structured perturbation of the brunovsky form a particular case we study the variation of the feedback invariants of a complex rectangular n x n m matrix when we make small additive perturbations on the last m columns in the particular case when the square n x n submatrix is similar to a block diagonal matrix with two blocks in the diagonal associated with the controllable part and the noncontrollable part of the pair a b respectively

application of a version of the rayleigh technique to problems of bars beams columns membranes and plates abstract a review is presented together with some new material of the application of a little known version of the rayleigh technique to a variety of problems in solid and structural mechanics this method was originally applied by rayleigh in 1894 to the one dimensional problem of determining the fundamental frequency of a stretched string undergoing small amplitude vibrations the essence of the method is that instead of using a specific trial function with an undetermined coefficient for the deflection as in the ordinary rayleigh method here one uses a function containing a power law term with an undetermined exponent application of the usual variational procedure for the quantity of interest such as the buckling load fundamental frequency or deflection results in a dimensionless quantity which is a function only of the undetermined exponent finally the best exponent is the one which minimizes the lagrangian and thus the buckling load natural frequency or reciprocal of deflection in the past five years the non integer exponent idea has been applied to elastic torsion buckling of columns and plates vibration of bars beams columns membranes and plates and deflection of beams and plates also it has been incorporated into the finite element method all of these applications are discussed and some additional original work of the present investigator is presented

sylvester matrix equation for matrix pencils abstract we study a homogeneous linear matrix equation system related to the strict equivalence of matrix pencils we obtain the dimension of the vector space of its solutions in terms of the invariants of the strict equivalence we give a characterization of the strict equivalence of matrix pencils by rank tests and we extend roth s criterion for the corresponding nonhomogeneous system

an example of explicit implementation strategy and preconditioning for the high order edge finite elements applied to the time harmonic maxwell s equations abstract in this paper we focus on high order finite element approximations of the electric field combined with suitable preconditioners to solve the time harmonic maxwell s equations in waveguide configurations the implementation of high order curl conforming finite elements is quite delicate especially in the three dimensional case here we explicitly describe an implementation strategy which has been embedded in the open source finite element software freefem http www freefem org ff in particular we use the inverse of a generalized vandermonde matrix to build basis functions in duality with the degrees of freedom resulting in an easy to use but powerful interpolation operator we carefully address the problem of applying the same vandermonde matrix to possibly differently oriented tetrahedra of the mesh over the computational domain we investigate the preconditioning for maxwell s equations in the time harmonic regime which is an underdeveloped issue in the literature particularly for high order discretizations in the numerical experiments we study the effect of varying several parameters on the spectrum of the matrix preconditioned with overlapping schwarz methods both for 2d and 3d waveguide configurations

the change of similarity invariants under row perturbations the problem of the possible similarity classes of all the matrices obtained by small perturbations on some rows of a given complex square matrix is under consideration some necessary conditions on these similarity classes are provided the near converse problem is also studied under what conditions can a given collection of polynomials or partitions be the invariant factors or the weyr characteristic of a matrix obtained by perturbing some rows of a given matrix a complete answer is provided in some particular cases the obtained conditions can all be derived from the previously mentioned necessary conditions

generic complete eigenstructures for sets of matrix polynomials with bounded rank and degree abstract the set pol d r m n of m n complex matrix polynomials of grade d and normal rank at most r in a complex d 1 m n dimensional space is studied for r 1 min m n 1 we show that pol d r m n is the union of the closures of the r d 1 sets of matrix polynomials with rank r degree exactly d and explicitly described complete eigenstructures in addition for the full rank rectangular polynomials i e r min m n and m n we show that pol d r m n coincides with the closure of a single set of the polynomials with rank r degree exactly d and the described complete eigenstructure these complete eigenstructures correspond to generic m n matrix polynomials of grade d and rank at most r

hierarchal vector basis functions of arbitrary order for triangular and tetrahedral finite elements new vector finite elements are proposed for electromagnetics the new elements are triangular or tetrahedral edge elements tangential vector elements of arbitrary polynomial order they are hierarchal so that different orders can be used together in the same mesh and p adaption is possible they provide separate representation of the gradient and rotational parts of the vector field explicit formulas are presented for generating the basis functions to arbitrary order the basis functions can be used directly or after a further stage of partial orthogonalization to improve the matrix conditioning matrix assembly for the frequency domain curl curl equation is conveniently carried out by means of universal matrices application of the new elements to the solution of a parallel plate waveguide problem demonstrates the expected convergence rate of the phase of the reflection coefficient for tetrahedral elements to order 4 in particular the full order elements have only the same asymptotic convergence rate as elements with a reduced gradient space such as the whitney element however further tests reveal that the optimum balance of the gradient and rotational components is problem dependent

