a solution procedure for second order difference equations and its application to electromagnetic wave diffraction in a wedge shaped region this paper proposes an efficient solution procedure for second order functional difference equations and outlines this procedure through investigating electromagnetic wave diffraction by a canonical structure comprising an impedance wedge and an impedance sheet bisecting the exterior region of the wedge applying the sommerfeld malyuzhinets technique to the original boundary value problem yields a linear system of equations for the two coupled spectral functions eliminating one spectral function leads to a second order difference equation for the other the chief steps in this work consist of transforming the second order equation into a simpler one by making use of a generalized malyuzhinets function and in expressing the solution to the latter in an integral form with help of the so called s integrals from this integral expression one immediately obtains a fredholm equation of the second kind for points on the imaginary axis of the complex plane solving this integral equation by means of the well known quadrature method enables us to calculate the sought for spectral function inside the basic strip via an interpolation formula and outside it via an analytic extension the second spectral function is obtained through its dependence upon the first the uniform asymptotic solution which is of particular interest in the geometrical theory of diffraction follows by evaluating the sommerfeld integrals in the far field from the exact one several examples demonstrate the efficiency and accuracy of the proposed procedure as well as typical behaviour of the far field solutions for such a canonical problem of diffraction theory

irreducible maps and bilinear forms abstract bautista showed in 1982 that the possible multiplicities of indecomposable summands of the domains and ranges of irreducible maps between modules over artin algebras are given by numerical invariants of certain bilinear forms associated with the algebra we obtain further information about these multiplicities by relating the forms to those studied elsewhere in algebra and geometry one spectacular result is that the allowable multiplicities for some algebras over the field of real numbers depend on j f adams determination of the number of linearly independent vector fields on a sphere

explicit solvability of a class of maliuzhinets equations with matrix coefficients a coupled system of maliuzhinets functional equations for a pair of unknown functions is studied it is shown that the system can be solved in an explicit form for a specific but non trivial class of its matrix coefficients two examples arising in electromagnetic diffraction theory in a wedge shaped region are briefly discussed

diffraction of flexural waves by cracks in orthotropic thin elastic plates i formal solution the problem of flexural wave diffraction by a semi infinite crack in an infinite orthotropic thin plate is considered such models have application to the ultrasonic non destructive inspection of thin components such as aeroplane wings for simplicity the plate is modelled using kirchhoff theory and the crack is chosen to be aligned along one of the principal directions of material orthotropy for incident plane waves an exact analytical expression for the scattered field is derived by means of the wiener hopf technique in this model problem the wiener hopf kernel is scalar and its factorization is expressed in terms of simple definite non singular contour integrals a detailed numerical evaluation of the solution will be provided in the second part of this work

the scattering of sound by two semi infinite parallel staggered plates ii evaluation of the velocity potential for an incident plane wave and an incident duct mode in a recent paper abrahams wickham showed how to exactly solve the diffraction problem for two parallel semi infinite plates that are arranged to form a duct the edges of the plates are misaligned so that the geometry is asymmetric with respect to the central line of the duct in this paper we examine in detail the effect of this asymmetry on the field radiated from the duct when a fundamental travelling mode is excited within it and the scattered field when a plane wave is excited at infinity to evaluate the field in either case it is first necessary to solve the integral equations derived in our paper here we show how this is effected using the expansion theorem also enunciated in that paper this solution is then used to calculate a number of physically interesting quantities such as the variation of the energy capture by the duct when irradiated by an incident plane wave as the stagger is increased and the asymmetric radiation pattern from an incident duct mode the numerical computations are checked using an energy balance argument and finally a simple heuristic approximation is given for the scattering problem that in many situations accurately reproduces the essential physical characteristics of the solution

arbitrarily oriented perfectly conducting wedge over a dielectric half space diffraction and total far field complex scattering targets are often made by structures constituted by wedges and penetrable substrates which may interact at near field in this paper we describe a complete procedure to study this problem with possible developments in radar technologies like gpr antenna development or electromagnetic compatibility tips near substrates the diffraction of an incident plane wave by a perfectly conducting pec wedge over a dielectric half space is studied using generalized wiener hopf equations gwhes and the solution is obtained using analytical and numerical analytical approaches that reduce the wiener hopf wh factorization to integral equations ies the mathematical aspects are described in a unified and consistent theory for angular and layered region problems the proposed procedure is valid for the general case and the paper focuses on e polarization at normal incidence the solutions are given in terms of geometrical uniform theory of diffraction gtd utd diffraction coefficients and total far fields for engineering applications this paper presents several numerical test cases that show the validity of the proposed method

