parametrization of compactly supported orthonormal wavelets the parametrization of two channel reconstruction quadrature mirror filter banks is extended to obtain a single characterization of discrete orthonormal wavelets in particular it is shown that discrete orthonormal wavelets of support less than 2n 1 where n is an even integer are parametrized by n 1 real parameters each taking values in 0 2 pi a parametrization of discrete orthonormal wavelets with a number of vanishing moments greater than one is also given

a nodal integration technique for meshfree radial point interpolation method ni rpim a novel nodal integration technique for the meshfree radial point interpolation method ni rpim is presented for solid mechanics problems in the ni rpim radial basis functions rbfs augmented with polynomials are used to construct shape functions that possess the delta function property galerkin weak form is adopted for creating discretized system equations in which nodal integration is used to compute system matrices a stable and simple nodal integration scheme is proposed to perform the nodal integration numerically the ni rpim is examined using a number of example problems including stress analysis of an automobile mechanical component the effect of shape parameters and dimension of local support domain on the results of the ni rpim is investigated in detail through these examples the numerical solutions show that the present method is a robust reliable stable meshfree method and possesses better computational properties compared with traditional linear fem and original rpim using gauss integration scheme

a gradient reproducing kernel collocation method for boundary value problems summary the earlier work in the development of direct strong form collocation methods such as the reproducing kernel collocation method rkcm addressed the domain integration issue in the galerkin type meshfree method such as the reproducing kernel particle method but with increased computational complexity because of taking higher order derivatives of the approximation functions and the need for using a large number of collocation points for optimal convergence in this work we intend to address the computational complexity in rkcm while achieving optimal convergence by introducing a gradient reproduction kernel approximation the proposed gradient rkcm reduces the order of differentiation to the first order for solving second order pdes with strong form collocation we also show that different from the typical strong form collocation method where a significantly large number of collocation points than the number of source points is needed for optimal convergence the same number of collocation points and source points can be used in gradient rkcm we also show that the same order of convergence rates in the primary unknown and its first order derivative is achieved owing to the imposition of gradient reproducing conditions the numerical examples are given to verify the analytical prediction copyright 2012 john wiley sons ltd

the natural element method in solid mechanics the application of the natural element method nem 1 2 to boundary value problems in two dimensional small displacement elastostatics is presented the discrete model of the domain consists of a set of distinct nodes n and a polygonal description of the boundary in the natural element method the trial and test functions are constructed using natural neighbour interpolants these interpolants are based on the voronoi tessellation of the set of nodes n the interpolants are smooth c 1 everywhere except at the nodes where they are c 0 in one dimension nem is identical to linear nite elements the nem interpolant is strictly linear between adjacent nodes on the boundary of the convex hull which facilitates imposition of essential boundary conditions a methodology to model material discontinuities and non convex bodies cracks using nem is also described a standard displacement based galerkin procedure is used to obtain the discrete system of linear equations application of nem to various problems in solid mechanics which include the patch test gradient problems bimaterial interface and a static crack problem are presented excellent agreement with exact analytical solutions is obtained which exemplies the accuracy and robustness of nem and suggests its potential application in the context of other classes of problems crack growth plates and large deformations to name a few 1998 john wiley sons ltd

nonlinear structures in electroencephalogram signals abstract we apply a nonlinear prediction algorithm to investigate the presence of nonlinear structure in electroencephalogram eeg recordings the eeg signal could be modeled as a realization of a nonlinear model plus a residual noise uncorrelated gaussian noise using linear and nonlinear models we analyze the statistical nature of these residual noises in the case of epileptic patients and normal subjects we found that the residual noise presents gaussian distribution for epileptic patients if a nonlinear model is used whereas in the case of normal subjects the residual noise will exhibit a gaussian distribution only if a linear model autoregressive is used these results provide another evidence of the nonlinear character of the epileptic seizure recordings while the normal eeg seems to be better described as linearly correlated noise

monitoring changing dynamics with correlation integrals case study of an epileptic seizure abstract we describe a procedure and the motivation behind it which rapidly and accurately tracks the onset and progress of an epileptic seizure roughly speaking one monitors changes in the relative dispersion of a re embedded time series the results are robust with respect to variation of adjustable parameters such as embedding dimension lag time and critical distances moreover the general method is virtually unaffected when the data are significantly corrupted by external noise when the information computed for the individual channels is displayed in an appropriate space time plot the progress and geometric location of the seizure are easily seen an interpretation of these results in terms of a cloud of particles moving in an abstract phase space is examined

a new cloud based hp finite element method a hybrid computational method for solving boundary value problems is introduced which combines features of the meshless hp cloud methods with features of conventional finite elements the method admits straightforward nonuniform hp type approximations easy implementation of essential boundary conditions is robust under severe distortions of the mesh and can deliver exponential rates of convergence results of numerical experiments are presented

a stabilized conforming nodal integration for galerkin mesh free methods domain integration by gauss quadrature in the galerkin mesh free methods adds considerable complexity to solution procedures direct nodal integration on the other hand leads to a numerical instability due to under integration and vanishing derivatives of shape functions at the nodes a strain smoothing stabilization for nodal integration is proposed to eliminate spatial instability in nodal integration for convergence an integration constraint ic is introduced as a necessary condition for a linear exactness in the mesh free galerkin approximation the gradient matrix of strain smoothing is shown to satisfy ic using a divergence theorem no numerical control parameter is involved in the proposed strain smoothing stabilization the numerical results show that the accuracy and convergent rates in the mesh free method with a direct nodal integration are improved considerably by the proposed stabilized conforming nodal integration method it is also demonstrated that the gauss integration method fails to meet ic in mesh free discretization for this reason the proposed method provides even better accuracy than gauss integration for galerkin mesh free method as presented in several numerical examples copyright 2001 john wiley sons ltd

mother wavelet functions generalized through q exponentials we generalize some widely used mother wavelets by means of the q exponential function exq 1 1 q x 1 1 q that emerges from nonextensive statistical mechanics in particular we define extended versions of the mexican hat and the morlet wavelets we also introduce new wavelets that are q generalizations of the trigonometric functions all cases reduce to the usual ones as q 1 within nonextensive statistical mechanics departures from unity of the entropic index q are expected in the presence of long range interactions long term memory multi fractal structures among others consistently the analysis of signals associated with such features is hopefully improved by proper tuning of the value of q we exemplify with the wavelet transform modulus maxima method for mono and multi fractal self affine signals

