calculation of cfie impedance matrix elements with rwg and n spl times rwg functions the method of moments mom solution of combined field integral equation cfie for electromagnetic scattering problems requires calculation of singular double surface integrals when galerkin s method with triangular vector basis functions rao wilton glisson functions and the cfie are applied to solve electromagnetic scattering by a dielectric object both rwg and n spl times rwg functions n is normal unit vector should be considered as testing functions robust and accurate methods based on the singularity extraction technique are presented to evaluate the impedance matrix elements of the cfie with these basis and test functions in computing the impedance matrix elements including the gradient of the green s function we can avoid the logarithmic singularity on the outer testing integral by modifying the integrand in the developed method all singularities are extracted and calculated in closed form and numerical integration is applied only for regular functions in addition we present compact iterative formulas for computing the extracted terms in closed form by these formulas we can extract any number of terms from the singular kernels of cfie formulations with rwg and n spl times rwg functions

on edge rupture degree of graphs the edge rupture degree of an incomplete connected graph g is defined as r g m a x g s s m g s s e g g s 1 where g s and m g s respectively denote the number of components and the order of a largest component in g s this is a reasonable parameter to measure the vulnerability of networks as it takes into account both the amount of work done to damage the network and how badly the network is damaged in this paper firstly the relationships between the edge rupture degree and some other graph parameters namely the edge connectivity edge integrity edge toughness edge tenacity diameter the algebraic connectivity and the minimum degree are established after that the edge rupture degree of the middle graphs of path and cycle are given then we introduced the concept of r maximal graph and give some basic results of such graphs finally we introduce the concept of edge ruptured and strictly edge ruptured graph and we establish necessary and sufficient conditions for a graph to be edge ruptured and strictly edge ruptured respectively

singular basis functions and curvilinear triangles in the solution of the electric field integral equation basis functions are formulated that account for singularities in the charge density near an edge on a conducting body the formulation is general and the basis functions are valid for planar as well curvilinear geometries in principle singularities of any order can be treated but best results are obtained for so called knife edge singularities results are compared with exact solutions or measurements where available for some simple problems

on the algebraic connectivity of graphs as a function of genus abstract we find an upper bound on the algebraic connectivity of graphs of various genus we begin by showing that for fixed k 1 the graph of genus k of largest algebraic connectivity is a complete graph we then find an upper bound for noncomplete graphs of a fixed genus k 1 and we determine the values of k for which the upper bound can be attained finally we find the upper bound of the algebraic connectivity of planar graphs graphs of genus zero and determine precisely which graphs attain this upper bound

projected single spin flip dynamics in the ising model we study transition matrices for projected dynamics in the energy magnetization magnetization and energy spaces several single spin flip dynamics are considered such as the glauber and metropolis canonical ensemble dynamics and the metropolis dynamics for three multicanonical ensembles the flat energy magnetization the flat energy and the flat magnetization histograms from the numerical diagonalization of the matrices for the projected dynamics we obtain the subdominant eigenvalues and the largest relaxation times for systems of varying size although the projected dynamics is an approximation to the full state space dynamics comparison with some available results obtained by other authors shows that projection in the magnetization space is a reasonably accurate method to study the scaling of relaxation times with system size for each system size the transition matrices for arbitrary single spin flip dynamics are obtained from a single monte carlo estimate of the infinite temperature transition matrix this makes the method an efficient tool for evaluating the relative performance of any arbitrary local spin flip dynamics we also present results for appropriately defined average tunneling times of magnetization and compare their finite size scaling exponents with results of energy tunneling exponents available for the flat energy histogram multicanonical ensemble

monte carlo sampling in fractal landscapes we design a random walk to explore fractal landscapes such as those describing chaotic transients in dynamical systems we show that the random walk moves efficiently only when its step length depends on the height of the landscape via the largest lyapunov exponent of the chaotic system we propose a generalization of the wang landau algorithm which constructs not only the density of states transient time distribution but also the correct step length as a result we obtain a flat histogram monte carlo method which samples fractal landscapes in polynomial time a dramatic improvement over the exponential scaling of traditional uniform sampling methods our results are not limited by the dimensionality of the landscape and are confirmed numerically in chaotic systems with up to 30 dimensions

a novel method for analyzing inverse problem of topological indices of graphs using competitive agglomeration a novel method for analyzing the inverse problem of topological indices of graph is proposed in this paper which applies the data mining technique to classification of graphs according to topological indices differing from the way of using topological indices to categorize graphs into different isomorphic classes this method aims at dividing molecular graphs with n vertices into several compact classes according to their topological indices then the chemical and physical features of each kind of graph can be researched and new compounds can be found which has great significance on bio medical filed in the experiment three classes of simple connected graphs with 5 6 and 7 vertices respectively are investigated and analyzed using the proposed method the experimental results show the validity of it however there are still some problems needed to be researched further such as selecting the vector of topological indices for clustering selecting the distance between the vectors of topological indices choosing the aspects to analyze the properties of each kind of graph after clustering etc

