a pyramid vector quantizer the geometric properties of a memoryless laplacian source are presented and used to establish a source coding theorem motivated by this geometric structure a pyramid vector quantizer pvq is developed for arbitrary vector dimension the pvq is based on the cubic lattice points that lie on the surface of an l dimensional pyramid and has simple encoding and decoding algorithms a product code version of the pvq is developed and generalized to apply to a variety of sources analytical expressions are derived for the pvq mean square error mse and simulation results are presented for pvq encoding of several memoryless sources for large rate and dimension pvq encoding of memoryless laplacian gamma and gaussian sources provides rose improvements of 5 64 8 40 and 2 39 db respectively over the corresponding optimum scalar quantizer although suboptimum in a rate distortion sense because the pvq can encode large dimensional vectors it offers significant reduction in rose distortion compared with the optimum lloyd max scalar quantizer and provides an attractive alternative to currently available vector quantizers

lattice basis reduction improved practical algorithms and solving subset sum problems we report on improved practical algorithms for lattice basis reduction we present a variant of the l3 algorithm with deep insertions and a practical algorithm for blockwise korkine zolotarev reduction a concept extending l3 reduction that has been introduced by schnorr 1987 empirical tests show that the strongest of these algorithms solves almost all subset sum problems with up to 58 random weights of arbitrary bit length within at most a few hours on a unisys 6000 70 or within a couple of minutes on a sparc 2 computer

selecting wavelengths for least squares range estimation we consider the problem of estimating the distance or range between two locations by measuring the phase of multiple sinusoidal signals transmitted between the locations traditional estimators developed for optical interferometry include the beat wavelength and excess fractions methods more recently estimators based on the chinese remainder theorem crt and least squares have appeared recent research suggests the least squares estimator to be the most accurate in many cases the accuracy of all of these range estimators depends upon the wavelengths chosen this leads to the problem of selecting wavelengths that maximize accuracy procedures for selecting wavelengths for the beat wavelength and excess fractions methods have previously been described but procedures for the crt and least squares estimators are yet to be developed in this paper we develop an algorithm to automatically select wavelengths for use with the least square range estimator the algorithm minimizes an optimization criterion connected with the mean square error interesting properties of a particular class of lattices simplify the criterion allowing minimization by depth first search monte carlo simulations indicate that wavelengths that minimize the criterion can result in considerably more accurate range estimates than wavelengths selected by ad hoc means

a generalized approach for implementing general boundary conditions in the gdq free vibration analysis of plates abstract a new approach for implementing general boundary conditions in the gdq free vibration analysis of rectangular plates is presented in this paper the proposed approach directly couples the boundary conditions with the governing equations and is referred to as the cbcge approach it can be applied to any combination of simply supported clamped and free boundary conditions the generality and accuracy of the cbcge approach are demonstrated through its application to the vibration analysis of a rectangular plate with various combinations of free edges and corners in particular the effect of grid point distribution on the numerical results for plates with free boundary conditions is discussed and a new grid point distribution is suggested

free vibration analysis of plates using least square based finite difference method in this paper we apply the two dimensional least square based finite difference lsfd method for solving free vibration problems of isotropic thin arbitrarily shaped plates with simply supported and clamped edges using the chain rule we show how the fourth order derivatives of the plate governing equation can be discretized in two or three steps as well as how the boundary conditions can be implemented directly into the governing equation by analyzing vibrating plates of various shapes and comparing the solutions obtained against existing results we clearly demonstrate the effectiveness of lsfd as a mesh free method for computing vibration frequencies of generally shaped plates accurately

an eight node curvilinear differential quadrature formulation for reissner mindlin plates a first endeavour to exploit the differential quadrature dq method as a simple accurate and efficient numerical technique for the bending of quadrilateral reissner mindlin plates with curvilinear boundaries is made the curvilinear quadrilateral is mapped onto a square domain by using the geometric coordinate transformation this geometric mapping technique is employed to transform the governing differential equations and boundary conditions of the problem from the physical domain onto the computational domain the dq procedures are then applied to discretise the transformed set of differential equations and boundary conditions into a set of linear algebraic equations the numerical solutions to the problem are obtained by solving this set of linear algebraic equations examples illustrating the accuracy and convergence of the dq method for reissner mindlin plates with curvilinear boundaries are presented the great development potential of the dq method as an alternative to other approximate techniques for solving problems related to engineering sciences and structural mechanics is exploited in the present study

weak form quadrature element method for accurate free vibration analysis of thin skew plates in this paper a novel weak form quadrature element method qem is proposed for accurate free transverse vibration analysis of thin isotropic skew plates with general boundary conditions in the formulation of the stiffness matrix the first and second order derivatives of the shape functions at integration points are computed explicitly by using the differential quadrature rule this leads to a great simplicity in formulating an n n node quadrature plate element and a large reduction of programming effort different from the existing weak form quadrature element method or differential quadrature finite element method the element nodes can be either the same or different from the integration points convergence studies are performed free vibration of skew thin plates with various skew angles and different combinations of boundary conditions is analyzed it is shown that although the assumed displacement field does not explicitly consider the bending moment singularities at the obtuse angles the proposed qem can yield accurate frequencies even for the thin isotropic skew plate with large skew angles

bending and buckling analysis of antisymmetric laminates using the moving least square differential quadrature method in this paper the moving least square differential quadrature mlsdq method is employed to bending and buckling analysis of antisymmetric thick laminates based on the first order shear deformation theory the displacement and rotation components of the plate are independently assumed with the centered moving least square mls approximation within each domain of influence the weighting coefficients are associated with the nodal partial derivatives which are calculated through a fast computation procedure together with the mls nodal shape functions numerical examples are illustrated to study the accuracy stability and convergence of the mlsdq method effects of support size order of the complete basis functions and node irregularity on the numerical accuracy are investigated typical computational results of displacements stresses and critical buckling loads of various laminated plates are presented and compared with the available exact and finite element solutions

a fast encoding method for lattice codes and quantizers in an earlier paper the authors described a very fast method which for the root lattices a n d n e n their duals and certain other lattices finds the closest lattice point to an arbitrary point of the underlying space if the lattices are used as codes for a gaussian channel the algorithm provides a fast decoding procedure or if they are used as vector quantizers the algorithm performs the analog to digital conversion efficiently the present paper offers a solution to the inverse problem for the same lattices the encoding problem for channel codes or the digital to analog part of quantizing namely given an integer k to find the kth code vector and to the closely related problem of finding the index k of a given code vector

