on the stability of perfect viscoelastic columns abstract the stability problem associated with an euler bernoulli beam made of an arbitrary linear viscoelastic material is formulated the three parameter and the kelvin voigt models are analyzed in the presence of constant as well as periodic loads the application of a finite time stability concept is shown for the constant loading case when the traditional stability criterion fails to make sense for the case of a periodic loading the stability diagrams are obtained through an application of floquet theory it is found that the addition of periodic loads may significantly alter the stability behavior of a column which is originally subjected to a constant load only

optimality approximation and complexity in set membership h sub spl infin identification investigates the set membership identification of time invariant discrete time exponentially stable possibly infinite dimensional linear systems from time or frequency domain data corrupted by deterministic noise the aim is to deliver not a single model but a set of models whose size in h sub spl infin norm measures the uncertainty in the identification the main focus of the note is on the optimality properties for finite data and on the tradeoff between optimality and complexity of approximated low order model sets a method is given for evaluating convergent and computationally efficient inner and outer approximations of the value set for a given frequency such approximations allow one to compute within any desired accuracy the identification error of any identified model and to evaluate an optimal model at any given number of frequencies by suitably approximating these values model sets with nominal models in rh sub spl infin are then derived whose order is selected by trading off between model set complexity and identification accuracy degradation this degradation is evaluated by computing the optimality level defined as the ratio between the reduced model identification error and the optimal one a numerical example demonstrates the effectiveness of the presented results

partitions of a simplex leading to accurate spectral finite volume reconstruction in this paper we present several systematic techniques based on the voronoi diagram and its variants to partition a one and two dimensional simplex the fekete points are used as input to generate the voronoi diagram as they concentrate near the edges and are almost optimal for polynomial interpolation in a simplex spectral finite volume reconstructions on the resultant partitions have small lebesgue constants when using the dubiner basis the reconstruction matrix is well conditioned moreover the total number of edges of the partitions the total work when being used in spectral volume methods is shown to be at most twice the minimum number of edges of all partitions for reconstructions of the same order accuracy these suggest that the obtained partitions are well suited for spectral volume methods and other numerical methods which rely on reconstructions from cell averages

space time tradeoffs for emptiness queries we develop the first nontrivial lower bounds on the complexity of online hyperplane and halfspace emptiness queries our lower bounds apply to a general class of geometric range query data structures called partition graphs informally a partition graph is a directed acyclic graph that describes a recursive decomposition of space we show that any partition graph that supports hyperplane emptiness queries implicitly defines a halfspace range query data structure in the fredman yao semigroup arithmetic model with the same asymptotic space and time bounds thus results of bronnimann chazelle and pach imply that any partition graph of size s that supports hyperplane emptiness queries in time t satisfies the inequality st d omega n log n d d 1 d 1 using different techniques we improve previous lower bounds for hopcroft s problem given a set of points and hyperplanes does any hyperplane contain a point in dimensions four and higher using this offline result we show that for online hyperplane emptiness queries omega n d mbox polylog n space is required to achieve polylogarithmic query time and omega n d 1 d mbox polylog n query time is required if only o n polylog n space is available these two lower bounds are optimal up to polylogarithmic factors for two dimensional queries we obtain an optimal continuous tradeoff st 2 omega n 2 between these two extremes finally using a lifting argument we show that the same lower bounds hold for both offline and online halfspace emptiness queries in mathbb r d d 3 2

the golden code a 2 spl times 2 full rate space time code with nonvanishing determinants in this paper the golden code for a 2 spl times 2 multiple input multiple output mimo system is presented this is a full rate 2 spl times 2 linear dispersion algebraic space time code with unprecedented performance based on the golden number 1 spl radic 5 2

fast decodable asymmetric space time codes from division algebras multiple input double output mido codes are important in the near future wireless communications where the portable end user device is physically small and will typically contain at most two receive antennas especially tempting is the 4 2 channel due to its immediate applicability in the digital video broadcasting dvb such channels optimally employ rate two space time st codes consisting of 4 4 matrices unfortunately such codes are in general very complex to decode hence setting forth a call for constructions with reduced complexity recently some reduced complexity constructions have been proposed but they have mainly been based on different ad hoc methods and have resulted in isolated examples rather than in a more general class of codes in this paper it will be shown that a family of division algebra based mido codes will always result in at least 37 5 worst case complexity reduction while maintaining full diversity and for the first time the nonvanishing determinant nvd property the reduction follows from the fact that similarly to the alamouti code the codes will be subsets of matrix rings of the hamiltonian quaternions hence allowing simplified decoding at the moment such reductions are among the best known for rate two mido codes 5 6 several explicit constructions are presented and shown to have excellent performance through computer simulations simulations

generalized silver codes for an nt transmit nr receive antenna system nt nr system a full rate space time block code stbc transmits at least nmin min nt nr complex symbols per channel use the well known golden code is an example of a full rate full diversity stbc for two transmit antennas its ml decoding complexity is of the order of m2 5 for square m qam the silver code for two transmit antennas has all the desirable properties of the golden code except its coding gain but offers lower ml decoding complexity of the order of m2 importantly the slight loss in coding gain is negligible compared to the advantage it offers in terms of lowering the ml decoding complexity for higher number of transmit antennas the best known codes are the perfect codes which are full rate full diversity information lossless codes for nr nt but have a high ml decoding complexity of the order of mntnmin for nr nt the punctured perfect codes are considered in this paper a scheme to obtain full rate stbcs for 2a transmit antennas and any nr with reduced ml decoding complexity of the order of mnt nmin 3 4 0 5 is presented the codes constructed are also information lossless for nr nt like the perfect codes and allow higher mutual information than the comparable punctured perfect codes for nr nt these codes are referred to as the generalized silver codes since they enjoy the same desirable properties as the comparable perfect codes except possibly the coding gain with lower ml decoding complexity analogous to the silver code and the golden code for two transmit antennas simulation results of the symbol error rates for four and eight transmit antennas show that the generalized silver codes match the punctured perfect codes in error performance while offering lower ml decoding complexity

reweighting to achieve elliptically contoured covariates in regression abstract we investigate a method of constructing weights to induce elliptically contoured covariates in regression analyses much recent work in regression has identified various data analytic and model robustness advantages associated with such covariates in particular new estimation methods like sir sirii save and phd have been built around the assumption of elliptically contoured covariates finite samples of regression covariates may deviate from this ideal in practice and the method developed here termed voronoi weighting can be used to induce elliptical symmetry in such samples in a number of examples we show that reweighting cases by the voronoi method can substantially enhance various procedures for covariates that deviate from elliptical symmetry we show that voronoi weighting in conjunction with some trimming via the minimum volume ellipsoid method can be effective