the tsallis entropy of natural information abstract estimating the information contained in natural data such as electroencephalography data is unusually difficult because the relationship between the physical data and the information that it encodes is unknown this unknown relationship is often called the encoding problem the present work provides a solution to this problem by deriving a method to estimate the tsallis entropy in natural data the method is based on two findings the first finding is that the physical instantiation of any information event that is the physical occurrence of a symbol of information must begin and end at a discontinuity or critical point maximum minimum or saddle point in the data the second finding is that in certain data types such as the encephalogram eeg the variance within of an eeg waveform event is directly proportional to its probability of occurrence these two outcomes yield two results the first is the easy binning of data into separate information events the second is the ability to estimate probabilities in two ways frequency counting and computing the variance within of an eeg waveform these results are used to derive a linear estimator of the tsallis entropy functional allowing it to be estimated without deducing the encoding this method for estimating the tsallis entropy is first used to estimate the information in simple signals the amount of information estimated is highly accurate the method is then applied to two problems in electroencephalography the first is distinguishing normal aging from very early alzheimer s disease mild cognitive impairment and the second is medication monitoring of alzheimer s disease treatment the former is done with an accuracy of 92 and the latter with an accuracy of 91 this detection accuracy is the highest published accuracy in the literature which suggests that this method for tsallis entropy estimation is both accurate and useful

a wavelet or lifting scheme based imputation method summary the paper proposes a new approach to imputation using the expected sparse representation of a surface in a wavelet or lifting scheme basis our method incorporates a bayesian mixture prior for these wavelet coefficients into a gibbs sampler to generate a complete posterior distribution for the variable of interest intuitively the estimator operates by borrowing strength from those observed neighbouring values to impute at the unobserved sites we demonstrate the strong performance of our estimator in both one and two dimensional imputation problems where we also compare its application with the standard imputation techniques of kriging and thin plate splines

tsallis conditional mutual information in investigating long range correlation in symbol sequences abstract the presence of long range correlation lrc as opposed to markov chain of a finite order is of primary interest in the study of symbol sequences such as dna among others methods using information theory have been developed for markov chain order estimation in this work we consider the tsallis entropy and define the tsallis conditional mutual information tcmi for markov chain order estimation which is more suitable to identify large orders suggesting effectively lrc the tcmi of order m for a tsallis parameter q is computed for increasing m and a significance test is performed for each m until tcmi is found non significant randomization and parametric significance tests for tcmi are developed for the latter the null distribution of tcmi is approximated with a gamma distribution deriving analytic expressions for its parameters we assess the accuracy of order estimation with the two tests for tcmi and compare them to the respective tests using the shannon entropy extended simulations on markov chains of different orders and structures of the transition probability matrix show that shannon and tsallis tests with parameter q 2 have similar performance for small orders when the problem becomes more demanding for higher markov chain orders and lrcs tsallis tests for q 2 have better approximation to the correct order with the sequence length n finally the tsallis tests are favorably compared to shannon tests to real dna sequences

dynamical aspects of tsallis entropy abstract the description of the temporal evolution of some physical systems is tackled from an information theory view point in which tsallis entropy is extremalized with the constraints posed by the knowledge of the generalized expectation values of a selected set of observables the concomitant tsallis density operator is shown to be a particular solution of von neumann s equation and tsallis entropy is seen to provide one with a constant of the motion the pertinent ideas are illustrated with reference to the larmor precession

meshfree and finite element nodal integration methods nodal integration can be applied to the galerkin weak form to yield a particle type method where stress and material history are located exclusively at the nodes and can be employed when using meshless or finite element shape functions this particle feature of nodal integration is desirable for large deformation settings because it avoids the remapping or advection of the state variables required in other methods to a lesser degree nodal integration can be desirable because it relies on fewer stress point evaluations than most other methods in this work aspects regarding stability consistency efficiency and explicit time integration are explored within the context of nodal integration both small and large deformation numerical examples are provided copyright 2007 john wiley sons ltd

phase space eigenfunctions of multidimensional quadratic hamiltonians abstract we obtain the explicit expressions for phase space eigenfunctions pse i e weyl s symbols of dyadic operators like n m n m being the solution of the schrodinger equation with the hamiltonian which is a quite arbitrary multidimensional quadratic form of the operators of cartesian coordinates and conjugated to them momenta with time dependent coefficients it is shown that for an arbitrary quadratic hamiltonian one can always construct the set of completely factorized pse which are products of n factors each factor being dependent only on two arguments for n m and on a single argument for n m these arguments are nothing but constants of motion of the correspondent classical system pse are expressed in terms of the associated laguerre polynomials in the case of a discrete spectrum and in terms of the airy functions in the continuous spectrum case three examples are considered a harmonic oscillator with a time dependent frequency a charged particle in a nonstationary uniform magnetic field and a particle in a time dependent uniform potential field

nonuniform multiresolution analyses and spectral pairs abstract a generalization of the notion of multiresolution analysis based on the theory of spectral pairs is considered in contrast to the standard setting the associated subspacev0ofl2 r has as an orthonormal basis a collection of translates of the scaling function of the form x where 0 r n 2z n 1 is an integer andris an odd integer with 1 r 2n 1 such thatrandnare relatively prime andzis the set of all integers furthermore the corresponding dilation factor is 2n the case wheren 1 corresponding to the usual definition of a multiresolution analysis with dilation factor 2 a necessary and sufficient condition for the existence of associated wavelets which is always satisfied whenn 1 or 2 is obtained and is shown to always hold if the fourier transform of is a constant multiple of the characteristic function of a set

normalized sph with stress points smoothed particle hydrodynamics is extended to a normalized staggered particle formulation with boundary conditions a companion set of interpolation points is introduced that carry the stress velocity gradient and other derived field variables the method is stable linearly consistent and has an explicit treatment of boundary conditions also a new method for finding neighbours is introduced which selects a minimal and robust set and is insensitive to anisotropy in the particle arrangement test problems show that these improvements lead to increased accuracy and stability published in 2000 by john wiley sons ltd

information theoretic wavelet noise removal for inverse elastic wave scattering theory a discussion of noise removal in ultrasound elastic wave scattering for nondestructive evaluation is given the methods used in this paper include a useful suboptimal wiener filter information theory and orthonormal wavelets the multiresolution analysis mra due to mallat is the key wavelet feature used here whereas fourier transforms have a translational symmetry wavelets have a dilation or affine symmetry which consists of the semi direct product of a translation with a change of scale of the variable the mra describes the scale change features of orthonormal wavelet families first an empirical method of noise removal from scattered elastic waves using wavelets is shown to markedly improve the l 1 and l 2 error norms this suggests that the wavelet scale can act as dial to tune out noise maximization of the kullback liebler information is also shown to provide a scale dependent noise removal technique that supports but does not prove the intuition that certain small energy coefficients that are retained contain large information content the wavelet mra thereby locates islands of information in the phase space of the signal it is conjectured that this method holds more generally