stratification of full rank polynomial matrices we show that perturbations of polynomial matrices of full normal rank can be analyzed viathe study of perturbations of companion form linearizations of such polynomial matrices it is proved that a

hierarchic finite element bases on unstructured tetrahedral meshes the problem of constructing hierarchic bases for finite element discretization of the spaces h1 h curl h div and l2 on tetrahedral elements is addressed a simple and efficient approach to ensuring conformity of the approximations across element interfaces is described hierarchic bases of arbitrary polynomial order are presented it is shown how these may be used to construct finite element approximations of arbitrary non uniform local order approximation on unstructured meshes of curvilinear tetrahedral elements

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

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

canonical form associated with the problem of perturbation of one column of a controllable pair abstract let a b be a completely controllable matrix pair when we consider the problem of characterizing the controllability indices of all the matrix pairs obtained by small perturbations on one column of b a new equivalence relation arises in a natural way this equivalence relation is a kind of partial feedback equivalence as a consequence the controllability indices are invariant but they do not form a complete system of invariants we have found new invariants for this equivalence relation and we have obtained a canonical form in terms of these invariants

higher order interpolatory vector bases for computational electromagnetics low order vector basis functions compatible with the nedelec 1980 representations are widely used for electromagnetic field problems higher order functions are receiving wider application but their development is hampered by the complex procedures used to generate them and lack of a consistent notation for both elements and bases in this paper fully interpolatory higher order vector basis functions of the nedelec type are defined in a unified and consistent manner for the most common element shapes it is shown that these functions can be obtained as the product of zeroth order nedelec representations and interpolatory polynomials with specially arranged arrays of interpolation points the completeness properties of the vector functions are discussed and expressions for the vector functions of arbitrary polynomial order are presented sample numerical results confirm the faster convergence of the higher order functions

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

a note on transverse vibrations of a rectangular plate with a free rectangular corner cut out abstract an approximate solution to the title problem is presented obtained by using the rayleigh ritz method the analysis is presented for the case of simply supported and clamped plates for the case of a rigidly clamped plate results are presented of numerical experiments on minimizing the calculated value of the fundamental frequency coefficient by using schmidt s approach an experimental investigation is described on a clamped square plate with a free square corner cut out which has led to the conclusion that the fundamental frequency coeficient remains practically invariant with respect to size when compared with the frequency coefficient of the fully clamped plate a similar conclusion is arrived at by means of the mathematical model the problem under consideration is important from a practical viewpoint since cut outs of the type considered here are quite common in engineering practice

perturbation theory of matrix pencils through miniversal deformations abstract we give new proofs of several known results about perturbations of matrix pencils in particular we give a direct and constructive proof of andrzej pokrzywa s theorem 1983 in which the closure of the orbit of each kronecker canonical matrix pencil is described in terms of inequalities for invariants of matrix pencils a more abstract description is given by klaus bongartz 1996 by methods of representation theory we formulate and prove pokrzywa s theorem in terms of successive replacements of direct summands in a kronecker canonical pencil first we show that it is sufficient to prove pokrzywa s theorem in two cases for matrices under similarity and for each matrix pencil p q that is a direct sum of two indecomposable pencils then we calculate the kronecker canonical form of pencils that are close to p q in fact the kronecker canonical form is calculated for only those pencils that belong to a miniversal deformation of p q this is sufficient since all pencils in a neighborhood of p q are reduced to them by a smooth strict equivalence transformation

nedelec spaces in affine coordinates in this note we provide a conveniently implementable basis for simplicial nedelec spaces of any order in any space dimension the main feature of the basis is that it is expressed solely in terms of the barycentric coordinates of the simplex

effect of geometric boundary perturbations on the natural frequencies and buckling loads of vibrating clamped circular plates abstract use is made of conformal mapping and the galerkin method coupled with rayleigh s optimization procedure to determine the fundamental frequency of vibration and the buckling coefficient of clamped circular plates with geometric boundary perturbations

second order nedelec curl conforming prismatic element for computational electromagnetics a systematic approach to obtaining mixed order curl conforming basis functions for a triangular prism is presented focus is made on the second order case space of functions for the prism is given basis functions are obtained as the dual basis with respect to suitably discretized nedelec degrees of freedom functionals acting on elements of the space thus the linear independence of the basis functions is assured while the belonging of the basis to the a priori given space of functions is guaranteed different strategies for the finite element assembly of the basis are discussed numerical results showing the verification procedure of the correctness of the implemented basis functions are given numerical results about sensibility of the condition number of the basis obtained concerning the quality of the elements of the mesh are also shown comparison with other representative sets of basis functions for prisms is included

