the mixed order bcgs fft method for the scattering of three dimensional inhomogeneous anisotropic magnetodielectric objects this paper presents the first fft based fast volume integral equation solver for inhomogeneous anisotropic magnetodielectric objects the volume integral equations are formulated by employing mixed order basis functions that expand the flux densities and vector potentials in the coupled field integral equations in terms of different sets of basis functions with different orders volumetric roof top basis functions are used for flux densities whereas second order curl conforming basis functions are used for vector potentials a fast volume integral equations solver namely the bcgs fft method is then applied to accelerate the solution of this mixed order weak form formulation examples show that the mixed order bcgs fft method has high accuracy compared to both analytical and numerical solutions examples also show the mixed order bcgs fft method has high computational efficiency compared to commercial software

induction of fuzzy rules and membership functions from training examples abstract most fuzzy controllers and fuzzy expert systems must predefine membership functions and fuzzy inference rules to map numeric data into linguistic variable terms and to make fuzzy reasoning work in this paper we propose a general learning method as a framework for automatically deriving membership functions and fuzzy if then rules from a set of given training examples to rapidly build a prototype fuzzy expert system based on the membership functions and the fuzzy rules derived a corresponding fuzzy inference procedure to process inputs is also developed

on the variance of the internal path length of generalized digital trees the mellin convolution approach abstract this paper studies a generalization of the internal path length of a random digital search tree to bucket trees which are capable of storing up to b records per node we show that under the assumption of the symmetric bernoulli probabilistic model the expected path length of a tree built from n records is asymptotically n log 2 n a 1 log 2 n n and the variance is c e 1 log 2 n n where a and c depend on b only the continuous functions 1 and e 1 are periodic with mean 0 and period 1 the proofs are analytical and make use of generating functions harmonic sums and the mellin integral transform an important and very general tool for the analysis is mellin s convolution integral these results and techniques are motivated by a paper by flajolet and richmond and by kirschenhofer prodinger and szpankowski

digital search trees revisited several algorithms have been proposed which build search trees using digital properties of the search keys a general approach to the study of the average case performance of such algorithms is discussed with particular attention to the analysis of the digital search tree structures of coflman and eve specifically the method leads to the solution of a problem left open by knuth finding the average number of nodes in digital search trees with both sons null the paper may be of interest as a survey and tutorial treatment of the analysis of the three primary digital tree search methods digital search trees radix search tries and patricia tries

asymptotic behavior of the lempel ziv parsing scheme and digital search trees the lempel ziv parsing scheme finds a wide range of applications most notably in data compression and algorithms on words it partitions a sequence of length n into variable phrases such that a new phrase is the shortest substring not seen in the past as a phase the parameter of interest is the number mn of phrases that one can construct from a sequence of length n in this paper for the memoryless source with unequal probabilities of symbols generation we derive the limiting distribution of mn which turns out to be normal this proves a long standing open problem in fact to obtain this result we solved another open problem namely that of establishing the limiting distribution of the internal path length in a digital search tree the latter is a consequence of an asymptotic solution of a multiplicative differential functional equation often arising in the analysis of algorithms on words interestingly enough our findings are proved by a combination of probabilistic techniques such as renewal equation and uniform integrability and analytical techniques such as mellin transform differential functional equations de poissonization and so forth in concluding remarks we indicate a possibility of extending our results to markovian models

on the balance property of patricia tries external path length viewpoint abstract in this paper we give exact and asymptotic approximations for the variance of the external path length in a symmetric patricia tree the problem was open up to now we prove that for the binary patricia tree the variance is asymptotically equal to 0 37 n np log 2 n where n is the number of stored records and p x is a periodic function with a very small amplitude this implies that the external path length is asymptotically equal to n log 2 n with probability one i e almost surely these results are next used to show that from the practical average viewpoint the patricia tree does not need to be restructured in order to keep it balanced in general we ask to what extent simpler and more direct algorithms for digital search trees can be expected in practice to match the performance of more complicated worst case asymptotically better ones

comments on application of fft and the conjugate gradient method for the solution of electromagnetic radiation from electrically large and small conducting bodies in the original paper ibid vol ap 34 no 5 p 635 40 1986 t k sarkar et al describe how the fast fourier transform fft and the conjugate gradient method cgm can be used to efficiently make use of the convolutional form of the electric field integral equation for straight wire antennas the commentators agree that the cgm converges more rapidly than the previously used method of steepest descent and the spectral iteration technique they also point out that experience with the dyadic green s function encountered in the two dimensional transverse electric te and three dimensional equations for dielectric bodies is not so encouraging it is found that for these cases the cgm convergence can be quite slow degrading the efficiency of the algorithm in addition the solution obtained using the pulse basis point matching method to discretize the integral equation are very inaccurate for the two dimensional te and three dimensional cases a reply to these comments from the authors of the original paper is included

