional bem formulations are developed for elastoplastic stress analysis these new procedures differ from previous work in that volume integration is not required to incorporate the nonlinear effects in the analysis instead initial stress rates are introduced in the bem system via particular integrals the present formulation is implemented in a general purpose multi region system and examples are presented to demonstrate the accuracy and versatility of the method

on the structure of vector quantizers vector quantization is intrinsically superior to predictive coding transform coding and other suboptimal and em ad hoc procedures since it achieves optimal rate distortion performance subject only to a constraint on memory or block length of the observable signal segment being encoded the key limitation of existing techniques is the very large randomly generated code books which must be stored and the computational complexity of the associated encoding procedures the quantization operation is decomposed into its rudimentary structural components this leads to a simple and elegant approach to derive analytical properties of optimal quantizers some useful properties of quantizers and algorithmic approaches are given which are relevant to the complexity of both storage and processing in the encoding operation highly disordered quantizers which have been designed using a clustering algorithm are considered finally lattice quantizers are examined which circumvent the need for a code book by using a highly structured code based on lattices the code vectors are algorithmically generated in a simple manner rather than stored in a code book and fast algorithms perform the encoding algorithm with negligible complexity

gradient algorithms for designing predictive vector quantizers a predictive vector quantizer pvq is a vector extension of a predictive quantizer it consists of two parts a conventional memoryless vector quantizer vq and a vector predictor two gradient algorithms for designing a pvq are developed in this paper the steepest descent sd algorithm and the stochastic gradient sg algorithm both have the property of improving the quantizer and the predictor in the sense of minimizing the distortion as measured by the average mean squared error the differences between the two design approaches are the period and the step size used in each iteration to update the codebook and predictor the sg algorithm updates once for each input training vector and uses a small step size while the sd updates only once for a long period possibly one pass over the entire training sequence and uses a relatively large step size code designs and tests are simulated for both gauss markov sources and for sampled speech waveforms and the results are compared to codes designed using techniques that attempt to optimize only the quantizer for the predictor and not vice versa

reusable intrinsic sample point risp algorithm for the efficient numerical integration of three dimensional curved boundary elements this paper presents an algorithm that provides an order of magnitude gain in the computational performance of the numerical integration of the boundary integral equations for three dimensional analysis existing algorithms for numerical integration have strategically clustered integration sample points based on the relative proximity of the load points to the boundary element being integrated using element subdivision or element co ordinate transformation the emphasis in these techniques has been on minimizing the number of sample points required to obtain a given level of accuracy the present algorithm while closely following the spirit of these earlier approaches employs a discrete number of sets of predetermined customized near optimum sample point quantities associated with the intrinsic boundary element the ability created by this approach to reuse sample point geometric information of the actual element allows for the realization of substantive computational economy this algorithm provides accurate and efficient numerical results both when load points are far from and when they are on the boundary element being integrated numerical results are provided to demonstrate the substantial economy achieved through the use of the present algorithm

predictive vector quantization of images the purpose of this paper is to present new image coding schemes based on a predictive vector quantization pvq approach the predictive part of the encoder is used to partially remove redundancy and the vq part further removes the residual redundancy and selects good quantization levels for the global waveform two implementations of this coding approach have been devised namely sliding block pvq and block tree pvq simulations on real images show significant improvement over the conventional dpcm and tree codes using these new techniques the strong robustness property of these coding schemes is also experimentally demonstrated

asymptotic quantization error of continuous signals and the quantization dimension extensions of the limiting qnanfizafion error formula of bennet are proved these are of the form d s k n f n beta b where n is the number of output levels d s k n f is the s th moment of the metric distance between quantizer input and output beta b 0 k s beta is the signal space dimension and f is the signal distribution if a suitably well behaved k dimensional signal density f x exists b b s k int f rho x dx 1 rho rho k s k and b s k does not depend on f for k 1 s 2 this reduces to bennett s formula if f is the cantor distribution on 0 1 0 and this k equals the fractal dimension of the cantor set 12 13 random quantization optimal quantization in the presence of an output information constraint and quantization noise in high dimensional spaces are also investigated

robust adaptive buffer instrumented entropy coded quantization of stationary sources an investigation is conducted of the effects of various source statistics and buffer management algorithms for a buffer of fixed size and an algorithm that provides particularly robust rate distortion performance for a wide range of stationary sources is described the performance of this scheme which is referred to as the adaptive entropy coded quantizer aecq is compared to that of conventional adaptive pulse code modulation apcm operating at the same fixed channel transmission rate experimental results indicate that the aecq can provide considerably improved performance and more robust operation than apcm the relative sensitivity of the aecq to scale mismatch is directly related to the rather simple memoryless feedback control strategy employed this might be remedied by alternative choices of the memoryless feedback control function or by introducing memory the former is much easier to implement and or analyze than the latter

adaptive entropy coded predictive vector quantization of images the authors consider 2 d predictive vector quantization pvq of images subject to an entropy constraint and demonstrate the substantial performance improvements over existing unconstrained approaches they describe a simple adaptive buffer instrumented implementation of this 2 d entropy coded pvq scheme which can accommodate the associated variable length entropy coding while completely eliminating buffer overflow underflow problems at the expense of only a slight degradation in performance this scheme called 2 d pvq aecq adaptive entropy coded quantization is shown to result in excellent rate distortion performance and impressive quality reconstructions of real world images indeed the real world coding results shown demonstrate little distortion at rates as low as 0 5 b pixel

classifying complements for associative algebras abstract for a given extension a e of associative algebras we describe and classify up to an isomorphism all a complements of e i e all subalgebras x of e such that e a x and a x 0 let x be a given complement and a x the canonical matched pair associated with the factorization e a x we introduce a new type of deformation of the algebra x by means of the given matched pair and prove that all a complements of e are isomorphic to such a deformation of x several explicit examples involving the matrix algebra are provided