a construction of a space time code based on number theory we construct a full data rate space time st block code over m 2 transmit antennas and t 2 symbol periods and we prove that it achieves a transmit diversity of 2 over all constellations carved from z i sup 4 further we optimize the coding gain of the proposed code and then compare it to the alamouti code it is shown that the new code outperforms the alamouti see ieee j select areas commun vol 16 p 1451 58 1998 code at low and high signal to noise ratio snr when the number of receive antennas n 1 the performance improvement is further enhanced when n or the size of the constellation increases we relate the problem of st diversity gain to algebraic number theory and the coding gain optimization to the theory of simultaneous diophantine approximation in the geometry of numbers we find that the coding gain optimization is equivalent to finding irrational numbers the furthest from any simultaneous rational approximations

signature recognition through spectral analysis abstract features such as shape motion and pressure minutiae details and timing and transformation methods such as hadamard and walsh have been used in handwritten signature recognition with various degrees of success others have successfully used nonlinear warping functions to optimally time match an unknown to a standard signature in this research a fast fourier transform is used to transform normalized signatures into the frequency domain fifteen harmonics having the largest magnitudes normalized by their corresponding variances were selected and used in a stepwise discriminant analysis it results in an error rate of 2 5 per cent with the generally more conservative jackknife procedure yielding the same small error rate

on the densest mimo lattices from cyclic division algebras it is shown why the discriminant of a maximal order within a cyclic division algebra must be minimized in order to get the densest possible matrix lattices with a prescribed nonvanishing minimum determinant using results from class field theory a lower bound to the minimum discriminant of a maximal order with a given center and index the number of tx rx antennas is derived also numerous examples of division algebras achieving the bound are given for example a matrix lattice with quadrature amplitude modulation qam coefficients that has 2 5 times as many codewords as the celebrated golden code of the same minimum determinant is constructed also a general algorithm due to ivanyos and ronyai for finding maximal orders within a cyclic division algebra is described and enhancements to this algorithm are discussed also some general methods for finding cyclic division algebras of a prescribed index achieving the lower bound are proposed

computing connectedness disconnectedness and discreteness we consider finite point set approximations of a manifold or fractal with the goal of determining topological properties of the underlying set we use the minimal spanning tree of the finite set of points to compute the number and size of its connected components by extrapolating the limiting behavior of these quantities as 0 we can say whether the underlying set appears to be connected totally disconnected or perfect we demonstrate the effectiveness of our techniques for a number of examples including a family of fractals related to the sierpinski triangle cantor subsets of the plane the henon attractor and cantori from four dimensional symplectic sawtooth maps for zero measure cantor sets we conjecture that the growth rate of the number of components as 0 is equivalent to the box counting dimension 2000 elsevier science b v all rights reserved

full diversity high rate space time block codes from division algebras we present some general techniques for constructing full rank minimal delay rate at least one space time block codes stbcs over a variety of signal sets for arbitrary number of transmit antennas using commutative division algebras field extensions as well as using noncommutative division algebras of the rational field spl qopf embedded in matrix rings the first half of the paper deals with constructions using field extensions of spl qopf working with cyclotomic field extensions we construct several families of stbcs over a wide range of signal sets that are of full rank minimal delay and rate at least one appropriate for any number of transmit antennas we study the coding gain and capacity of these codes using transcendental extensions we construct arbitrary rate codes that are full rank for arbitrary number of antennas we also present a method of constructing stbcs using noncyclotomic field extensions in the later half of the paper we discuss two ways of embedding noncommutative division algebras into matrices left regular representation and representation over maximal cyclic subfields the 4 spl times 4 real orthogonal design is obtained by the left regular representation of quaternions alamouti s 1998 code is just a special case of the construction using representation over maximal cyclic subfields and we observe certain algebraic uniqueness characteristics of it also we discuss a general principle for constructing cyclic division algebras using the nth root of a transcendental element and study the capacity of the stbcs obtained from this construction another family of cyclic division algebras discovered by brauer 1933 is discussed and several examples of stbcs derived from each of these constructions are presented

discrete voronoi diagrams and the skiz operator a dynamic algorithm the voronoi diagram vd is a popular tool for partitioning the support of an image an algorithm is presented for constructing vd when the seed set which determines the voronoi regions can be modified by adding and removing seeds the number of pixels and seeds revisited for updating the diagram and the neighbor relationships among seeds is minimized a result on cocircular seeds is presented the adjacency or dual graph of the vd is readily obtained the use of the algorithm for constructing skeletons by influence zones skiz is demonstrated

polygon based contact resolution for superquadrics the representation of discrete objects in the discrete element modelling is a fundamental issue which has a direct impact on the efficiency of discrete element implementation and the dynamic behaviour of particulate systems disks and spheres are the most commonly used geometric shapes due to their geometric simplicity and computational efficiency but they are unable to provide resistance to rolling motion for this reason some non circular spherical objects such as polygons polyhedrons superquadrics or the clustering of disks spheres to form irregular shapes are introduced when superquadrics are used as discrete elements the bottleneck of contact resolution is associated with the searching for intersections of two non linear functions which is a very expensive operation and may sometimes fail in finding the solution in this work an efficient and robust algorithm is proposed for contact resolution of 2d superquadrics in which any superquadric is approximated with a convex polygon through adaptive sampling then by clipping two polygons an efficient linear algorithm is performed to search for intersections and overlap area of the polygons the contact forces and directions are determined by employing a newly established corner corner contact model it is important to highlight that the proposed methodology can also be extended to general non circular discrete object cases the performance of the algorithm is demonstrated via numerical examples copyright 2005 john wiley sons ltd

generators extremals and bases of max cones abstract max cones are max algebraic analogs of convex cones in the present paper we develop a theory of generating sets and extremals of max cones in r n this theory is based on the observation that extremals are minimal elements of max cones under suitable scalings of vectors we give new proofs of existing results suitably generalizing restating and refining them of these it is important that any set of generators may be partitioned into the set of extremals and the set of redundant elements we include results on properties of open and closed cones on properties of totally dependent sets and on computational bounds for the problem of finding the essentially unique basis of a finitely generated cone

