prime boolean matrices and factorizations abstract we study the prime boolean matrices in the semigroup of boolean matrices we also study the factorization properties of the n n rank n boolean matrices in terms of elementary matrices and prime matrices finally we consider the minimal and the maximal prime boolean matrices

index and stability of symmetric periodic orbits in hamiltonian systems with application to figure eight orbit in this paper using the maslov index theory in symplectic geometry we build up some stability criteria for symmetric periodic orbits in a hamiltonian system which is motivated by the recent discoveries in the n body problem the key ingredient is a generalized bott type iteration formula for periodic solution in the presence of finite group action on the orbit for second order system we prove under general boundary conditions the close formula for the relationship between the morse index of an orbit in a lagrangian system and the maslov index of the fundamental solution for the corresponding orbit in its hamiltonian system counterpart and the boundary conditions cover the cases which appeared in the n body problem as an application we consider the stability problem of the celebrated figure eight orbit due to chenciner and montgomery in the planar three body problem with equal masses and we clarify the relationship between linear stability and its variational nature on various loop spaces the basic idea is as follows the variational characterization of the figure eight orbit provides information about its morse index based on its relation to the maslov index our stability criteria come into play

the determinant and adjoint of a square fuzzy matrix abstract the theory of the determinant and adjoint of a square fuzzy matrix will be studied also we define the circular fuzzy matrices and show that some properties of a square fuzzy matrix such as reeflexivity transitivity circularity are carried over to the adjoint of the matrix finally we show how to construct a transitive fuuzzy matrix from a given one through the adjoint matrix

exact solutions for the longitudinal vibration of non uniform rods abstract the objective of this paper is to present exact analytical solutions for the longitudinal vibration of rods with non uniform cross section using appropriate transformation the equation of motion of axial vibration of a rod with varying cross section is reduced to analytically solvable standard differential equations whose form depends upon the specific area variation solutions are obtained for a rod with a polynomial area vibration and for a sinusoidal rod the solutions are obtained in terms of special functions such as bessel and neumann as well as trigonometric functions simple formulas to predict the natural frequencies of non uniform rods with various end conditions are presented the natural frequencies of non uniform rods for these end conditions are calculated and their dependence on taper is discussed the governing equation for the problem is the same as that of wave propagation through ducts with non uniform cross sections therefore solutions presented here can be used to investigate such problems

unique solvability of max min fuzzy equations and strong regularity of matrices over fuzzy algebra a matrix is said to have strongly linearly independent columns or in the case of square matrices to be strongly regular if for some vector b the system a x b has a unique solution we formulate a necessary and sufficient condition for a linear system of equations over a fuzzy algebra to have a unique solution and prove the equivalence of strong regularity and trapezoidal property moreover an algorithm for testing these properties is reviewed

eigen fuzzy sets and fuzzy relations abstract when r is a fuzzy relation between the elements of a finite set x the fuzzy sets a of x such that r a a max min composition are called eigen fuzzy sets the main result of this paper is the determination of the greatest eigen fuzzy set associated with a given fuzzy relation and we give three methods illustrated by an example we then state that the greatest eigen fuzzy set associated with r the transitive closure of r is exactly the one associated with r finally we describe how to obtain all fuzzy relations keeping invariant a given fuzzy set

factorizations in upper triangular matrices over information semialgebras abstract an integral domain or a commutative cancellative monoid is atomic if every nonzero nonunit element factors into atoms and it satisfies the accp if every ascending chain of principal ideals eventually stabilizes the interplay between these two properties has been investigated since the 1970s an atomic domain or monoid satisfies the finite factorization property ffp if every element has only finitely many factorizations and it satisfies the bounded factorization property bfp if for each element there is a bound for the number of atoms in each of its factorizations these two properties have been systematically studied since they were introduced by anderson anderson and zafrullah in 1990 noetherian domains satisfy the bfp while dedekind domains satisfy the ffp it is well known that for commutative cancellative monoids and in particular for multiplicative monoids of integral domains ffp bfp accp atomic for n 2 we show that each of these four properties transfers back and forth between an information semialgebra s certain commutative cancellative semiring and the multiplicative monoid t n s consisting of n n upper triangular matrices over s we also show that a similar transfer behavior takes place if one replaces t n s by its submonoid u n s consisting of upper triangular matrices with units along their main diagonals as a consequence we find that the atomic chain ffp bfp accp atomic also holds for the two classes comprising the noncommutative monoids t n s and u n s finally we construct various rational information semialgebras to verify that in general none of the established implications is reversible

a two point boundary value problem with a rapidly oscillating solution some numerical methods are developed for a two point boundary value problem with a rapidly oscillating solution the two point boundary value problem is chosen to model some of the difficulties that may be expected to occur in solving the reduced wave equation at moderately high frequencies

a galerkin least squares finite element method for the two dimensional helmholtz equation in this paper a galerkin least squares gls finite element method in which residuals in least squares form are added to the standard galerkin variational equation is developed to solve the helmholtz equation in two dimensions an important feature of gls methods is the introduction of a local mesh parameter that may be designed to provide accurate solutions with relatively coarse meshes previous work has accomplished this for the one dimensional helmholtz equation using dispersion analysis in this paper the selection of the gls mesh parameter for two dimensions is considered and leads to elements that exhibit improved phase accuracy for any given direction of wave propagation an optimal gls mesh parameter is determined using two dimensional fourier analysis in general problems the direction of wave propagation will not be known a priori in this case an optimal gls parameter is found which reduces phase error for all possible wave vector orientations over elements the optimal gls parameters are derived for both consistent and lumped mass approximations several numerical examples are given and the results compared with those obtained from the galerkin method the extension of gls to higher order quadratic interpolations is also presented

morse index and stability of elliptic lagrangian solutions in the planar three body problem abstract we illustrate a new way to study the stability problem in celestial mechanics in this paper using the variational nature of elliptic lagrangian solutions in the planar three body problem we study the relation between morse index and its stability via maslov type index theory of periodic solutions of hamiltonian system for elliptic lagrangian solutions we get an estimate of the algebraic multiplicity of unit eigenvalues of its monodromy matrix in terms of the morse index which is the key to understand the stability problem as a special case we provide a criterion to spectral stability of relative equilibrium

