minimal symmetric factorizations of symmetric real and complex rational matrix functions abstract real and complex rational matrix functions w such that w w t where 1 and their minimal factorizations w l t d l are studied such factorizations are described geometrically in terms of invariant subspaces with certain isotropic properties with respect to a quadratic form using results partly known and partly proved in the paper concerning the dimensions of such subspaces upper bounds are given for the degrees of the rational matrix functions l in the above factorizations

an alternative approach to unitoidness abstract in a recent paper c r johnson s furtado a generalization of sylvester s law of inertia linear algebra appl 338 2001 287 290 sylvester s law of inertia is generalized to any matrix that is congruent to a diagonal matrix such a matrix is called unitoid in the present paper an alternative approach to the subject of unitoidness is offered specifically sylvester s law of inertia states that a hermitian n n matrix of rank r with inertia p q n r is congruent to the direct sum e i 0 i p e i i q 0 i n r it is demonstrated herein that a unitoid matrix a of rank r is congruent to a direct sum of diagonal blocks of the form e i i p e i i q together with the zero block 0in r moreover the s together with the multiplicities p and q are specified in terms of the eigenvalues and eigenvectors of a a where a is the moore penrose inverse of a

pairs of hermitian and skew hermitian quaternionic matrices canonical forms and their applications canonical forms are described for pairs of quaternionic matrices or equivalently matrix pencils where each one of the matrices is either hermitian or skew hermitian under strict equivalence and simultaneous congruence several applications are developed including structure of selfadjoint and skew adjoint matrices with respect to a regular sesquilinear quaternion valued form or with respect to a regular skew sesquilinear form another application gives canonical forms for quaternionic matrices under congruence

the transfer function analysis of various schemes for the two dimensional shallow water equations in this paper various finite difference and finite element approximations to the linearized two dimensional shallow water equations are analyzed this analysis complements previous results for the one dimensional case

the solution of the equation ax x b 0 abstract we describe how to find the general solution of the matrix equation ax x b 0 where a c m n and b c n m are arbitrary matrices x c n m is the unknown and x denotes either the transpose or the conjugate transpose of x we first show that the solution can be obtained in terms of the kronecker canonical form of the matrix pencil a b and the two nonsingular matrices which transform this pencil into its kronecker canonical form we also give a complete description of the solution provided that these two matrices and the kronecker canonical form of a b are known as a consequence we determine the dimension of the solution space of the equation in terms of the sizes of the blocks appearing in the kronecker canonical form of a b the general solution of the homogeneous equation ax x b 0 is essential to finding the general solution of ax x b c which is related to palindromic eigenvalue problems that have attracted considerable attention recently

precise and fast computation of jacobian elliptic functions by conditional duplication we developed a new method to compute the cosine amplitude function c equiv mathrm cn u m by using its double argument formula the accumulation of round off errors is effectively suppressed by the introduction of a complementary variable b equiv 1 c and a conditional switch between the duplication of b and c the sine and delta amplitude functions s equiv mathrm sn u m and d equiv mathrm dn u m are evaluated from thus computed b or c by means of their identity relations the new method is sufficiently precise as its errors are less than a few machine epsilons also it is significantly faster than the existing procedures in case of single precision computation it runs more than 50 times faster than bulirsch s sncndn based on the gauss transformation and 2 7 times faster than our previous method based on the simultaneous duplication of s c and d the ratios change to 7 6 and 3 5 respectively in case of the double precision environment

dispersion and group velocity in numerical schemes for three dimensional hydrodynamic equations we investigate dispersion and group velocity relations of numerical schemes for the three dimensional hydrodynamic equations by using the transform function method the numerical methods are developed for two of arakawa s spatial grid types namely b grid and c grid using a spectral method in the vertical direction one of our results is that if the horizontal scale length is greater than the rossby radius of deformation then the b grid is more appropriate than the commonly used c grid

a note on canonical forms for matrix congruence abstract canonical forms for matrix congruence for general matrices are exhibited as an easy consequence of results presented in a recent paper by r c thompson on pencils of symmetric and skew symmetric matrices we also show how to exhibit standard equivalence class representatives in the orbits of the semiorthogonal groups acting naturally on skew symmetric forms

a generalization of sylvester s law of inertia abstract we introduce the class of unitoid matrices as those that are diagonalizable by congruence the nondegenerate canonical angles of a unitoid matrix a are the directions of the nonzero entries of a diagonal matrix congruent to a which are shown to be unique sylvester s law states that two hermitian matrices of the same size are congruent if and only if they have the same numbers of positive respectively negative eigenvalues two unitoid matrices are congruent if and only if they have the same nondegenerate canonical angles

the solution of the equation xa axt 0 and its application to the theory of orbits abstract describe how to find the general solution of the matrix equation xa ax t 0 with a c n n which allows us to determine the dimension of its solution space this result has immediate applications in the theory of congruence orbits of matrices in c n n because the set xa ax t x c n n is the tangent space at a to the congruence orbit of a hence the codimension of this orbit is precisely the dimension of the solution space of xa ax t 0 as a consequence we also determine the generic canonical structure of matrices under the action of congruence all these results can be directly extended to palindromic pencils a a t

topological classification of systems of bilinear and sesquilinear forms abstract let a and b be two systems consisting of the same vector spaces c n 1 c n t and bilinear or sesquilinear forms a i b i c n k i c n l i c for i 1 s we prove that a is transformed to b by homeomorphisms within c n 1 c n t if and only if a is transformed to b by linear bijections within c n 1 c n t

cmv block matrices for symmetric matrix measures on the unit circle abstract in this contribution we study the relationship between the cmv block matrices associated with two symmetric positive definite matrix measures supported on the unit circle we also consider certain transformations of an orthogonality matrix measure supported on the unit circle and deduce connection formulas for the corresponding orthogonal families and their verblunsky matrix coefficients