linearization and completeness results for terminating transitive closure queries on spatial databases we study queries to spatial databases where spatial data are modeled as semi algebraic sets using the relational calculus with polynomial inequalities as a basic query language we work with the extension of the relational calculus with terminating transitive closures the main result is that this language can express the linearization of semialgebraic databases we also show that the sublanguage with linear inequalities only can express all computable queries on semilinear databases as a consequence of these results we obtain a completeness result for topological queries on semialgebraic databases

an optimal algorithm for intersecting line segments in the plane the main contribution of this work is an o n log n k time algorithm for computing all k intersections among n line segments in the plane this time complexity is easily shown to be optimal within the same asymptotic cost our algorithm can also construct the subdivision of the plane defined by the segments and compute which segment if any lies right above or below each intersection and each endpoint the algorithm has been implemented and performs very well the storage requirement is on the order of n k in the worst case but it is considerably lower in practice to analyze the complexity of the algorithm an amortization argument based on a new combinatorial theorem on line arrangements is used

tomographic image reconstruction based on a content adaptive mesh model in this paper we explore the use of a content adaptive mesh model camm for tomographic image reconstruction in the proposed framework the image to be reconstructed is first represented by a mesh model an efficient image description based on nonuniform sampling in the camm image samples represented as mesh nodes are placed most densely in image regions having fine detail tomographic image reconstruction in the mesh domain is performed by maximum likelihood ml or maximum a posteriori map estimation of the nodal values from the measured data a camm greatly reduces the number of unknown parameters to be determined leading to improved image quality and reduced computation time we demonstrated the method in our experiments using simulated gated single photon emission computed tomography spect cardiac perfusion images a channelized hotelling observer cho was used to evaluate the detectability of perfusion defects in the reconstructed images a task based measure of image quality a minimum description length mdl criterion was also used to evaluate the effect of the representation size in our application both mdl and cho suggested that the optimal number of mesh nodes is roughly five to seven times smaller than the number of projection bins when compared to several commonly used methods for image reconstruction the proposed approach achieved the best performance in terms of defect detection and computation time the research described in this paper establishes a foundation for future development of a four dimensional space time reconstruction framework for image sequences in which a built in deformable mesh model is used to track the image motion

an enhanced dmt optimality criterion for stbc schemes for asymmetric mimo systems for any nt transmit nr receive antenna nt nr multiple input multiple output mimo system in a quasi static rayleigh fading environment it was shown by elia that linear space time block code schemes lstbc schemes that have the nonvanishing determinant nvd property are diversity multiplexing gain tradeoff dmt optimal for arbitrary values of nr if they have a code rate of nt complex dimensions per channel use however for asymmetric mimo systems where with the exception of a few lstbc schemes it is unknown whether general lstbc schemes with nvd and a code rate of nr complex dimensions per channel use are dmt optimal in this paper an enhanced sufficient criterion for any stbc scheme to be dmt optimal is obtained and using this criterion it is established that any lstbc scheme with nvd and a code rate of min nt nr complex dimensions per channel use is dmt optimal this result settles the dmt optimality of several well known low ml decoding complexity lstbc schemes for certain asymmetric mimo systems

solution methods for one dimensional viscoelastic problems a recently developed differential methodology for solution of one dimensional nonlinear viscoelastic problems is presented itsing the example of an eccentrically loaded cantilever beamcolunm the results from the differential formulation are compared to results generated using a previously published integral solution technique it is shown that the results obtained from these distinct methodologies exhibit a surprising high degree of correlation with one another a discussion of the various factors affecting the numerical accuracy and rate of convergence of these two procedures is also included finally the influences of some higher order effects such as straining along the centroidal axis are discussed i introduction a number of nlethods are available to solve viscoelastic problenls in which the behavior of the material may be adequately characterized by a linear viscoelastic operator and where the deformation of the body is sufficiently small to allow the use of a linear kinematic formulation 1 2 commonly integral transform nethods separation of variables series expansions or other such techniques provide methodologies wherein exact closed form solutions may be derived when exact solutions cannot be obtained approximate techniques such as one proposed hy schapery 3j may provide an alternate approach the inclusion of nonlinear effects in the analysis significantly reduces the mathematical tractability of the problem these nonlinear influences can be induced by geometric factors resulting from the magnitude of the deformation or from gross rotation of cross sections alternatively nonlinearities in the material response may need be included to provide an accurate

the koiter newton approach using von karman kinematics for buckling analyses of imperfection sensitive structures the koiter newton approach is a reduced basis method for nonlinear structural analyses the method combines koiter s approach as a predictor step and the newton arc length technique as a corrector step to trace the entire load displacement equilibrium path of a structure in a step by step manner this computationally highly efficient and accurate solution method has recently demonstrated a superior performance in nonlinear analyses compared to standard techniques in this paper we propose an extension to buckling imperfection sensitivity analyses exploiting the method s stability and reliability we present two different modeling techniques for imperfection loads that both profit from the koiter newton approach reducing the initial computation steps to a small fraction thereof we introduce von karman kinematics which neglect some nonlinear terms of the green strain displacement relations to further increase the computational efficiency of the analysis by an essential reduction of the computation effort required to obtain higher order derivatives of the strain energy using various numerical examples and benchmark tests we demonstrate the overall high quality and performance of the proposed method

vibration and dynamic stability of a traveling sandwich beam the vibration and dynamic stability of a traveling sandwich beam are studied using the finite element method the damping layer is assumed to be linear viscoelastic and almost incompressible the extensional and shear moduli of the viscoelastic material are characterized by complex quantities complex eigenvalue problems are solved by the state space method and the natural frequencies and modal loss factors of the composite beam are extracted the effects of stiffness and thickness ratio of the viscoelastic and constrained layers on natural frequencies and modal loss factors are reported tension fluctuations are the dominant source of excitation in a traveling sandwich material and the regions of dynamic instability are determined by modified bolotin s method numerical results show that the constrained damping layer stabilizes the traveling sandwich beam

an optimal parallel algorithm for the visibility of a simple polygon from a point we present a parallel algorithm for computing the visible portion of a simple polygonal chain with 7i vertices from a point in the plane the algorithm runs in o logn time using 0 nf log n processors in the crew pram computational model and hence is asymptotically optimal