average profile of the generalized digital search tree and the generalized lempel ziv algorithm the goal of this research is threefold i to analyze generalized digital search trees ii to derive the average profile i e phrase length of a generalization of the well known parsing algorithm due to lempel and ziv and iii to provide analytic tools to analyze asymptotically certain partial differential functional equations often arising in the analysis of digital trees in the generalized lempel ziv parsing scheme one partitions a sequence of symbols from a finite alphabet into phrases such that the new phrase is the shortest substring seen in the past by at most b 1 phrases b 1 corresponds to the original lempel ziv scheme such a scheme can be analyzed through a generalized digital search tree in which every node is capable of storing up to b strings in this paper we investigate the depth of a randomly selected node in such a tree and the length of a randomly selected phrase in the generalized lempel ziv scheme these findings and some recent results allow us to compute the average redundancy of the generalized lempel ziv code and compare it to the ordinary lempel ziv code leading to an optimal value of b analytic techniques of precise analyses of algorithms are used to establish most of these conclusions

volume surface integral equation method based on higher order hierarchical vector basis functions for em scattering and radiation from composite metallic and dielectric structures this paper presents a novel galerkin type method of moments solution of the volume surface integral equation vsie which is developed for the analysis of electromagnetic scattering and radiation from composite metallic and dielectric structures our new scheme utilizes curved tetrahedral elements and curved triangular patches for geometric modeling and the associated higher order hierarchical vector hohv basis functions for volume surface current modeling in contrast to the previous studies on higher order vsie our scheme which is reported for the first time in vsie is much more flexible in comparison with the conventional low order scheme our scheme requires much less memory and computational time for the same accuracy level numerical results are provided to demonstrate the accuracy efficiency and flexibility of our scheme

phase transition in a generalized eden growth model on a tree we study analytically the late time statistics of the number of particles in an eden growth model on a tree in this model a cluster grows in continuous time on a binary cayley tree starting from the root by absorbing new particles at the empty perimeter sites at a rate proportional to c l where c is a positive parameter and l is the distance of the perimeter site from the root for c 1 this model corresponds to random binary search trees and for c 2 it corresponds to digital search trees in computer science by introducing a backward fokker planck approach we calculate the mean and the variance of the number of particles at large times and show that the variance undergoes a phase transition at a critical value c sqrt 2 while for c sqrt 2 the variance is proportional to the mean and the distribution is normal for c sqrt 2 the variance is anomalously large and the distribution is non gaussian due to the appearance of extreme fluctuations the model is generalized to one where growth occurs on a tree with m branches and in this more general case we show that the critical point occurs at c sqrt m

laws of large numbers and tail inequalities for random tries and patricia trees we consider random tries and random patricia trees constructed from n independent strings of symbols drawn from any distribution on any discrete space if hn is the height of this tree we show that hn e hn tends to one in probability additional tail inequalities are given for the height depth size internal path length and profile of these trees and ordinary tries that apply without any conditions on the string distributions they need not even be identically distributed

modified spherical wave functions with anisotropy ratio application to the analysis of scattering by multilayered anisotropic shells we describe a novel and rigorous vector eigenfunction expansion of electric type green s dyadics for radially multi layered uniaxial anisotropic media in terms of the modified spherical vector wave functions which can take into account the effects of anisotropy ratio systematically in each layer the material constitutions e and epsiv macrmu macr are tensors and distribution of sources is arbitrary both the unbounded and scattering dyadic green s functions dgfs for rotationally uniaxial anisotropic media are derived in spherical coordinates r thetas phi the coefficients of scattering dgfs based on the coupling recursive algorithm satisfied by the coefficient matrix are derived and expressed in a compact form with these dgfs obtained the electromagnetic fields in each layer are straightforward once the current source is known a specific model is proposed for the scattering and absorption characteristics of multilayered uniaxial anisotropic spheres and some novel performance regarding anisotropy effects is revealed