canonical matrices of isometric operators on indefinite inner product spaces abstract we give canonical matrices of a pair a b consisting of a nondegenerate form b and a linear operator a satisfying b ax ay b x y on a vector space over f in the following cases f is an algebraically closed field of characteristic different from 2 or a real closed field and b is symmetric or skew symmetric f is an algebraically closed field of characteristic 0 or the skew field of quaternions over a real closed field and b is hermitian or skew hermitian with respect to any nonidentity involution on f these classification problems are wild if b may be degenerate we use a method that admits to reduce the problem of classifying an arbitrary system of forms and linear mappings to the problem of classifying representations of some quiver this method was described in v v sergeichuk classification problems for systems of forms and linear mappings math ussr izv 31 3 1988 481 501

precise and fast computation of the general complete elliptic integral of the second kind we developed an efficient procedure to evaluate two auxiliary complete elliptic integrals of the second kind b m and d m by using their taylor series expansions the definition of jacobi s nome and legendre s relation the developed procedure is more precise than the existing ones in the sense that the maximum relative errors are 1 3 machine epsilons and it runs drastically faster around 5 times faster than bulirsch s cel2 and 16 times faster than carlson s rf and rd

fast computation of a general complete elliptic integral of third kind by half and double argument transformations we developed a novel method to calculate an associate complete elliptic integral of the third kind j n m p n m k m n the key idea is the double argument formula of j n m with respect to n we derived it from the not so popular addition theorem of jacobi s complete elliptic integral of the third kind p 1 a m with respect to a which is a real or pure imaginary argument connected with n and m as n msn 2 a m repeatedly using the half argument transformation fukushima 2010 28 of a new variable y n m or its complement x m n m we reduce y sufficiently small say less than 0 3 or so then we evaluate the integral for the reduced variable by its maclaurin series expansion the coefficients of the series expansion are recursively computed from two other associate complete elliptic integrals b m e m 1 m k m m and d m k m e m m the precise and fast computation of these two integrals is found in our previous work fukushima 2011 17 finally we recover the integral value for the original n by successively applying the double argument formula of j n m the new method is sufficiently precise in the sense that the maximum errors are less than around 10 machine epsilons for the sole computation of j n m the new method runs 1 2 1 5 and 4 7 5 5 times faster than bulirsch s cel and carlson s r j respectively in the simultaneous computation of three associate complete integrals the new method runs 1 6 1 7 and 5 3 8 0 times faster than cel and carlson s r d and r j respectively

on the sign characteristics of hermitian matrix polynomials the sign characteristics of hermitian matrix polynomials are discussed and in particular an appropriate definition of the sign characteristics associated with the eigenvalue infinity the concept of sign characteristic arises in different forms in many scientific fields and is essential for the stability analysis in hamiltonian systems or the perturbation behavior of eigenvalues under structured perturbations we extend classical results by gohberg lancaster and rodman to the case of infinite eigenvalues we derive a systematic approach studying how sign characteristics behave after an analytic change of variables including the important special case of mobius transformations and we prove a signature constraint theorem we also show that the sign characteristic at infinity stays invariant in a neighborhood under perturbations for even degree hermitian matrix polynomials while it may change for odd degree matrix polynomials we argue that the non uniformity can be resolved by introducing an extra zero leading matrix coefficient

hybrid eulerian lagrangian semi implicit time integrators hybrid eulerian lagrangian semi implicit time integrators helsi are presented which use the standard semi implicit formulation as their starting point the advantage of such an approach is that existing models which employ eulerian semi implicit time integrators can easily be augmented to use the helsi approach that is provided that either a series of advection problems can be solved efficiently as in the operator integration factor splitting oifs method or that the grid and underlying numerics allow the construction of efficient search and interpolation stencils typically required of semi lagrangian methods once the advection problem is solved the approach then follows the standard semi implicit approach an important caveat of this approach is that the semi implicit time integrator must be constructed around the implicit backward difference formulas

pattern formation of a spatial predator prey system abstract there are random and directed movements of predator and prey populations in many natural systems which are strongly influenced and modified by spatial factors to investigate how these migration directed movement and diffusion random movement affect predator prey systems we have studied the spatiotemporal complexity in a predator prey system with holling tanner form a theoretical analysis of emerging spatial pattern is presented and wavelength and pattern speed are calculated at the same time we present the properties of pattern solutions the results of numerical simulations show that migration has prominent effect on the pattern formation of the population i e changing turing pattern into traveling pattern this study suggests that modelling by migration and diffusion in predator prey systems can account for the dynamical complexity of ecosystems

canonical forms for symmetric skew symmetric real matrix pairs under strict equivalence and congruence a systematic development is made of the simultaneous reduction of pairs of quadratic forms over the reals one of which is skew symmetric and the other is either symmetric or skew symmetric these reductions are by strict equivalence and by congruence over the reals or over the complex numbers and essentially complete proofs are presented the proofs are based on canonical forms attributed to jordan and kronecker some closely related results which can be derived from the canonical forms of pairs of symmetric skew symmetric real forms are also included they concern simultaneously neutral subspaces hamiltonian and skew hamiltonian matrices and canonical structures of real matrices which are selfadjoint or skew adjoint in a regular skew symmetric indefinite inner product and real matrices which are skew adjoint in a regular symmetric indefinite inner product the paper is largely expository and continues the comprehensive account of the reduction of pairs of matrices started in p lancaster l rodman canonical forms for hermitian matrix pairs under strict equivalence and congruence siam rev in press

congruences of a square matrix and its transpose abstract it is known that any square matrix a over any field is congruent to its transpose at stas for some nonsingular s moreover s can be chosen such that s2 i that is s can be chosen to be involutory we show that a and at are congruent over any field f of characteristic not two with involution a a the involution can be the identity a t s t as for some nonsingular s moreover s can be chosen such that s s i that is s can be chosen to be coninvolutory the short and simple proof is based on sergeichuk s canonical form for congruence math ussr izvestiya 31 3 1988 481 it follows that any matrix a over f can be represented as a eb in which e is coninvolutory and b is symmetric