dispersion analysis and error estimation of galerkin finite element methods for the helmholtz equation when applying numerical methods for the computation of stationary waves from the helmholtz equation one obtains numerical waves that are dispersive also in non dispersive media the numerical wave displays a phase velocity that depends on the parameter k of the helmholtz equation in dispersion analysis the phase difference between the exact and the numerical solutions is investigated in this paper the authors recent result on the phase difference for one dimensional problems is numerically evaluated and discussed in the context of other work directed to this topic it is then shown that previous error estimates in h1 norm are of nondispersive character but hold for medium or high wavenumber on extremely refined mesh only on the other hand recently proven error estimates for constant resolution contain a pollution term with certain assumptions on the exact solution this term is of the order of the phase difference thus a link is established between the results of dispersion analysis and the results of numerical analysis throughout the paper the presentation and discussion of theoretical results is accompanied by numerical evaluation of several model problems special attention is given to the performance of the galerkin method with a higher order of polynomial approximation p h p version

primes in the semigroup of boolean matrices abstract let a b c be n n matrices of zeros and ones using boolean addition and multiplication we say that a is prime if it is not a permutation matrix and if a bc implies that b or c must be a permutation matrix conditions sufficient for a matrix to be prime are provided and a characterization of primes in terms of a nation of rank is given finally an order property of primes is used to obtain a result on prime factors

efficient numerical continuation and stability analysis of spatiotemporal quadratic optical solitons a numerical method is set out which efficiently computes stationary z independent two and three dimensional spatiotemporal solitons in second harmonic generating media the method relies on a chebyshev decomposition with an infinite mapping bunching the collocation points near the soliton core known results for the type i interaction are extended and a stability boundary is found by two parameter continuation as defined by the vakhitov kolokolov criteria the validity of this criterion is demonstrated in 2 1 dimensions by simulation and direct calculation of the linear spectrum the method has wider applicability for general soliton bearing equations in 2 1 and 3 1 dimensions

numerical computation of nonlinear normal modes in mechanical engineering this paper reviews the recent advances in computational methods for nonlinear normal modes nnms different algorithms for the computation of undamped and damped nnms are presented and their respective advantages and limitations are discussed the methods are illustrated using various applications ranging from low dimensional weakly nonlinear systems to strongly nonlinear industrial structures

generalized inverses of substochastic matrices abstract this paper looks at the question of when a substochastic matrix has a substochastic generalized inverse this question is answered for several generalized inverses including semiinverses the moore penrose inverse and the group inverse methods for constructing all such inverses are given

a high order purely frequency based harmonic balance formulation for continuation of periodic solutions the case of non polynomial nonlinearities abstract in this paper we extend the method proposed by cochelin and vergez a high order purely frequency based harmonic balance formulation for continuation of periodic solutions journal of sound and vibration 324 2009 243 262 to the case of non polynomial nonlinearities this extension allows for the computation of branches of periodic solutions of a broader class of nonlinear dynamical systems the principle remains to transform the original ode system into an extended polynomial quadratic system for an easy application of the harmonic balance method hbm the transformation of non polynomial terms is based on the differentiation of state variables with respect to the time variable shifting the nonlinear non polynomial nonlinearity to a time independent initial condition equation not concerned with the hbm the continuation of the resulting algebraic system is here performed by the asymptotic numerical method high order taylor series representation of the solution branch using a further differentiation of the non polynomial algebraic equation with respect to the path parameter a one dof vibro impact system is used to illustrate how an exponential nonlinearity is handled showing that the method works at very high order 1000 in that case various kinds of nonlinear functions are also treated and finally the nonlinear free pendulum is addressed showing that very accurate periodic solutions can be computed with the proposed method

time reversal symmetry in dynamical systems a survey abstract in this paper we survey the topic of time reversal symmetry in dynamical systems we begin with a brief discussion of the position of time reversal symmetry in physics after defining time reversal symmetry as it applies to dynamical systems we then introduce a major theme of our survey namely the relation of time reversible dynamical sytems to equivariant and hamiltonian dynamical systems we follow with a survey of the state of the art on the theory of reversible dynamical systems including results on symmetric periodic orbits local bifurcation theory homoclinic orbits and renormalization and scaling some areas of physics and mathematics in which reversible dynamical systems arise are discussed in an appendix we provide an extensive bibliography on the topic of time reversal symmetry in dynamical systems

morse index and linear stability of the lagrangian circular orbit in a three body type problem via index theory it is well known that the linear stability of the lagrangian elliptic solutions in the classical planar three body problem depends on a mass parameter and on the eccentricity e of the orbit we consider only the circular case e 0 but under the action of a broader family of singular potentials homogeneous potentials for alpha in 0 2 and the logarithmic one it turns out indeed that the lagrangian circular orbit persists also in this more general setting we discover a region of linear stability expressed in terms of the homogeneity parameter and the mass parameter then we compute the morse index of this orbit and of its iterates and we find that the boundary of the stability region is the envelope of a family of curves on which the morse indices of the iterates jump in order to conduct our analysis we rely on a maslov type index theory devised and developed by y long x hu and s sun a key role is played by an appropriate index theorem and by some precise computations of suitable maslov type indices

a heuristic method to compute the approximate postinverses of a fuzzy matrix this paper considers the problem of computing the approximate inverses of a fuzzy matrix under max min composition a polynomial time algorithm is proposed to construct an approximate postinverse by minimizing an evaluation function which balances two different distance measures the obtained approximate postinverse contains no zero columns unless the given fuzzy matrix itself contains zero rows subsequently the techniques for solving fuzzy relational equations are applied to obtain all the approximate postinverses of equal quality

continuation of periodic orbits in conservative and hamiltonian systems we introduce and justify a computational scheme for the continuation of periodic orbits in systems with one or more first integrals and in particular in hamiltonian systems having several independent symmetries our method is based on a generalization of the concept of a normal periodic orbit as introduced by sepulchre and mackay nonlinearity 10 1997 679 we illustrate the continuation method on some integrable hamiltonian systems with two degrees of freedom and briefly discuss some further applications