diffraction from the juncture of pressure release and locally reacting half planes solutions to two coupled half plane problems are derived in the first an exact solution to the canonical problem of acoustic plane wave diffraction from a pressure release zero impedance surface coupled to a perfectly rigid infinite impedance surface is given using the wiener hopf method the solution is expressed in terms of an angular spectrum of plane waves in the second the rigid surface is replaced by one characterized by a locally reacting finite impedance here an approximate kernel to the wiener hopf functional equation is used which leads to a complete and readily interpretable solution this solution is expressed in terms of the canonical angular spectrum for the first problem multiplied by a function that depends on material parameters of the locally reacting surface polar plots of the far field amplitude level for the diffracted field are presented

differential calculus for p norms of complex valued vector functions with applications for complex valued n dimensional vector functions t s t supposed to be sufficiently smooth the differentiability properties of the mapping t s t p at every point t t0 r 0 t r t 0 are investigated where p is the usual vector norm in cn resp rn for p 1 moreover formulae for the first three right derivatives d k s t p k 1 2 3 are determined these formulae are applied to vibration problems by computing the best upper bounds on s t p in certain classes of bounds these results cannot be obtained by the methods used so far the systematic use of the differential calculus for vector norms as done here for the first time could lead to major advances also in other branches of mathematics and other sciences

diffraction at skew incidence by an anisotropic impedance wedge in electromagnetism theory a new class of canonical cases an original mathematical approach for the diffraction of an electromagnetic skew incident wave by a wedge allows us to reduce such a problem to new elementary functional equations using this method we present an exact analytical solution with closed form expressions for a class of wedge of any angle with a certain type of anisotropic boundary condition which does not require any field component to vanish on the faces we thereby consider the important case where the principal axes of anisotropy are along and normal to the edge the relative impedance matrix attached to each face being diagonal with its determinant equal to unity

transformation techniques towards the factorization of non rational 2 2 matrix functions abstract for the wiener hopf factorization of 2 2 matrix functions g defined on a closed carleson curve transformations g ugv where u and v are invertible rational 2 2 matrix functions are important in the first part of this paper we establish a classification scheme for 2 2 matrix functions which is based on such transformations we determine invariants under these transformations and describe those matrix functions which can be transformed to triangular or daniele khrapkov form in the second part we consider special rational transformations and study the same problem for instance we consider transformations where u and v are rational matrix functions that are analytic and invertible on an open neighborhood of in the more complicated but for factorization theory important case where u and v are rational matrix functions that are analytic and invertible on an open neighborhood of the closure of the domain inside of or outside of respectively the answer is slightly different

on the kernel of an irreducible map let k be the kernel of an irreducible epimorphism f which is not a simk map if the almost split sequence starting at k has decomposable middle term then k is a simple module this generalises a theorem of krause which requires either the domain or range of f to be indecomposable

general wiener hopf factorization of matrix kernels with exponential phase factors the factorization of wiener hopf matrices with exponentially growing elements has long remained an unsolved problem such matrices often occur in the calculation of scattering from complex canonical geometries abrahams and wickham proc roy soc london ser a pp 131 156 and other problems in mathematical physics here a general method is given for resolving the difficulty and obtaining a single scalar integral equation the solution of which generates the required factors it is proven that this solution has a simple infinite series representation the method is illustrated by reference to three particular examples the first is a model matrix arising from a system of simple delay differential equations and the second occurs in the diffraction of sound by two knife edges finally a heat conduction problem is solved for an infinite wedge with asymmetric mixed boundary conditions on its faces

on the commutative factorization of n n matrix wiener hopf kernels with distinct eigenvalues in this article we present a method for factorizing n x n matrix wiener hopf kernels where n 2 and the factors commute we are motivated by a method posed by jones jones 1984a proc r soc a 393 185 192 to tackle a narrower class of matrix kernels however no matrix of jones form has yet been found to arise in physical wiener hopf models in contrast the technique proposed herein should find broad application to illustrate the approach we consider a 3 x 3 matrix kernel arising in a problem from elastostatics while this kernel is not of jones form we shall show how it can be factorized commutatively we discuss the essential difference between our method and that of jones and explain why our method is a generalization the majority of wiener hopf kernels that occur in canonical diffraction problems are however strictly non commutative for 2x2 matrices abrahams has shown that one can overcome this difficulty using pade approximants to rearrange a non commutative kernel into a partial commutative form an approximate factorization can then be derived by considering the dynamic analogue of antipov s model we show for the first time that abrahams pade approximant method can also be employed within a 3x3 commutative matrix form