gp coach genetic programming based learning of compact and accurate fuzzy rule based classification systems for high dimensional problems in this paper we propose gp coach a genetic programming based method for the learning of compact and accurate fuzzy rule based classification systems for high dimensional problems gp coach learns disjunctive normal form rules generated by means of a context free grammar coded as one rule per tree the population constitutes the rule base so it is a genetic cooperative competitive learning approach gp coach uses a token competition mechanism to maintain the diversity of the population and this obliges the rules to compete and cooperate among themselves and allows the obtaining of a compact set of fuzzy rules the results obtained have been validated by the use of non parametric statistical tests showing a good performance in terms of accuracy and interpretability

average profile and limiting distribution for a phrase size in the lempel ziv parsing algorithm consider the parsing algorithm developed by lempel and ziv 1978 that partitions a sequence of length n into variable phrases blocks such that a new block is the shortest substring not seen in the past as a phrase in practice the following parameters are of interest number of phrases the size of a phrase the number of phrases of given size and so forth in this paper we focus on the size of a randomly selected phrase and the average number of phrases of a given size the so called average profile of phrase sizes these parameters can be efficiently analyzed through a digital search tree representation for a memoryless source with unequal probabilities of symbols generation the so called asymmetric bernoulli model we prove that the size of a typical phrase is asymptotically normally distributed with mean and variance explicitly computed in terms of digital search trees we prove the normal limiting distribution of the typical depth i e the length of a path from the root to a randomly selected node the latter finding is proved by a technique that belongs to the toolkit of the analytical analysis of algorithms and it seems to be novel in the context of data compression

distributed representation of fuzzy rules and its application to pattern classification abstract this paper introduces the concept of distributed representation of fuzzy rules and applies it to classification problems distributed representation is implemented by superimposing many fuzzy rules corresponding to different fuzzy partitions of a pattern space this means that we simulatenously employ many fuzzy rule tables corresponding to different fuzzy partitions in fuzzy inference in order to apply distributed representation of fuzzy rules to pattern classification problems we first propose an algorithm to generate fuzzy rules from numerical data next we propose a fuzzy inference method using the generated fuzzy rules the classification power of distributed representation is compared with that of ordinary fuzzy rules which can be viewed as local representation

the biconjugate gradient method for electromagnetic scattering the biconjugate gradient bcg method for solving linear systems is shown to be more efficient than the conjugate gradient cg method for several examples from electromagnetic scattering a remedy for the occasional stagnation of the algorithm is proposed the potential flaw in the bcg algorithm may be avoided when encountered by restarting the algorithm with a perturbed estimate of the solution

integral equation based analysis of scattering from 3 d inhomogeneous anisotropic bodies this paper presents an integral equation based scheme to analyze scattering from inhomogeneous bodies with anisotropic electromagnetic properties both the permittivity and permeability are assumed to be generalized tensors requisite integral equations are derived using volume equivalence theorem with the electric and magnetic flux densities being the unknown quantities matrix equations are derived by discretizing these unknowns using three dimensional rao wilton glisson basis functions reduction of the integral equation to a corresponding matrix equation is considerably more involved due to the presence of anisotropy and the use of vector basis function methods for evaluation of the integrals involved in the construction of this matrix is elucidated in detail the method of moments technique is augmented with the fast multipole method and a compression scheme the latter two enable large scale analysis finally several numerical results are presented and compared against analytical solutions to validate the proposed scheme an appendix provides analytical derivations for the formulae that are used to validate numerical method and the necessary formulae that extends the approach presented herein to the analysis of scattering bianisotropic bodies

asymptotic behavior of the height in a digital search tree and the longest phrase of the lempel ziv scheme we study the height of a digital search tree dst built from n random strings generated by an unbiased memoryless source i e all symbols are equally likely we shall argue that the height of such a tree is equivalent to the length of the longest phrase in the lempel ziv parsing scheme that partitions a random sequence into n phrases we also analyze the longest phrase in the lempel ziv scheme in which a string of fixed length m is parsed into a random number of phrases in the course of our analysis we shall identify four natural regions of the height distribution and characterize them asymptotically for large n in particular for the region where most of the probability mass is concentrated the asymptotic distribution of the height exhibits an exponential of a gaussian distribution with an oscillating term around the most probable value k 1 lfloor log 2 n sqrt 2 log 2 n log 2 sqrt 2 log 2 n frac 1 log 2 frac 1 2 rfloor 1 more precisely we shall prove that the asymptotic distribution of a dst is concentrated on either the one point k1 or the two points k1 1 and k1 which actually proves slightly modified kesten s conjecture quoted in probab theory related fields 79 1988 pp 509 542 finally we compare our findings for dst with the asymptotic distributions of the height for other digital trees such as tries and patricia tries we derive these results by a combination of analytic methods such as generating functions laplace transform the saddle point method and ideas of applied mathematics such as linearization asymptotic matching and the wkb method our analysis makes certain assumptions about the forms of some of the asymptotic expansions as well as their asymptotic matching we also present detailed numerical verification of our results