the discontinuous enrichment method for medium frequency helmholtz problems with a spatially variable wavenumber numerical dispersion or what is often referred to as the pollution effect presents a challenge to an efficient finite element discretization of the helmholtz equation in the medium frequency regime to alleviate this effect and improve the unsatisfactory pre asymptotic convergence of the classical galerkin finite element method based on piecewise polynomial basis functions several discretization methods based on plane wave bases have been proposed among them is the discontinuous enrichment method that has been shown to offer superior performance to the classical galerkin finite element method for a number of constant wavenumber helmholtz problems and has also outperformed two representative methods that use plane waves the partition of unity and the ultra weak variation formulation methods in this paper the discontinuous enrichment method is extended to the variable wavenumber helmholtz equation to this effect the concept of enrichment functions based on free space solutions of the homogeneous form of the governing differential equation is enlarged to include free space solutions of approximations of this equation obtained in this case by successive taylor series expansions of the wavenumber around a reference point this leads to plane wave enrichment functions based on the piece wise constant approximation of the wavenumber and to airy wave enrichment functions several elements based on these enrichment functions are constructed and evaluated on benchmark problems modeling sound hard scattering by a disk submerged in an acoustic fluid where the speed of sound varies in space all these elements are shown to outperform by a substantial margin their continuous polynomial counterparts

fuzzy matrices of permanent one abstract we study the n n fuzzy matrices of permanent one hall matrices which are generalizations of the reflexive fuzzy matrices we give some necessary and sufficient conditions for a hall matrix to have a generalized inverse in the semigroup hn of n n hall matrices as applications we also study the fuzzy linear maps and the principal left ideals of hn

invertible incline matrices and cramer s rule over inclines inclines are the additively idempotent semirings in which the products are less than or equal to factors thus inclines generalize boolean algebra fuzzy algebra and distributive lattice and the boolean matrices the fuzzy matrices and the lattice matrices are the prototypical examples of the incline matrices i e the matrices over inclines in this paper the complete description of the invertible incline matrices is given some necessary and sufficient conditions for an incline matrix to be invertible are studied cramer s rule over inclines is presented and the group of invertible incline matrices is investigated the main results in the present paper generalize and develop the corresponding results in the literatures for the boolean matrices the fuzzy matrices and the lattice matrices

factorizations of matrices over semirings in this paper as a generalization of prime boolean matrices and prime fuzzy matrices we study semiprime matrices over chain semirings using the relationship between semiprime matrices and their row spaces we show that a nonmonomial matrix with full semiring rank can be expressed as a product of elementary matrices and semiprime matrices furthermore we show that a nonmonomial regular matrix with full semiring rank can be expressed as a product of elementary matrices

regular matrices in the semigroup of hall matrices abstract we study the regular matrices in the semigroup h n of hall matrices boolean matrices with positive permanent we study some necessary and sufficient conditions for a hall matrix to be regular in h n in terms of idempotent matrices adjoint matrices and identifying permutation matrices

robust heteroclinic cycles in the one dimensional complex ginzburg landau equation numerical evidence is presented for the existence of stable heteroclinic cycles in large parameter regions of the one dimensional complex ginzburg landau equation cgl on the unit spatially periodic domain these cycles connect difierent spatially and temporally inhomogeneous time periodic solutions as t 1 a careful analysis of the connections is made using a projection onto 5 complex fourier modes it is shown flrst that the time periodic solutions can be treated as relative equilibria after consideration of the symmetries of the cgl second the cycles are shown to be robust since the individual heteroclinic connections exist in invariant

standard eigenvectors of incline matrices abstract inclines are the additively idempotent semirings in which products are less than or equal to either factor thus inclines generalize boolean algebra fuzzy algebra and distributive lattice this paper studies the standard eigenvectors of incline matrices i e matrices over inclines in detail as the main results for an incline matrix with index and for a lattice matrix the structures of all the standard eigenvectors are given and the greatest standard eigenvector is determined also it is proved that for an incline matrix with index the set of all the standard eigenvectors of it coincides with the set of all the standard eigenvectors of its transitive closure the power convergence of an incline matrix with index is characterized in terms of the sets of standard eigenvectors the results in the present paper include some previous results for the boolean matrices the fuzzy matrices and the lattice matrices among their special cases

morse index and bifurcation for figure eight choreographies of the equal mass three body problem we report on the morse index and periodic solutions bifurcating from the figure eight choreography for the equal mass three body problem under homogeneous potential 1 r a for and under lennard jones lj type potential where r is a distance between bodies it is shown that the morse index changes at a bifurcation point and all solutions bifurcating are approximated by variational functions responsible for the change of the morse index inversely we observed bifurcation occurs at every point where the morse index changes for the figure eight choreography under 1 r a and for solution under the lj type potential where solution is a figure eight choreography tending to that under 1 r 6 for an infinitely large period thus to our numerical studies change of the morse index is not only necessary but also a sufficient condition for bifurcation for these choreographies further we observed that the change of the morse index is equal to the number of bifurcated solutions regarding solutions with congruent orbits as the same solution

on nilpotent incline matrices inclines are additively idempotent semirings in which products are less than or equal to either factor thus they generalize boolean algebra fuzzy algebra and distributive lattice this paper studies the nilpotent incline matrices in detail it is proved that an incline matrix is nilpotent if and only if it has index and the zero vector is its unique standard eigenvector the nilpotent matrices over an incline without nilpotent elements are characterized in terms of principal minors main diagonals nilpotent indices and adjoint matrices also some properties of the reduction of nilpotent matrices over an additively residuated incline without nilpotent elements are established the results obtained here generalize the corresponding ones on fuzzy matrices and lattice matrices shown in the references

an accelerated approach for solving fuzzy relation equations with a linear objective function in literature the optimization model with a linear objective function subject to fuzzy relation equations has been converted into a 0 1 integer programming problem by fang and li 1999 they proposed a jump tracking branch and bound method to solve this 0 1 integer programming problem in this paper we propose an upper bound for the optimal objective value based on this upper bound and rearranging the structure of the problem we present a backward jump tracking branch and bound scheme for solving this optimization problem a numerical example is provided to illustrate our scheme furthermore testing examples show that the performance of our scheme is superior to the procedure in the paper by fang and li several testing examples show that our initial upper bound is sharp