reproducing kernel particle methods for large deformation analysis of non linear structures abstract large deformation analysis of non linear elastic and inelastic structures based on reproducing kernel particle methods rkpm is presented the method requires no explicit mesh in computation and therefore avoids mesh distortion difficulties in large deformation analysis the current formulation considers hyperelastic and elasto plastic materials since they represent path independent and path dependent material behaviors respectively in this paper a material kernel function and an rkpm material shape function are introduced for large deformation analysis the support of the rkpm material shape function covers the same set of particles during material deformation and hence no tension instability is encountered in the large deformation computation the essential boundary conditions are introduced by the use of a transformation method the transformation matrix is formed only once at the initial stage if the rkpm material shape functions are employed the appropriate integration procedures for the moment matrix and its derivative are studied from the standpoint of reproducing conditions in transient problems with an explicit time integration method the lumped mass matrices are constructed at nodal coordinate so that masses are lumped at the particles several hyperelasticity and elasto plasticity problems are studied to demonstrate the effectiveness of the method the numerical results indicated that rkpm handles large material distortion more effectively than finite elements due to its smoother shape functions and consequently provides a higher solution accuracy under large deformation unlike the conventional finite element approach the nodal spacing irregularity in rkpm does not lead to irregular mesh shape that significantly deteriorates solution accuracy no volumetric locking is observed when applying non linear rkpm to nearly incompressible hyperelasticity and perfect plasticity problems further model adaptivity in rkpm can be accomplished simply by adding more points in the highly deformed areas without remeshing

human brain dynamics the analysis of eeg signals with tsallis information measure we undertake the study of human eeg signals by recourse to a wavelet based multiresolution analysis as adapted to an information measure scenario dierent information measures are employed it is shown that non extensive ones seem to be of particular usefulness their use opens up perspectives of building up automatic detection devices conjectures concerning general characteristics of focal epilepsy are formulated on the basis of a tsallis type of analysis c 1999 elsevier science b v all rights reserved

evidence of self organization in brain electrical activity using wavelet based informational tools abstract in the present work we show that appropriate information theory tools based on the wavelet transform relative wavelet energy normalized total wavelet entropy h generalized wavelet complexity c w when applied to tonic clonic epileptic eeg data provide one with valuable insights into the dynamics of neural activity twenty tonic clonic secondary generalized epileptic records pertaining to eight patients have been analyzed if the electromyographic activity is excluded the difference between the ictal and pre ictal mean entropic values h h ictal h pre ictal is negative in 95 of the cases p 0 0001 and the mean complexity variation c w c w ictal c w pre ictal is positive in 85 of the cases p 0 0002 thus during the seizure entropy diminishes while complexity grows this is construed as evidence supporting the conjecture that an epileptic focus in this kind of seizures triggers a self organized brain state characterized by both order and maximal complexity

power law sensitivity to initial conditions in a time series with applications to epileptic seizure detection recently the concept of exponential sensitivity to initial conditions esic of deterministic chaos is generalized to power law sensitivity to initial conditions psic we describe a general computational procedure to examine psic from a time series study of noise free and noisy logistic and henon maps at the edge of chaos finds that psic cannot be shown from clean scalar time series however when there is dynamic noise motions around the edges of chaos all collapse onto the psic attractor regardless whether they are simply regular or truly chaotic when noise is absent hence dynamic noise makes psic observable the psic concept is further applied to the analysis of long continuous eeg signals with epileptic seizures it is shown that measures from the psic framework is quite effective in detecting seizures often better than the lyapunov exponent based methods from the conventional esic framework

a strong form meshfree collocation method for frictional contact on a rigid obstacle abstract in this work we present a strong form meshfree collocation method for frictional contact problems frictional contact constraints are included in the strong form force balance equation as part of neumann boundary conditions the equation is then directly discretized with the precomputed derivative operators obtained from moving least squares approximation using taylor expansion of the displacement field through point wise computations at collocation points the accuracy and effectiveness of the method are examined via several benchmark frictional and frictionless contact problems using non uniformly distributed collocation points numerical results are also compared with analytical and finite element solutions

wavelet methods for pdes some recent developments abstract this paper is concerned with recent developments of wavelet schemes for the numerical treatment of operator equations with special emphasis on two issues adaptive solution concepts and nontrivial domain geometries after describing a general multiresolution framework the key features of wavelet bases are highlighted namely locality norm equivalences and cancellation properties assuming first that wavelet bases with these properties are available on the relevant problem domains the relevance of these features for a wide class of stationary problems is explained in subsequent sections the main issues are preconditioning and the efficient adaptive application of wavelet representations of the involved operators we indicate then how these ingredients combined with concepts from nonlinear or best n term approximation culminate in an adaptive wavelet scheme for elliptic selfadjoint problems covering boundary value problems as well as boundary integral equations these schemes can be shown to exhibit convergence rates that are in a certain sense asymptotically optimal we conclude this section with some brief remarks on data structures and implementation interrelations with regularity in a certain scale of besov spaces and strategies of extending such schemes to unsymmetric or indefinite problems we address then the adaptive evaluation of nonlinear functionals of wavelet expansions as a central task arising in connection with nonlinear problems wavelet bases on nontrivial domains are discussed next the main issues are the development of fourier free construction principles and criteria for the validity of norm equivalences finally we indicate possible combinations of wavelet concepts with conventional discretizations such as finite element or finite volume schemes in connection with convection dominated and hyperbolic problems

computational applications of nonextensive statistical mechanics computational applications of the nonextensive entropy s q and nonextensive statistical mechanics a current generalization of the boltzmann gibbs bg theory are briefly reviewed the corresponding bibliography is provided as well

non linear version of stabilized conforming nodal integration for galerkin mesh free methods a stabilized conforming sc nodal integration which meets the integration constraint in the galerkin meshfree approximation is generalized for nonlinear problems in sc nodal integration a strain smoothing stabilization is introduced and applied to the deformation gradient using a lagrangian meshfree discretization the assumed strain method expressed in terms of deformation gradient is employed to formulate an efficient meshfree discrete equation for nonlinear problems a significant gain in computational efficiency is achieved as well as enhanced accuracy in comparison to the solution of meshfree method using gauss integration method the performance of the proposed method appears to be quite robust when dealing with irregular discretization