a new approach to the differential quadrature method for fourth order equations a generalized and more complete methodology for treating boundary conditions in the differential quadrature method dqm is presented this improved approach eliminates the deficiencies of the type grid arrangement which represents an approximation by applying the boundary conditions exactly two kinds of basis functions chebyshev and lagrange are used for concept demonstration it is found that the new approach cures most deficiencies of the current dqm 1997 by john wiley sons ltd

partial marginalization soft mimo detection with higher order constellations a new method for multiple input multiple output mimo detection with soft output the partial marginalization pm algorithm was recently proposed advantages of the method are that it is straightforward to parallelize and that it offers a fully predictable runtime pm trades performance for computational complexity via a user defined parameter in the limit of high computational complexity the algorithm becomes the map demodulator the pm algorithm also works with soft input but until now it has been unclear how to apply it for other modulation formats than binary phase shift keying bpsk per real dimension in this correspondence we explain how to extend pm with soft input to general signaling constellations while maintaining the low complexity advantage of the original algorithm

the hardness of the closest vector problem with preprocessing we give a new simple proof of the np hardness of the closest vector problem in addition to being much simpler than all previously known proofs the new proof yields new interesting results about the complexity of the closest vector problem with preprocessing this is a variant of the closest vector problem in which the lattice is specified in advance and can be preprocessed for an arbitrarily long amount of time before the target vector is revealed we show that there are lattices for which the closest vector problem remains hard regardless of the amount of preprocessing

periodic fixed points of backlund transformations and the korteweg de vries equation a new method for studying integrable systems based on the periodic fixed points of backlund transformations bt s is presented normally the bt maps an old solution into a new solution and requires a known seed solution to get started besides this limitation it can also be difficult to qualitatively classify the result of applying the bt several times to a known solution by studying the periodic fixed points of the bt regarded as a nonlinear map in a function space integrable systems of equations of finite degree equal to the order of the fixed point and a method for the systematic classification of the solutions of the original system are obtained

exact solutions of the multidimensional derivative nonlinear schrodinger equation for many body systems of criticality the authors investigate the problem of strongly interacting many body systems near criticality as recently described in terms of nonlinear dynamics by dixon and tuszynski 1989 they show that in the first order of approximation the equation of motion for the order parameter can be mapped on the derivative nonlinear schrodinger equation and thus can be solved exactly they solve the equations both in the absence and presence of current densities

elastoplastic buckling analysis of thick rectangular plates by using the differential quadrature method the paper investigates the elastoplastic buckling behavior of thick rectangular plates by using the differential quadrature dq method mindlin plate theory is adopted to take the transverse shear effect into considerations both incremental theory and deformation theory are employed due to the material non linearity iteration processes are involved for obtaining solutions detailed methodology and procedures are derived the elastoplastic buckling behavior of thick rectangular plates with ten combinations of boundary conditions and under various loadings is studied to verify the dq solution procedures dq results are compared with existing analytical solutions for plates with two boundaries simply supported and the others simply supported clamped or free then the dq method is used to obtain solutions of rectangular thick plates with other combinations of boundary conditions since no analytical solutions for such cases are available the buckling loads obtained by the dq method could serve as a reference the phenomenon reported in the literature namely the deformation theory generally gives consistently lower buckling loads than the incremental theory and large discrepancy in predictions between the two theories exists with increasing of plate thickness e s 0 and c in the ramberg osgood relations is also observed for the cases studied herein apart from the phenomenon reported earlier by durban that deformation theory predicted a progressively lower in plane shear modulus as the level of plasticity increased thus predicted lower buckling loads another reason is given herein to explain the large discrepancy in predictions for thicker plates with the deformation theory and incremental theory

mimo detection for high order qam based on a gaussian tree approximation this paper proposes a new detection algorithm for mimo communication systems employing high order qam constellations the factor graph that corresponds to this problem is very loopy in fact it is a complete graph hence a straightforward application of the belief propagation bp algorithm yields very poor results our algorithm is based on an optimal tree approximation of the gaussian density of the unconstrained linear system the finite set constraint is then applied to obtain a cycle free discrete distribution simulation results show that even though the approximation is not directly applied to the exact discrete distribution applying the bp algorithm to the cycle free factor graph outperforms current methods in terms of both performance and complexity the improved performance of the proposed algorithm is demonstrated on the problem of mimo detection

on random rotations diversity and minimum mse decoding of lattices we establish a simple relation between high diversity multidimensional rotations obtained from totally complex cyclotomic fields and the discrete fourier transform the diversity distribution of an hadamard like random rotation is derived analytically it is shown that a random multidimensional rotation exhibits an excellent diversity distribution and can be combined to quadrature amplitude modulation qam constellations to combat channel fading we also describe a mean square error mse universal lattice decoder suitable for large dimensions up to 1024 the mse criterion treats the lattice structure as intersymbol interference the universal decoder is applied to both gaussian and rayleigh fading channels to decode dense lattice sphere packings and rotated cubic constellations respectively

the icosian code and the e_8 lattice a new 4 times 4 space time code with nonvanishing determinant this paper introduces a new rate 2 full diversity space time code for four transmit antennas and one receive antenna the 4times4 codeword matrix consists of four 2times2 alamouti blocks with entries from q i radic5 and these blocks can be viewed as quaternions which in turn represent rotations in r 3 the alamouti blocks that appear in a codeword are drawn from the icosian ring consisting of all linear combinations of 120 basic rotations corresponding to symmetries of the icosahedron this algebraic structure is different from the golden code but the complex entries are taken from a common underlying field the minimum determinant is bounded below by a constant that is independent of the signal constellation and the new code admits a simple decoding scheme that makes use of a geometric correspondence between the icosian ring and the e 8 lattice

mathematical modeling of vibration problem of nano sized annular sector plates using the nonlocal continuum theory via eight node discrete singular convolution transformation abstract in the present study free vibration analysis of nano sized annular sector plate is analyzed using the nonlocal continuum theory the method of discrete singular convolution dsc is used for numerical computations firstly equation of motion of thin plates is formulated via nonlocal elasticity then irregular physical domain is transformed into a rectangular domain by using geometric coordinate transformation the dsc procedures are then applied to discretization of the transformed set of governing equations and related boundary conditions the effects of nonlocal parameter mode numbers sector angle and radius ratio on the vibration frequencies are investigated in detail it is seen that the size effects are significant in vibration analysis of nano scaled annular sector plates and need to be included in the mechanical model