analysis of the turkel zwas scheme for the two dimensional shallow water equations in spherical coordinates a linear analysis of the shallow water equations in spherical coordinates for the turkel zwas t z explicit large time step scheme is presented this paper complements the results of schoenstadt neta and navon and others in 1 d and of neta and devito in 2 d but applied to the spherical coordinate case of the t z scheme this coordinate system is more realistic in meteorology and more complicated to analyze since the coefficients are no longer constant the analysis suggests that the t z scheme must be staggered in a certain way in order to get eigenvalues and eigenfunctions approaching those of the continuous case the importance of such an analysis is the fact that it is also valid for nonconstant coefficients and thereby applicable to any numerical scheme numerical experiments comparing the original unstaggered and staggered versions of the t z scheme are presented these experiments corroborate the analysis by showing the improvements in accuracy gained by staggering the turkel zwas scheme

on the matrix equation xa ax t 0 abstract the matrix equation xa ax t 0 which has relevance to the study of lie algebras was recently studied by de teran and dopico linear algebra appl 434 2011 44 67 they reduced the study of this equation to several special cases and produced explicit solutions in most instances in this note we obtain an explicit solution in one of the difficult cases for which only the dimension of the solution space and an algorithm to find a basis of this space were known previously

a regularization algorithm for matrices of bilinear and sesquilinear forms abstract over a field or skew field f with an involution a a possibly the identity involution each singular square matrix a is congruent to a direct sum s as b j n 1 j n p 1 n 1 n p inwhich s is nonsingular and s s t b is nonsingular and is determined by a up to congruence and the ni by ni singular jordan blocks j n i and their multiplicities are uniquely determined by a we give a regularization algorithm that needs only elementary row operations to construct such a decomposition if f c respectively f r we exhibit a regularization algorithm that uses only unitary respectively real orthogonal transformations and a reduced form that can be achieved via a unitary congruence or congruence respectively a real orthogonal congruence the selfadjoint matrix pencil a a is decomposed by our regularization algorithm into the direct sum s a a s b b j n 1 j n 1 j n p j n p with selfadjoint summands

classification of sesquilinear forms with the first argument on a subspace or a factor space abstract let v be a vector space over a field or skew field f and let u be its subspace we study the canonical form problem for bilinear or sesquilinear forms u v f v u v f and linear mappings u v v u v u v v v u we solve it over f c and reduce it over all f to the canonical form problem for ordinary linear mappings w w and bilinear or sesquilinear forms w w f moreover we give an algorithm that realizes this reduction the algorithm uses only unitary transformations if f c which improves its numerical stability for linear mapping this algorithm can be derived from the algorithm by nazarova et al l a nazarova a v roiter v v sergeichuk v m bondarenko application of modules over a dyad for the classification of finite p groups possessing an abelian subgroup of index p and of pairs of mutually annihilating operators zap nauchn sem leningrad otdel mat inst steklov lomi 28 1972 69 92 translation in j soviet math 3 5 1975 636 654

a spectral element semi lagrangian sesl method for the spherical shallow water equations a spectral element semi lagrangian sesl method for the shallow water equations on the sphere is presented the sphere is discretized using a hexahedral grid although any grid imaginable can be used as long as it is comprised of quadrilaterals the equations are written in cartesian coordinates to eliminate the pole singularity which plagues the equations in spherical coordinates in a previous paper int j numer methods fluids 35 2001 869 we showed how to construct an explicit eulerian spectral element se model on the sphere we now extend this work to a semi lagrangian formulation the novelty of the lagrangian formulation presented is that the high order se basis functions are used as the interpolation functions for evaluating the values at the lagrangian departure points this makes the method not only high order accurate but quite general and thus applicable to unstructured grids and portable to distributed memory computers the equations are discretized fully implicitly in time in order to avoid having to interpolate derivatives at departure points by incorporating the coriolis terms into the lagrangian derivative the block lu decomposition of the equations results in a symmetric positive definite pseudo helmholtz operator which we solve using the generalized minimum residual method gmres with a fast projection method comput methods appl mech eng 163 1998 193 results for eight test cases are presented to confirm the accuracy and stability of the method these results show that sesl yields the same accuracy as an eulerian spectral element semi implicit sesi while allowing for time steps 10 times as large and being up to 70 more efficient

analysis of the turkel zwas scheme for the shallow water equations abstract a transfer function analysis is used to analyze the turkel zwas explicit large time step scheme applied to the shallow water equations the transfer function concept leads to insight into the behavior of this discretization scheme in terms of comparison between continuous and discrete amplitude phase and group velocity coefficients the dependence of the distortion increases with the increase in the time step size taken for the turkel zwas scheme which depends on the ratio between a coarse and a fine mesh a comparison with earlier results of schoenstadt naval post graduate school report nps 53 79 001 1978 unpublished shows the turkel zwas scheme to give reasonable results up to time steps three times larger than the cfl limit

finite element schemes for extended integrations of atmospheric models abstract the effect of conservation of integral invariants by finite element discretization schemes of the shallow water equations as a model for long term integrations of atmospheric models is investigated two finite element models are used the first uses rectangular elements and conserves total energy using an intrinsic method the second model uses triangular elements and a high accuracy two stage numerov galerkin method it well conserves total energy and potential enstrophy by applying a periodical shuman filter every 48 or 96 time steps different conserving and non conserving versions of these finite element schemes are compared in terms of their conservation of the integral invariants of the shallow water equations for long term integrations 20 100 model days critical times for numerical non linear instability are investigated along with the determination of the critical amount of dissipation required to achieve stable long term integrations comparisons of the two finite element schemes namely the rectangular and the triangular in terms of their relative computational efficiency and accuracy are also provided similarities and differences in the behavior of finite element schemes and finite difference schemes for long term stable integrations of the shallow water equations are finally discussed