a unified stability analysis of meshless particle methods a uni ed stability analysis of meshless methods with eulerian and lagrangian kernels is presented three types of instabilities were identi ed in one dimension an instability due to rank de ciency a tensile instability and a material instability which is also found in continua the stability of particle methods with eulerian and lagrangian kernels is markedly di erent lagrangian kernels do not exhibit the tensile instability in both kernels the instability due to rank de ciency can be suppressed by stress points in two dimensions the stabilizing e ect of stress points is dependent on their locations it was found that the best approach to stable particle discretizations is to use lagrangian kernels with stress points the stability of the least squares stabilization was also studied copyright 2000 john wiley sons ltd

the wavelet transform of stochastic processes with stationary increments and its application to fractional brownian motion the wavelet transform of random processes with wide sense stationary increments is shown to be a wide sense stationary process whose correlation function and spectral distribution are determined the second order properties of the coefficients in the wavelet orthonormal series expansion of such processes is obtained applications to the spectral analysis and to the synthesis of fractional brownian motion are given

superconvergent gradient smoothing meshfree collocation method abstract a superconvergent meshfree collocation method with smoothed nodal gradients is presented in this method the first order smoothed gradients of meshfree shape function are constructed through a meshfree interpolation of the standard derivatives of meshfree shape function while the second order smoothed gradients are computed through directly differentiating the first order smoothed gradients it is noted that the second order smoothed gradients evaluated at meshfree nodes can be conveniently expressed as two successive first order gradient smoothing operations on the meshfree shape function which facilitates a trivial numerical implementation subsequently an employment of the second order smoothed gradients in the strong form of a given problem leads to the gradient smoothing meshfree collocation method purely using nodes as the collocation points based upon a local truncation error analysis it is systematically shown that the proposed meshfree collocation method yields superconvergent solutions for odd degree basis functions a key ingredient attributed to this superconvergence property is that the second order smoothed gradients meet the consistency conditions which go one order beyond the original basis degree of meshfree approximation another interesting fact is that the present formulation enables a convergent collocation scheme when the linear basis function is used in meshfree approximation which is non feasible in the conventional collocation formulation the effectiveness of the proposed methodology is validated by numerical examples for both potential and elasticity problems numerical results well demonstrate the superconvergence and higher efficiency of the present gradient smoothing meshfree collocation method

algebraic quantisation with indefinite metric a real symplectic flow may be viewed as being pseudo unitary provided there exists a symplectic complex structure j which commutes with it the author determines the class of dynamical systems for which j exists such systems may be quantised algebraically but the state space may have indefinite metric this procedure admits a larger class of classical systems than does segal s algebraic quantisation but there are problems in interpreting the indefinite metric and these are listed and discussed

a four point integration scheme with quadratic exactness for three dimensional element free galerkin method based on variationally consistent formulation abstract the formulation of three dimensional element free galerkin efg method based on the hu washizu three field variational principle is described the orthogonality condition between stress and strain difference is satisfied by correcting the derivatives of the nodal shape functions this leads to a variationally consistent formulation which has a similar form as the formulation of standard galerkin weak form based on this formulation an integration scheme which employs only four cubature points in each background tetrahedral element cell is rationally developed for three dimensional efg with quadratic approximation the consistency of the corrected nodal derivatives and the satisfaction of patch test conditions for the developed integration scheme are theoretically proved extension of the proposed method to small strain elastoplasticity is also presented the proposed method can exactly pass quadratic patch test that is quadratic exactness is achieved and thus it is named as quadratically consistent 4 point qc4 integration method in contrast efg with standard tetrahedral cubature and the existing linearly consistent 1 point lc1 integration fail to exactly pass quadratic patch test numerical results of elastic examples demonstrate the superiority of the proposed method in accuracy convergence efficiency and stability the capability of the proposed qc4 scheme in solving elastoplastic problems is also demonstrated by numerical examples

a particle partition of unity method part ii efficient cover construction and reliable integration in this paper we present a meshfree discretization technique based only on a set of irregularly spaced points x i in mathbb r d and the partition of unity approach in this sequel to m griebel and m a schweitzer siam j sci comput 22 2000 pp 853 890 we focus on the cover construction and its interplay with the integration problem arising in a galerkin discretization we present a hierarchical cover construction algorithm and a reliable decomposition quadrature scheme here we decompose the integration domains into disjoint cells on which we employ local sparse grid quadrature rules to improve computational efficiency the use of these two schemes already reduces the operation count for the assembly of the stiffness matrix significantly now the overall computational costs are dominated by the number of the integration cells we present a regularized version of the hierarchical cover construction algorithm which reduces the number of integration cells even further and subsequently improves the computational efficiency in fact the computational costs during the integration of the nonzeros of the stiffness matrix are comparable to that of a finite element method yet the presented method is completely independent of a mesh moreover our method is applicable to general domains and allows for the construction of approximations of any order and regularity

finite element stabilization matrices a unification approach abstract in this paper we show that the stabilization vector can be obtained naturally by taking the partial derivatives with respect to the natural coordinates hence the components of the strains and stresses can be expressed in terms of a set of orthogonal coordinates with these definitions of stresses and strains a general form of the finite element stabilization matrices is then constructed for underintegrated irregular shape and anisotropic elements the explicit expressions of the stabilization matrices for the 4 9 and 8 node laplace and continuum elements and the 4 node plate element are then derived from the general form these are sufficiently general for the developments herein the generalization of the stabilization matrices concept to nonlinear problems which usually require the numerical integration of a fairly general class of nonlinear finite deformation constitutive equations is also presented the computer implementation aspects and numerical evaluation of these stabilized elements are also considered numerical tests confirm the stability and accuracy characteristics of the resulting elements in particular the numerical experiments reported here also show that the rates of convergence in the l2 norm and in the energy norm agree well with the expected convergence rates