quadrature over a pyramid or cube of integrands with a singularity at a vertex a simple transformation is introduced which facilitates the evaluation of integrals over certain square based pyramids or cubes in cases where the integrand has a singularity at a vertex

rcs computation of large inhomogeneous objects using a fast integral equation solver in this paper we apply a fast fourier transform fft accelerated volume integral equation solver to compute the radar cross section of large scale inhomogeneous objects this method is related to bojarski s k space method and the subsequent conjugate cg and biconjugate gradient bcg fft algorithms the method developed here combines the weak form discretization with the bcg and stabilized bcg bcgs solvers we show that this method has marked improvements over related algorithms the numerical method has been validated by mie series for multilayer spheres and applied to some practical problems with this method we are currently able to solve a three dimensional problem with a volume of size 3648 spl lambda sup 3 21 23 million unknowns on a single workstation

a characterization of digital search trees from the successful search viewpoint this paper studies the average complexity of digital search trees from the successful search point of view the average value of the successful search is used to evaluate the search time for a given record the number of comparisons to insert a record etc the average value however is rather a poor measure and the need for higher moments of the successful search is obvious for example the variance provides information on how well a digital tree is balanced the third centralized moment is a measure of the skewness property of the distribution etc in this paper we concentrate on an open problem how to evaluate all moments of the successful search in an asymmetric multiway digital search tree we prove that the mth moment esmn of the successful search sn where n is the number of stored records satisfies esmn lnmn 1 hm1 as n where h1 is the entropy of the alphabet in particular it is shown that the variance of sn is var sn c ln n o 1 for the asymmetric case and var sn o 1 for the symmetric case this paper completes a study undertaken by the author to characterize some digital trees such as regular tries patricia tries and digital search trees from the average viewpoint

universal limit laws for depths in random trees random binary search trees b ary search trees median of 2k 1 trees quadtrees simplex trees tries and digital search trees are special cases of random split trees for these trees we offer a universal law of large numbers and a limit law for the depth of the last inserted point as well as a law of large numbers for the height

on the variance of a class of inductive valuations of data structures for digital search let an inductive valuation l on the family of binary tries or patricia tries or digital search trees be defined in the following way l t l tl l tr r t where tl and tr denote the left and right subtrees of t and r depends only on the size the number of records t of t let ln denote l restricted to the trees of size n in theorem 1 we give sufficient conditions on the sequence r t r t for the variance var ln to be of exact order n if the family of tries resp patricia tries resp digital search trees is equipped with the bernoulli model for the symmetric bernoulli model we prove the existence of a continuous periodic function with period 1 such that var ln log2 n n holds

meshfree solutions of volume integral equations for electromagnetic scattering by anisotropic objects volume integral equations vies are indispensable for solving inhomogeneous or anisotropic electromagnetic em problems by integral equation approach the solution of vies strongly relies on the discretization of volume integral domains and tetrahedral elements in discretization are usually preferred for arbitrary geometric shapes unlike discretizing a surface domain discretizing a volume domain is very inconvenient in practice and special commercial software is needed in general even for a simple and regular geometry to release the burden of descretizing volume domains especially to remove the constraint of mesh conformity in the traditional method of moments mom we propose a novel meshfree scheme for solving vies in this paper the scheme is based on the transformation of volume integrals into boundary integrals through the green gauss theorem when integral kernels are regularized by excluding a small cylinder enclosing the observation node the original integral domain represented by the object is also expanded to a cylindrical domain circumscribing the object to facilitate the evaluation of boundary integrals the singular integrals over the small cylinder are specially handled with singularity subtraction techniques numerical examples for em scattering by inhomogeneous or anisotropic objects are presented to illustrate the scheme and good results are observed