change of the congruence canonical form of 2 by 2 and 3 by 3 matrices under perturbations and bundles of matrices under congruence we construct the hasse diagrams g2 and g3 for the closure ordering on the sets of congruence classes of 2 2 and 3 3 complex matrices in other words we construct two directed graphs whose vert

lagrange galerkin methods on spherical geodesic grids the weak lagrange galerkin finite element method for the 2d shallow water equations on the sphere is presented this method offers stable and accurate solutions because the equations are integrated along the characteristics the equations are written in 3d cartesian conservation form and the domains are discretized using linear triangular elements the use of linear triangular elements permits the construction of accurate by virtue of the second order spatial and temporal accuracies of the scheme and efficient by virtue of the less stringent cfl condition of lagrangian methods schemes on unstructured domains using linear triangles in 3d cartesian space allows for the explicit construction of area coordinate basis functions thereby simplifying the calculation of the finite element integrals the triangular grids are constructed by a generalization of the icosahedral grids that have been typically used in recent papers an efficient searching strategy for the departure points is also presented for these generalized icosahedral grids which involves very few floating point operations in addition a high order scheme for computing the characteristic curves in 3d cartesian space is presented a general family of runge kutta schemes results for six test cases are reported in order to confirm the accuracy of the scheme

miniversal deformations of matrices under congruence and reducing transformations arnold 1971 1 constructed a miniversal deformation of a square complex matrix under similarity that is a simple normal form to which not only a given square matrix a but all matrices b close to it can be reduced by similarity transformations that smoothly depend on the entries of b we give miniversal deformations of matrices of sesquilinear forms that is of square complex matrices under congruence and construct an analytic reducing transformation to a miniversal deformation analogous results for matrices under congruence were obtained by dmytryshyn futorny and sergeichuk 2012 11

instability of periodic traveling wave solutions in a modified fitzhugh nagumo model for excitable media we introduce a two variable system of reaction diffusion equations for excitable media we numerically investigate the existence and stability of periodic traveling wave solutions in a two dimensional parameter plane our results based on the method of continuation show a stability change of eckhaus type there are two families of periodic traveling waves fast and slow the fast family is stable in the case of standard fitzhugh nagumo excitable system however we observe that the fast family becomes unstable in our model consequently it bifurcates to an oscillating wave we explain this phenomenon by numerically calculating the essential spectra of the periodic traveling wave solutions moreover we study the stability of the periodic traveling wave solutions for the aliev panfilov excitable system and compare its results with the proposed model we show that the stability boundary of our model agrees qualitatively with that of the aliev panfilov model whereas it is different in the fitzhugh nagumo model in addition our results based on the stability boundary of the models show oscillating periodic traveling waves except for the fitzhugh nagumo model

two applications of the theory of primary matrix functions among all the ways one might define f a for a square complex matrix a and a given function f c c the notion of a primary matrix function is perhaps the most useful and natural using only basic conceptual properties of primary matrix functions we consider whether f a can have rank one and whether for a given b there is a unique a such that f a b

canonical forms for normal matrices that commute with their complex conjugate we consider the class of normal complex matrices that commute with their complex conjugate we show that such matrices are real orthogonally similar to a canonical direct sum of 1 by 1 and certain 2 by 2 matrices a canonical form for quasi real normal matrices is obtained as a special case we also exhibit a special form of the spectral theorem for normal matrices that commute with their conjugate

pencils of complex and real symmetric and skew matrices abstract this expository paper establishes the canonical forms under congruence for pairs of complex or real symmetric or skew matrices the treatment is in the spirit of the well known book of gantmacher on matrix theory and may be regarded as a supplement to gantmacher s chapters on pencils of matrices

numerical analysis of periodic solutions and bifurcations in the planetary annulus problem this paper discusses the dynamics of particles orbiting planetary rings under a general central potential starting with the mathematical description of the dynamical system we analyze the motion of a particle with infinitesimal mass as attracted by a central body surrounded by a homogeneous circular annular disk throughout the paper we carry out an analytic search of the most relevant equilibria solutions and based on that we investigate numerically the stability matrix of the system to find stability inequalities in this way we describe the in plane and out of plane motion by means of the numerical continuation of a wide number of uni parametric families of planar and spatial periodic orbits we present a description of the main families of periodic orbits encountered their bifurcations and linear stability with the aim of reproducing a more realistic scenario we analyse different mass proportions between the annulus and the central body we consider an oblate planet and we also include a composition of rings in the dynamical model

numerical continuation methods for studying periodic travelling wave wavetrain solutions of partial differential equations abstract periodic travelling waves wavetrains are an important solution type for many partial differential equations in this paper i review the use of numerical continuation for studying these solutions i discuss the calculation of the form and stability of a given periodic travelling wave and the calculation of boundaries in a two dimensional parameter plane for wave existence and stability i also describe the automated implementation of these numerical continuation procedures via the software package wavetrain http www ma hw ac uk wavetrain i conclude by discussing ongoing work on numerical continuation methods for determining the absolute stability of periodic travelling waves

computing spectra of linear operators using the floquet fourier hill method in order to establish the stability of an equilibrium solution u of an infinite dimensional dynamical system u x u one is interested in the spectrum of the linear operator lu obtained by linearizing the dynamical system around u we use a spectrally accurate method for the computation of the spectrum of the maximal extension of the operator lu the method is particularly well suited to the case of periodic u although no periodic boundary conditions on the perturbations are imposed by incorporating the fundamentals of floquet theory an almost uniform approximation to the entire spectrum of the maximal extension is obtained as opposed to an approximation of a few selected elements the numerical component of the method is limited to i choosing the size of the matrices to be used and ii an eigenvalue solver such as the qr algorithm compared to often used finite difference approaches the method is an order of magnitude faster for comparable accuracy we illustrate that the method is efficiently extended to problems defined on the whole line