time discretized variational formulation of non smooth frictional contact the present work extends the non smooth contact class of algorithms introduced by kane et al to the case of friction the formulation specifically addresses contact geometries e g involving multiple collisions between tightly packed non smooth bodies for which neither normals nor gap functions can be properly defined a key aspect of the approach is that the incremental displacements follow from a minimum principle the objective function comprises terms which account for inertia strain energy contact friction and external forcing the euler lagrange equations corresponding to this extended variational principle are shown to be consistent with the equations of motion of solids in frictional contact in addition to its value as a basis for formulating numerical algorithms the variational framework offers theoretical advantages as regards the selection of trajectories in cases of non uniqueness we present numerical and analytical examples which demonstrate the good momentum and energy conservation characteristics of the numerical algorithms as well as the ability of the approach to account for stick and slip conditions

esflib a library to compute the element free galerkin shape functions we present a library for the computation of the shape functions for the element free galerkin efg method while the efg method is in many respects strikingly similar to the nite element method the construction of the shape functions for the efg method is much more complicated than in the fem thus the question arises whether it is possible to encapsulate the complexity of the shape function construction process while providing a simple interface to the shape function data we present a shape function library design which proved unrestrictive it can be used with almost any programming language without any special requirements on application program data structures highly congurable and robust

handwritten character recognition using a 2 layer random graph model by relaxation matching abstract a handwritten character recognizer is proposed in this paper first using components and strokes of a character as primitives a 2 layer attributed graph is constructed to represent the character next a relaxation matching technique is used in the learning stage to synthesize different attributed graphs which are due to writing variations and belong to a same character into a 2 layer random graph as a reference data model of the character and the relaxation matching technique is also used in the recognition stage to match the attributed graph of an input character to the random graph of each reference character then a similarity measure between an attributed graph and a random graph based on whether the attributed graph is a possible outcome of the random graph is presented in the recognition stage according to the similarity measure an input character can be classified into correct reference character experimental results are finally provided to show the effectiveness of the proposed approach

on line signature verification based on logarithmic spectrum in this paper an on line signature verification scheme based on similarity measurement of logarithmic spectrum is proposed the principal components of the logarithmic spectrum of each signature are extracted we then compute the similarity of logarithmic spectrum between input signature and the reference template by comparing the similarity of logarithmic spectrum with the verification threshold we can determine the authenticity of the input signature based on the experimentation the rates of false rejection errors and false acceptance errors are as low as 1 4 and 2 8 respectively this demonstrates the effectiveness of the proposed scheme

numerical differentiation for use in integrating unified constitutive equations the nonss method for numerically integrating time dependent constitutive equations requires evaluation of the partial derivatives of the constitutive equations a numerical method for evaluating these partial derivatives has been developed thus avoiding the cumbersome task of analytically re deriving the partial derivatives when improvements to the constitutive model are evaluated this method is based upon the adaptive finite difference algorithm published by stepleman and winarsky modifications of their algorithm are described that allow its efficient use within an extended version of nonss called nonss nd using the matmod constitutive equations the capabilities of nonss nd are demonstrated by comparing the results of a wide variety of deformation simulations with those of nonss for which the derivatives are computed from analytical solutions the numerical differentiation within nonss nd is shown to be sufficiently accurate for use in integrating time dependent constitutive equations but results in a 30 100 per cent increase in computation time multi element structural calculations are therefore found to be uneconomical but nonss nd is well suited for testing alternative constitutive models owing to its accuracy and its general applicability to highly non linear systems of differential equations

perfect space time codes for any number of antennas in a recent paper perfect n times n space time codes were introduced as the class of linear dispersion space time st codes having full rate nonvanishing determinant a signal constellation isomorphic to either the rectangular or hexagonal lattices in 2n 2 dimensions and uniform average transmitted energy per antenna consequence of these conditions include optimality of perfect codes with respect to the zheng tse diversity multiplexing gain tradeoff dmt as well as excellent low signal to noise ratio snr performance yet perfect space time codes have been constructed only for two three four and six transmit antennas in this paper we construct perfect codes for all channel dimensions present some additional attributes of this class of st codes and extend the notion of a perfect code to the rectangular case

the hamiltonian properties of supergrid graphs in this paper we first introduce a novel class of graphs namely supergrid supergrid graphs include grid graphs and triangular grid graphs as their subgraphs the hamiltonian cycle and path problems for grid graphs and triangular grid graphs were known to be np complete however they are unknown for supergrid graphs the hamiltonian cycle path problem on supergrid graphs can be applied to control the stitching traces of computerized sewing machines in this paper we will prove that the hamiltonian cycle problem for supergrid graphs is np complete it is easily derived from the hamiltonian cycle result that the hamiltonian path problem on supergrid graphs is also np complete we then show that two subclasses of supergrid graphs including rectangular parallelism and alphabet always contain hamiltonian cycles

computing farthest neighbors on a convex polytope let n be a set of n points in convex position in r3 the farthest point voronoi diagram of n partitions r3 into n convex cells we consider the intersection g n of the diagram with the boundary of the convex hull of n we give an algorithm that computes an implicit representation of g n in expected o n log2 n time more precisely we compute the combinatorial structure of g n the coordinates of its vertices and the equation of the plane defining each edge of g n the algorithm allows us to solve the all pairs farthest neighbor problem for n in expected time o n log2 n and to perform farthest neighbor queries on n in o log2 n time with high probability

icp registration using invariant features investigates the use of euclidean invariant features in a generalization of iterative closest point icp registration of range images pointwise correspondences are chosen as the closest point with respect to a weighted linear combination of positional and feature distances it is shown that under ideal noise free conditions correspondences formed using this distance function are correct more often than correspondences formed using the positional distance alone in addition monotonic convergence to at least a local minimum is shown to hold for this method when noise is present a method that automatically sets the optimal relative contribution of features and positions is described this method trades off the error in feature values due to noise against the error in positions due to misalignment experimental results suggest that using invariant features decreases the probability of being trapped in a local minimum and may be an effective solution for difficult range image registration problems where the scene is very small compared to the model

bounding solutions of geometrically nonlinear viscoelastic problems integral transform techniques such as the laplace transform provide simple and direct methods for solving viscoelastic problems formulated within a context of linear material response and using linear measures for deformation application of the transform operator reduces the governing linear integro differential equations to a set of algebraic relations between the transforms of the unknown functions the viscoelastic operators and the initial and boundary conditions inversion either directly or through the use of the appropriate convolution theorem provides the time domain response once the unknown functions have been expressed in terms of sums products or ratios of known transforms when exact inversion is not possible approximate techniques may provide accurate results the overall problem becomes substantially more complex when nonlinear effects must be included situations where a linear material constitutive law can still be productively employed but where the magnitude of the resulting time dependent deformations warrants the use of a nonlinear kinematic analysis are considered the governing equations will be nonlinear integro differential equations for this class of problems thus traditional as well as approximate techniques such as cited above cannot be employed since the transform of a nonlinear function is not explicitly expressible