nearly hypersingular integrals by double arctan transformation in higher order geometry modeling in order to directly utilize the dyadic green s functions in surface integral equations sies a novel double arctan transformation for nearly hypersingular integrals is proposed in this communication this new transformation is flexible and applicable to nearly hypersingular integrals in the forms of hat r hat r r 3 hat r r 3 and 1 r 3 over the curved surfaces by a fully numerical method with the help of the sigmoidal transformation to improve the stability of this new singularly handling method there results an efficient solution for the third order near singularity problems in sies moreover the proposed method is also effective for the lower orders of the nearly singular integral kernels with typical testing cases the performance of this method is fairly evaluated and its validity and stability is well demonstrated

application of sigmoidal transformations to weakly singular and near singular boundary element integrals accurate numerical determination of line integrals is fundamental to reliable implementation of the boundary element method for a source point distant from a particular element standard gaussian quadrature is adequate as well as being the technique of choice however when the integrals are weakly singular or nearly singular source point near the element this technique is no longer adequate here a co ordinate transformation technique based on sigmoidal transformations is introduced to evaluate weakly singular and near singular integrals a sigmoidal transformation has the effect of clustering the integration points towards the endpoints of the interval of integration the degree of clustering is governed by the order of the transformation comparison of this new method with existing co ordinate transformation techniques shows that more accurate evaluation of these integrals can be obtained based on observations of several integrals considered guidelines are suggested for the order of the sigmoidal transformations copyright 1999 john wiley sons ltd

a reverse operation self consistent evaluation approach for singular integrals in the sie analysis of pec targets discussed herein is a novel approach entitled reverse operation self consistent evaluation rose specifically we applied rose for the treatment of weak singular and hypersingular kernels in surface integral equations sies for the computation of electromagnetic em scattering from perfect electrical conductors pecs different from the existing forward methods rose is based on reverse operations the singular entries are not evaluated directly by analytical or numerical evaluation instead they are introduced as unknowns and are solved algebraically for em scattering from pec sies can be set up by using the stratton chu representation formula an auxiliary problem with the same meshed boundary is then formulated of which any plane waves are the self consistent solutions it is shown that by applying a consistent discretization the matrix of the auxiliary problem can be made identical with the original one as a consequence these self consistent solutions can be utilized to solve the singular entries of the original system explicitly a set of plane waves are interpolated using the appropriate basis functions subsequently the corresponding coefficient vectors are utilized to construct small linear systems to solve the troublesome singular entries in the underlying matrix the proposed rose approach is applicable to different singular kernels as well as different discretization schemes in this paper we restrict our discussion to the weak singular and hypersingular kernels using respectively div conformal rao wilton glisson basis functions and nonconformal piecewise constant vector basis functions accelerated by the multilevel fast multipole algorithm the proposed method is applicable for real life applications

improved double arctan transformation for the stable evaluation of nearly hypersingular integrals in sie the evaluation of singular or near singular integrals is one of the key techniques in the integral equation method based on the previously proposed double arctan transformation dat for the nearly hypersingular integrals in curved surface modeling this work presents new transformation methods to improve the accuracy and stability of dat for the following two situations i e the field point is 1 extremely close to the source surface and 2 extremely close to the boundary of the source surface the original dat fails in these two situations and hence limits its applications the reasons that cause such an instability are studied two additional transformations are introduced to make the dat stable for these two cases for situation i the exponential transformation et is adopted to optimize the distribution of the gaussian quadrature points for situation ii the shape function transformation sft is further proposed to remove the nearly angular singularity we denote the newly developed dat with et and sft as the improved dat idat in this communication the typical testing cases are provided to validate the proposed method

numerical evaluation via singularity cancellation schemes of near singular integrals involving the gradient of helmholtz type potentials in this paper we present a purely numerical procedure to evaluate strongly near singular integrals involving the gradient of helmholtz type potentials for observation points at finite arbitrarily small distances from the source domain in the proposed approach the source domain is subdivided into a disc plus truncated subtriangles and proper variable transformations are applied in each integration domain to exactly cancel the kernel singularity a novel feature of the proposed angular transform is that required discrete values of the inverse transform which is transcendental are determined via a root finding procedure the same idea can also be applied to other transforms that arise in singularity cancellation methods the resulting integral may then evaluated via a low order gauss legendre quadrature scheme

considerations on double exponential based cubatures for the computation of weakly singular galerkin inner products highly accurate and efficient cubatures based on the double exponential quadrature rules are presented for the computation of weakly singular integrals arising in galerkin mixed potential integral equation formulations due to their unique ability to handle non smooth kernels the proposed integration schemes can safely replace in a plug n play sense the traditional gauss legendre rules in the existing singularity cancellation and singularity subtraction methods numerical examples using rwg basis functions confirm the excellent performance of the proposed method

advanced numerical schemes for the accurate evaluation of 4 d reaction integrals in the method of moments in this paper we present a simple and efficient fully numerical technique for the evaluation of 4 d reaction integrals arising in the method of moments in the proposed approach the accuracy of the source integration which uses a singularity cancellation scheme is retained in the final 4 d reaction integral through the application of an appropriate variable transformation and numerical quadrature scheme to the test integral several numerical results compared to the usual gauss legendre test quadrature demonstrate the accuracy of the method for both planar and curvilinear discretizations moreover the proposed scheme is independent of test basis functions hence it can be used for higher order basis functions as well as curl conforming or buffa christiansen test basis functions

numerical evaluation of singular and near singular potential integrals a simple and efficient numerical procedure using a singularity cancellation scheme is presented for evaluating singular and near singular potential integrals with 1 r singularities the procedure not only has several advantages over singularity subtraction methods but also improves on some aspects of other singularity cancellation methods such as polar and duffy transformations a theoretical analysis is presented for triangles quadrilaterals tetrahedrons bricks and prisms and numerical results are presented for triangles and prisms