absolute and convective instabilities of waves on unbounded and large bounded domains abstract instabilities of nonlinear waves on unbounded domains manifest themselves in different ways an absolute instability occurs if the amplitude of localized wave packets grows in time at each fixed point in the domain in contrast convective instabilities are characterized by the fact that even though the overall norm of wave packets grows in time perturbations decay locally at each given point in the unbounded domain wave packets are convected towards infinity in experiments as well as in numerical simulations bounded domains are often more relevant we are interested in the effects that the truncation of the unbounded to a large but bounded domain has on the aforementioned in stability properties of a wave these effects depend upon the boundary conditions that are imposed on the bounded domain we compare the spectra of the linearized evolution operators on unbounded and bounded domains for two classes of boundary conditions it is proved that periodic boundary conditions reproduce the point and essential spectrum on the unbounded domain accurately spectra for separated boundary conditions behave in quite a different way firstly separated boundary conditions may generate additional isolated eigenvalues secondly the essential spectrum on the unbounded domain is in general not approximated by the spectrum on the bounded domain instead the so called absolute spectrum is approximated that corresponds to the essential spectrum on the unbounded domain calculated with certain optimally chosen exponential weights we interpret the difference between the absolute and the essential spectrum in terms of the convective behavior of the wave on the unbounded domain in particular it is demonstrated that the stability of the absolute spectrum implies convective instability of the wave but that convectively unstable waves can destabilize under domain truncation the theoretical predictions are compared with numerical computations

precise and fast computation of a general incomplete elliptic integral of second kind by half and double argument transformations this is a continuation of our works to compute the incomplete elliptic integrals of the first and second kind fukushima 2010 2011 21 23 we developed a method to compute an associate incomplete elliptic integral of the third kind j f n m p f n m f f m n by the half argument formulas of the sine and cosine amplitude functions and the double argument transformation of the integral the relative errors of j f n m computed by the new method are sufficiently small as less than 20 machine epsilons meanwhile the simplicity of the adopted algorithm makes the new method run 1 5 to 3 7 times faster than carlson s duplication method the combination of the new method and that to compute simultaneously two other associate incomplete elliptic integrals of the second kind b f m e f m 1 m f f m m and d f m f f m e f m m which we established recently 23 enables a precise and fast computation of arbitrary linear combination of legendre s incomplete elliptic integrals of all three kinds f f m e f m and p f n m these new procedures share the same device the half argument transformations while the double argument transformation of j f n m includes those of b f m and d f m as its sub component as a result the simultaneous computation of the three associate integrals is significantly faster than computing them separately in fact our combined procedure is 2 7 to 5 9 times faster than the combination of carlson s duplication method to compute r d and r j

classification of linear mappings between indefinite inner product spaces abstract let a u v be a linear mapping between vector spaces u and v over a field or skew field f with symmetric or skew symmetric or hermitian forms b u u f and c v v f we classify the triples a b c if f is r or c or the skew field of quaternions h we also classify the triples a b c up to classification of symmetric forms and hermitian forms if the characteristic of f is not 2

lagrange galerkin methods on spherical geodesic grids the shallow water equations abstract the weak lagrange galerkin finite element method for the 2d shallow water equations on the sphere is presented this method offers stable and accurate solutions because the equations are integrated along the characteristics the equations are written in 3d cartesian conservation form and the domains are discretized using linear triangular elements the use of linear triangular elements permits the construction of accurate by virtue of the second order spatial and temporal accuracies of the scheme and efficient by virtue of the less stringent cfl condition of lagrangian methods schemes on unstructured domains using linear triangles in 3d cartesian space allows for the explicit construction of area coordinate basis functions thereby simplifying the calculation of the finite element integrals the triangular grids are constructed by a generalization of the icosahedral grids that have been typically used in recent papers an efficient searching strategy for the departure points is also presented for these generalized icosahedral grids which involves very few floating point operations in addition a high order scheme for computing the characteristic curves in 3d cartesian space is presented a general family of runge kutta schemes results for six test cases are reported in order to confirm the accuracy of the scheme

dynamics in a predator prey model with space and noise in this paper we presented a predator prey model with spatial diffusion and density dependent noise it was found that the spatial model without noise has spotted pattern however combined with density dependent noise the predator prey model exhibits transition from spotted to labyrinth pattern furthermore when noise intensity and temporal correlation are in appropriate levels the populations will extinct the obtained results suggest that stochastic factors may play an important role in populations dynamics which well enrich the findings in predator prey models

topological classification of sesquilinear forms reduction to the nonsingular case abstract two sesquilinear forms c m c m c and c n c n c are called topologically equivalent if there exists a homeomorphism c m c n i e a continuous bijection whose inverse is also a continuous bijection such that x y x y for all x y c m r a horn and v v sergeichuk in 2006 constructed a regularizing decomposition of a square complex matrix a that is a direct sum s a s r j n 1 j n p in which s and r are nonsingular and each j n i is the n i by n i singular jordan block in this paper we prove that and are topologically equivalent if and only if the regularizing decompositions of their matrices coincide up to permutation of the singular summands j n i and replacement of r c r r by a nonsingular matrix r c r r such that r and r are the matrices of topologically equivalent forms c r c r c analogous results for bilinear forms over c and over r are also obtained

exact series solution to the epstein hubbell generalized elliptic type integral using complex variable residue theory in this paper complex variable residue theory is used to arrive at an exact infinite series solution to the epstein hubbell integral this method of solution is different from those used to derive previous solutions to the epstein hubbell integral but produces exactly equivalent results several sets of numerical data are included that verify the results additional information on the improvement of convergence will be forthcoming at a later date