diffraction of a normally incident plane wave by an impedance wedge with its exterior bisected by a semi infinite impedance sheet this paper reports an application of a previously proposed procedure to diffraction of a normally incident arbitrarily polarized plane electromagnetic wave by a canonical structure which consists of a wedge with different face impedances and a semi infinite impedance sheet bisecting the exterior of the wedge the use of the sommerfeld malyuzhinets technique converts the original boundary value problem into a system of linear equations for two coupled spectral functions eliminating one of them we get a second order difference equation for the other spectral function from this function and the boundary condition on the upper wedge face we construct an even and in the basic strip regular new spectral function then we transform the second order equation into a simpler one by means of a generalized malyuzhinets function spl chi sub spl phi spl alpha and express the solution to the latter in an integral form with help of the so called s integrals solving a fredholm equation of the second kind for points on the imaginary axis of the complex plane which follows from the integral representation enables one to compute the sought for function the second spectral function is obtained via its dependence upon the first one we present a first order uniform asymptotic solution as well as numerical results

on the solution of two coupled wiener hopf equations vector wiener hopi equations previously solved in closed form by using different techniques involve matrices having a particular structure a general procedure is given to factorize these matrices it leads to an approach which even if sometimes cumbersome is uniform and allows the solution of new problems in addition a new method is proposed which transforms simultaneous wiener hopf equations in the form of fredholm integral equation of the second kind

auslander reiten components containing modules with bounded betti numbers let r be a connected selfinjective artin algebra and m an indecomposable nonprojective r module with bounded betti numbers lying in a regular component of the auslander reiten quiver of r we prove that the auslander reiten sequence ending at m has at most two indecomposable summands in the middle term furthermore we show that the component of the auslander reiten quiver containing m is either a stable tube or of type za we use these results to study modules with eventually constant betti numbers and modules with eventually periodic betti numbers

the wiener hopf technique for impenetrable wedges having arbitrary aperture angle diffraction by impenetrable wedges having arbitrary aperture angle is studied by means of the wiener hopf w h technique a system of functional equations called generalized wiener hopf equations gwhe is obtained only for certain values of the aperture angle are these equations recognizable as standard or classical wiener hopf equations cwhe however in all cases a mapping is found that reduces the gwhe to cwhe it means that the diffraction by an impenetrable wedge always reduces to a standard w h factorization the solution for the diffraction by a wedge with given face impedances the malyuzhinets problem is obtained in closed form by an explicit factorization of the kernel

wiener hopf solution for impenetrable wedges at skew incidence a new wiener hopf approach for the solution of impenetrable wedges at skew incidence is presented mathematical aspects are described in a unified and consistent theory for angular region problems solutions are obtained using analytical and numerical analytical approaches several numerical tests from the scientific literature validate the new technique and new solutions for anisotropic surface impedance wedges are solved at skew incidence the solutions are presented considering the geometrical and uniform theory of diffraction coefficients total fields and possible surface wave contributions

factorization in a torus and riemann hilbert problems abstract a new concept of meromorphic factorization for holder continuous functions defined on a contour that is the pullback of r or the unit circle in a riemann surface of genus 1 is introduced and studied and its relations with holomorphic factorization are discussed it is applied to study and solve some scalar riemann hilbert problems in and vectorial riemann hilbert problems in c including wiener hopf matrix factorization as well as to study some properties of a class of toeplitz operators with 2 2 matrix symbols

a general multisegment artificial neural network architecture for the efficient evaluation of electromagnetic plane wave wedge diffraction a multisegment artificial neural network ann is proposed as an interpolation technique for the evaluation of the electromagnetic field diffracted at the edge of anisotropic impedance wedges under plane wave illumination at oblique incidence multisegmentation is needed as the high frequency wedge diffracted field is characterized by a number of discontinuities at the shadow boundaries of the geometrical optics and surface wave fields the proposed approach is applied as a test case to the problem of an anisotropic impedance right angled wedge illuminated by a skewly incident plane wave some exact analytical solutions valid for specific surface impedance tensors are used to obtain numerical data for the ann training phase as well as to show the interpolation capabilities of the implemented ann nevertheless the proposed ann structure is general and can be trained with data obtained from other available solutions analytical perturbative numerical valid for more general wedge configurations eventually leading to a single software tool encompassing all of them and providing accurate approximations of the wedge diffracted field in a relatively short time comparable to that of a closed form analytical solution