space time codes from structured lattices we present constructions of space time st codes based on lattice coset coding first we focus on st code constructions for the short block length case i e when the block length is equal to or slightly larger than the number of transmit antennas we present constructions based on dense lattice packings and nested lattice voronoi shaping our codes achieve the optimal diversity multiplexing tradeoff dmt of quasi static multiple input multiple output mimo fading channels for any fading statistics and perform very well also at practical moderate values of signal to noise ratios snr then we extend the construction to the case of large block lengths by using trellis coset coding we provide constructions of trellis coded modulation tcm schemes that are endowed with good packing and shaping properties both short block and trellis constructions allow for a reduced complexity decoding algorithm based on minimum mean squared error generalized decision feedback equalizer mmse gdfe lattice decoding and a combination of this with a viterbi tcm decoder for the tcm case beyond the interesting algebraic structure we exhibit codes whose performance is among the state of the art considering codes with similar encoding decoding complexity

on fast decodable space time block codes we focus on full rate fast decodable space time block codes stbcs for 2 times 2 and 4times2 multiple input multiple output mimo transmission we first derive conditions and design criteria for reduced complexity maximum likelihood ml decodable 2times2 stbcs and we apply them to two families of codes that were recently discovered next we derive a novel reduced complexity 4times2 stbc and show that it outperforms all previously known codes with certain constellations

structure of a simple scheduling polyhedron in a one machine nonpreemptive scheduling problem the feasible schedules may be defined by the vector of the corresponding job completion times for given positive processing times the associated simple scheduling polyhedronp is the convex hull of these feasible completion time vectors the main result of this paper is a complete description of the minimal linear system definingp we also give a complete combinatorial description of the face lattice ofp and a simple o n logn separation algorithm this algorithm has potential usefulness in cutting plane type algorithms for more difficult scheduling problems

single symbol maximum likelihood decodable linear stbcs space time block codes stbcs from orthogonal designs ods and coordinate interleaved orthogonal designs ciod have been attracting wider attention due to their amenability for fast single symbol maximum likelihood ml decoding and full rate with full rank over quasi static fading channels however these codes are instances of single symbol decodable codes and it is natural to ask if there exist codes other than stbcs form ods and ciods that allow single symbol decoding in this paper the above question is answered in the affirmative by characterizing all linear stbcs that allow single symbol ml decoding not necessarily full diversity over quasi static fading channels calling them single symbol decodable designs sdd the class sdd includes ods and ciods as proper subclasses further among the sdd a class of those that offer full diversity called full rank sdd fsdd are characterized and classified we then concentrate on square designs and derive the maximal rate for square fsdds using a constructional proof it follows that 1 except for n 2 square complex ods are not maximal rate and 2 a rate one square fsdd exist only for two and four transmit antennas for nonsquare designs generalized coordinate interleaved orthogonal designs a superset of ciods are presented and analyzed finally for rapid fading channels an equivalent matrix channel representation is developed which allows the results of quasi static fading channels to be applied to rapid fading channels using this representation we show that for rapid fading channels the rate of single symbol decodable stbcs are independent of the number of transmit antennas and inversely proportional to the block length of the code significantly the ciod for two transmit antennas is the only stbc that is single symbol decodable over both quasi static and rapid fading channels

optimal separable algorithms to compute the reverse euclidean distance transformation and discrete medial axis in arbitrary dimension in binary images the distance transformation dt and the geometrical skeleton extraction are classic tools for shape analysis in this paper we present time optimal algorithms to solve the reverse euclidean distance transformation and the reversible medial axis extraction problems for d dimensional images we also present a d dimensional medial axis filtering process that allows us to control the quality of the reconstructed shape

voronoi diagram and medial axis algorithm for planar domains with curved boundaries i theoretical foundations in this first installment of a two part paper the underlying theory for an algorithm that computes the voronoi diagram and medial axis of a planar domain bounded by free form polynomial or rational curve segments is presented an incremental approach to computing the voronoi diagram is used wherein a single boundary segment is added to an existing boundary segment set at each step the introduction of each new segment entails modifying the voronoi regions of the existing boundary segments and constructing the voronoi region of the new segment we accomplish this by i computing the bisector of the new segment with each of the current boundary segments ii updating the voronoi regions of the current boundary segments by partitioning them with these bisectors and iii constructing the voronoi region of the new segment as a union of regions obtained from the partitioning in ii when all boundary segments are included and their voronoi regions have been constructed the voronoi diagram of the boundary is obtained as the union of the voronoi polygons for each boundary segment to construct the medial axis of a planar domain we first compute the voronoi diagram of its boundary the medial axis is then obtained from the voronoi diagram by i removing certain edges of the voronoi diagram that do not belong to the medial axis and ii adding certain edges that do belong to the medial axis but are absent from the voronoi diagram unambiguous characterizations for edges in both these categories are given details of algorithms based on this theory are deferred to the second installment of this two part paper

fast decodable mido codes with large coding gain in this paper a new method is proposed to obtain full diversity rate 2 rate of two complex symbols per channel use space time block codes stbcs that are full rate for multiple input double output mido systems using this method rate 2 stbcs for 4 2 6 2 8 2 and 12 2 systems are constructed and these stbcs are fast ml decodable have large coding gains and stbc schemes consisting of these stbcs have a non vanishing determinant nvd so that they are dmt optimal for their respective mido systems it is also shown that the srinath rajan code for the 4 2 system which has the lowest ml decoding complexity among known rate 2 stbcs for the 4 2 mido system with a large coding gain for 4 16 qam has the same algebraic structure as the stbc constructed in this paper for the 4 2 system this also settles in positive a previous conjecture that the stbc scheme that is based on the srinath rajan code has the nvd property and hence is dmt optimal for the 4 2 system

surface bounded growth modeling applied to human mandibles from a set of longitudinal three dimensional scans of the same anatomical structure the authors have accurately modeled the temporal shape and size changes using a linear shape model on a total of 31 computed tomography scans of the mandible from six patients 14 851 semilandmarks are found automatically using shape features and a new algorithm called geometry constrained diffusion the semilandmarks are mapped into procrustes space principal component analysis extracts a one dimensional subspace which is used to construct a linear growth model the worst case mean modeling error in a cross validation study is 3 7 mm