a genetic algorithm for generating fuzzy classification rules abstract fuzzy classification systems based on fuzzy logic are capable of dealing with cognitive uncertainties such as vagueness and ambiguity involved in classification problems to build a fuzzy classification system the key is to find an optimal set of fuzzy rules machine learning methods such as fuzzy neural networks and fuzzy decision tree induction have been applied to learn fuzzy rules but they may be trapped into local optimal based on the principle of natural evolution and global searching a genetic algorithm is promising in obtaining better results but it is usually less efficient in this paper a fuzzy genetic algorithm fga is developed to generate fuzzy classification rules several techniques such as multi value logic coding composite fitness function viability check and rule extraction are used to improve the efficiency and the effectiveness of the algorithm two experiments are conducted to demonstrate the performance of fga and to compare fga with other machine learning algorithms

on the average redundancy rate of the lempel ziv code with the k error protocol abstract in this paper we examine the average redundancy rate of a lempel ziv 78 code with the k error protocol storer and reif have studied this modification of the lempel ziv lz scheme and showed that it provides an efficient protection against error propagation while preserving the asymptotic optimality of the code we refine this result by providing an asymptotic expression for the average redundancy rate of this code for memoryless sources using analytic methods we establish our result by exploiting a relationship between a parsing scheme of the lz encoder with the k error protocol and a generalization of the digital search tree dst structure we accompany our analysis with a number of experiments that test the validity of our theoretical result and demonstrate the effects of various additional modifications of the lz algorithm

combining gp operators with sa search to evolve fuzzy rule based classifiers abstract the genotype phenotype encoding of fuzzy rule bases in ga along with their corresponding crossover and mutation operators can be used by other search schemes improving the behavior of these last ones as a practical consequence of this a simulated annealing based method for inducting both parameters and structure of a fuzzy classifier has been developed the adjacency operator in sa has been replaced with a macromutation taken from tree shaped genotype gas we will show that results of sa search are similar to those of gp in both the efficiency of the learned classifiers and in its linguistic interpretability while the memory consumption of the learning process is lower

linguistic recognition system based on approximate reasoning a linguistic recognition system based on approximate reasoning has been described which is capable of handling various imprecise input patterns and of providing a natural decision the input feature is considered to be of either linguistic form or quantitative form or mixed form or set form an input has been viewed as consisting of various combinations of the three primary properties small medium and high possessed by its different features to some degree the various uncertainty ambiguity in the input statement has been managed by providing modifying membership values heuristically to a great extent unlike the conventional fuzzy set theoretic approach the sets small and high have been represented here by functions the weight matrices corresponding to various properties and classes have been taken into account in the composition rule of inference in order to make the analysis more effective the natural output decision is associated with a confidence factor denoting the degree of certainty of the decision thus providing a low rate of misclassification as compared to the conventional two state system the effectiveness of the algorithm has been demonstrated on the speech recognition problem

a simple simulated annealing algorithm for the maximum clique problem abstract simulated annealing technique has mostly been used to solve various optimization and learning problems and it is well known that the maximum clique problem is one of the most studied np hard optimization problems owing to its numerous applications in this note a simple simulated annealing algorithm for the maximum clique problem is proposed and tested on all 80 dimacs maximum clique instances although it is simple the proposed simulated annealing algorithm is efficient on most of the dimacs maximum clique instances the simulation results show that the proposed simulated annealing algorithm outperforms a recent efficient simulated annealing algorithm proposed by xu and ma and the solutions obtained by the proposed simulated annealing algorithm have the equal quality with those obtained by a recent trust region heuristic algorithm of stanislav busygin

cfvie formulation for em scattering on inhomogeneous anisotropic metallic objects the coupled field volume integral equation dini series expansion cfvie dse method is employed for electromagnetic scattering on inhomogeneous anisotropic metallic objects the anisotropic coating is both of gyroelectric and gyromagnetic type unlike traditional methods that describe the metallic core through a surface integral a full wave method is developed based on vies only in conjunction with modified tensorial green s functions taking into account the electric boundary condition bc on core s perfect electric conducting surface cfvie dse is then solved using entire domain basis functions of dini type new orthogonal sets in the domain of anisotropy are developed for the proper satisfaction of the physical bcs along with the reduction of the cfvie to algebraic form the method is fully validated and numerical results regarding radar cross section rcs computations for various anisotropic configurations are presented

potential integrals for uniform and linear source distributions on polygonal and polyhedral domains formulas for the potentials due to uniform and linearly varying source distributions defined on simply shaped domains are systematically developed and presented domains considered are infinite planar strips infinite cylinders of polygonal cross sections planar surfaces with polygonal boundaries and volumetric regions with polyhedral boundaries the expressions obtained are compact in form and their application in the numerical solution of electromagnetics problems by the method of moments is illustrated

