localization of laplacian eigenfunctions in circular spherical and elliptical domains we consider laplacian eigenfunctions in circular spherical and elliptical domains in order to discuss three kinds of high frequency localization whispering gallery modes bouncing ball modes and focusing modes although the existence of these modes has been known for a class of convex domains the separation of variables for circular spherical and elliptical domains helps us to better understand the mechanism of localization i e how an eigenfunction is getting distributed in a small region of the domain and decays rapidly outside this region using the properties of bessel and mathieu functions we derive inequalities which imply and clearly illustrate localization moreover we provide an example of a nonconvex domain an elliptical annulus for which the high frequency localized modes are still present at the same time we show that there is no localization in most rectangle like domains this observation leads us to formulate an open problem of localization in polygonal domains and more ge

maps on idempotent matrices over division rings abstract let d be an arbitrary division ring and p n d the set of all n n idempotent matrices over d under some mild conditions we give a complete description of maps on p n d that preserve either commutativity or order or orthogonality we give examples showing that our assumptions cannot be relaxed much further as an application we will prove a quaternionic analogue of ovchinnikov s result that is important in quantum mechanics other applications of our theorems include results on automorphisms of operator and matrix semigroups local automorphisms linear preserver problems and geometry of matrices and grassmannians

mesh refinement processes based on the generalized bisection of simplices mesh refinement processes based on the generalized bisection of simplices are discussed and characterized in the context of the general theory of rheinboldt siam j numer anal 17 1980 pp 766 778 then it is proved that such processes allow the construction of sequences of naturally smooth conforming and nested nonuniform meshes in fact it is shown that there exists a generalized mesh refinement operator that for any conforming triangulation delta and for any v subset delta produces a nested smooth conforming triangulation delta containing successors of all elements of v and such that the minimum cell size of delta is bounded from below by half the minimum cell size of delta two conforming mesh refinement algorithms that allow the selective refinement of computational triangulations are explicitly given

selective refinement derefinement algorithms for sequences of nested triangulations the construction of changing sequences of irregular and nested triangulations based on the use of conforming refinement derefinement algorithms for triangulations is presented and discussed this strategy is particularly appropriate to combine adaptivity and full multigrid algorithms for dealing with moving fronts or fluid dynamics problems it is shown that the quality of all the triangulations iteratively generated depends only on the geometric characteristics of the initial grid a data structure suitable to create manage and modify series of nested triangulations as well as the main features of the deref prototype package are described and numerical examples are given

homomorphisms of matrix semigroups over division rings from dimension two to four abstract let d be an arbitrary division ring and m n d the multiplicative semigroup of all n n matrices over d we study non degenerate injective homomorphisms from m 2 d to m 4 d in particular we present a structural result for the case when d is the ring of quaternions

matrix semigroup homomorphisms into higher dimensions abstract we study non degenerate irreducible homomorphisms from the multiplicative semigroup of all n by n matrices over an algebraically closed field of characteristic zero to the semigroup of m by m matrices over the same field we prove that every non degenerate homomorphism from the multiplicative semigroup of all n by n matrices to the semigroup of n 1 by n 1 matrices when n 3 is reducible and that every non degenerate homomorphism from the multiplicative semigroup of all 3 by 3 matrices to the semigroup of 5 by 5 matrices is reducible

more on matrix semigroup homomorphisms abstract we study non degenerate irreducible homomorphisms from the multiplicative semigroup of all 2 by 2 complex matrices to the semigroup of n by n complex matrices if such a homomorphism maps a cyclic unipotent to a cyclic unipotent it is the composition of a symmetric power a field homomorphism used entrywise and a matrix conjugation in the case n 4 we characterise all non degenerate irreducible homomorphisms

using longest side bisection techniques for the automatic refinement of delaunay triangulations in this paper we discuss study and compare two linear algorithms for the triangulation refinement problem the known longest side triangle bisection refinement algorithm as well as a new algorithm that uses longest side bisection techniques for refining delaunay triangulations we show that the automatic point insertion criterion taken from the fractal property of optimal linear longest side bisection algorithms assures the construction of good quality delaunay triangulations in linear time numerical evidence showing that the practical behaviour of the new algorithm is in complete agreement with the theory is included 1997 by john wiley sons ltd