improved perfect space time block codes perfect space time block codes stbcs are based on four design criteria full rateness nonvanishing determinant cubic shaping and uniform average transmitted energy per antenna per time slot cubic shaping and transmission at uniform average energy per antenna per time slot are important from the perspective of energy efficiency of stbcs the shaping criterion demands that the generator matrix of the lattice from which each layer of the perfect stbc is carved be unitary in this paper it is shown that unitariness is not a necessary requirement for energy efficiency in the context of space time coding with finite input constellations and an alternative criterion is provided that enables one to obtain full rate rate of nt complex symbols per channel use for an nt transmit antenna system stbcs with larger normalized minimum determinants than the perfect stbcs further two such stbcs one each for 4 and 6 transmit antennas are presented and they are shown to have larger normalized minimum determinants than the comparable perfect stbcs which hitherto had the best known normalized minimum determinants

computational geometry an introduction from the reviews this book offers a coherent treatment at the graduate textbook level of the field that has come to be known in the last decade or so as computational geometry the book is well organized and lucidly written a timely contribution by two founders of the field it clearly demonstrates that computational geometry in the plane is now a fairly well understood branch of computer science and mathematics it also points the way to the solution of the more challenging problems in dimensions higher than two

face recognition approach based on rank correlation of gabor filtered images face recognition is challenging because variations can be introduced to the pattern of a face by varying pose lighting scale and expression a new face recognition approach using rank correlation of gabor filtered images is presented using this technique gabor filters of different sizes and orientations are applied on images before using rank correlation for matching the face representation the representation used for each face is computed from the gabor filtered images and the original image although training requires a fairly substantial length of time the computation time required for recognition is very short recognition rates ranging between 83 5 and 96 are obtained using the at t formerly orl database using different permutations of 5 and 9 training images per subject in addition the effect of pose variation on the recognition system is systematically determined using images from the umist database

construction methods for asymmetric and multiblock space time codes in this paper the need for the construction of asymmetric and multiblock space time codes is discussed above the trivial puncturing method i e switching off the extra layers in the symmetric multiple input multiple output mimo setting two more sophisticated asymmetric construction methods are proposed the first method called the block diagonal method bdm can be converted to produce multiblock space time codes that achieve the diversity multiplexing tradeoff dmt it is also shown that maximizing the density of the newly proposed block diagonal asymmetric space time ast codes is equivalent to minimizing the discriminant of a certain order a result that also holds as such for the multiblock codes an implicit lower bound for the density is provided and made explicit for an important special case that contains e g the systems equipped with 4tx 2rx antennas further an explicit scheme achieving the bound is given another method proposed here is the smart puncturing method spm that generalizes the subfield construction method proposed in earlier work by hollanti and ranto and applies to any number of transmitting and lesser receiving antennas the use of the general methods is demonstrated by building explicit sphere decodable codes using different cyclic division algebras cdas computer simulations verify that the newly proposed methods can compete with the trivial puncturing method and in some cases clearly outperform it the conquering construction exploiting maximal orders improves upon the punctured perfect code and the djabba code as well as the icosian code also extensive dmt analysis is provided

geometric classification of triangulations and their enumeration in a convex polygon abstract triangulation of polygons is a classical problem in computational geometry for an arbitrary polygon the triangulation may depend on its shape in this paper we describe a new geometric classification of triangulations of a convex polygon and then derive expressions for counting the number of nonisomorphic triangulations in each class

the multipoint morisita index for the analysis of spatial patterns abstract in many fields the spatial clustering of sampled data points has significant consequences therefore several indices have been proposed to assess the degree of clustering affecting datasets e g the morisita index ripley s k function and renyi s information the classical morisita index measures how many times it is more likely to randomly select two sampled points from the same quadrat the dataset is covered by a regular grid of changing size than it would be in the case of a random distribution generated from a poisson process the multipoint version takes into account m points with m 2 the present research deals with a new development of the multipoint morisita index m morisita which is directly related to multifractality this relationship to multifractality is first demonstrated and highlighted on a mathematical multifractal set then the new version of the m morisita index is adapted to the characterization of environmental monitoring network clustering and finally an additional extension the functional m morisita index is presented for the detection of structures in monitored phenomena

reduction of bivariate polynomials from convex dense to dense with application to factorizations in this article we present a new algorithm for reducing the usual sparse bivariate factorization problems to the dense case this reduction simply consists in computing an invertible monomial transformation that produces a polynomial with a dense size of the same order of magnitude as the size of the integral convex hull of the support of the input polynomial this approach turns out to be very efficient in practice as demonstrated with our implementation

the stability of viscoelastic perfect columns a dynamic approach abstract in this article the dynamic approach for stability analysis is used to obtain an approximate closed form expression for the viscoelastic critical load of perfect columns made of a linear three element model material it is shown that a critical time for such structures cannot be defined independently of the type and nature of the disturbance causing instability in the sense of lyapunov criteria are introduced to define a critical time that is independent of the magnitude of the disturbance the nature of the disturbance whether it is dynamic or static however still has a significant influence on the value of such a critical time

multigroup decodable stbcs from clifford algebras a space time block code stbc in k symbols variables is called a g group decodable stbc if its maximum likelihood ml decoding metric can be written as a sum of g terms for some positive integer g greater than one such that each term is a function of a subset of the k variables and each variable appears in only one term in this paper we provide a general structure of the weight matrices of multigroup decodable codes using clifford algebras without assuming that the number of variables in each group is the same a method of explicitly constructing the weight matrices of full diversity delay optimal multigroup decodable codes is presented for arbitrary number of antennas for the special case of 2 a number of transmit antennas we construct two subclass of codes 1 a class of 2 a group decodable codes with rate a 2 a 1 which is equivalently a class of single symbol decodable codes and 2 a class of 2a 2 group decodable codes with rate a 1 2 a 2 i e a class of double symbol decodable codes