on the equivalence of generalized differential quadrature and highest order finite difference scheme abstract it is shown in this paper that the global method of generalized differential quadrature gdq is equivalent to the highest order finite difference scheme which is the extension of low order finite difference schemes the effect of grid point distribution on the accuracy of numerical results is studied by two numerical examples the results showed that the ability to alter the grid point distribution in the gdq method is an important feature to obtain accurate numerical solutions in certain problems

static analysis of structures by the quadrature element method qem abstract the differential quadrature method dqm is an alternative discrete approach to solving directly the governing equations of engineering and mathematical physics since the dqm does not require the derivation of the weak forms of the governing equations like the fem it can greatly reduce formulation efforts in higher order approximation the dqm has been applied in the past to the analyses of various single structural components such as bars beams membranes and plates all of the component analyses yielded good to excellent results however previous studies were limited to simple geometries and simple boundary conditions due to the limitation of using global basis functions in the present study a domain decomposition technique for the dqm is proposed to analyse truss and frame structures where the whole structural domain is represented by a collection of simple element subdomains connected together at specific nodal points this method is named the quadrature element method qem

free vibration analyses of thin sector plates by the new version of differential quadrature method abstract the new version of differential quadrature dq method is extended to analyze the free vibration of thin sector plates with various sector angles and six combinations of boundary conditions detailed formulations are given various numerical examples are studied comparisons are made with existing analytical and or numerical data numerical results indicate that convergence can be achieved with increasing in number of grid points and accurate results could be obtained with 9 9 gird or even higher grid by using the extended dq method for the cases considered

dynamical analysis of kirchhoff plates by an explicit time integration morley element method an explicit time integration finite element method is proposed to investigate dynamical analysis of kirchhoff plates where the morley element is used for spatial discretization and the second order central scheme for time discretization certain error estimates in the energy norm are achieved a number of numerical results are included to show computational performance of the method

signal codes convolutional lattice codes the coded modulation scheme proposed in this paper has a simple construction an integer sequence representing the information is convolved with a fixed continuous valued finite impulse response fir filter to generate the codeword a lattice point due to power constraints the code construction includes a shaping mechanism inspired by precoding techniques such as the tomlinson harashima filter we naturally term these codes convolutional lattice codes or alternatively signal codes due to the signal processing interpretation of the code construction surprisingly properly chosen short fir filters can generate good codes with large minimal distance decoding can be done efficiently by sequential decoding or for better performance by bidirectional sequential decoding error analysis and simulation results indicate that for the additive white gaussian noise awgn channel convolutional lattice codes with computationally reasonable decoders can achieve low error rate close to the channel capacity

a constructive approach to the soliton solutions of integrable quadrilateral lattice equations scalar multidimensionally consistent quadrilateral lattice equations are studied we explore a confluence between the superposition principle for solutions related by the backlund transformation and the method of solving a riccati map by exploiting two known particular solutions this leads to an expression for the n soliton type solutions of a generic equation within this class as a particular instance we give an explicit n soliton solution for the primary model which is adler s lattice equation or q4

integer forcing linear receivers linear receivers are often used to reduce the implementation complexity of multiple antenna systems in a traditional linear receiver architecture the receive antennas are used to separate out the codewords sent by each transmit antenna which can then be decoded individually although easy to implement this approach can be highly suboptimal when the channel matrix is near singular this paper develops a new linear receiver architecture that uses the receive antennas to create an effective channel matrix with integer valued entries rather than attempting to recover transmitted codewords directly the decoder recovers integer combinations of the codewords according to the entries of the effective channel matrix the codewords are all generated using the same linear code which guarantees that these integer combinations are themselves codewords provided that the effective channel is full rank these integer combinations can then be digitally solved for the original codewords this paper focuses on the special case where there is no coding across transmit antennas and no channel state information at the transmitter s which corresponds either to a multiuser uplink scenario or to single user v blast encoding in this setting the proposed integer forcing linear receiver significantly outperforms conventional linear architectures such as the zero forcing and linear minimum mean squared error receiver in the high signal to noise ratio regime the proposed receiver attains the optimal diversity multiplexing tradeoff for the standard multiple input multiple output mimo channel with no coding across transmit antennas it is further shown that in an extended mimo model with interference the integer forcing linear receiver achieves the optimal generalized degrees of freedom

a new reduced complexity conditional mean based mimo signal detection using symbol distribution approximation technique in this correspondence we propose a new conditional mean based approach for a reduced complexity suboptimal multiple input multiple output mimo detector in general the optimal error minimizing metric should take all possible symbol candidates into account and thus exhaustive computations are required on the other hand our new approach makes use of approximating the distributions of distinct symbol candidates by a continuous random variable such that the exhaustive summation is replaced by an integration that can be reduced to a simple closed form the resulting metric depends only on a subset of symbol candidates and thus the overall complexity is reduced considerably it is found that uniform ring approximation in combination with phase shift keying psk and amplitude modulated phase shift keying apsk achieves a performance close to that of maximum likelihood detection mld while its complexity is a linear order of the modulation multiplicity when the number of transmit antennas is two

static analysis of frame structures by the differential quadrature element method in this paper a new version of differential quadrature method dqm is proposed and then extended to analyse frame structures the new method called the differential quadrature element method dqem retains all advantages of the earlier version of the differential quadrature method and overcomes some critical shortcomings existing in the original dqm the proposed method is however different from the quadrature element method qem proposed earlier by striz et al the methodology is explained in details herein via a differential quadrature beam element and some numerical examples are given to show the efficiency of the present method 1997 by john wiley sons ltd

new similarity reductions and painleve analysis for the symmetric regularised long wave and modified benjamin bona mahoney equations the author discusses similarity reductions and painleve analysis for the symmetric regularised long wave and modified benjamin bona mahoney equations 1972 both of which arise in several physical applications including shallow water waves both equations are thought to non integrable i e not solvable by inverse scattering since numerical studies show that the interaction of solitary waves in inelastic in particular we determine some new similarity reductions of the symmetric regularised long wave equation these new similarity reductions are not obtainable using the classical lie group method for finding group invariant solutions of partial differential equations they are determined using a new and direct method which involves no group theoretical techniques it is shown that every similarity reduction of both the symmetric regularised long wave and modified benjamin bona mahoney equations obtained using the classical lie group method reduces the associated partial differential equation to an ordinary differential equation of painleve type whereas the new similarity solution of the symmetric regularised long wave equation reduces it to an ordinary differential equation which is not of painleve type it is also shown that neither the symmetric regularised long wave equation nor the modified benjamin bona mahoney equation possesses the painleve property for partial differential equations as defined by weiss et al 1983