music type electromagnetic imaging of a collection of small three dimensional inclusions in this paper we consider the localization of a collection of small three dimensional bounded homogeneous inclusions via time harmonic electromagnetic means typically using arrays of electric or magnetic dipole transmitters and receivers with given polarization s at some distance from the collection possibly also lying in the far field the inclusions somewhat apart or closely spaced are buried within a homogeneous medium and are of arbitrary contrast of permittivity conductivity and permeability vis ag vis this embedding medium the problem is formulated as an inverse scattering problem for the full maxwell equations and it involves a robust asymptotic modeling of the multistatic response matrix no specific application is studied at this stage but characterization of obstacles in subsoils nondestructive evaluation of man made structures and medical imaging are primary fields of application envisaged the proposed approach uses a music multiple signal classification type algorithm and it yields fast numbering accurate localization and estimates of the electromagnetic and geometric parameters polarization tensors of the inclusions the mathematical machinery is detailed first some specific attention being given to triaxial ellipsoidal inclusions and degenerate spherical shapes for the latter known results are retrieved then the viability of this algorithm which would be easily extended to planarly layered environments by introduction of their green s functions is documented by a variety of numerical results from synthetic noiseless and severely noisy field data

numerical resolution of an electromagnetic inverse medium problem at fixed frequency the aim of this paper is to solve numerically the inverse problem of determining the complex refractive index of an electromagnetic medium from partial boundary field measurements at a fixed frequency the governing equations are the time harmonic maxwell equations formulated in electric field in a two dimensional bounded domain we express the inverse problem as the minimization of a cost function representing the difference between the measured and predicted fields our numerical reconstruction algorithm combines the bfgs method and an iterative process called the adaptive eigenspace inversion the unknown complex coefficient is expanded in terms of eigenfunctions of an elliptic operator both the eigenspace and the mesh are iteratively adapted during the minimization procedure numerical experiments illustrate the performance of the reconstruction for various configurations

mesh generation refinement using fractal concepts and iterated function systems a novel method of mesh generation is proposed which is based on the use of fractal concepts to derive contractive affine transformations the transformations are constructed in such a manner that the attractors of the resulting maps are a union of the points lines and surfaces in the domain in particular the mesh nodes may be generated recursively as a sequence of points which are obtained by applying the transformations to a coarse background mesh constructed from the given boundary data a delaunay triangulation or similar edge connection approach can then be performed on the resulting set of nodes in order to generate the mesh local refinement of an existing mesh can also be performed using the procedure the method is easily extended to three dimensions in which case the delaunay triangulation is replaced by an analogous 3 d tesselation

automorphisms of the poset of skew projections abstract any automorphism of the poset b of all idempotents of the algebra b h of all bounded linear operators in a hilbert space h dim h 3 is shown to be of the form p apa 1 or p ap a 1 with a c where c is the set of all semi linear bijections a h h if h is finite dimensional and the set of all continuous invertible linear or conjugate linear operators in h if h is infinite dimensional further applications of the method are outlined

orthogonality preserving transformations on indefinite inner product spaces generalization of uhlhorn s version of wigner s theorem abstract we present an analogue of uhlhorn s version of wigner s theorem on symmetry transformations for the case of indefinite inner product spaces this significantly generalizes a result of van den broek the proof is based on our main theorem which describes the form of all bijective transformations on the set of all rank one idempotents of a banach space which preserve zero products in both directions

recent developments in inverse acoustic scattering theory we survey some of the highlights of inverse scattering theory as it has developed over the last 15 years with emphasis on uniqueness theorems and reconstruction algorithms for time harmonic acoustic waves included in our presentation are numerical experiments using real data and numerical examples of the use of inverse scattering methods to detect buried objects

inverse problems for the anisotropic maxwell equations we prove that the electromagnetic material parameters are uniquely determined by boundary measurements for the time harmonic maxwell equations in certain anisotropic settings we give a uniqueness result in the inverse problem for maxwell equations on an admissible riemannian manifold and a uniqueness result for maxwell equations in euclidean space with admissible matrix coefficients the proofs are based on a new fourier analytic construction of complex geometrical optics solutions on admissible manifolds and involve a proper notion of uniqueness for such solutions

adjacency preserving maps on matrices and operators we characterize injective continuous maps on the space of real or complex rectangular matrices preserving adjacent pairs of matrices we also extend hua s fundamental theorem of the geometry of rectangular matrices to the infinite dimensional case an application in the theory of local automorphisms is presented

applying projective geometry to transformations on rank one idempotents abstract we present a short proof of molnar s characterization of bijective transformations on the set of all rank one idempotent operators on a banach space which preserve zero products in both directions an improvement in the finite dimensional case is given we apply these results to describe automorphisms of standard operator semigroups and to improve uhlhorn s version of wigner s theorem