the design and analysis of the generalized finite element method abstract in this paper we introduce the generalized finite element method gfem as a combination of the classical finite element method fem and the partition of unity method pum the standard finite element spaces are augmented by adding special functions which reflect the known information about the boundary value problem and the input data e g the singular functions obtained from the local asymptotic expansion of the exact solution in the neighborhood of a corner point etc the special functions are multiplied with the partition of unity corresponding to the standard linear vertex shape functions and pasted together with the existing finite element basis to construct an augmented conforming finite element space in this way the local approximability afforded by the special functions is included in the approximation while maintaining the existing infrastructure of finite element codes the major features of the gfem are 1 the essential boundary conditions can be imposed exactly as in the standard fem unlike other partition of unity based methods where this is a major issue 2 the accuracy of the numerical integration of the entries of the stiffness matrix and load vector is controlled adaptively so that the errors in integration of the special functions do not affect the accuracy of the constructed approximation this issue also has not been sufficiently addressed in other implementations of partition of unity or meshless methods and 3 linear dependencies in the system of equations are resolved by employing an easy modification of the direct linear solver the power of the gfem for solving problems in domains with complex geometry with less error and less computer resources than the standard fem is illustrated by numerical examples

the helmholtz equation in heterogeneous media a priori bounds well posedness and resonances abstract we consider the exterior dirichlet problem for the heterogeneous helmholtz equation i e the equation a u k 2 n u f where both a and n are functions of position we prove new a priori bounds on the solution under conditions on a n and the domain that ensure nontrapping of rays the novelty is that these bounds are explicit in k a n and geometric parameters of the domain we then show that these a priori bounds hold when a and n are l and satisfy certain monotonicity conditions and thereby obtain new results both about the well posedness of such problems and about the resonances of acoustic transmission problems i e a and n discontinuous where the transmission interfaces are only assumed to be c 0 and star shaped the novelty of this latter result is that until recently the only known results about resonances of acoustic transmission problems were for c convex interfaces with strictly positive curvature

the smallest solution of max min fuzzy equations abstract the object of this paper is to study the smallest solution of a max min fuzzy equation we give some necessary and sufficient conditions for the existence of a unique minimal solution of the equation also a sufficient and necessary condition for the equation to have only one solution these results offer some solutions to an unsolved problem posed by guo si zhong et al 1

further contributions to the study of finite fuzzy relations equations abstract an interesting property of the schein rank of a fuzzy matrix stressed by k h kim and f w roush is formalized as definition of the basis of a set of fuzzy vectors defined over a commutative semiring with respect to another set of fuzzy vectors we apply this definition and another related concept in the setting of finite fuzzy relation equations in particular we give a sufficient condition for the existence of the smallest solution of a max min fuzzy equation

ranks of boolean matrices abstract we denote by m n 0 1 the set of all n n boolean matrices for any boolean matrix x in m n 0 1 has row rank r and column rank s and we denote it by rank k r s let v be the set of all x in m n 0 1 such that rank x n s where n is fixed and s is not fixed we find the lower bound s for x in v

a comparison of high order polynomial and wave based methods for helmholtz problems the application of computational modelling to wave propagation problems is hindered by the dispersion error introduced by the discretisation two common strategies to address this issue are to use high order polynomial shape functions e g hp fem or to use physics based or trefftz methods where the shape functions are local solutions of the problem typically plane waves both strategies have been actively developed over the past decades and both have demonstrated their benefits compared to conventional finite element methods but they have yet to be compared in this paper a high order polynomial method p fem with lobatto polynomials and the wave based discontinuous galerkin method are compared for two dimensional helmholtz problems a number of different benchmark problems are used to perform a detailed and systematic assessment of the relative merits of these two methods in terms of interpolation properties performance and conditioning it is generally assumed that a wave based method naturally provides better accuracy compared to polynomial methods since the plane waves or bessel functions used in these methods are exact solutions of the helmholtz equation results indicate that this expectation does not necessarily translate into a clear benefit and that the differences in performance accuracy and conditioning are more nuanced than generally assumed the high order polynomial method can in fact deliver comparable and in some cases superior performance compared to the wave based dgm in addition to benchmarking the intrinsic computational performance of these methods a number of practical issues associated with realistic applications are also discussed

finite element solution of the helmholtz equation with high wave number part i the h version of the fem abstract the paper addresses the properties of finite element solutions for the helmholtz equation the h version of the finite element method with piecewise linear approximation is applied to a one dimensional model problem new results are shown on stability and error estimation of the discrete model in all propositions assumptions are made on the magnitude of hk only where k is the wavelength and h is the stepwidth of the fe mesh previous analytical results had been shown with the assumption that k 2 h is small for medium and high wavenumber these results do not cover the meshsizes that are applied in practical applications the main estimate reveals that the error in h 1 norm of discrete solutions for the helmholtz equation is polluted when k 2 h is not small the error is then not quasioptimal i e the relation of the fe error to the error of best approximation generally depends on the wavenumber k it is noted that the pollution term in the relative error is of the same order as the phase lead of the numerical solution in the result of this analysis thorough and rigorous understanding of error behavior throughout the range of convergence is gained numerical results are presented that show sharpness of the error estimates and highlight some phenomena of the discrete solution behavior the h p version of the fem is studied in part ii

approximation of periodic solutions for a dissipative hyperbolic equation this paper studies the numerical approximation of periodic solutions for an exponentially stable linear hyperbolic equation in the presence of a periodic external force f these approximations are obtained by combining a fixed point algorithm with the galerkin method it is known that the energy of the usual discrete models does not decay uniformly with respect to the mesh size our aim is to analyze this phenomenon s consequences on the convergence of the approximation method and its error estimates we prove that under appropriate regularity assumptions on f the approximation method is always convergent however our error estimates show that the convergence s properties are improved if a numerically vanishing viscosity is added to the system the same is true if the nonhomogeneous term f is monochromatic to illustrate our theoretical results we present several numerical simulations with finite element approximations of the wave equation in one or two dimensional domains and with different forcing terms

the generalized finite element method for helmholtz equation part ii effect of choice of handbook functions error due to absorbing boundary conditions and its assessment abstract in part i t strouboulis i babuska r hidajat the generalized finite element method for helmholtz equation theory computation and open problems comput methods appl mech engrg 195 2006 4711 4731 we introduced the q version of the generalized finite element method gfem for the helmholtz equation and we addressed its a pollution error due to the wave number b exponential q convergence c robustness to perturbations of the mesh the roundoff and numerical quadrature errors and d a posteriori error estimation here we continue the development of the gfem for helmholtz and we address the effects of a alternative handbook functions and mesh types b the error due to the artificial truncation boundary conditions and its assessment the conclusions are 1 the employment of plane wave wave band and vekua handbook functions lead to equivalent results and 2 for high q the most significant component of error may be the one due to the artificial truncation boundary conditions a rather straightforward approach for assessing this error is proposed