elastoviscoplastic buckling behavior of simply supported columns the elastoviscoplastic buckling behavior of simply supported columns is investigated the analysis is based on an incremental formulation that can deal with nonlinear geometrical and constitutive relations both vis coelastic and viscoplastic models are discussed and bodner partom s constitutive equations are used in the viscoplastic case a numerical example is given to demonstrate the buckling behavior of columns under various temperature and loading levels he problem of stability of columns is one of the fun damental problems in mechanics the earliest work can be traced to euler who first introduced the elastic critical load for slender columns that is still widely used in engineer ing in the last four decades considerable attention has been focused on the buckling behavior of viscoelastic materials the reason for continuing research is that current engineering requires not only the establishment of critical loads but also the details of the response of the structure under complex loading conditions which include elevated temperature time dependent behavior etc in the high temperature case life prediction is of primary importance with regard to solution methodology stubstad and simitses1 proposed a differential formulation for one dimensional prob lems that involves nonlinear kinematic and nonlinear material effects the method can decouple the temporal and spatial dependence so that the general solution can be treated as a sequential combination of solutions of a nonlinear boundary value problem and a nonlinear initial value problem the two groups can then be solved separately using an integral trans form method dost and glockner2 investigated the dynamic stability of viscoelastic perfect columns subsequently szysz kowski and glockner3 studied the dynamic response of im perfect viscoelastic columns with laplace transforms being used in both references 2 3 because of the complexity of the inverse of the transform some numerical calculations were employed in order to obtain the final results vinogradov4 suggested a quasielastic method to study the behavior of vis coelastic beam columns and gave some comparisons with an alytical results in order to solve the viscoelastic problem efficiently stubstad and simitses5 introduced a bounding so lution scheme for geometrically nonlinear viscoelastic prob lems both upper and lower bounds were established by bounding the convolution integral of the governing nonlinear volterra type integral equation this method can substan tially reduce the computational effort required for numerical evaluation

on pointers versus addresses summary what is the cost of random access to memory we address this fnndamental problem by studying the simulation of random addressing by a machine which lacks it a pointer machine the model we define allows the use of a data type of our choice a ram program of time i and space s can be simulated in o t log a time using a tree however this is not an obvious lower bound since a high level data type may allow us to encode the data in a more economic way our major contribution is the formalization of incompressibility for general data types the definition extends a similar property 01 strings that underlies the theory of kolmogorov complexity the main theorem states that for all incompressible data types an q t log s lower bound holds incompressibility trivially holds for strings but is harder to prove for a powerlul data type we prove incompressibility for the real numbers with a set of primitives which includes all functions which are continuously differentiable except on a countable closed set this may be the richest set of operations considered in a lower bound proof the prooi relies on the implicit functions theorem and baire s category theorem we also show that the integers with arithmetic x and lz zj any boolean operations and left shift are incompressible the inclusion of right shift reverses the situation and we obtain an o tq s upper bound where s is a functional inverse of ackermann s function

iterated space time code constructions from cyclic algebras we propose a full rate iterative space time code construction which uses an algebraic space time code in order to design a new space time code of double the code rank we give a condition for determining whether the resulting codes satisfy the full diversity property and study their maximum likelihood decoding complexity with respect to sphere decoding particular emphasis is given to the asymmetric multiple input double output codes in the process we derive an interesting way of obtaining division algebras and study their centers and maximal subfields

optimal expected case planar point location point location is the problem of preprocessing a planar polygonal subdivision s of size n into a data structure in order to determine efficiently the cell of the subdivision that contains a given query point we consider this problem from the perspective of expected query time we are given the probabilities p z that the query point lies within each cell z in s the entropy h of the resulting discrete probability distribution is the dominant term in the lower bound on the expected case query time we show that it is possible to achieve query time h o sqrt h 1 with space o n which is optimal up to lower order terms in the query time we extend this result to subdivisions with convex cells assuming a uniform query distribution within each cell in order to achieve space efficiency we introduce the concept of entropy preserving cuttings

close point spatial tests and their application to random number generators we study statistical tests of uniformity based on thel p distances between them nearest pairs of points forn points generated uniformly over thek dimensional unit hypercube or unit torus the number of distinct pairs at distance no more thant fort 0 is a stochastic process whose initial part after an appropriate transformation and asn 8 is asymptotically a poisson process with unit rate convergence to this asymptotic is slow in the hypercube as soon ask exceeds 2 or 3 due to edge effects but is reasonably fast in the torus we look at the quality of approximation of the exact distributions of the tests statistics by their asymptotic distributions discuss computational issues and apply the tests to random number generators linear congruential generators fail decisively certain variants of the tests as soon asn approaches the square root of the period length

four group decodable space time block codes two new rate one full diversity space time block codes stbcs are proposed they are characterized by the lowest decoding complexity among the known rate one stbc arising due to the complete separability of the transmitted symbols into four groups for maximum likelihood detection the first and the second codes are delay optimal if the number of transmit antennas is a power of 2 and even respectively the exact pairwise error probability is derived to allow for the performance optimization of the two codes compared with existing low decoding complexity stbc the two new codes offer several advantages such as higher code rate lower encoding decoding delay and complexity lower peak to average power ratio and better performance

a simple algorithm for nearest neighbor search in high dimensions the problem of finding the closest point in high dimensional spaces is common in pattern recognition unfortunately the complexity of most existing search algorithms such as k d tree and r tree grows exponentially with dimension making them impractical for dimensionality above 15 in nearly all applications the closest point is of interest only if it lies within a user specified distance spl epsiv we present a simple and practical algorithm to efficiently search for the nearest neighbor within euclidean distance spl epsiv the use of projection search combined with a novel data structure dramatically improves performance in high dimensions a complexity analysis is presented which helps to automatically determine spl epsiv in structured problems a comprehensive set of benchmarks clearly shows the superiority of the proposed algorithm for a variety of structured and unstructured search problems object recognition is demonstrated as an example application the simplicity of the algorithm makes it possible to construct an inexpensive hardware search engine which can be 100 times faster than its software equivalent a c implementation of our algorithm is available

perfect space time block codes in this paper we introduce the notion of perfect space time block codes stbcs these codes have full rate full diversity nonvanishing constant minimum determinant for increasing spectral efficiency uniform average transmitted energy per antenna and good shaping we present algebraic constructions of perfect stbcs for 2 3 4 and 6 antennas