reproducing kernel hierarchical partition of unity part ii applications in this part of the work the meshless hierarchical partition of unity proposed in 1 referred here as part i is used as a multiple scale basis in numerical computations to solve practical problems the applications discussed in the present work fall into two categories 1 a wavelet adaptivity refinement procedure and 2 a so called wavelet petrov galerkin procedure in the applications of wavelet adaptivity the hierarchical reproducing kernels are used as a multiple scale basis to compute the numerical solutions of the helmholtz equation a model equation of wave propagation problems and to simulate shear band formation in an elasto viscoplastic material a problem dictated by the presence of the high gradient deformation in both numerical experiments numerical solutions with high resolution are obtained by inserting the wavelet like basis into the primary interpolation function basis a process that may be viewed as a spectral p type refinement by using the interpolant that has synchronized convergence property as a weighting function a wavelet petrov galerkin procedure is proposed to stabilize computations of some pathological problems in numerical computations such as advection diffusion problems and stokes flow problem it offers an alternative procedure in stablized methods and also provides some insight or new interpretation of the method detailed analysis has been carried out on the stability and convergence of the wavelet petrov galerkin method copyright 1999 john wiley sons ltd

effect of numerical integration on meshless methods abstract in this paper we present the effect of numerical integration on meshless methods with shape functions that reproduce polynomials of degree k 1 the meshless method was used on a second order neumann problem and we derived an estimate for the energy norm of the error between the exact solution and the approximate solution from the meshless method under the presence of numerical integration this estimate was obtained under the assumption that the numerical integration scheme satisfied a form of green s formula we also indicated how to obtain numerical integration schemes satisfying this property

joint similarities and parameterizations for naimark complementary frames abstract it is known that the naimark complementary frames for a given frame are not necessarily unique up to the similarity in this paper we introduce the concept of joint complementary frame pairs for a given dual frame pair and prove that they are unique up to the joint similarity as an application we give a necessary and sufficient condition under which two naimark complementary frames are similar for different pairs of dual frames we present an operator parameterization for their joint complementary frame pairs

possible generalization of boltzmann gibbs statistics with the use of a quantity normally scaled in multifractals a generalized form is postulated for entropy namelys q k 1 i 1 w p i q q 1 whereq characterizes the generalization andp i are the probabilities associated withw microscopic configurations w the main properties associated with this entropy are established particularly those corresponding to the microcanonical and canonical ensembles the boltzmann gibbs statistics is recovered as theq 1 limit

multiresolution expansion approximation order and quasiasymptotic behavior of tempered distributions abstract multiresolution analysis of tempered distributions is studied through multiresolution analysis on the corresponding test function spaces s r r r n 0 for a function h which is smooth enough and of appropriate decay it is shown that the derivatives of its projections to the corresponding spaces v j j z in a regular multiresolution analysis of l 2 r denoted by h j multiplied by a polynomial weight converge in sup norm i e h j h in s r r as j analogous result for tempered distributions is obtained by duality arguments the analysis of the approximation order of the projection operator within the framework of the theory of shift invariant spaces gives a further refinement of the results the order of approximation is measured with respect to the corresponding space of test functions as an application we give abelian and tauberian type theorems concerning the quasiasymptotic behavior of a tempered distribution at infinity

a maximum likelihood digital receiver using coordinate ascent and the discrete wavelet transform in this paper a maximum likelihood ml method is presented for joint estimation of amplitude phase time delay and data symbols in a single user direct sequence spread spectrum communication system since maximization of the likelihood function is analytically intractable a novel coordinate ascent algorithm is used to obtain sequential updates of the data symbols and all unknown nuisance parameters the novelty of the algorithm is due to the use of a multiresolution expansion of the received signal and the use of polynomial rooting in the complex plane in place of a line search over the signal delay parameter the multiresolution structure of the algorithm is exploited to reduce sensitivity to impulsive noise via wavelet thresholding computer simulations of the single user system show that the algorithm has fast convergence and comparison with theoretical lower bounds establishes that the algorithm achieves nearly optimal error performance

a generalization of the inhomogeneity measure for point distributions to the case of finite size objects abstract the statistical measure of spatial inhomogeneity for n points placed in cells each of size k k is generalized to incorporate finite size objects like black pixels for binary patterns of size l l as a function of length scale k the measure is modified in such a way that it relates to the smallest realizable value for each considered scale to overcome the limitation of pattern partitions to scales with k being integer divisors of l we use a sliding cell sampling approach for given patterns particularly in the case of clusters polydispersed in size the comparison between the statistical measure and the entropic one reveals differences in detection of the first peak while at other scales they well correlate the universality of the two measures allows both a hidden periodicity traces and attributes of planar quasi crystals to be explored

local maximum entropy approximation schemes a seamless bridge between finite elements and meshfree methods we present a one parameter family of approximation schemes which we refer to as local maximum entropy approximation schemes that bridges continuously two important limits delaunay triangulation and maximum entropy max ent statistical inference local max ent approximation schemes represent a compromise in the sense of pareto optimality between the competing objectives of unbiased statistical inference from the nodal data and the definition of local shape functions of least width local max ent approximation schemes are entirely defined by the node set and the domain of analysis and the shape functions are positive interpolate affine functions exactly and have a weak kronecker delta property at the boundary local max ent approximation may be regarded as a regularization or thermalization of delaunay triangulation which effectively resolves the degenerate cases resulting from the lack or uniqueness of the triangulation local max ent approximation schemes can be taken as a convenient basis for the numerical solution of pdes in the style of meshfree galerkin methods in test cases characterized by smooth solutions we find that the accuracy of local max ent approximation schemes is vastly superior to that of finite elements

spline prewavelets for non uniform knots in this paper we develop a framework suitable for performing a multiresolution analysis using univariate spline spaces of arbitrary degree and with non uniform knot sequences to this end we show among other things the existence of compactly supported prewavelets and of prewavelets that are globally supported but decay exponentially in each case we obtain a decomposition of a fine spline space as a sum of a coarse spline space plus a spline space spanned by prewavelets

construction of polygonal interpolants a maximum entropy approach in this paper we establish a link between maximizing information theoretic entropy and the construction of polygonal interpolants the determination of shape functions on n gons n 3 leads to a non unique under determined system of linear equations the barycentric co ordinates i which form a partition of unity are associated with discrete probability measures and the linear reproducing conditions are the counterpart of the expectations of a linear function the i are computed by maximizing the uncertainty h 1 2 n i log i subject to the above constraints the description is expository in nature and the numerical results via the maximum entropy maxent formulation are compared to those obtained from a few distinct polygonal interpolants the maximum entropy formulation leads to a feasible solution for i in any convex or non convex polygon this study is an instance of the application of the maximum entropy principle wherein least biased inference is made on the basis of incomplete information copyright 2004 john wiley sons ltd