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 novel approach for evaluating hypersingular and strongly singular surface integrals in electromagnetics solving electromagnetic em problems by integral equation methods requires an accurate and efficient treatment for the singular integral kernels related to the green s function for surface integral equations sies there are l and k operators which include hypersingular integrals hsis and strongly singular integrals ssis respectively the hsis are generated from the double gradient of the green s function while the ssis come from the single gradient of the green s function although the hsis could be reduced to weakly singular integrals wsis in the method of moments mom implementation with divergence conforming basis function such as the rao wilton glisson rwg basis function they do appear in nystrom method nm or boundary element method bem and one has to tackle them the ssis always exist in the k operator and could also exist in the l operator when the testing function is not the rwg like basis function the treatment for the hsis and ssis is essential because they have a significant influence on the numerical solutions there have been many publications dealing with the singular integrals but they mainly focus on the wiss or ssis and the hsis were seldom addressed in this work we develop a novel approach for evaluating those hsis and ssis based on the stokes theorem the derived formulas are much simpler and more friendly in implementation since no polar coordinates or extra coordinate transformation are involved numerical experiments are presented to demonstrate the effectiveness of the approach

optimized multicanonical simulations a proposal based on classical fluctuation theory we propose a recursive procedure to estimate the microcanonical density of states in multicanonical monte carlo simulations which relies only on measurements of moments of the energy distribution avoiding entirely the need for energy histograms this method yields directly a piecewise analytical approximation to the microcanonical inverse temperature e and allows improved control over the statistics and efficiency of the simulations we demonstrate its utility in connection with recently proposed schemes for improving the efficiency of multicanonical sampling either with adjustment of the asymptotic energy distribution or with the replacement of single spin flip dynamics with collective updates

generalized gaussian quadrature rules for systems of arbitrary functions a numerical algorithm is presented for the construction of generalized gaussian quadrature rules originally introduced by s karlin and w studden over three decades ago the quadrature rules to be discussed possess most of the desirable properties of the classical gaussian integration formulae such as positivity of the weights rapid convergence mathematical elegance etc the algorithm is applicable to a wide class of functions including smooth functions not necessarily polynomials as well as functions with end point singularities such as those encountered in the solution of integral equations complex analysis potential theory and several other areas the performance of the algorithm is illustrated with several numerical examples

a projection height independent adaptive radial angular r 2 transformation for singular integrals a new radial angular r 2 adaptive singularity cancellation transformation is proposed this new transformation is flexible and applicable to singular integrals over triangular domains when the height tends to zero and the observation point is in the plane of the source domain the transformation remains stable and approaches the corresponding transformation within the source plane usually the green s function and its gradient appear in the integral kernels leading to first order and second order singularities the newly derived transformation formula provides an efficient solution for second order singular coupling integral kernels and the formula is in particular more efficient than alternative schemes for singular vector integral kernels however it is also effective for the lower order singular integral kernels in electromagnetic boundary integral equation formulations the proposed transformation is efficient for all types of singular integrals

a family of augmented duffy transformations for near singularity cancellation quadrature a new family of systematically constructed near singularity cancellation transformations is presented yielding quadrature rules for integrating near singular kernels over triangular surfaces this family results from a structured augmentation of the well known duffy transformation the benefits of near singularity cancellation quadrature are that no analytical integral evaluations are required and applicability in higher order basis function and curvilinear settings six specific transformations are constructed for near singularities of orders one two and three two of these transformations are found to be equivalent to existing ones the performance of the new schemes is thoroughly assessed and compared with that of existing schemes results for the gradient of the scalar green function are also presented for simplicity static kernel results are shown the new schemes are competitive with and in some cases superior to the existing schemes considered

note on two generalizations of the randic index for a given graph g the well known randic index of g introduced by milan randic in 1975 is defined as r g u v e g d u d v 1 2 where the sum is taken over all edges uv and du denotes the degrees of u bollobas and erdos generalized this index by replacing 1 2 with any real number which is called the general randic index dvo ak et al introduced a modified version of randic index r g u v e g max d u d v 1 based on this recently knor et al introduced two generalizations r g u v e g min d u d v and r g u v e g max d u d v for any real number clearly the former is a lower bound for the general randic index and the latter is its upper bound knor et al studied extremal values of r g and r g and concluded some open problems in this paper we consider the open problems and give some comments and results some results for chemical trees are obtained