an application of incline matrices in dynamic analysis of generalized fuzzy bidirectional associative memories this paper studies an application of the incline matrix theory in the dynamic analysis of incline valued fuzzy bidirectional associative memories l fbams for short it is proved that the strong convergence and the strong stability of an l fbam are equivalent to the existence of indices and the convergence in finite steps of the product matrices of connection incline matrices of the l fbam respectively it is shown that the convergence index the period of limit cycles the stable states and equilibria of a strongly convergent l fbam are expressed by the indices the periods and the standard eigenvectors of the product matrices of connection incline matrices of the l fbam respectively

determinant theory for d 01 lattice matrices abstract we study determinant theory for d01 lattice matrices and consider the cayley hamilton theorem for these matrices a distributive lattice l is said to be d01 lattice if 0 1 l and 0 x 1 for all x l we prove that det ab det a det b det aadj a det a det adj a a where adj a denotes the adjoint matrix of a d01 lattice matrice a

a comparative study of the harmonic balance method and the orthogonal collocation method on stiff nonlinear systems the high order purely frequency based harmonic balance method hbm presented by cochelin and vergez 2009 1 and extended by karkar et al 2013 2 now allows to follow the periodic solutions of regularized non smooth systems stiff systems this paper compares its convergence property to a reference method in applied mathematics orthogonal collocation with piecewise polynomials a first test is conducted on a nonlinear smooth 2 degree of freedom spring mass system showing better convergence of the hbm the second test is conducted on a one degree of freedom vibro impact system with a very stiff regularization of the impact law the hbm continuation of the nonlinear mode was found to be very robust even with a very large number of harmonics surprisingly the hbm was found to have a better convergence than the collocation method for this vibro impact system

unidimensional models of acoustic propagation in axisymmetric waveguides this paper presents a model of the linear acoustic propagation in axisymmetric waveguides the pressure depending on a single space variable the approach consists of writing the wave equation and the boundary conditions for a coordinate system rectifying the isobaric map at each time the two dimensional dependence of the problem is thus transferred from the pressure to the coefficients of the wave equation from this result an exclusively geometrical necessary condition is deduced for the admissibility of isobaric maps however the knowledge of the waveguide geometry is not sufficient to separate the pressure and the isobaric map solutions in order to develop a unidimensional wave equation a geometrical hypothesis is discussed for lossless and motionless rigid waveguides the deduced equation leads to exact results for tubes and cones it may be interpreted as a webster equation for a particular coordinate system so that the particular profiles for which analytical solutions of the pressure exist are redefined the wave equation is also established for pipes with visco thermal losses and more generally for mobile walls having a small admittance the compatibility of the geometrical hypothesis with the exact model is specified for this general case

ranks of fuzzy matrices applications in state reduction of fuzzy automata abstract in this paper we consider different types of ranks of fuzzy matrices over residuated lattices we investigate relations between ranks and prove that row rank column rank and schein rank of idempotent fuzzy matrices are equal in particular ranks and corresponding decompositions of fuzzy matrices representing fuzzy quasi orders are studied in detail we show that fuzzy matrix decomposition by ranks can be used in the state reduction of fuzzy automata moreover we prove that using rank decomposition of fuzzy matrices improves results of any state reduction method based on merging indistinguishable states of fuzzy automata

a residual based finite element method for the helmholtz equation a new residual based finite element method for the scalar helmholtz equation is developed this method is obtained from the galerkin approximation by appending terms that are proportional to residuals on element interiors and inter element boundaries the inclusion of residuals on inter element boundaries distinguishes this method from the well known galerkin least squares method and is crucial to the resulting accuracy of this method in two dimensions and for regular bilinear quadrilateral finite elements it is shown via a dispersion analysis that this method has minimal phase error numerical experiments are conducted to verify this claim as well as test and compare the performance of this method on unstructured meshes with other methods it is found that even for unstructured meshes this method retains a high level of accuracy copyright 2000 john wiley sons ltd

semiring fank versus column rank abstract this paper concerns two notions of rank of matrices over semirings semiring rank and column rank these two rank functions are the same over fields and euclidean rings but differ for matrices over many combinatorially interesting semirings including the nonnegative integer matrices the fuzzy matrices and the binary boolean matrices we investigate the largest value of r for which the column rank and semiring rank of all m n matrices over a given semiring are both r this value is determined for the semirings mentioned above as well as many others

inverses of l fuzzy matrices abstract inverses and generalized inverses of matrices over a brouwerian lattice are considered necessary and sufficient conditions for the existence of inverses g inverses thierrin vagner inverses minimum norm g inverses least squares g inverses and moore penrose inverses of matrices are obtained algorithms for finding all these types of generalized inverses are given

iteration algorithm for solving ax b in max min algebra abstract for solving the linear system a x b we introduce some iteration algorithms where a aij and b bi a given matrix and a column vector with elements from a brouwerian lattice and is a max min composite operation

sets of solution set equivalent coefficient matrices of fuzzy relation equations abstract a set of coefficient matrices which are equivalent to the solution set of a fuzzy relation equation is considered it is also shown that the cardinal numbers of solution sets of fuzzy relation equations for two constant vectors with the same partition on the index set are equal to each other

the semigroup of incline hall matrices this paper investigates the semigroup of incline hall matrices an incline hall matrix means an incline matrix which is greater than or equal to an incline invertible matrix it is proved that any incline hall matrix has index some necessary and sufficient conditions for an incline hall matrix to be regular in the semigroup of incline hall matrices are given and green s relations on the such semigroup are described also the sandwich semigroup of incline hall matrices is considered the main results in the present paper generalize some previous results on boolean matrices fuzzy matrices and lattice matrices