maps on matrix spaces it is well known that every automorphism of the full matrix algebra is inner we give a short proof of this statement and discuss several extensions of this theorem including structural results for multiplicative maps on matrix algebras characterizations of monotone and orthogonality preserving maps on idempotent matrices some nonlinear preserver results and some recent theorems concerning geometry of matrices we show that all these topics are closely related and point out the connections with physics and geometry several open problems are posed

endomorphisms of the poset of idempotent matrices abstract let d be a division ring n 3 an integer and p n d the poset of all n n idempotent matrices over d with the partial order defined by p q if p q q p p let t m n d be an invertible matrix and d d an endomorphism d d an anti endomorphism for any p p n d we denote by p p the idempotent matrix obtained from p by applying entrywise the map p n d p n d defined by p t p t 1 p p n d p t t p t 1 p p n d is an injective map preserving order in both directions every such map is called a standard map it has been known before that if d is an eas division ring then every injective order preserving map p n d p n d is either standard or of a very special degenerate form in this paper we use some ideas from geometry of algebraic homogeneous spaces and elementary field theory to give examples showing that the eas assumption is indispensable then we define generalized standard maps and using them we describe the general form of injective order preservers on p n d for an arbitrary division ring d our proof is shorter than the original one for the special case of eas division rings under somewhat stronger assumptions we get two characterizations of standard maps all the results are optimal as shown by counterexamples

on hua s fundamental theorem of the geometry of rectangular matrices abstract we show that hua s fundamental theorem of the geometry of rectangular matrices can be proved without the bijectivity assumption when the underlying field is the field of real numbers we also give a counterexample showing that this generalization is not possible in the complex case

generalized symmetry transformations on quaternionic indefinite inner product spaces an extension of quaternionic version of wigner s theorem uhlhorn s extension of wigner s unitary antiunitary theorem has been recently generalized by molnar to indefinite inner product spaces we present the quaternionic version of this result

analysis and comparison of geometric and algebraic multigrid for convection diffusion equations the discrete convection diffusion equations obtained from streamline diffusion finite element discretization are solved on both uniform meshes and adaptive meshes estimates of error reduction rates for both geometric multigrid gmg and algebraic multigrid amg are established on uniform rectangular meshes for a model problem our analysis shows that gmg with line gauss seidel smoothing and bilinear interpolation converges if h gg epsilon 2 3 and amg with the same smoother converges more rapidly than gmg if the interpolation constant beta in the approximation assumption of amg satisfies beta ll frac h sqrt epsilon alpha hs 1mm mbox where alpha big begin smallmatrix 1 h sqrt varepsilon 2 h ge sqrt varepsilon end smallmatrix on unstructured triangular meshes the performance of gmg and amg both as solvers and as preconditioners for gmres are evaluated numerical results show that gmres with amg preconditioning is a robust and reliable solver on both type of meshes

sharp sufficient conditions for the observation control and stabilization of waves from the boundary for the observation or control of solutions of second order hyperbolic equation in mathbb r t times omega ralston s construction of localized states comm pure appl math 22 1969 pp

an algorithm for adaptive refinement of triangular element meshes a simple algorithm is developed for adaptive and automatic h refinement of two dimensional triangular finite element meshes the algorithm is based on an element refinement ratio that can be calculated from an a posteriori error indicator the element subdivision algorithm is robust and recursive smooth transition between large and small elements is achieved without significant degradation of the aspect ratio of the elements in the mesh several example problems are presented to illustrate the utility of the approach

matrix semigroup homomorphisms from dimension two to three abstract we characterise all non degenerate homomorphisms from the multiplicative semigroup of all 2 2 matrices over an arbitrary field to the semigroup of 3 3 matrices over the same field in the case of a field of real numbers every irreducible non degenerate homomorphism is a conjugation of the symmetric square

adaptive eigenspace for multi parameter inverse scattering problems abstract a nonlinear optimization method is proposed for inverse scattering problems in the frequency domain when the unknown medium is characterized by one or several spatially varying parameters the time harmonic inverse medium problem is formulated as a pde constrained optimization problem and solved by an inexact truncated newton type method combined with frequency stepping instead of a grid based discrete representation each parameter is projected to a separate finite dimensional subspace which is iteratively adapted during the optimization each subspace is spanned by the first few eigenfunctions of a linearized regularization penalty functional chosen a priori the small and slowly increasing finite number of eigenfunctions effectively introduces regularization into the inversion and thus avoids the need for standard tikhonov type regularization numerical results illustrate the accuracy and efficiency of the resulting adaptive eigenspace regularization for single and multi parameter problems including the well known marmousi model from geosciences