an arbitrary order variationally consistent integration for galerkin meshfree methods author s hillman michael charles advisor s chen jiun shyan abstract the rate of convergence in galerkin methods for solving boundary value problems is determined by the order of completeness in the trial space and order of accuracy in the domain integration if insufficiently accurate domain integration is employed the optimal convergence rate cannot be attained for meshfree methods accurate domain integration is difficult to achieve without costly high order quadrature and the lack of accuracy in the domain integration may lead to sub optimal convergence or even solutions that diverge with refinement the difficulty in domain integration is due to the overlap of the shape function supports and the rational nature of the shape functions themselves this dissertation introduces a general framework to achieve the optimal order of convergence consistent with the order of trial space without high order quadrature first the conditions for achieving arbitrary order exactness in a boundary value problem using the galerkin approximation with quadrature are derived the conditions are derived in a general form and are applicable to all types of problems the test function gradients in the galerkin approximation should be consistent with the chosen numerical integration and this is termed variational consistency specifically integration by parts of the inner product between the test function and differential operator acting on the desired exact solution should hold when evaluated with the chosen quadrature specific problems are then considered with the conditions explicitly stated including elasticity the euler bernoulli beam the kirchhoff love plate and the non linear formulation of solid mechanics treating the type of numerical integration as a given test function gradients are then constructed to satisfy this condition the resulting method is arbitrarily high order exact and applicable to all types of integration the method is then used as a correction to several commonly used numerical integration methods and applied to the various boundary value problems it is demonstrated that the error induced by numerical integration is greatly reduced and optimal convergence is either partially or fully restored further it is shown that the variationally consistent integration methods are more effective than their standard counterparts in terms of computing time and solution error

tsallis entropy and cortical dynamics the analysis of eeg signals abstract we undertake the study of eeg signals by recourse to a wavelet based multiresolution analysis as adapted to an information measure scenario different information measures are employed it is shown that non extensive ones seem to be of particular usefulness

traveling salesman problem and tsallis statistics a generalization of the stochastic method of simulated annealing algorithm based on tsallis statistics is proposed this algorithm is considerably faster than the traditional ones in solving the traveling salesman problem acceptable solutions are found in fewer steps and higher temperatures than both the classical and the fast simulated annealings recent developments in solving np complete problems can be incorporated and improve the performance even more

generalized mutual information tests applied to fmri analysis abstract to analyze functional magnetic resonance imaging fmri data sets the physiologically induced signals have to be separated from noise or from artifacts as a result of involuntary patient motion or mri detection techniques in the past years various methods of analysis of fmri data were proposed however one of the unsolved problems is related to dependence of the results with the threshold applied within each analysis an alternative fmri time series analysis method based on information theory and tsallis entropy is proposed to decrease this dependency the method applies the generalized mutual information gmi concept within the fmri framework the gmi between the hemodynamic response function of a given voxel and the paradigm slope is evaluated as a measure of the amount of information that the paradigm could be encoded by a given brain region or voxel this value represented the activation of the voxels however the analysis relies heavily on the tsallis q parameter allowing for a broad spectrum of results as a test of applicability of the method real fmri data were obtained in a simple motor paradigm and subsequently analyzed with gmi for several values of a q as well as with classical methods cross correlation and student s t test the simulated data were used to compare quantitatively the presented method with the classical ones by means of the construction of receiver operating characteristic roc curve the results showed the applicability of the gmi method as well as the range of optimum q values

minimax estimation via wavelet shrinkage we attempt to recover an unknown function from noisy sampled data using orthonormal bases of compactly supported wavelets we develop a nonlinear method which works in the wavelet domain by simple nonlinear shrinkage of the empirical wavelet coefficients the shrinkage can be tuned to be nearly minimax over any member of a wide range of triebel and besov type smoothness constraints and asymptotically minimax over besov bodies with p q linear estimates cannot achieve even the minimax rates over triebel and besov classes with p 2 so the method can significantly outperform every linear method e g kernel smoothing spline sieve in a minimax sense variants of our method based on simple threshold nonlinear estimators are nearly minimax our method possesses the interpretation of spatial adaptivity it reconstructs using a kernel which may vary in shape and bandwidth from point to point depending on the data least favorable distributions for certain of the triebel and besov scales generate objects with sparse wavelet transforms many real objects have similarly sparse transforms which suggests that these minimax results are relevant for practical problems sequels to this paper which was first drafted in november 1990 discuss practical implementation spatial adaptation properties universal near minimaxity and applications to inverse problems

second order accurate derivatives and integration schemes for meshfree methods summary the consistency condition for the nodal derivatives in traditional meshfree galerkin methods is only the differentiation of the approximation consistency dac one missing part is the consistency between a nodal shape function and its derivatives in terms of the divergence theorem in numerical forms in this paper a consistency framework for the meshfree nodal derivatives including the dac and the discrete divergence consistency ddc is proposed the summation of the linear ddc over the whole computational domain leads to the so called integration constraint in the literature a three point integration scheme using background triangle elements is developed in which the corrected derivatives are computed by the satisfaction of the quadratic ddc we prove that such smoothed derivatives also meet the quadratic dac and therefore the proposed scheme possesses the quadratic consistency that leads to its name qc3 numerical results show that qc3 is the only method that can pass both the linear and the quadratic patch tests and achieves the best performances for all the four examples in terms of stability convergence accuracy and efficiency among all the tested methods particularly it shows a huge improvement for the existing linearly consistent one point integration method in some examples copyright 2012 john wiley sons ltd

dynamic multiscale spatiotemporal models for gaussian areal data summary we introduce a new class of dynamic multiscale models for spatiotemporal processes arising from gaussian areal data specifically we use nested geographical structures to decompose the original process into multiscale coefficients which evolve through time following state space equations our approach naturally accommodates data that are observed on irregular grids as well as heteroscedasticity moreover we propose a multiscale spatiotemporal clustering algorithm that facilitates estimation of the nested geographical multiscale structure in addition we present a singular forward filter backward sampler for efficient bayesian estimation our multiscale spatiotemporal methodology decomposes large data analysis problems into many smaller components and thus leads to scalable and highly efficient computational procedures finally we illustrate the utility and flexibility of our dynamic multiscale framework through two spatiotemporal applications the first example considers mortality ratios in the state of missouri whereas the second example examines agricultural production in espirito santo state brazil

quantum field theory in ninety minutes on presente divers aspects de la theorie quantique des champs la theorie p 2 a 2 dimensions la theorie 3 4 a 3 dimensions la theorie de yang mills a 4 dimensions