electromagnetic diffraction by arbitrary angle impedance wedges the problem of the diffraction of a plane electromagnetic wave incident at an oblique angle on a wedge of arbitrary angle with general tensor impedance boundary conditions is solved using a semi an

embedding formulae for diffraction by rational wedge and angular geometries embedding is the process of taking the far field directivity from a diffraction problem or problems involving line sources or multipoles placed at a sharp edge and then constructing the far field for the same geometry for more general incidence using only this canonical problem s thus far embedding has been limited to planar parallel scattering surfaces for instance collections of parallel cracks or slits it appeared that there was a fundamental limitation to embedding disallowing its use for angular structures in this article we overcome this limitation and demonstrate the use of embedding upon wedge diffraction problems and upon a simple polygonal shape a limitation is that the final formulae are for rational wedge angles

on the non commutative factorization of wiener hopf kernels of khrapkov type there are a great many problems in mathematical physics and engineering which when modelled mathematically are reduced to wiener hopf functional equations defined in some region of a complex plane in simple models this equation is scalar but in more complicated situations inherent coupling etc can often give rise to coupled systems of equations and so the resulting wiener hopf kernels are of matrix form the crucial step in the solution of any wiener hopf equation is to decompose the kernel which can be a quite general function of the complex transform variable into a product of two factors which are regular in overlapping half planes this can be accomplished explicitly for scalar kernels and may be generalized to a particular class of matrix kernels namely those which permit a commutative decomposition although this class is wide and therefore contains some valuable physical problems it is found that many such kernels yield factorization elements with exponentially large behaviour at infinity in their half planes of analyticity this prevents a later step in the wiener hopf procedure being carried through and therefore introduces a serious limitation to the usefulness of khrapkov s approach previous researchers have offered schemes to eliminate the exponential behaviour of the khrapkov factors but for reasons discussed herein these have problems in their successful implementation this article offers a new procedure for quenching the exponential growth which is based on a method proposed recently by the author for fully coupled kernels which do not have commutative factors the approach is two step first a commutative factorization is obtained which is algebraic at infinity as required but which has spurious branch cuts in the regions of intended analyticity second an approximate procedure is used to eliminate the offending branch points by introduction of pade approximants for a certain scalar function and by inclusion of a regularizing meromorphic matrix for one particular example kernel in which an exact factorization happens to be known the explicit approximate kernel factors thus derived are shown to yield very close agreement to the exact factors the application of this work to problems in the area of fluid structural interaction is discussed

on the solution of wiener hopf problems involving noncommutative matrix kernel decompositions many problems in physics and engineering with semi infinite boundaries or interfaces are exactly solvable by the wiener hopf technique it has been used successfully in a multitude of different disciplines when the wiener hopf functional equation contains a single scalar kernel for complex boundary value problems however the procedure often leads to coupled equations which therefore have a kernel of matrix form the key step in the technique is to decompose the kernel into a product of two functions one analytic in an upper region of a complex transform plane and the other analytic in an overlapping lower half plane this is straightforward for scalar kernels but no method has yet been proposed for general matrices in this article a new procedure is introduced whereby pade approximants are employed to obtain an approximate but explicit noncommutative factorization of a matrix kernel as well as being simple to apply the use of approximants allows the accuracy of the factorization to be increased

an iterative wiener hopf method for triangular matrix functions with exponential factors this paper introduces a new method for constructing approximate solutions to a class of wiener hopf equations this is particularly useful since exact solutions of this class of wiener hopf equations currently cannot be obtained the proposed method could be considered as a generalization of the pole removal technique and schwarzschild s series the criteria for convergence is proved the error in the approximation is explicitly estimated and by a sufficient number of iterations it could be made arbitrarily small typically only a few iterations are required for practical purposes the theory is illustrated by numerical examples that demonstrate the advantages of the proposed procedure this method was motivated by and successfully applied to problems in acoustics