algorithms to construct minkowski reduced and hermite reduced lattice bases abstract up to now the problem of constructing minkowski reduced lattice bases has been solved only for the two and three dimensional case this paper presents an algorithm to solve the problem for arbitrary dimension for fixed dimension the runtime is polynomial the algorithm hinges on the previous reduction algorithms of lenstra lenstra and lovasz 1982 and kannan 1983 moreover we shall improve kannan s algorithm to construct hermite reduced lattice bases

a hierarchy of polynomial time lattice basis reduction algorithms we present a hierarchy of polynomial time lattice basis reduction algorithms that stretch from lenstra lenstra lovasz reduction to korkine zolotareff reduction let l be the length of a shortest nonzero element of a lattice l we present an algorithm which for k n finds a nonzero lattice vector b so that b 2 6k2 nk l 2 this algorithm uses o n2 kk o k n2 log b arithmetic operations on o n log b bit integers this holds provided that the given basis vectors b1 bn zn are integral and have the length bound b this algorithm successively applies korkine zolotareff reduction to blocks of length k of the lattice basis we also improve kannan s algorithm for korkine zolotareff reduction

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

good lattice constellations for both rayleigh fading and gaussian channels recent work on lattices matched to the rayleigh fading channel has shown how to construct good signal constellations with high spectral efficiency we present a new family of lattice constellations based on complex algebraic number fields which have good performance on rayleigh fading channels some of these lattices also present a reasonable packing density and thus may be used at the same time over a gaussian channel conversely we show that particular versions of the best lattice packings d sub 4 e sub 6 e sub 8 k sub 12 spl lambda sub 16 spl lambda sub 24 constructed from totally complex algebraic cyclotomic fields present better performance over the rayleigh fading channel the practical interest in such signal constellations rises from the need to transmit information at high rates over both terrestrial and satellite links some further results in algebraic number theory related to ideals and their factorization are presented and the decoding algorithm used with these lattice constellations are illustrated together with practical results

on the voronoi neighbor ratio for binary linear block codes soft decision decoding of block codes is regarded as the geometrical problem of identifying the voronoi region within which a given input vector lies a measure called the neighbor ratio is proposed to characterize how many facets a voronoi region has theory and algorithms are presented to determine the neighbor ratio for binary linear block codes and results are given for several types of codes an asymptotic analysis for long codes reveals that the neighbor ratio depends on whether the code rate is less than 1 2 or not for rates below this threshold all pairs of codewords tend to share a voronoi facet for higher rates a relatively small fraction of them do

identifiability and aliasing in polynomial phase signals polynomial phase signals have numerous applications including radar sonar geophysics and radio communication many techniques for estimating the parameters of polynomial phase signals have been described in the literature despite the significant interest aliasing of polynomial phase parameters has not been fully clarified we address the problem of identifiability and aliasing in polynomial phase signals we fully describe the region in which aliasing does not occur for polynomial phase signals of any order we call this the identifiable region we find that this region is the voronoi region of a lattice generated by the coefficients of a set of polynomials known as the integer valued polynomials we show how aliasing can be resolved by solving the nearest lattice point problem we discuss some of the consequences of these results on a popular estimator for polynomial phase signals that is based on the discrete polynomial phase transform dppt it is shown that the range of parameters suitable for the dppt estimator is very small compared to the identifiable region

factoring polynomials with rational coefficients in this paper we present a polynomial time algorithm to solve the following problem given a non zero polynomial fe q x in one variable with rational coefficients find the decomposition of f into irreducible factors in q x it is well known that this is equivalent to factoring primitive polynomials fez x into irreducible factors in z x here we call f z x primitive if the greatest common divisor of its coefficients the content of f is 1 our algorithm performs well in practice cf 8 its running time measured in bit operations is o nl2 n9 log fd3

painleve analysis of the generalized burgers huxley equation a complete painleve test 1900 is applied to the generalized burgers huxley equation using the version of the painleve analysis recently developed by weiss tabor and carnevale 1983 for partial nonlinear differential equations in so doing the authors are able to find a complete set of new solutions as well as recovering some previous particular solutions already found by using ad hoc methods which have been recently published

coset codes ii binary lattices and related codes for pt i see ibid vol 34 no 5 p 1123 51 1988 the family of barnes wall lattices including d sub 4 and e sub 8 of lengths n 2 sup n and their principal sublattices which are useful in constructing coset codes are generated by iteration of a simple construction called the squaring construction the closely related reed muller codes are generated by the same construction the principal properties of these codes and lattices are consequences of the general properties of iterated squaring constructions which also exhibit the interrelationships between codes and lattices of different lengths an extension called the cubing construction generates good codes and lattices of lengths n 3 2 sup n including the golay code and leech lattice with the use of special bases for 8 space another related construction generates the nordstrom robinson code and an analogous 16 dimensional nonlattice packing these constructions are represented by trellis diagrams that display their structure and interrelationships and that lead to efficient maximum likelihood decoding algorithms

lattice decoding for joint detection in direct sequence cdma systems a new joint detection method based on sphere packing lattice decoding is presented in this paper the algorithm is suitable for both synchronous and asynchronous multiple access direct sequence code division multiple access ds cdma systems and it may jointly detect up to 64 users with a reasonable complexity the detection complexity is independent of the modulation size and large m pam or m qam constellations can be used furthermore a theoretical gain analysis is performed in which the multiple access system performance is derived from the lattice parameters

closest point search in lattices in this semitutorial paper a comprehensive survey of closest point search methods for lattices without a regular structure is presented the existing search strategies are described in a unified framework and differences between them are elucidated an efficient closest point search algorithm based on the schnorr euchner 1995 variation of the pohst 1981 method is implemented given an arbitrary point x spl isin spl ropf sup m and a generator matrix for a lattice spl lambda the algorithm computes the point of spl lambda that is closest to x the algorithm is shown to be substantially faster than other known methods by means of a theoretical comparison with the kannan 1983 1987 algorithm and an experimental comparison with the pohst 1981 algorithm and its variants such as the viterbo boutros see ibid vol 45 p 1639 42 1999 decoder modifications of the algorithm are developed to solve a number of related search problems for lattices such as finding a shortest vector determining the kissing number computing the voronoi 1908 relevant vectors and finding a korkine zolotareff 1873 reduced basis