wave train selection behind invasion fronts in reaction diffusion predator prey models abstract wave trains or periodic travelling waves can evolve behind invasion fronts in oscillatory reaction diffusion models for predator prey systems although there is a one parameter family of possible wave train solutions in a particular predator invasion a single member of this family is selected sherratt 1998 13 has predicted this wave train selection using a system that is a valid approximation near a supercritical hopf bifurcation in the corresponding kinetics and when the predator and prey diffusion coefficients are nearly equal away from a hopf bifurcation or if the diffusion coefficients differ somewhat these predictions lose accuracy we develop a more general wave train selection prediction for a two component reaction diffusion predator prey system that depends on linearizations at the unstable homogeneous steady states involved in the invasion front this prediction retains accuracy farther away from a hopf bifurcation and can also be applied when the predator and prey diffusion coefficients are unequal we illustrate the selection prediction with its application to three models of predator invasions

stable invariant lagrangian subspaces factorization of symmetric rational matrix functions and other applications several applications of earlier results by the authors concerning various notions of stability of invariant lagrangian subspaces are studied the applications include stability of symmetric minimal factorizations of real symmetric rational matrix functions stability of factorizations of certain classes of symmetric matrix polynomials and stably well posed matricial boundary value problems with symmetries an interpretation of many stability results in terms of equilibria of lie group actions on subspaces is given as well

generalized elliptic type integrals and their representations epstein hubbell l f epstein j h hubbell evaluation of a generalized elliptic type integral j res nbs 67b 1963 1 17 elliptic type integrals occur in radiation field problems in this paper we consider a generalization 10 of the elliptic type integrals introduced by kalla and tuan s l kalla vu kim tuan asymptotic formulas for generalized elliptic type integrals comput math appl 32 1996 49 55 many generalizations of elliptic type integrals studied earlier by several authors can be derived as particular cases of our unified form we study the uniform convergence of the integral representation 10 we derive the power series representations which are valid in different domains also we obtain some relationships between this generalized form and laurecella s hypergeometric function of three variables f d 3 appell s hypergeometric functions f 1 and f 3 and gauss hypergeometric function 2f 1 some important particular cases of these representations are derived

leibniz algebras with small derived ideal abstract a nilpotent leibniz algebra l is called extra special if dim z l dim l 2 1 in this paper the structure the second homology and the capability of extra special leibniz algebras are completely determined

asymptotic formulas for generalized elliptic type integrals abstract epstein hubbell 1 elliptic type integrals occur in radiation field problems the object of the present paper is to consider a unified form of different elliptic type integrals defined and developed recently by several authors we obtain asymptotic formulas for the generalized elliptic type integrals

opuc cmv matrices and perturbations of measures supported on the unit circle abstract let us consider a hermitian linear functional defined on the linear space of laurent polynomials with complex coefficients in the literature canonical spectral transformations of this functional are studied the aim of this research is focused on perturbations of hermitian linear functionals associated with a positive borel measure supported on the unit circle some algebraic properties of the perturbed measure are pointed out in a constructive way we discuss the corresponding sequences of orthogonal polynomials as well as the connection between the associated verblunsky coefficients then the structure of the matrices of the perturbed linear functionals which is the main tool for the comparison of their corresponding cmv matrices is deeply analyzed from the comparison between different cmv matrices other families of perturbed verblunsky coefficients will be considered we introduce a new matrix named fundamental matrix that is a tridiagonal symmetric unitary matrix containing basic information about the family of orthogonal polynomials however we show that it is connected to another family of orthogonal polynomials through the takagi decomposition

the spectral approximation of linear operators with applications to the computation of eigenelements of differential and integral operators we are concerned with the numerical solution of the eigenvalue problem t varphi lambda varphi varphi ne 0 where t is a linear operator in a bana ch space t may represent a bounded integral operator or a closed differential operator with bounded inverse the linear operator t and its approximation t n are defined in the same space perturbation theory is then a suitable framework for our problem we present in the first part a systematic study of the various notions of convergence t n to t which imply the convergence of the eigenvalues with preservation of the multiplicities the results are then applied to practical methods for integral and differential operators in the second part we present convergence rates analytic perturbation theory is used to refine on the computed eigenelements of an integral operator and to produce localization results on the eigenelements finally superconvergence results are discussed both for integral and differential operators

canonical forms for hamiltonian and symplectic matrices and pencils abstract we study canonical forms for hamiltonian and symplectic matrices or pencils under equivalence transformations which keep the class invariant in contrast to other canonical forms our forms are as close as possible to a triangular structure in the same class we give necessary and sufficient conditions for the existence of hamiltonian and symplectic triangular jordan kronecker and schur forms the presented results generalize results of lin and ho w w lin t c ho on schur type decompositions for hamiltonian and symplectic pencils technical report institute of applied mathematics national tsing hua university taiwan 1990 and simplify the proofs presented there

matrices that are self congruent only via matrices of determinant one abstract ocovic and szechtman d ocovic f szechtman characterization of bilinear spaces with unimodular isometry group proc amer math soc 133 2005 2853 2863 considered a vector space v endowed with a bilinear form they proved that all isometries of v over a field f of characteristic not 2 have determinant 1 if and only if v has no orthogonal summands of odd dimension the case of characteristic 2 was also considered their proof is based on riehm s classification of bilinear forms coakley et al e s coakley f m dopico c r johnson matrices with orthogonal groups admitting only determinant one linear algebra appl 428 2008 796 813 gave another proof of this criterion over r and c using thompson s canonical pairs of symmetric and skew symmetric matrices for congruence let m be the matrix of the bilinear form on v we give another proof of this criterion over f using our canonical matrices for congruence and obtain necessary and sufficient conditions involving canonical forms of m for congruence of m t m for equivalence and of m t m if m is nonsingular for similarity