nodal integration of the element free galerkin method abstract spatial integration of the element free galerkin efg method is achieved by evaluating the integrals of the weak form only at the nodes in previous efg formulations a grid of cells was used to perform gaussian quadrature over the domain the absence of a cell structure for nodal integration results in a completely meshless method similar in simplicity to particle methods such as smooth particle hydrodynamics sph it is shown that nodal integration like sph suffers from spurious singular modes this spatial instability results from underintegration of the weak form and it is treated by the addition to the potential energy functional of a stabilization term which contains the square of the residual of the equilibrium equation example problems illustrate the effect of the stabilization and provide the basis for convergence studies

an implicit gradient model by a reproducing kernel strain regularization in strain localization problems abstract a reproducing kernel strain regularization rksr as a mathematical generalization of gradient theory and non local theory for strain localization problems is presented rksr introduces a correction of the weight function in the non local strain by imposition of gradient reproducing conditions both continuum and discrete forms of rksr are presented and they lead to an implicit representation of gradient models as such rksr provides a gradient type regularization to the localization problem without increasing the order of differentiation in the governing equations hence no additional boundary conditions are required and the need for higher order continuity for the approximation of unknowns in the governing equations is no longer an issue a von neumann spectral analysis is employed to study the spectral properties of rksr of various orders in one dimension it is shown that rksr almost duplicates the spectral properties of second and fourth order gradient theories in summary rksr reproduces the regularization properties of gradient methods without dealing with additional boundary conditions or higher order continuity issues

a block spin construction of ondelettes part i lemarie functions using block spin assignments we construct anl2 orthonormal basis consisting of dyadic scalings and translates of just a finite number of functions these functions have exponential localization and for even values of a construction parameterm one can make them classcm 1 with vanishing moments up to orderm inclusive such a basis has an important application to phase cell cluster expansions in quantum field theory

refinement of operator valued reproducing kernels this paper studies the construction of a refinement kernel for a given operator valued reproducing kernel such that the vector valued reproducing kernel hilbert space of the refinement kernel contains that of the given kernel as a subspace the study is motivated from the need of updating the current operator valued reproducing kernel in multi task learning when underfitting or overfitting occurs numerical simulations confirm that the established refinement kernel method is able to meet this need various characterizations are provided based on feature maps and vector valued integral representations of operator valued reproducing kernels concrete examples of refining translation invariant and finite hilbert schmidt operator valued reproducing kernels are provided other examples include refinement of hessian of scalar valued translation invariant kernels and transformation kernels existence and properties of operator valued reproducing kernels preserved during the refinement process are also investigated

orthonormal bases of compactly supported wavelets we construct orthonormal bases of compactly supported wavelets with arbitrarily high regularity the order of regularity increases linearly with the support width we start by reviewing the concept of multiresolution analysis as well as several algorithms in vision decomposition and reconstruction the construction then follows from a synthesis of these different approaches

generalized relative entropy in functional magnetic resonance imaging abstract the generalized kullback leibler distance d q q is the tsallis parameter is shown to be an useful measure for analysis of functional magnetic resonance imaging fmri data series this generalized form of entropy is used to evaluate the distance between the probability functions p 1 and p 2 of the signal levels related to periods of stimulus and non stimulus in event related fmri experiments the probability densities of the mean distance d q averaged over the n epochs of the entire experiment are obtained through numerical simulations for different values of signal to noise ratio snr and found to be fitted very well by gamma distributions 2 0 0008 for small values of n n 30 these distributions allow us to determine the sensitivity and specificity of the method by construction of the receiver operating characteristic roc curves the performance of the method is also investigated in terms of the parameters q and l number of signal levels and our results indicate that the optimum choice is q 0 8 and l 3 the entropic index q is found to exert control on both sensitivity and specificity of the method as q q 0 is raised sensitivity increases but specificity decreases finally the method is applied in the analysis of a real event related fmri motor stimulus experiment and the resulting maps show activation in primary and secondary motor brain areas

wavelets of multiplicity r a multiresolution approximation km m z of l2 r is of multi plicity r 0 if there are r functions x r whose translates form a riesz basis for v in the general theory we derive necessary and sufficient conditions for the translates of x r y x y r to form a riesz basis for v the resulting reconstruction and decomposition sequences lead to the construction of dual bases for v0 and its orthogonal complement w0 in vx the general theory is applied in the construction of spline wavelets with mul tiple knots algorithms for the construction of these wavelets for some special cases are given

quadrature for meshless methods in this paper we discuss quadrature for meshless methods we present results of two analyses the first of these was developed in 1 and the second was developed in 2 using a simple example we show that a natural quadrature scheme that satisfies no special properties leads to severe inaccuracies in the approximate solution we then show in the first analysis that a simple correction to the stiffness matrix controls the effect of the quadrature in the second analysis we show that if the quadrature scheme satisfies green s formula then the effect of the quadrature is controlled

non extensive block entropy statistics of cantor fractal sets abstract by using non extensive block entropy statistics we demonstrate analytically that the static structures of deterministic cantor sets with fractal dimension d f are characterised by a non extensive q exponent q 1 d f d for d f d where d is the embedding dimensions of the fractal set to calculate the s q entropy we use the block entropy method based on non overlapping windows and standard exact enumeration on cantor sets this result indicates that fractal structures are formed via dynamical processes which operate at the edge of chaos

convergence and stabilization of stress point integration in mesh free and particle methods stress point integration provides significant reductions in the computational effort of mesh free galerkin methods by using fewer integration points and thus facilitates the use of mesh free methods in applications where full integration would be prohibitively expensive the influence of stress point integration on the convergence and stability properties of mesh free methods is studied it is shown by numerical examples that for regular nodal arrangements good rates of convergence can be achieved for non uniform nodal arrangements stress point integration is associated with a mild instability which is manifested by small oscillations addition of stabilization improves the rates of convergence significantly the stability properties are investigated by an eigenvalue study of the laplace operator it is found that the eigenvalues of the stress point quadrature models are between those of full integration and nodal integration stabilized stress point integration is proposed in order to improve convergence and stability properties copyright 2007 john wiley sons ltd