the painleve property for partial differential equations ii backlund transformation lax pairs and the schwarzian derivative in this paper we investigate the painleve property for partial differential equations by application to several well known partial differential equations burgers kdv mkdv bousinesq higher order kdv and kp equations it is shown that consideration of the singular manifold leads to a formulation of these equations in terms of the schwarzian derivative this formulation is invariant under the moebius group acting on dependent variables and is shown to obtain the appropriate lax pair linearization for the underlying nonlinear pde

free vibration analysis of curvilinear quadrilateral plates by the differential quadrature method a methodology for applying the differential quadrature dq method to the free vibration analysis of arbitrary quadrilateral plates is developed in our approach the irregular physical domain is transformed into a rectangular domain in the computational space the governing equation and the boundary conditions are also transformed into relevant forms in the computational space then all the computations are based on the computational domain as compared to the approach proposed by c w bert and m malik int j mech sci 38 589 1996 the present approach requires much less computational effort and virtual storage in addition the present work uses a simple and convenient way to implement clamped and simply supported boundary conditions an exact mapping technique is used to perform the coordinate transformation in this study some numerical examples are provided to show the computational efficiency of the present scheme

two new approximate methods for analyzing free vibration of structural components two approximate methods which have not previously been used for structural dynamics problems are applied to the free vibration analysis of various structural components the first method is a new version of the complementary energy method it is shown to be considerably more accurate than the conventional rayleigh and rayleigh schmidt methods when applied to spatially one dimensional free vibration problems prismatic and tapered bars prismatic beams and axisymmetric motion of circular membranes the second method is the differential quadrature method introduced by bellman and his associates it is applied successfully here to all of the problems mentioned plus square membranes and circular and square plates

glrt optimal noncoherent lattice decoding this paper presents new low complexity lattice decoding algorithms for noncoherent block detection of quadrature amplitude modulation qam and pulse amplitude modulation pam signals over complex valued fading channels the algorithms are optimal in terms of the generalized likelihood ratio test glrt the computational complexity is polynomial in the block length making glrt optimal noncoherent detection feasible for implementation we also provide even lower complexity suboptimal algorithms simulations show that the suboptimal algorithms have performance indistinguishable from the optimal algorithms finally we consider block based transmission and propose to use noncoherent detection as an alternative to pilot assisted transmission pat the new technique is shown to outperform pat

vector perturbation based on symbol scaling for limited feedback miso downlinks we propose a low complexity vector precoding vp scheme for the downlink of multi user multiple input single output mu miso systems with limited feedback conventional vp requires the use of modulo operation and knowledge of the scaling factor used at the transmitter in order to remove the perturbation quantity at the receiver the latter may be problematic in certain limited feedback scenarios where only quantized versions of the scaling factor can be made available at the receiver to circumvent this shortcoming we propose a modified vp technique where the search of perturbing vectors is limited to the area in the symbol constellation which is constrictive to the information symbols i e the area where the distances from the decision thresholds are increased with respect to a distance threshold by doing this the perturbation quantities can only enhance detection and need not be removed at the receiver successful detection can therefore be done without the use of the modulo operation and the scaling factor which makes the proposed vp scheme applicable to limited feedback scenarios analytical and simulation results show that the performance of the proposed scheme is free of the error floor typically encountered with conventional vp in the presence of limited feedback

vibration analysis of kirchhoff plates by the morley element method vibration analysis of kirchhoff plates is of great importance in many engineering fields the semi discrete and the fully discrete morley element methods are proposed to solve such a problem which are effective even when the region of interest is irregular the rigorous error estimates in the energy norm for both methods are established some reasonable approaches to choosing the initial functions are given to keep the good convergence rate of the fully discrete method a number of numerical results are provided to illustrate the computational performance of the method in this paper

low complexity decoding via reduced dimension maximum likelihood search in this paper we consider a low complexity detection technique referred to as a reduced dimension maximum likelihood search rd mls rd mls is based on a partitioned search which approximates the maximum likelihood ml estimate of symbols by searching a partitioned symbol vector space rather than that spanned by the whole symbol vector the inevitable performance loss due to a reduction in the search space is compensated by 1 the use of a list tree search which is an extension of a single best searching algorithm called sphere decoding and 2 the recomputation of a set of weak symbols i e those ignored in the reduced dimension search for each strong symbol candidate found during the list tree search through simulations on m quadrature amplitude modulation qam transmission in frequency nonselective multi input multioutput mimo channels we demonstrate that the rd mls algorithm shows near constant complexity over a wide range of bit error rate ber 10 1 10 4 while limiting performance loss to within 1 db from ml detection

periodic solutions of the kdv equation in this paper we construct a large family of special solutions of the kdv equation which are periodic in x and almost periodic in t these solutions lie on n dimensional tori very likely they are dense among all solutions the special solutions are characterized variationally they minimize fn u subject to the constraints fj u aj j 1 n 1 here fj denote the remarkable sequence of conserved functionals discovered by kruskal and zabusky the above minimum problem was originally suggested by them in exploring the manifold of solutions of this minimum problem we make essential use of gardner s discovery that these functionals are in involution with respect to a suitable poisson bracket gardner greene kruskal and miura have shown that the eigenvalues of the schrodinger operator are conserved functionals if the potential is a function of t and satisfies the kdv equation in section 6 a new set of conserved quantities is constructed which serve as a link between the eigenvalues of the schrodinger operator and the fj another result in section 6 is a slight sharpening of an earlier result of the author and j moser for the special solutions constructed above all but 2n 1 eigenvalues of the schrodinger operator are double the simplest class of special solutions n 1 are cnoidal waves in an appendix m hyman describes the results of computing numerically the next simplest case n 2 these calculations show that the shape of these solutions recurs exactly after a finite time in a shifted position the theory verifies this fact

differential quadrature analysis of free vibration of rhombic plates with free edges an accurate free vibration analysis of rhombic plates with free edges is presented by using the modified differential quadrature method dqm fourteen combinations of boundary conditions with free edges and five skew angles for each plate are analyzed it is found for the first time that the problem is extremely sensitive to grid spacing and the method to applying the multiple boundary conditions when the skew angle is very large various existing non uniform grid spacing and methods for applying the multiple boundary conditions are tested it is surprising to see that dqm with only one grid spacing can yield accurate and reliable solutions besides applying boundary conditions correctly at free corners is also important for success for verifications the dq results are compared with either existing accurate solutions or data obtained by finite element method due to bending stress singularities occurred at corners having obtuse angles the rate of convergence is low for a few cases and quite different from rhombic plates without free edges some new results are presented and waiting for comparison with future data obtained by other investigators