canonical forms for hermitian matrix pairs under strict equivalence and congruence after a brief historical review and an account of the canonical forms attributed to jordan and kronecker a systematic development is made of the simultaneous reduction of pairs of quadratic forms over the complex numbers and over the reals these reductions are by strict equivalence and by congruence and essentially complete proofs are presented some closely related results which can be derived from the canonical forms are also included they concern simultaneous diagonalization a new criterion for the existence of positive definite linear combinations of a pair of hermitian matrices and the canonical structures of matrices which are self adjoint in an indefinite inner product

canonical matrices of bilinear and sesquilinear forms abstract canonical matrices are given for i bilinear forms over an algebraically closed or real closed field ii sesquilinear forms over an algebraically closed field and over real quaternions with any nonidentity involution and iii sesquilinear forms over a field f of characteristic different from 2 with involution possibly the identity up to classification of hermitian forms over finite extensions of f the canonical matrices are based on any given set of canonical matrices for similarity over f a method for reducing the problem of classifying systems of forms and linear mappings to the problem of classifying systems of linear mappings is used to construct the canonical matrices this method has its origins in representation theory and was devised in v v sergeichuk classification problems for systems of forms and linear mappings math ussr izv 31 1988 481 501

computing absolute and essential spectra using continuation a continuation approach to the computation of essential and absolute spectra of dierential operators on the real line is presented the advantage of this approach compared with direct eigenvalue computations for the discretized operator are the ecient and accurate computation of selected parts of the spectrum typically those near the imaginary axis and the option to compute nonlinear travelling waves and selected eigenvalues or other stability indicators simultaneously in order to locate accurately the onset to instability we also discuss the implementation and usage of this approach with the software package auto and provide example computations for the fitzhugh nagumo and the complex ginzburg landau equation

miniversal deformations of matrices of bilinear forms arnold v i arnold on matrices depending on parameters russian math surveys 26 2 1971 29 43 constructed miniversal deformations of square complex matrices under similarity that is a simp

canonical forms for complex matrix congruence and congruence abstract canonical forms for congruence and congruence of square complex matrices were given by horn and sergeichuk in linear algebra appl 389 2004 347 353 based on sergeichuk s paper math ussr izvestiya 31 3 1988 481 501 which employed the theory of representations of quivers with involution we use standard methods of matrix analysis to prove directly that these forms are canonical our proof provides explicit algorithms to compute all the blocks and parameters in the canonical forms we use these forms to derive canonical pairs for simultaneous congruence of pairs of complex symmetric and skew symmetric matrices as well as canonical forms for simultaneous congruence of pairs of complex hermitian matrices

congruence of multilinear forms abstract let f u u k g v v k be two n linear forms with n 2 on finite dimensional vector spaces u and v over a field k we say that f and g are symmetrically equivalent if there exist linear bijections 1 n u v such that f u 1 u n g i 1 u 1 i n u n for all u 1 u n u and each reordering i 1 i n of 1 n the forms are said to be congruent if 1 n let f and g be symmetrically equivalent we prove that i if k c then f and g are congruent ii if k r f f 1 f s 0 g g 1 g r 0 and all summands f i and g j are nonzero and direct sum indecomposable then s r and after a suitable reindexing f i is congruent to g i

a numerov galerkin technique applied to a finite element shallow water equations model with enforced conservation of integral invariants and selective lumping abstract a new two stage numerov galerkin method is presented and applied following a suggestion by cullen and morton 1 to achieve higher accuracy by combining the galerkin product with a high order compact numerov difference approximation to derivatives in the nonlinear advection operator of the shallow water equations the practical applicability of the new method is supported by computational examples involving medium 10 days and longterm 20 days integrations of the nonlinear shallow water equations small scale noise was eliminated by periodical application of a shuman filter to only one component of the velocity conservation of integral invariants is ensured by using a new augmented lagrangian multiplier penalty method in the numerov galerkin method experiments were performed using the new method with differently weighted selective lumping schemes for the mass matrix the new method exhibits a consistently higher accuracy than the single stage galerkin method and requires much less computational effort a hypothesis is offered to explain the increased accuracy obtained by selective lumping schemes in the long term numerical integrations 10 20 days as compared to the consistent mass approach in numerov galerkin schemes

local normal forms for geodesically equivalent pseudo riemannian metrics this is an extended version of the submitted paper it contains all proofs with all details and also an appendix where we explain how can one construct a complex structure by a 1 1 tensor such that all its eigenvalues are not real and such that the nijenhuis torsion vanishes

tridiagonal canonical matrices of bilinear or sesquilinear forms and of pairs of symmetric skew symmetric or hermitian forms abstract tridiagonal canonical forms of square matrices under congruence or congruence pairs of symmetric or skew symmetric matrices under congruence and pairs of hermitian matrices under congruence are given over an algebraically closed field of characteristic different from 2

recursive computation of derivatives of elliptic functions and of incomplete elliptic integrals presented are the recurrence formulas to compute the derivatives of a general elliptic function weierstrass s function the jacobian elliptic functions and the incomplete elliptic integrals in the forms of jacobi and legendre with respect to the argument or the amplitude the double precision computation by the formulas is correct with 15 digits or so for the first 10 orders of differentiation at least

on the matrix equation xa axt 0 ii type 0 i interactions abstract the matrix equation x a a x t 0 was recently introduced by de teran and dopico 2011 3 to study the dimension of congruence orbits they reduced the study of this equation to a number of special cases several of which have not been explicitly solved in this note we obtain an explicit closed form solution in the difficult type 0 i interaction case