a note on the height of suffix trees consider a random word in which the individual symbols are drawn from a finite or infinite alphabet with symbol probabilities p i and let h n be the height of the suffix tree constructed from the first n suffixes of this word it is shown that h n is asymptotically close to 2 log n log 1 sum i p i 2 in many respects the difference is o log log n in probability and the ratio tends to one almost surely and in the mean

patricia tries again revisited the patricia trie is a simple modification of a regular trie by eliminating unary branching nodes the patricia achieves better performance than regular tries however the question is how much on the average is the patricia better this paper offers a thorough answer to this question by considering some statistics of the number of nodes examined in a successful search and an unsuccessful search in the patricia tries it is shown that for the patricia containing n records the average of the successful search length s n asymptotically becomes 1 h 1 ln n o 1 and the variance of s n is either var s n c ln n 0 1 for an asymmetric patricia or var s n 0 1 for a symmetric patricia where h 1 is the entropy of the alphabet over which the patricia is built and c is an explicit constant higher moments of s n are also assessed the number of nodes examined in an unsuccessful search u n is studied only for binary symmetric patricia tries we prove that the m th moment of the unsuccessful search length eu m n satisfies lim n eu m n log m 2 n 1 and the variance of u n is var u n 0 87907 these results suggest that patricia tries are very well balanced trees in the sense that a random shape of patriciatries resembles the shape of complete trees that are ultimately balanced trees

analysis of volume integral equation formulations for scattering by high contrast penetrable objects the volume integral equation method is applied in electromagnetic scattering from arbitrarily shaped three dimensional inhomogeneous objects the properties of the volume electric and magnetic field integral equations vefie and vmfie are investigated numerical experiments show that if the galerkin s method with the lowest mixed order basis functions is used to discretize the equations the accuracy of the vmfie can be significantly poorer than the accuracy of the vefie in particular for high contrast objects at high frequencies the accuracy of the vmfie can be essentially improved with full first order linear basis functions the linear basis functions are found to be useful also when a single volume integral equation is used to model a general scatterer where both permittivity and permeability differ from the background

further results on digital search trees abstract in this paper distribution results are proved on the cost of insertion in digital search trees binary tries and patricia tries a method from the calculus of finite differences is used to achieve asymptotic results

some results on tries with adaptive branching we study a modification of digital trees or tries with adaptive multi digit branching such tries can dynamically adjust degrees of their nodes by choosing the number of digits to be processed per lookup while we do not specify any particular method for selecting the degrees of nodes we assume that such selection can be accomplished by examining the number of strings remaining in each sub tree and or estimating parameters of the input distribution we call this class of digital trees adaptive multi digit tries or amd tries and provide a preliminary analysis of their expected behavior in a memoryless model we establish the following results 1 there exist amd tries attaining a constant expected time of a successful search 2 there exist amd tries consuming a linear in the number of strings inserted amount of space 3 both constant search time and linear space usage can be attained if the memoryless source is symmetric we accompany our analysis with a brief survey of several known types of adaptive trie structures and show how our analysis extends and or complements previous results

nonconformal discretization of electric current volume integral equation with higher order hierarchical vector basis functions a nonconformal discretization of the electric current volume integral equation jvie is presented for the electromagnetic scattering analysis of inhomogeneous dielectric objects this jvie is based on higher order geometrical and current modeling where curved tetrahedral elements and higher order hierarchical vector hohv basis functions are adopted in order to implement the nonconformal discretization the half face based hohv basis function is introduced to address the nonconformal volumetric elements compared to the conventional low order conformal nonconformal vie our nonconformal hohv based jvie has the advantage of improving the computational efficiency and accuracy compared to the previous vie discretized by other higher order functions our nonconformal jvie is based on the higher order geometric modeling and the hohv bases in order to improve the efficiency the basis expansion and recombination technique is introduced to significantly accelerate the matrix filling moreover the flexibility of basis order selection is further enhanced by the mixed order schemes numerical results are given to demonstrate its accuracy efficiency and flexibility

mellin transforms and asymptotics finite differences and rice s integrals high order differences of simple number sequences may be analysed asymptotically by means of integral representations residue calculus and contour integration this technique akin to mellin transform asymptotics is put in perspective and illustrated by means of several examples related to combinatorics and the analysis of algorithms like digital tries digital search trees quadtrees and distributed leader election