improved methods for calculating vectors of short length in a lattice including a complexity analysis the standard methods for calculating vectors of short length in a lattice use a reduction procedure followed by enumerating all vectors of z in a suitable box however it suffices to consider those x e z which lie in a suitable ellipsoid having a much smaller volume than the box we show in this paper that searching through that ellipsoid is in many cases much more efficient if combined with an appropriate reduction procedure our method allows to do computations in lattices of much higher dimensions several randomly constructed numerical examples illustrate the superiority of our new method over the known ones

performance of sphere decoding of block codes a sphere decoder searches for the closest lattice point within a certain search radius the search radius provides a tradeoff between performance and complexity we focus on analyzing the performance of sphere decoding of linear block codes we analyze the performance of soft decision sphere decoding on awgn channels and a variety of modulation schemes a hard decision sphere decoder is a bounded distance decoder with the corresponding decoding radius we analyze the performance of hard decision sphere decoding on binary and q ary symmetric channels an upper bound on the performance of maximum likelihood decoding of linear codes defined over fq e g reed solomon codes and transmitted over q ary symmetric channels is derived and used in the analysis we then discuss sphere decoding of general block codes or lattices with arbitrary modulation schemes the tradeoff between the performance and complexity of a sphere decoder is then discussed

trellis complexity and minimal trellis diagrams of lattices this paper presents results on trellis complexity and low complexity trellis diagrams of lattices we establish constructive upper bounds on the trellis complexity of lattices these bounds both improve and generalize the similar results of tarokh and vardy see ibid vol 43 p 1294 1300 1997 we also construct trellis diagrams with minimum number of paths for some important lattices such trellises are called minimal the constructed trellises which are novel in many cases can be employed to efficiently decode the lattices via the viterbi algorithm in particular a general structure for minimal trellis diagrams of d sub n lattices is obtained this structure corresponds to a new code formula for d sub n moreover we develop some important duality results which are used in both deriving the upper bounds and finding the minimal trellises all the discussions are based on a universal approach to the construction and analysis of trellis diagrams of lattices using their bases

soliton solutions for abs lattice equations i cauchy matrix approach in recent years there have been new insights into the integrability of quadrilateral lattice equations i e partial difference equations which are the natural discrete analogues of integrable partial differential equations in 1 1 dimensions in the scalar i e single field case there now exist classification results by adler bobenko and suris abs leading to some new examples in addition to the lattice equations of kdv type that were known since the late 1970s and early 1980s in this paper we review the construction of soliton solutions for the kdv type lattice equations and use those results to construct n soliton solutions for all lattice equations in the abs list except for the elliptic case of q4 which is left to a separate treatment

on classes of integrable systems and the painleve property the caudrey dodd gibbon equation is found to possess the painleve property investigation of the backlund transformations for this equation obtains the kuperschmidt equation a certain transformation between the kuperschmidt and caudrey dodd gibbon equation is obtained this transformation is employed to define a class of p d e s that identically possesses the painleve property for equations within this class backlund transformations and rational solutions are investigated in particular the sequences of higher order kdv caudrey dobb gibbon and kuperschmidt equations are shown to possess the painleve property

postbuckling behavior of bi axially compressed arbitrarily straight sided quadrilateral functionally graded material plates abstract in this paper the problem of the postbuckling behavior of biaxial compressed straight sided functionally graded material fgm plates of quadrilateral shape is studied the plate considered is subjected to in plane loads on all four edges a computational framework based on an improved moving least squares imls approximation for the field variables is developed for the analysis the solution procedures involved a transformation from the physical domain to a computational domain and then discrete the nonlinear governing equation using the imls ritz method the first order shear deformation theory fsdt with the von karman nonlinearity is employed a nonlinear solution to the postbuckling of quadrilateral fgm plates is computed through the modified newton raphson method combined with the arc length iterative algorithm a stabilized conforming nodal integration scheme is employed to improve computational efficiency and eliminate shear and membrane locking the validity and accuracy of the numerical results are established through convergence studies to the best of the authors knowledge the problem has not been attempted in the open literature

implementation of clamped and simply supported boundary conditions in the gdq free vibration analysis of beams and plates abstract in this paper a new methodology for implementing the clamped and simply supported boundary conditions is presented for the free vibration analysis of beams and plates using the generalized differential quadrature gdq method the proposed approach directly substitutes the boundary conditions into the governing equations and is referred to as sbcge approach the sbcge approach is presented to overcome the drawbacks of previous approaches in treating the boundary conditions a comparison of the sbcge approach with the method of modifying weighting coefficient matrices mwcm is made by their application to the vibration analysis of beams and plates with combinations of simply supported and clamped boundary conditions some details of the gdq method are also described in the paper

periodic fixed points of backlund transformations the periodic fixed points of the backlund transformations are finite dimensional invariant manifolds for the flow of the system the dynamics occur as commuting hamiltonian flows on this finite dimensional manifold we examine the flow of the kdv periodic fixed points in the neighborhood of steady states and reductions these are analogous to a flow in the neighborhood of a sequence of heteroclinic points

on the complexity of decoding lattices using the korkin zolotarev reduced basis upper and lower bounds are derived for the decoding complexity of a general lattice l the bounds are in terms of the dimension n and the coding gain spl gamma of l and are obtained based on a decoding algorithm which is an improved version of kannan s 1983 method the latter is currently the fastest known method for the decoding of a general lattice for the decoding of a point x the proposed algorithm recursively searches inside an n dimensional rectangular parallelepiped cube centered at x with its edges along the gram schmidt vectors of a proper basis of l we call algorithms of this type recursive cube search rcs algorithms it is shown that kannan s algorithm also belongs to this category the complexity of rcs algorithms is measured in terms of the number of lattice points that need to be examined before a decision is made to tighten the upper bound on the complexity we select a lattice basis which is reduced in the sense of korkin zolotarev 1873 it is shown that for any selected basis the decoding complexity using rcs algorithms of any sequence of lattices with possible application in communications spl gamma spl ges 1 grows at least exponentially with n and spl gamma it is observed that the densest lattices and almost all of the lattices used in communications e g barnes wall lattices and the leech lattice have equal successive minima esm for the decoding complexity of esm lattices a tighter upper bound and a stronger lower bound result are derived