fast numerical solution of the electromagnetic medium scattering problem and applications to the inverse problem we propose a method for computing the scattering of a time harmonic electromagnetic wave in a medium with a locally perturbed refractive index we prove that it converges of arbitrarily high order depending on the smoothness of the refractive index and that the time complexity is o n 3log n as an application we solve the corresponding inverse problem using a preconditioned newton method and discuss how to compute the frechet derivative of the solution operator and its adjoint by the proposed method the performance of both the forward and the inverse solver is illustrated in a number of numerical experiments

numerical solution of an inverse medium scattering problem for maxwell s equations at fixed frequency consider a time harmonic electromagnetic plane wave incident on a medium enclosed by a bounded domain in r 3 in this paper well posedness of the variational problem for the direct scattering is examined an energy estimate for the scattered field is obtained on which the born approximation is based a regularized recursive linearization method for the inverse medium scattering which reconstructs the scatterer of an inhomogeneous medium from the boundary measurements of the scattered field is developed the algorithm requires only single frequency data using an initial guess from the born approximation each update is obtained via continuation on the spatial frequency of a two parameter family of plane waves by solving one direct problem and one adjoint problem of the maxwell equation

algorithms for refining triangular grids suitable for adaptive and multigrid techniques two general algorithms for refining triangular computational meshes based on the bisection of triangles by the longest side are presented and discussed the algorithms can be applied globally or locally for selective refinement of any conforming triangulation and always generate a new conforming triangulation after a finite number of interactions even when locally used the algorithms also ensure that all angles in subsequent refined triangulations are greater than or equal to half the smallest angle in the original triangulation the shape regularity of all triangles is maintained and the transition between small and large triangles is smooth in a natural way proofs of the above properties are presented the second algorithm is a simpler improved version of the first which retains most of the properties of the latter the algorithms can be used either for constructing irregular computational meshes or for locally refining any given triangulation in this sense they can be adequately combined with adaptive and or multigrid techniques for solving finite element systems examples of the application of the algorithms are given and two possible generalizations are pointed out

hua s fundamental theorem of the geometry of matrices abstract let m n f be the space of all n n matrices over the field f n 2 two matrices a b m n f are adjacent if rank a b 1 hua s fundamental theorem of the geometry of square matrices characterizes bijective maps on m n f that preserve adjacency in both directions in this paper we treat a long standing open problem whether the result of hua holds true under the weaker assumption of preserving the adjacency in one direction only we answer this question in the affirmative in the case that every nonzero homomorphism f f f is surjective for example the field of real numbers has this property in order to prove this result we have to improve ovchinnikov s characterization of automorphisms of the poset of idempotent matrices

a grid generator based on 4 triangles conforming mesh refinement algorithms a procedure to generate 2d meshes of triangles in a quite general fashion allowing for local and selective mesh refinement is presented and discussed the grid generator is based on the iterative application of 4 triangles conforming mesh refinement algorithms for triangulations which are also introduced in this paper these algorithms are modified versions of those proposed in int j numer methods eng 20 745 756 1984 and they can be used for global refinement of a grid as well as for local refinement the grid generator works in the following way given any initial coarse triangulation which properly defines the geometry of the problem a set of user defined refinement subregions ri i 1 2 n and an associated set of tolerance parameters hi an irregular and conforming final triangulation is automatically constructed in such a way that the diameter of all the triangles contained in ri is smaller than hi i 1 2 n moreover all angles in the final triangulation are greater than or equal to half the smallest angle in the initial coarse one the refinement is propagated only to assure the conformity and smoothness of the grid and consequently the number of involved nodes will be minimized the refinement of the final mesh will be determined by the subregions ri and the parameters ni and will be essentially independent of the initial coarse grid the 4 triangles conforming mesh refinement algorithms are presented and their properties discussed the implementation of these techniques is discussed and examples of the application of the grid generator are given

transformations on the set of all n dimensional subspaces of a hilbert space preserving principal angles wigner s classical theorem on symmetry transformations plays a fundamental role in quantum mechanics it can be formulated for example in the following way every bijective transformation on the set of all 1 dimensional subspaces of a hilbert space h which preserves the angle between the elements of is induced by either a unitary or an antiunitary operator on h the aim of this paper is to extend wigner s result from the 1 dimensional case to the case of n dimensional subspaces of h with n fixed