a fuzzy approach to visibility maps creation over digital terrains the construction of inter visibility models for a terrain is an important tool for environmental and animal studies efficient crisp algorithms are known to address this problem directly on a raster digital elevation map dem of the study area randomness plays an important role several factors like vegetation height of the observer presence of unmapped human artifacts etc cannot be modeled in advance into a traditional dem model we propose the use of fuzzy terrain models to incorporate uncertainty and unpredictable variability of the landscape accordingly we propose a suitable definition of visibility to obtain intervisibility models an algorithm that uses approximation of fuzzy numbers with a discrete family of intervals is proposed the proposed model produces sound results and can be extended to model complex situations like triangulating a radio source using two directional receivers all of these situations are difficult to manage with a precise stochastic model and are a frequent occurrence in animal behavior studies under field conditions the proposed techniques have been implemented inside a user friendly software package that may easily exchange data with the most common giss we report also some experimental results obtained using over artificial terrain under controlled noise and under sampling conditions

maximal orders in the design of dense space time lattice codes in this paper we construct explicit rate one full diversity geometrically dense matrix lattices with large nonvanishing determinants nvds for four transmit antenna multiple input single output miso space time st applications the constructions are based on the theory of rings of algebraic integers and related subrings of the hamiltonian quaternions and can be extended to a larger number of tx antennas the usage of ideals guarantees an nvd larger than one and an easy way to present the exact proofs for the minimum determinants the idea of finding denser sublattices within a given division algebra is then generalized to a multiple input multiple output mimo case with an arbitrary number of tx antennas by using the theory of cyclic division algebras cdas and maximal orders it is also shown that the explicit constructions in this paper all have a simple decoding method based on sphere decoding related to the decoding complexity the notion of sensitivity is introduced and experimental evidence indicating a connection between sensitivity decoding complexity and performance is provided simulations in a quasi static rayleigh fading channel show that our dense quaternionic constructions outperform both the earlier rectangular lattices and the rotated quasi orthogonal abba lattice as well as the diagonal algebraic space time dast lattice we also show that our quaternionic lattice is better than the dast lattice in terms of the diversity multiplexing gain tradeoff dmt

multigroup ml decodable collocated and distributed space time block codes in this paper collocated and distributed space time block codes dstbcs which admit multigroup maximum likelihood ml decoding are studied first the collocated case is considered and the problem of constructing space time block codes stbcs which optimally tradeoff rate and ml decoding complexity is posed recently sufficient conditions for multigroup ml decodability have been provided in the literature and codes meeting these sufficient conditions were called clifford unitary weight cuw stbcs an algebraic framework based on extended clifford algebras ecas is proposed to study cuw stbcs and using this framework the optimal tradeoff between rate and ml decoding complexity of cuw stbcs is obtained for few specific cases code constructions meeting this tradeoff optimally are also provided the paper then focuses on multigroup ml decodable dstbcs for application in synchronous wireless relay networks and three constructions of four group ml decodable dstbcs are provided finally the orthogonal frequency division multiplexing ofdm based alamouti space time coded scheme proposed by li xia for a 2 relay asynchronous relay network is extended to a more general transmission scheme that can achieve full asynchronous cooperative diversity for arbitrary number of relays it is then shown how differential encoding at the source can be combined with the proposed transmission scheme to arrive at a new transmission scheme that can achieve full cooperative diversity in asynchronous wireless relay networks with no channel information and also no timing error knowledge at the destination node four group decodable dstbcs applicable in the proposed ofdm based transmission scheme are also given

a local convergence proof for the minvar algorithm for computing continuous piecewise linear approximations the class of continuous piecewise linear pl functions represents a useful family of approximants because invertibility can be readily imposed and if a pl function is invertible then it can be inverted in closed form many applications arising for example in control systems and robotics involve the simultaneous construction of a forward and inverse system model from data most approximation techniques require that separate forward and inverse models be trained whereas an invertible continuous pl affords simultaneously the forward and inverse system model in a single representation the minvar algorithm computes a continuous pl approximation to data local convergence of minvar is proven for the case when the data generating function is itself a pl function and available directly rather than through data

a consistent formulation for the integration of combined plasticity and creep within the context of the consistent tangent update this paper outlines a stress update algorithm for combined creep and plasticity the algorithm is implicit providing unconditional stability and utilizes local newton iteration to solve scalar forms of the coupled constitutive equations for the creep and plastic strain increments the tangent for the local iteration is obtained accurately providing quadratic convergence at the gauss point level quadratic convergence of the global iteration procedure is also maintained using an explicitly derived consistent tangent for combined plasticity and creep further combination with an automatic time stepping scheme provides an efficient stable accurate and robust computational algorithm the algorithm has been implemented in the general purpose fe package lusas1

explicit space time codes achieving the diversity multiplexing gain tradeoff a recent result of zheng and tse states that over a quasi static channel there exists a fundamental tradeoff referred to as the diversity multiplexing gain d mg tradeoff between the spatial multiplexing gain and the diversity gain that can be simultaneously achieved by a space time st code this tradeoff is precisely known in the case of independent and identically distributed i i d rayleigh fading for tgesnt nr 1 where t is the number of time slots over which coding takes place and nt nr are the number of transmit and receive antennas respectively for t nt case we present two general techniques for building d mg optimal rectangular st codes from their square counterparts a byproduct of our results establishes that the d mg tradeoff for all tgesnt is the same as that previously known to hold for tgesnt n r 1

multidimensional search trees that provide new types of memory reductions an orthogonal query that asks to aggregate the set of records in k dimensional box regions is studied and it is shown that space o n log n log log n k 1 makes possible a combined time complexity o log k n for retrievals insertions and deletions

poisson polyhedra in high dimensions abstract the zero cell of a parametric class of random hyperplane tessellations depending on a distance exponent and an intensity parameter is investigated as the space dimension tends to infinity the model includes the zero cell of stationary and isotropic poisson hyperplane tessellations as well as the typical cell of a stationary poisson voronoi tessellation as special cases it is shown that asymptotically in the space dimension with overwhelming probability these cells satisfy the hyperplane conjecture if the distance exponent and the intensity parameter are suitably chosen dimension dependent functions also the high dimensional limits of the mean number of faces are explored and the asymptotic behaviour of an isoperimetric ratio is analysed in the background are new identities linking the f vector of the zero cell to certain dual intrinsic volumes

on the dynamic stability of viscoelastic perfect columns abstract the dynamic stability of simple supported perfect columns made of a linearly viscoelastic material and subjected to an axial compressive load p smaller than the classical euler elastic buckling load pe is examined the solution to the integro differential equation is obtained by means of laplace transforms in addition an approximate solution is also derived by adopting the approximation technique introduced in 1 the results are applied to a simple three element model viscoelastic column