an adaptive generalized finite element method applied to free vibration analysis of straight bars and trusses in this paper a first application of an adaptive generalized finite element method to free longitudinal vibration analysis of straight bars and trusses is presented the generalized finite element method is developed by enriching the standard finite element method space whose basis performs a partition of unity with knowledge about the differential equation being solved the enrichment functions used are dependent on the geometric and mechanical properties of the element the proposed approach converges very fast and is able to approximate the frequency related to any vibration mode the variational problem of free vibration is formulated and the main aspects of the adaptive generalized finite element method are presented and discussed the efficiency and convergence of the proposed method in vibration analysis of uniform and non uniform straight bars are checked the application of this technique in a truss is also presented the frequencies obtained by the adaptive generalized finite element method are compared with those obtained by the analytical solution the composite element method and the h version of finite element method

operators that preserve semiring matrix functions abstract characterizations are obtained of those linear operators over certain semirings that preserve 1 the rth coefficient of the rook polynomial those that preserve 2 the term rank of matrices with term rank r and those that preserve 3 the rth elementary symmetric permanental function our results apply to many algebraic systems of combinatorial interest including the nonnegative integer matrices boolean matrices and fuzzy matrices

a geometric treatment of generalized inverses and semigroups of nonnegative matrices abstract the purpose of this paper is to provide a unified treatment from the geometric viewpoint of the following closely related aspects of nonnegative matrices nonnegative matrices with nonnegative generalized inverses of various kinds nonnegative rank factorization regular elements green s relations and maximal subgroups of the semigroups of nonnegative matrices stochastic matrices column stochastic matrices and doubly stochastic matrices

a remarkable periodic solution of the three body problem in the case of equal masses using a variational method we exhibit a surprisingly simple periodic orbit for the newtonian problem of three equal masses in the plane the orbit has zero angular momentum and a very rich symmetry pattern its most surprising feature is that the three bodies chase each other around a flxed eight shaped curve setting aside collinear motions the only other known motion along a flxed curve in the inertial plane is the lagrange relative equilibrium in which the three bodies form a rigid equilateral triangle which rotates at constant angular velocity within its circumscribing circle our orbit visits in turns every euler conflguration in which one of the bodies sits at the midpoint of the segment deflned by the other two figure 1 numerical computations

indices and periods of incline matrices abstract inclines are the additively idempotent semirings in which products are less than or equal to factors thus inclines generalize boolean algebra fuzzy algebra and distributive lattice and the boolean matrices the fuzzy matrices and the lattice matrices are the prototypical examples of the incline matrices i e the matrices over inclines this paper studies the power sequence of incline matrices in detail a necessary and sufficient condition for the incline matrix to have index is given and the indices of some incline matrices with indices are estimated it is proved that the period of n n incline matrix with index is a divisor of n and that the set of periods of n n incline matrices with indices is not bounded from above in the sense of a power of n for all n an equivalent condition and some sufficient conditions for the incline matrix to converge in finite steps are established the stability of the orbits of an incline matrix is considered and a theorem in fuzzy sets and systems 81 1996 227 is pointed out being false the results in the present paper include some previous results in the literatures which were obtained for the boolean matrices the fuzzy matrices and the lattice matrices among their special cases

a purely frequency based floquet hill formulation for the efficient stability computation of periodic solutions of ordinary differential systems abstract since the founding theory established by g floquet more than a hundred years ago computing the stability of periodic solutions has given rise to various numerical methods mostly depending on the way the periodic solutions are themselves determined either in the time domain or in the frequency domain in this paper we address the stability analysis of branches of periodic solutions that are computed by combining a pure harmonic balance method hbm with an asymptotic numerical method anm hbm is a frequency domain method for determining periodic solutions under the form of fourier series and anm is continuation technique that relies on high order taylor series expansion of the solutions branches with respect to a path parameter it is well established now that this hbm anm combination is efficient and reliable provided that the system of ode is first of all recasted with quadratic nonlinearities allowing an easy manipulation of both the taylor and the fourier series in this context hill s method a frequency domain version of floquet theory is revisited so as to become a by product of the hbm applied to a quadratic system allowing the stability analysis to be implemented in an elegant way and with good computing performances the different types of stability changes of periodic solutions are all explored and illustrated through several academic examples including systems that are autonomous or not conservative or not free or forced

inclines and incline matrices a survey inclines are additively idempotent semirings in which products are less than or equal to either factor thus they generalize boolean and fuzzy algebra we outline results obtained in a book with cao cao kim roush incline algebra and applications john wiley sons new york 1984 on algebraic structure of inclines themselves matrices over inclines topological and convergence results and some applications this survey paper represents our talk at the 10th annual ilas conference at auburn university june 2002

wavenumber explicit convergence analysis for galerkin discretizations of the helmholtz equation we develop a stability and convergence theory for a class of highly indefinite elliptic boundary value problems bvps by considering the helmholtz equation at high wavenumber k as our model problem the key element in this theory is a novel k explicit regularity theory for helmholtz bvps that is based on decomposing the solution into two parts the first part has the sobolev regularity properties expected of second order elliptic pdes but features k independent regularity constants the second part is an analytic function for which k explicit bounds for all derivatives are given this decomposition is worked out in detail for several types of bvps namely the helmholtz equation in bounded smooth domains or convex polygonal domains with robin boundary conditions and in exterior domains with dirichlet boundary conditions we present an error analysis for the classical hp version of the finite element method hp fem where the dependence on the mesh width h the approximation order p and the wavenumber k is given explicitly in particular under the assumption that the solution operator for helmholtz problems is polynomially bounded in k it is shown that quasi optimality is obtained under the conditions that kh p is sufficiently small and the polynomial degree p is at least o log k

determinant theory for fuzzy matrices abstract we study determinant theory for fuzzy square matrices including some boolean matrices and consider the cayley hamilton theorem for these matrices we prove that det a adj a det a det adj a a where adj a denotes the adjoint matrix of a fuzzy matrix a

finite element solution of the helmholtz equation with high wave number part ii the h p version of the fem in this paper which is part ii in a series of two the investigation of the galerkin finite element solution to the helmholtz equation is continued while part i contained results on the h version with piecewise linear approximation the present part deals with approximation spaces of order p ge 1 as in part i the results are presented on a one dimensional model problem with dirichlet robin boundary conditions in particular there are proven stability estimates both with respect to data of higher regularity and data that is bounded in lower norms the estimates are shown both for the continuous and the discrete spaces under consideration further there is proven a result on the phase difference between the exact and the galerkin finite element solutions for arbitrary p that had been previously conjectured from numerical experiments these results and further preparatory statements are then employed to show error estimates for the galerkin finite element method fem it becomes evident that the error estimate for higher approximation can with certain assumptions on the data be written in the same form as the piecewise linear case namely as the sum of the error of best approximation plus a pollution term that is of the order of the phase difference the paper is concluded with a numerical evaluation