symplectic structures their backlund transformations and hereditary symmetries abstract it is shown that compatible symplectic structures lead in a natural way to hereditary symmetries we recall that a hereditary symmetry is an operator valued function which immediately yields a hierarchy of evolution equations each having infinitely many commuting symmetries all generated by this hereditary symmetry furthermore this hereditary symmetry usually describes completely the soliton structure and the conservation laws of these equations this result then provide us with a method for constructing hereditary symmetries and hence exactly solvable evolution equations in addition we show how symplectic structures transform under backlund transformations this leads to a method for generating a whole class of symplectic structures from a given one several examples and applications are given illustrating the above results also the connection of our results with those of gelfand and dikii and of magri is briefly pointed out

minkowski s convex body theorem and integer programming the paper presents an algorithm for solving integer programming problems whose running time depends on the number n of variables as non this is done by reducing an n variable problem to 2n5i 2 problems in n i variables for some i greater than zero chosen by the algorithm the factor of on5 2 per variable improves the best previously known factor which is exponential in n minkowski s convex body theorem and other results from geometry of numbers play a crucial role in the algorithm several related algorithms for lattice problems are presented the complexity of these problems with respect to polynomial time reducibilities is studied

on lattice sequential decoding for the unconstrained awgn channel in this paper the performance limits and the computational complexity of the lattice sequential decoder are analyzed for the unconstrained additive white gaussian noise channel the performance analysis available in the literature for such a channel has been studied only under the use of the minimum euclidean distance decoder that is commonly referred to as the lattice decoder lattice decoders based on solutions to the np hard closest vector problem are very complex to implement and the search for low complexity receivers for the detection of lattice codes is considered a challenging problem however the low computational complexity advantage that sequential decoding promises makes it an alternative solution to the lattice decoder in this work we characterize the performance and complexity tradeoff via the error exponent and the decoding complexity respectively of such a decoder as a function of the decoding parameter the bias term for the above channel we derive the cut off volume to noise ratio that is required to achieve a good error performance with low decoding complexity

chinese remaindering with errors the chinese remainder theorem states that a positive integer m is uniquely specified by its remainder module k relatively prime integers p sub 1 spl middot spl middot spl middot p sub k provided m spl pi sub i 1 sup k p sub i thus the residues of m module relatively prime integers p sub 1 p sub 2 spl middot spl middot spl middot p sub n form a redundant representation of m if m spl pi sub i 1 sup k p sub i and k n this gives a number theoretic construction of an error correcting code that has been considered often in the past in this code a message integer m spl pi sub i 1 sup k p sub i is encoded by the list of its residues module p sub 1 spl middot spl middot spl middot p sub n by the chinese remainder theorem if a codeword is corrupted in e n k 2 coordinates then there exists a unique integer m whose corresponding codesword differs from the corrupted word in at most e places furthermore mandelbaum 1976 1978 shows how m can be recovered efficiently given the corrupted word provided that the p sub i s are very close to one another to deal with arbitrary p sub i s we present a variant of his algorithm that runs in almost linear time and recovers from e log p sub 1 log p sub 1 log p sub n spl middot n k errors our main contribution is an efficient decoding algorithm for the case in which the error e may be larger than n k 2 specifically given n residues r sub 1 spl middot spl middot spl middot r sub n and an agreement parameter t we find a list of all integers m spl pi sub i 1 sup k p sub i such that m mod p sub i r sub i for at least t values of i spl isin 1 spl middot spl middot spl middot n provided t spl omega spl radic kn log p sub n log p sub 1 for n spl gt k and p sub n spl les p sub 1 sup o 1 the fraction of error corrected by the algorithm is almost twice that corrected by the previous work more significantly the algorithm recovers the message even when the amount of agreement between the received word and the codeword is much smaller than the number of errors

vibration analysis of symmetrically laminated plates based on fsdt using the moving least squares differential quadrature method in this paper we adopt the first order shear deformation theory in the moving least squares differential quadrature mlsdq procedure for predicting the free vibration behavior of moderately thick symmetrically laminated composite plates the transverse deflection and two rotations of the laminate are independently approximated with the moving least squares mls approximation the weighting coefficients used in the mlsdq approximation are obtained through the fast computation of the mls shape functions and their partial derivatives the natural frequencies of vibration are computed for various laminated plates and compared with the available published results through numerical experiments the capability and efficiency of the mlsdq method for eigenvalue problems are demonstrated and the numerical accuracy and convergence are thoughtfully examined effects of the size of support order of completeness of the basis functions and node irregularity on the numerical accuracy are also investigated

high accuracy differential quadrature finite element method and its application to free vibrations of thin plate with curvilinear domain summary based on the differential quadrature dq rule the gauss lobatto quadrature rule and the variational principle a dq finite element method dqfem is proposed for the free vibration analysis of thin plates the dqfem is a highly accurate and rapidly converging approach and is distinct from the differential quadrature element method dqem and the quadrature element method qem by employing the function values themselves in the trial function for the title problem the dqfem without using shape functions essentially combines the high accuracy of the differential quadrature method dqm with the generality of the standard finite element formulation and has superior accuracy to the standard fem and fdm and superior efficiency to the p version fem and qem in calculating the stiffness and mass matrices by incorporating the reformulated dq rules for general curvilinear quadrilaterals domains into the dqfem a curvilinear quadrilateral dq finite plate element is also proposed the inter element compatibility conditions as well as multiple boundary conditions can be implemented simply and conveniently as in fem through modifying the nodal parameters when required at boundary grid points using the dq rules thus the dqfem is capable of constructing curvilinear quadrilateral elements with any degree of freedom and any order of inter element compatibilities a series of frequency comparisons of thin isotropic plates with irregular and regular planforms validate the performance of the dqfem copyright q 2009 john wiley sons ltd

uniform and piecewise uniform lattice vector quantization for memoryless gaussian and laplacian sources lattice vector quantizer design procedures for nonuniform sources are presented the procedures yield lattice vector quantizers with excellent performance and retaining the structure required for fast quantization analytical methods for truncating and scaling lattices to be used in vector quantizations are given and their utility is demonstrated for independent and identically distributed i i d gaussian and laplacian sources an analytical technique for piecewise linear multidimensional compandor designs is evaluated for i i d gaussian and laplacian sources by comparing its performance to that of the other vector quantizers