a contribution to the quarter plane problem in diffraction theory abstract the three dimensional analog of sommerfeld s famous half plane problem for a screen 1 x1 0 x2 0 x3 0 is treated by an operator theoretic approach this paper shows that the dirichlet problem is well posed for any complex wave number k r if we look for a solution u h1 r3 of the helmholtz equation k2 u 0 in e 7mid 1 g h 1 2 1 with u 1 g h 1 2 1 an explicit representation formula for the solution is established by use of general wiener hopf operator theory sobolev space properties and hilbert space methods the corresponding neumann problem can be solved by analogy

solving wiener hopf problems without kernel factorization a new approach to solving problems of wiener hopf type is expounded by showing its implementation in two concrete and typical examples from fluid mechanics the new method adapts mathematical ideas underlying the so called unified transform method due to a s fokas and collaborators in recent years the method has the key advantage of avoiding what is usually the most challenging part of the usual wiener hopf approach the factorization of kernel functions into sectionally analytical functions two example boundary value problems involving both harmonic and biharmonic fields are solved in detail the approach leads to fast and accurate schemes for evaluation of the solutions

extension and further development of the differential calculus for matrix norms with applications in this paper the differential calculus for the operator norms p p 1 2 of the fundamental matrix or evolution t e at t 0 of a complex n n matrix a introduced by the author in a former paper is extended to m times continuously differentiable matrix functions t t 0 and developed further for other p norms p 1 1 p results similar to those for t are obtained in addition for this function t formulae for the first two logarithmic derivatives d 1 0 p and d 2 0 p 1 p are obtained as special cases also upper bounds on the discrete evolution t t 0 that is a matrix power function and on the difference or remainder r t t t t 0 are derived the discrete evolution occurs when a step by step method is employed to approximate the exact solution of the initial value problem x t ax t x 0 x0 which here models a vibration problem the results are applied to the computation of the optimal upper bounds on r t r t 2 and r t 2

the gabriel roiter measure for representation finite hereditary algebras abstract assume is a representation finite hereditary algebra over an algebraically closed field k we provide a method to find orthogonal exceptional pairs for an indecomposable module and study the almost split sequences terminating at a gabriel roiter factor module

electromagnetic diffraction of an obliquely incident plane wave by a right angled anisotropic impedance wedge with a perfectly conducting face the diffraction of an arbitrarily polarized electromagnetic plane wave obliquely incident on the edge of a right angled anisotropic impedance wedge with a perfectly conducting face is analyzed the impedance tensor on the loaded face has its principal anisotropy axes along directions parallel and perpendicular to the edge exhibiting arbitrary surface impedance values in these directions the proposed solution procedure applies both to the exterior and the interior right angled wedges the rigorous spectral solution for the field components parallel to the edge is determined through the application of the sommerfeld maliuzhinets technique a uniform asymptotic solution is provided in the framework of the uniform geometrical theory of diffraction utd the diffracted field is expressed in a simple closed form involving ratios of trigonometric functions and the utd transition function samples of numerical results are presented to demonstrate the effectiveness of the asymptotic expressions proposed and to show that they contain as limit cases all previous three dimensional 3 d solutions for the right angled impedance wedge with a perfectly conducting face

commutative wiener hopf factorization of a matrix a class of n x n matrices is given for which commutative wiener hopf factorization is possible and the explicit factors are obtained it is also shown that there are no other matrices with factors which commute if the factors possess distinct eigenvalues

wiener hopf splitting of a 2 x 2 matrix explicit expressions for the wiener hopf factors of a 2 x 2 matrix are determined in terms of a single scalar examples of the application to various matrices are given

on the scattering of sound by two semi infinite parallel staggered plates i explicit matrix wiener hopf factorization this paper discusses the two dimensional scattering of sound waves by two semi infinite rigid parallel plates the plates are staggered so that a line in the plane of the motion passing through both edges is not in general perpendicular to the plane of either plate the problem is formulated as a matrix wiener hopf functional equation which exhibits the difficulty of a kernel containing exponentially growing elements we show how this difficulty may be overcome by constructing an explicit product decomposition of the matrix kernel with both factors having algebraic behaviour at infinity this factorization is written in terms of a single entire auxiliary function that has a simple infinite series representation the wiener hopf equation is solved for arbitrary incident wave fields and we derive an asymptotic expression for the field scattered to infinity the latter includes the possibility of propagating modes in the region between the plates in part ii of this work we will evaluate our solution numerically and obtain some analytical estimates in a number of physically interesting limits