a new jameson schmidt turkel smooth particle hydrodynamics algorithm for large strain explicit fast dynamics this paper presents a new smooth particle hydrodynamics sph computational framework for large strain explicit solid dynamics a mixed based set of total lagrangian conservation laws bonet et al 2015 gil et al 2016 is presented in terms of the linear momentum and an extended set of geometric strain measures comprised of the deformation gradient its co factor and the jacobian taking advantage of this representation the main aim of this paper is the adaptation of the very efficient jameson schmidt turkel jst algorithm jameson et al 1981 extensively used in computational fluid dynamics to a sph based discretisation of the mixed based set of conservation laws with three key distinct novelties first a conservative jst based sph computational framework is presented with emphasis in nearly incompressible materials second the suppression of numerical instabilities associated with the non physical zero energy modes is addressed through a well established stabilisation procedure third the use of a discrete angular momentum projection algorithm is presented in conjunction with a monolithic total variation diminishing runge kutta time integrator in order to guarantee the global conservation of angular momentum for completeness exact enforcement of essential boundary conditions is incorporated through the use of a lagrange multiplier projection technique a series of challenging numerical examples e g in the near incompressibility regime are examined in order to assess the robustness and accuracy of the proposed algorithm the obtained results are benchmarked against a wide spectrum of alternative numerical strategies

wavelet transform based adaptive filters analysis and new results in this paper the wavelet transform is used in an adaptive filtering structure the coefficients of the adaptive filter are updated by the help of the least mean square lms algorithm first the wavelet transform based adaptive filter wtaf is described and it is analyzed for its wiener optimal solution then the performance of the wtaf is studied by the help of learning curves for three different convergence factors 1 constant convergence factor 2 time varying convergence factor and 3 exponentially weighted convergence factor the exponentially weighted convergence factor is proposed to introduce scale based variation to the weight update equation it is shown for two different sets of data that the rate of convergence increases significantly for all three wtaf structures as compared to that of time domain lms the high convergence rates of the wtaf give us reason to expect that it will perform well in tracking rapid changes in a signal

stable particle methods based on lagrangian kernels abstract a large deformation particle method based on the krongauz belytschko corrected gradient meshfree method with lagrangian kernels is developed in this form the gradient is corrected by a linear transformation so that linear completeness is satisfied for the test functions shepard functions are used this guarantees that the patch test is met lagrangian kernels are introduced to eliminate spurious distortions of the domain of material stability a mass allocation scheme is developed that captures correct reflection of waves without any explicit application of traction boundary conditions in addition the lagrangian kernel versions of various forms of smooth particle methods sph including the standard forms and the randles libersky modification are presented and studied results are obtained for a variety of problems that compare this method to standard forms of sph the randles libersky correction and large deformation versions of the element free galerkin method

stabilized conforming nodal integration in the natural element method a stabilized conforming nodal integration scheme is implemented in the natural neighbour method in conjunction with non sibsonian interpolation in this approach both the shape functions and the integration scheme are defined through use of first order voronoi diagrams the method illustrates improved performance and significant advantages over previous natural neighbour formulations the method also shows substantial promise for problems with large deformations and for the computation of higher order gradients copyright 2004 john wiley sons ltd

canonical forms for quadratic hamiltonians abstract our analysis of the applicable representations of the group of bogoliubov transformations shows that the diagonalization of a quadratic fermion hamiltonian with arbitrary complex coefficients is equivalent to the reduction of a skew symmetric matrix to secondary diagonal form by an orthogonal transformation which we construct explicitly similarly the diagonalization of a positive definite boson hamiltonian with complex coefficients is equivalent to whittaker s diagonalization of a symmetric matrix by a symplectic transformatio both results are shown to follow from a general spectral theorem for indefinite inner product space a recent extension of which allows us to block diagonalize a positive definite quadratic boson hamiltonian with complex coefficients and infinite degrees of freedom and thereby provide a counterpart to araki s result for fermi fields

wavelet analysis and nonlinear dynamics in a nonextensive setting abstract we undertake the study of signals originated in time dependent nonlinear systems by recourse to a wavelet based multiresolution analysis as adapted to a nonextensive tsallis scenario diverse applications are discussed it is shown that the tsallis environment provides one with more detailed information than the conventional shannon one

characterization of strange attractors a new measure of strange attractors is introduced which offers a practical algorithm to determine their character from the time series of a single observable the relation of this new measure to fractal dimension and information theoretic entropy is discussed

determining lyapunov exponents from a time series we present the first algorithms that allow the estimation of non negative lyapunov exponents from an experimental time series lyapunov exponents which provide a qualitative and quantitative characterization of dynamical behavior are related to the exponentially fast divergence or convergence of nearby orbits in phase space a system with one or more positive lyapunov exponents is defined to be chaotic our method is rooted conceptually in a previously developed technique that could only be applied to analytically defined model systems we monitor the long term growth rate of small volume elements in an attractor the method is tested on model systems with known lyapunov spectra and applied to data for the belousov zhabotinskii reaction and couette taylor flow

reproducing kernel hierarchical partition of unity part i formulation and theory this work is concerned with developing the hierarchical basis for meshless methods a reproducing kernel hierarchical partition of unity is proposed in the framework of continuous representation as well as its discretized counterpart to form such hierarchical partition a class of basic wavelet functions are introduced based upon the built in consistency conditions the differential consistency conditions for the hierarchical kernel functions are derived it serves as an indispensable instrument in establishing the interpolation error estimate which is theoretically proven and numerically validated for a special interpolant with different combinations of the hierarchical kernels a synchronized convergence effect may be observed being different from the conventional legendre function based p type hierarchical basis the new hierarchical basis is an intrinsic pseudo spectral basis which can remain as a partition of unity in a local region because the discrete wavelet kernels form a partition of nullity these newly developed kernels can be used as the multi scale basis to solve partial differential equations in numerical computation as a p type refinement copyright 1999 john wiley sons ltd

a tsallis statistics based neural network model for novel word learning abstract we invoke the tsallis entropy formalism a nonextensive entropy measure to include some degree of non locality in a neural network that is used for simulation of novel word learning in adults a generalization of the gradient descent dynamics realized via nonextensive cost functions is used as a learning rule in a simple perceptron the model is first investigated for general properties and then tested against the empirical data gathered from simple memorization experiments involving two populations of linguistically different subjects numerical solutions of the model equations corresponded to the measured performance states of human learners in particular we found that the memorization tasks were executed with rather small but population specific amounts of nonextensivity quantified by the entropic index q our findings raise the possibility of using entropic nonextensivity as a means of characterizing the degree of complexity of learning in both natural and artificial systems

a new support integration scheme for the weakform in mesh free methods a new support integration technique is proposed which is similar to those used in truly mesh free methods the contribution of this paper is to exploit the divergence free condition for the support integrals to construct quadrature formulas that only require three integration points per particle in two dimensions numerical examples show that the proposed integration method can achieve results that agree with manufactured closed form solutions copyright 2009 john wiley sons ltd