vibration of non uniform rods and beams we show that for some non uniform rods and beams the equation of motion can be transformed into the equation of motion for a uniform rod or beam then when the ends are completely fixed the eigenvalues of the non uniform continuum are the same as those of uniform rods or beams for other end support conditions exact solutions are obtained an efficient procedure is used to analyze the free vibration of non uniform beams with general shape and arbitrary boundary conditions simple formulas are presented for predicting the fundamental natural frequency of non uniform beams with various end support conditions

fuzzy relation equations theory as a basis of fuzzy modelling an overview abstract we give a wide overview on the applications of fuzzy relation equations theory to decision making processes to the construction of preference relations and to knowledge engineering mainly fuzzy control and fuzzy pattern recognition general theoretical results and methodological aspects are pointed out

bifurcation structures and dominant modes near relative equilibria in the one dimensional discrete nonlinear schrodinger equation abstract we investigate the bifurcation structure of a family of relative equilibria of a ring of seven oscillators described by the discrete nonlinear schrodinger equation dnlse when the period of these orbits and a suitable defect act as bifurcation parameters we find a reduced hamiltonian that gives substantial insight into the dynamics of this system the convexity of this hamiltonian at given nonresonant equilibria supports the stability of nearby quasiperiodic solutions we show that the local loss of convexity in the reduced hamiltonian is determined by the hessian of its integrable part in the family of relative equilibria under study stable quasiperiodic solutions are studied by considering the power spectral densities of a set of suitable fast and slow actions whose origin is suggested by the averaging principle we also show that the return times form an optimal embedding to characterize the system dynamics we show that the power spectral density of a suitable interference signal arising from a ring of bose einstein condensates and described by the dnlse has a single prominent peak at the breather like relative equilibria

a fourth order compact scheme for the helmholtz equation alpha interpolation of fem and fdm stencils we propose a fourth order compact scheme on structured meshes for the helmholtz equation given by r f x 2 0 the scheme consists of taking the alpha interpolation of the galerkin finite element method and the classical central finite difference method in 1d this scheme is identical to the alpha interpolation method j comput appl math 1982 8 1 15 19 and in 2d making the choice 0 5 we recover the generalized fourth order compact pade approximation j comput phys 1995 119 252 270 comput meth appl mech engrg 1998 163 343 358 therein using the parameter 2 we follow siam rev 2000 42 3 451 484 comput meth appl mech engrg 1995 128 325 359 for the analysis of this scheme and its performance on square meshes is compared with that of the quasi stabilized fem comput meth appl mech engrg 1995 128 325 359 in particular we show that the relative phase error of the numerical solution and the local truncation error of this scheme for plane wave solutions diminish at the rate o l 4 where l represent the wavenumber and the mesh size respectively an expression for the parameter is given that minimizes the maximum relative phase error in a sense that will be explained in section 4 5 convergence studies of the error in the l2 norm the h1 semi norm and the l euclidean norm are done and the pollution effect is found to be small copyright 2010 john wiley sons ltd

a discontinuous finite element formulation for helmholtz equation abstract a discontinuous finite element formulation for helmholtz problems is presented with c 0 continuity across inter element boundaries enforced in a weak sense depending on two parameters and whose choice is crucial for the performance of the proposed method these parameters are chosen through one dimension numerical experiments aiming at minimizing the pollution error optimal values of these parameters are then adopted in more general situations the accuracy and stability of the proposed formulation for linear and bilinear shape functions are demonstrated in several numerical examples in one and two dimensions

staggered discontinuous galerkin methods for the helmholtz equation with large wave number abstract in this paper we investigate staggered discontinuous galerkin method for the helmholtz equation with large wave number on general polygonal meshes the method is highly flexible by allowing rough grids such as the trapezoidal grids and highly distorted grids and at the same time is numerical flux free furthermore it allows hanging nodes which can be simply treated as additional vertices by exploiting a modified duality argument the stability and convergence can be proved under the condition that h c 0 1 1 m 1 where m is the polynomial order is the wave number h is the mesh size and c 0 is a positive constant independent of h error estimates for both the scalar and vector variables in l 2 norm are established several numerical experiments are tested to verify our theoretical results and to present the capability of our method for capturing singular solutions

the generalized finite element method this paper describes a pilot design and implementation of the generalized finite element method gfem as a direct extension of the standard finite element method sfem or fem which makes possible the accurate solution of engineering problems in complex domains which may be practically impossible to solve using the fem the development of the gfem is illustrated for the laplacian in two space dimensions in domains which may include several hundreds of voids and or cracks for which the construction of meshes used by the fem is practically impossible the two main capabilities are 1 it can construct the approximation using meshes which may overlap part or all of the domain boundary 2 it can incorporate into the approximation handbook functions which are known analytically or are generated numerically and approximate well the solution of the boundary value problem in the neighborhood of corner points voids cracks etc the main tool is a special integration algorithm which we call the fast remeshing approach which is robust and works for any domain with arbitrary complexity the incorporation of the handbook functions into the gfem is done by employing the partition of unity method pum the presented formulations and implementations can be easily extended to the multi material medium where the voids are replaced by inclusions of various shapes and sizes and to the case of the elasticity problem this work can also be understood as a pilot study for the feasibility and demonstration of the capabilities of the gfem which is needed before analogous implementations are attempted in the three dmensional and nonlinear cases which are the cases of main interest for future work

stability of discrete breathers in nonlinear klein gordon type lattices with pure anharmonic couplings we consider the discrete breathers in one dimensional nonlinear klein gordon type lattices with pure anharmonic couplings a discrete breather in the limit of vanishing couplings i e the anti continuous limit consists of a number of in phase or anti phase excited particles separated by particles at rest existence of the discrete breathers is proved for weak couplings by continuation from the anti continuous limit we prove a theorem which determines the linear stability of the discrete breathers the theorem shows that the stability or instability of a discrete breather depends on the phase difference and distance between the two sites in each pair of adjacent excited sites in the anti continuous solution it is shown that there are two types of the dependence determined by the sign of where and are parameters such that positive respectively negative represents hard respectively soft on site nonlinearity and positive respectively negative represents attractive respectively repulsive couplings