maximum likelihood decoding of the leech lattice an algorithm for maximum likelihood decoding of the leech lattice is presented the algorithm involves projecting the points of the leech lattice directly onto the codewords of the 6 3 4 quaternary code the hexacode projection on the hexacode induces a partition of the leech lattice into four cosets of a certain sublattice 24 such a partition into cosets enables maximum likelihood decoding of the leech lattice with 3595 real operations in the worst case and only 2955 operations on the average this is about half the worst case and the average complexity of the best previously known algorithm

optimization of lattices for quantization a training algorithm for the design of lattices for vector quantization is presented the algorithm uses a steepest descent method to adjust a generator matrix in the search for a lattice whose voronoi regions have minimal normalized second moment the numerical elements of the found generator matrices are interpreted and translated into exact values experiments show that the algorithm is stable in the sense that several independent runs reach equivalent lattices the obtained lattices reach as low second moments as the best previously reported lattices or even lower specifically we report lattices in nine and ten dimensions with normalized second moments of 0 0716 and 0 0708 respectively and nonlattice tessellations in seven and nine dimensions with 0 0727 and 0 0711 which improves on previously known values the new nine and ten dimensional lattices suggest that conway and sloane s 1993 conjecture on the duality between the optimal lattices for packing and quantization might be false a discussion of the application of lattices in vector quantizer design for various sources uniform and nonuniform is included

integrability characteristics of two dimensional generalizations of nls type equations a recent procedure based on truncated painleve expansions is used to derive lax pairs darboux transformations and various soliton solutions for integrable 2 1 generalizations of nls type equations in particular diverse classes of solutions are found analogous to the dromion instanton lump and ring soliton solutions derived recently for 2 1 korteweg de vries type equations the nizhnik novikov veselov equation and the 2 1 broer kaup system

on the use of isospectral eigenvalue problems for obtaining hereditary symmetries for hamiltonian systems we present an algorithmic method for obtaining an hereditary symmetry the generalized squared eigenfunction operator from a given isospectral eigenvalue problem this method is applied to the n n eigenvalue problem considered by ablowitz and haberman and to the eigenvalue problem considered by alonso the relevant hamiltonian formulations are also determined finally an alternative method is presented in the case two evolution equations are related by a miura type transformation and their hamiltonian formulations are known

on the diversity multiplexing tradeoff under groupwise successive interference cancelation in this paper we study the diversity multiplexing tradeoff assuming groupwise successive interference cancelation gsic gsic with layered codes in multiple input multiple output mimo can vary from the plain sic to the maximum likelihood ml decoding according to the group sizes thus is flexible in both complexity and performance the maximum group size is a dominating factor for the computation complexity of gsic when the processing complexity is polynomially dependent on the group size with a high order in this paper we show the tradeoff among the maximum group size diversity gain and multiplexing gain gdmt under the gsic and under the gsic combined with antenna selection gsicas for a given group size bound c we show that the number of groups achieving the optimal dmt under the gsic with selected transmit antennas is equal to c and a larger group should have a higher priority in decoding order the optimal grouping is obtained in a systematic way based on these results we find a limited set of groupings that contains the grouping achieving the optimal gdmt under gsic and under gsicas the optimal multiplexing gain allocation for the optimal grouping is then found systematically and efficiently within the limited set

a universal lattice code decoder for fading channels we present a maximum likelihood decoding algorithm for an arbitrary lattice code when used over an independent fading channel with perfect channel state information at the receiver the decoder is based on a bounded distance search among the lattice points falling inside a sphere centered at the received point by judicious choice of the decoding radius we show that this decoder can be practically used to decode lattice codes of dimension up to 32 in a fading environment

detection of gaussian constellations in mimo systems under imperfect csi this paper considers the problem of gaussian symbols detection in mimo systems in the presence of channel estimation errors under this framework we develop a computationally efficient approximations of the map detector the new detectors are based on a relaxation of the discrete nature of the digital constellation and on the channel estimation error statistics this leads to a non convex program that is solved efficiently via a hidden convexity minimization approach additionally we show that using a bayesian em approach comparable ber performance to that of the map detector can be achieved next we extend the detection scheme to the case where the noise variance is unknown we present a modified bayesian em approach with annealed gibbs sampling to perform joint noise variance estimation and symbols detection simulation results in a random mimo system show that the proposed algorithm outperforms the linear mmse receiver in terms of ber

maximum likelihood sequence estimation from the lattice viewpoint considers the problem of data detection in multilevel lattice type modulation systems in the presence of intersymbol interference and additive white gaussian noise the conventional maximum likelihood sequence estimator using the viterbi algorithm has a time complexity of o m sup spl nu 1 operations per symbol and a space complexity of o spl delta m sup spl nu storage elements where m is the size of input alphabet spl nu is the length of channel memory and spl delta is the truncation depth by revising the truncation scheme and viewing the channel as a linear transform the authors identify the problem of maximum likelihood sequence estimation with that of finding the nearest lattice point from this lattice viewpoint the lattice sequence estimator for pam systems is developed which has the following desired properties 1 its expected time complexity grows as spl delta sup 2 as snr spl rarr spl infin 2 its space complexity grow as spl delta and 3 its error performance is effectively optimal for sufficiently large m a tight upper bound on the symbol error probability of the new estimator is derived and is confirmed by the simulation results of an example channel it turns out that the estimator is effectively optimal for m spl ges 4 and the loss in signal to noise ratio is less than 0 5 db even for m 2 finally limitations of the proposed estimator are also discussed

accurate free vibration analysis of rhombic plates with simply supported and fully clamped edge conditions abstract the superposition method is used to obtain a highly accurate analytical type solution for the free vibration of rhombic plates with simply supported and with fully clamped boundary conditions recent modifications to the superposition method have been incorporated into the analysis and are described in some detail these modifications have led to vast increases in the efficiency of the computational procedure accurate eigenvalues are tabulated for numerous vibration modes and for a large range of plate geometries

direct expansion method of boundary condition for solving 3d elliptic equations with small parameters in the irregular domain in this article a new methodology direct expansion method of boundary condition dembc is developed to solve 3d elliptic equations in the irregular domain first the previous rational differential quadrature method rational spectral collocation method in berrut et al 2005 8 developed by berrut et al 2005 8 has been generalized to solve 3d elliptic equations second it is showed that direct expansion method of boundary condition is capable of handling boundary problems with higher efficiency finally with the help of conformal mapping tee and trefethen 2006 9 and domain decomposition method dembc and 3d rdqm are able solve three kinds of 3d elliptic equations with small parameters in the irregular domain numerous test results justify the accuracy and efficiency of our approach