the effects of unequal diffusion coefficients on periodic travelling waves in oscillatory reaction diffusion systems abstract many oscillatory biological systems show periodic travelling waves these are often modelled using coupled reaction diffusion equations however the effects of different movement rates diffusion coefficients of the interacting components on the predictions of these equations are largely unknown here we investigate the ways in which varying the diffusion coefficients in such equations alters the wave speed time period wavelength amplitude and stability of periodic wave solutions we focus on two sets of kinetics that are commonly used in ecological applications lambda omega equations which are the normal form of an oscillatory coupled reaction diffusion system close to a supercritical hopf bifurcation and a standard predator prey model our results show that changing the ratio of the diffusion coefficients can significantly alter the shape of the one parameter family of periodic travelling wave solutions the position of the boundary between stable and unstable waves also depends on the ratio of the diffusion coefficients in all cases stability changes through an eckhaus sideband instability these effects are always symmetrical in the two diffusion coefficients for the lambda omega equations but are asymmetric in the predator prey equations especially when the limit cycle of the kinetics is of large amplitude in particular there are two separate regions of stable waves in the travelling wave family for some parameter values in the predator prey scenario our results also show the existence of a one parameter family of travelling waves but not necessarily a hopf bifurcation for all values of the diffusion coefficients simulations of the full partial differential equations reveals that varying the ratio of the diffusion coefficients can significantly change the properties of periodic travelling waves that arise from particular wave generation mechanisms and our analysis of the travelling wave families assists in the understanding of these effects

numerical inversion of a general incomplete elliptic integral we present a numerical method to invert a general incomplete elliptic integral with respect to its argument and or amplitude the method obtains a solution by bisection accelerated by half argument formulas and addition theorems to evaluate the incomplete elliptic integrals and jacobian elliptic functions required in the process if faster execution is desirable at the cost of complexity of the algorithm the sequence of bisection is switched to allow an improvement by using newton s method halley s method or higher order schroder methods in the improvement process the elliptic integrals and functions are computed by using maclaurin series expansion and addition theorems based on the values obtained at the end of the bisection also the derivatives of the elliptic integrals and functions are recursively evaluated from their values by adopting 0 2 as the critical value of the length of the solution interval to shift to the improvement process we suppress the expected number of bisections to be as low as four on average the typical number of applications of update formulas in the double precision environment is three for newton s method and two for halley s method or higher order schroder methods whether the improvement process is added or not our method requires none of the procedures to compute the incomplete elliptic integrals and jacobian elliptic functions but only those to evaluate the complete elliptic integrals once at the beginning as a result it runs fairly quickly in general for example when using the improvement process it is around 2 5 times faster than newton s method using boyd s starter boyd 2012 25 in inverting e f m legendre s incomplete elliptic integral of the second kind

more on regular subgroups of the affine group abstract this paper is a new contribution to the study of regular subgroups of the affine group agl n f for any field f in particular we associate to each partition 1 n 1 of n 1 abelian regular subgroups in such a way that different partitions define non conjugate subgroups moreover we classify the regular subgroups of certain natural types for n 4 our classification is equivalent to the classification of split local algebras of dimension n 1 over f our methods based on classical results of linear algebra are computer free

hermitian matrix polynomials with real eigenvalues of definite type part i classification the spectral properties of hermitian matrix polynomials with real eigenvalues have been extensively studied through classes such as the definite or definitizable pencils definite hyperbolic or quasihyperbolic matrix polynomials and overdamped or gyroscopically stabilized quadratics we give a unified treatment of these and related classes that uses the eigenvalue type or sign characteristic as a common thread equivalent conditions are given for each class in a consistent format we show that these classes form a hierarchy all of which are contained in the new class of quasidefinite matrix polynomials as well as collecting and unifying existing results we make several new contributions we propose a new characterization of hyperbolicity in terms of the distribution of the eigenvalue types on the real line by analyzing their effect on eigenvalue type we show that homogeneous rotations allow results for matrix polynomials with nonsingular or definite leading coefficient to be translated into results with no such requirement on the leading coefficient which is important for treating definite and quasidefinite polynomials we also give a sufficient condition for a quasihyperbolic matrix polynomial to be diagonalizable by structure preserving congruence and show that this condition is always satisfied in the quadratic case and for any hyperbolic matrix polynomial thereby identifying an important new class of diagonalizable matrix polynomials

fast computation of incomplete elliptic integral of first kind by half argument transformation we developed a new method to calculate the incomplete elliptic integral of the first kind f varphi m by using the half argument formulas of jacobian elliptic functions the method reduces the magnitude of varphi by repeated usage of the formulas while fixing m the method is sufficiently precise in the sense that the maximum relative error is 3 5 machine epsilons at most thanks to the simplicity of the half argument formulas the new procedure is significantly faster than the existing procedures for example it runs 20 60 faster than bulirsch function el1 and 1 9 2 2 times faster than the method using carlson s function r f

pattern solutions of the klausmeier model for banded vegetation in semi arid environments iii the transition between homoclinic solutions abstract self organised patterns of vegetation are a characteristic feature of semi deserts on hillsides these typically comprise vegetation bands running parallel to the contours separated by regions of bare ground tiger bush the present study concerns the klausmeier mathematical model for this phenomenon c a klausmeier regular and irregular patterns in semiarid vegetation science 284 1999 1826 1828 which is one of the earliest and most influential of the various theoretical models for banded vegetation the model is a system of reaction diffusion advection equations and after rescaling it contains three dimensionless parameters one of which the slope parameter is much larger than the other two the present study is the third in a series of papers in which the author exploits the large value of the slope parameter to obtain leading order approximations to the parameter regions in which patterns exist and to the form of these patterns the boundary of the parameter region giving patterns consists in part of two loci of homoclinic solutions that are homoclinic to different steady states the present paper concerns behaviour for parameters close to the intersection point of these loci the author shows that this part of parameter space divides naturally into three regions with a different solution structure in each in one region the solution corresponds to a limit cycle of a reduced system of ordinary differential equations the other two regions involve multiple matched layers as part of the analysis the author derives formulae for the homoclinic solution loci and for the location of their intersection all of the results are valid to leading order for large values of the slope parameter the author presents a detailed numerical verification of his analytical results the paper concludes with discussions of the ecological implications of the results and the main outstanding mathematical questions