a tutorial in elementary finite element error analysis a systematic presentation of a priori and a posteriori error estimates abstract this paper provides a tutorial on the derivation of a priori and explicit residual based a posteriori error estimates for galerkin finite element discretizations of general linear elliptic operators for simplicity the presentation is in one dimension although extensions to multidimensions should be straightforward the structures of explicit residual based a posteriori error estimators and a priori error estimators are very similar they are each in the form of an upper bound on the error measured in some norm the structures of their derivations also follow similar lines it is therefore appropriate to present and discuss a priori and a posteriori error estimators in parallel in this paper error bounds are obtained for the error measured in the energy norm and the l2 norm in addition symmetric nonsymmetric positive definite and indefinite operators are discussed a priori estimates for nonsymmetric operators are treated with a novel approach involving the introduction of the skew norm special attention is given to the advection diffusion equation a nonsymmetric operator and the helmholtz equation an indefinite operator we systematically describe the necessary steps in deriving a priori and a posteriori error estimates and provide a simplified understanding of their differences as well as their similarities

maximum order index of matrices over commutative inclines an answer to an open problem abstract this paper proves that the maximum order index of n n matrices over an arbitrary commutative incline equals n 1 2 1 this is an answer to an open problem compute the maximum order index of a member of m n l proposed by cao kim and roush in a monograph incline algebra and applications 1984 where m n l is the set of all n n matrices over an incline l

extended finite element method for three dimensional crack modelling an extended finite element method x fem for three dimensional crack modelling is described a discontinuous function and the two dimensional asymptotic crack tip displacement fields are added to the finite element approximation to account for the crack using the notion of partition of unity this enables the domain to be modelled by finite elements with no explicit meshing of the crack surfaces computational geometry issues associated with the representation of the crack and the enrichment of the finite element approximation are discussed stress intensity factors sifs for planar three dimensional cracks are presented which are found to be in good agreement with benchmark solutions copyright 2000 john wiley sons ltd

a fast algorithm for the electromagnetic scattering from a large cavity a fast algorithm is presented for solving electromagnetic scattering from a rectangular open cavity embedded in an infinite ground plane the medium inside the cavity is assumed to be vertically layered by introducing a transparent artificial boundary condition the problem in the open cavity is reduced to a bounded domain problem a simple finite difference method is then applied to solve the model helmholtz equation the fast algorithm is designed for solving the resulting discrete system in terms of the discrete fourier transform in the horizontal direction a gaussian elimination in the vertical direction and a preconditioning conjugate gradient method with a complex diagonal preconditioner for the indefinite interface system the existence and uniqueness of the finite difference solution are established for arbitrary wave numbers our numerical experiments for large numbers of mesh points up to 16 million unknowns and for large wave numbers e g between 100 and 200 wavelengths show that the algorithm is extremely efficient the cost for calculating the radar cross section which is of significant interest in practice is o m2 for an m times m mesh the proposed algorithm may be extended easily to solve discrete systems from other discretization methods of the model problem

enriched finite elements for time harmonic webster s equation this work presents an enriched finite element method fem dedicated to the numerical resolution of webster s equation in the time harmonic regime which models many physical configurations e g wave propagation in acoustic waveguides or vibration of bars with varying cross section building on the wave based methods existing in the literature we present new enriched finite element bases that account for both the frequency of the problem and the heterogeneity of the coefficients of the equation the enriched method is compared to the classical fifth order polynomial fem and we show they share the same asymptotic convergence order present the same easiness of implementation and have similar computational costs the main improvement brought by these enriched bases relies on the convergence threshold the mesh size at which the convergence regime begins and the convergence multiplicative constant which are observed to be i better than the ones associated with polynomial bases and ii not only dependent on the resolution the number of elements per wavelength but also on the frequency for a fixed resolution making the method we propose well adapted to high frequency regimes moreover taking into account the heterogeneity of the coefficient in webster s equation by using an element dependent enrichment leads to a significant decrease of the approximation error on the considered examples compared to a uniform enrichment again with almost no additional cost several possible extensions of this work are finally discussed

stability and bifurcations of the figure 8 solution of the three body problem the stability properties of a recently discovered solution of the general three body problem with equal masses and the shape of a figure 8 are analyzed as the masses are varied it is shown by numerical continuation and the evaluation of the characteristic multipliers that the solution is stable only in a narrow mass interval other less symmetrical and unstable solutions with equal masses in the same homotopy class as the figure 8 orbit have been found the branching behavior is also analyzed

regular fuzzy matrices and fuzzy equations abstract there have been many researches on fuzzy relational equations in this paper under the general framework of fuzzy matrix theory we first introduce ranks and regularity properties of fuzzy matrices then for a regular matrix a mn f we study the generalized inverses of a and the solutions of a fuzzy equation x a b

stratified symplectic spaces and reduction let m w be a hamiltonian g space with proper momentum map j m g it is well known that if zero is a regular value of j and g acts freely on the level set j 0 then the reduced space mo j 0 g is a symplectic manifold we show that if the regularity assumptions are dropped the space mo is a union of symplectic manifolds i e it is a stratified symplectic space arms et al 2 proved that mo possesses a natural poisson bracket using their result we study hamiltonian dynamics on the reduced space in particular we show that hamiltonian flows are strata preserving and give a recipe for lifting a reduced hamiltonian flow to the level set j 0 finally we give a detailed description of the stratification of mo and prove the existence of a connected

exchange orbits in the planar 1 4 body problem abstract we study some doubly symmetric orbits in the planar 1 2 n body problem that is the mass of the central body is significantly bigger than the other 2 n equal masses the necessary and sufficient conditions for periodicity of the orbits are discussed we also study numerically these kinds of orbits for the case n 2 the system under study corresponds to one conformed by a planet and four satellites of equal mass we determine a 1 parameter family of time reversible invariant tori related with the reversing symmetries of the equations of motion the initial conditions of the orbits were determined by means of solving a boundary value problem with one free parameter the numerical solution of the boundary value problem was obtained using the software auto for the numerical analysis we have used the value of 3 5 1 0 4 as mass ratio of some satellite and the planet in the computed solutions the satellites are in mean motion resonance 1 1 and they librate around a relative equilibria that is a solution where the distances between the bodies remain constant for all time