generation of higher order polynomial basis of nedelec h curl finite elements for maxwell s equations the goal of this study is the automatic construction of a vectorial polynomial basis for nedelec mixed finite elements j c nedelec 1980 in particular the generation of finite elements without the expression of the polynomial basis functions using symbolic calculus the exhibition of basis functions has no practical interest

on perturbations and the equivalence orbit of a matrix pencil abstract two theorems on the canonical kronecker form of a perturbed matrix pencil and the characterization of the closure of the set of all matrix pencils with a fixed kronecker canonical form are given

an explicit description of the irreducible components of the set of matrix pencils with bounded normal rank abstract the set of m n singular matrix pencils with normal rank at most r is an algebraic set with r 1 irreducible components these components are the closure of the orbits under strict equivalence of r 1 matrix pencils which are in kronecker canonical form in this paper we provide a new explicit description of each of these irreducible components which is a parametrization of each component therefore one can explicitly construct any pencil in each of these components the new description of each of these irreducible components consists of the sum of r rank 1 matrix pencils namely a column polynomial vector of degree at most 1 times a row polynomial vector of degree at most 1 where we impose one of these two vectors to have degree zero the number of row vectors with zero degree determines each irreducible component

the algebraic structure of pencils and block toeplitz matrices abstract we restate several results on the kronecker structure obtainable after a given matrix pencil is affected by an arbitrarily small perturbation and state some new results on the kronecker structure after a new row is appended or a column deleted the results are simple consequences of a theory of majorization for semi infinite integer sequences the bounds after a new row is appended correspond to bounding the controllability indices after adding a single input to a linear time invariant dynamical system we illustrate the known relations between the kronecker indices the segre characteristics and the nullities of certain block toeplitz matrices constructed from the original pencils and discuss how those nullities change when the pencil is affected by a perturbation or a new row

non linear optimization of the shape function when solving the 2 d helmholtz equation by means of the finite element method abstract lord rayleigh suggested in 1984 in connection with his now classical principle the use of trial functions containing an undetermined exponential parameter k to be optimized by minimization of the eigenvalues with respect to itself in the past five years several authors have basically used rayleigh s optimization concept to solve a variety of problems of elastic stability vibrations heat conduction etc and more recently implemented it in finite element formulations the present paper deals with an extension of a previous study performed by the authors named the k optimization of the shape functions in the fem to 2 d eigenproblems governed by the helmholtz differential equation it is concluded that the method leads to considerable economies both in computer memory requirements and in cpu time three node triangular and four node quadrilateral elements are investigated in the present study

vibrations of plates of arbitrary shape by a modified galerkin method this paper deals with a modification of galerkin s method which allows for the optimization of the fundamental eigenvalue when a single co ordinate function is used use of the conformal mapping method allows for the consideration of two dimensional domains of complicated boundary shape

incompressible finite elements via hybridization part ii the stokes system in three space dimensions in this paper we introduce a new and efficient way to compute exactly divergence free velocity approximations for the stokes equations in two space dimensions we begin by considering a mixed method that provides an exactly divergence free approximation of the velocity and a continuous approximation of the vorticity we then rewrite this method solely in terms of the tangential fluid velocity and the pressure on mesh edges by means of a new hybridization technique this novel formulation bypasses the difficult task of constructing an exactly divergence free basis for velocity approximations moreover the discrete system resulting from our method has fewer degrees of freedom than the original mixed method since the pressure and the tangential velocity variables are defined just on the mesh edges once these variables are computed the velocity approximation satisfying the incompressibility condition exactly as well as the continuous numerical approximation of the vorticity can at once be obtained locally moreover a discontinuous numerical approximation of the pressure within elements can also be obtained locally we show how to compute the matrix system for our tangential velocity pressure formulation on general meshes and present in full detail such computations for the lowest order case of our method

similarity and block similarity we present some equivalent characterizations of the block similarity of matrix pairs with an application to a characterization of the similarity of two square matrices also a survey of some generalizations of properties of the invariants of similarity to the invariants of block similarity is presented

minimal singularities in orbit closures of matrix pencils abstract we show that all singularities occurring in minimal degenerations of matrix pencils are cohen macaulay and regular in codimension 1