hierarchical divergence conforming bases for edge singularities in quadrilateral cells singular divergence conforming bases have been proposed for the solution of integral equations although they have seen only occasional use in practical applications the existing singular bases are not hierarchical which prevents their use in adaptive p refinement applications in this paper a new family of singular hierarchical basis functions is proposed for quadrilateral cells these functions model the singularities associated with current and charge density at edges and are more convenient for modeling such singularities than triangular bases of the same kind the basis functions are of the additive kind and combine a hierarchical polynomial representation on quadrilaterals with linearly independent singular terms that incorporate general exponents that may be adjusted for the specific wedge angle of interest moreover the added singular basis functions are computed on the fly on the basis of various reported numerical results this paper also illustrates the difficulties the advantages the accuracy and the cost of using such bases in the method of moment solutions of integral equations

scattering by a semi infinite sandwich panel perforated on one side the scattering of sound waves by the edge of a sandwich panel which consists of two thin elastic plates containing a light interior cellular structure is analysed the wiener hopf method is used to examine the interaction effects of a semi infinite panel clamped to a semi infinite rigid screen one major difficulty is that the presence of two different velocity potentials on either side of the plane y 0 results in a matrix wiener hopf equation an approximate solution is given in the asymptotic limit of small values of a parameter tau which accounts for the perforations

wave diffraction by wedges having arbitrary aperture angle abstract the problem of plane wave diffraction by a wedge sector having arbitrary aperture angle has a very long and interesting research background in fact we may recognize significant research on this topic for more than one century despite this fact up to now no clear unified approach was implemented to treat such a problem from a rigourous mathematical way and in a consequent appropriate sobolev space setting in the present paper we are considering the corresponding boundary value problems for the helmholtz equation with complex wave number admitting combinations of dirichlet and neumann boundary conditions the main ideas are based on a convenient combination of potential representation formulas associated with weighted mellin pseudo differential operators in appropriate sobolev spaces and a detailed fredholm analysis thus we prove that the problems have unique solutions with continuous dependence on the data which are represented by the single and double layer potentials where the densities are solutions of derived pseudo differential equations on the half line

numerical solution of scattering problems using a riemann hilbert formulation a fast and accurate numerical method for the solution of scalar and matrix wiener hopf wh problems is presented the wh problems are formulated as riemann hilbert problems on the real line and a numerical approach developed for these problems is used it is shown that the known far field behaviour of the solutions can be exploited to construct numerical schemes providing spectrally accurate results a number of scalar and matrix wh problems that generalize the classical sommerfeld problem of diffraction of plane waves by a semi infinite plane are solved using the approach

an asymptotic method of factorization of a class of matrix functions a novel method of asymptotic factorization of nn matrix functions is proposed the considered class of matrices is motivated by certain problems originated from the elasticity theory an example is

lyapunov type operators for numerical solutions of pdes in the present paper numerical methods are developed to approximate the solutions of some evolutionary nonlinear problems the continuous problems are transformed into some lyapunov type equations and then analysed for existence uniqueness convergence stability and error estimates the main idea consists in applying fourier analysis and von neumann criterion acting translation and scaling parameter methods to obtain contractive operators leading to fixed point theory

asymmetric channel divider in stokes flow this article examines the classic problem of stokes flow into a divided channel with in contrast to previous literature the divider barrier asymmetrically placed with respect to the moving parallel channel walls the boundary value problem is reduced to a wiener hopf equation that is of matrix form and of a class for which no exact solution is known an explicit approximate solution in general accurate to any specified degree is obtained by a recent method which employs pade approximants numerical results exhibit the flows due to moving walls or various combinations of downstream pressure gradients

the effective boundary conditions for a perforated elastic sandwich panel in a compressible fluid the transmission of sound through a sandwich panel consisting of a honeycomb cellular structure bounded by two thin elastic plates is considerably reduced at a certain frequency if one or both of the plates is perforated so as to link a significant number of the cells to the exterior fluid this effect occurs at or near the helmholtz resonance frequency for the cells with the acoustic wavelength large compared with the cell dimensions an analysis is given for the problem of plane waves incident upon a plane sandwich plate of infinite extent using matched expansions for the cases of acoustically hard or acoustically transparent cell walls the compound panel is shown to be acoustically equivalent to that of a hypothetical surface with different normal velocities on either side and effective boundary conditions are derived with generalizations to deal with more complicated structures finite plates and more general incident fields

exact solution to diffraction problem by wedges with a class of anisotropic impedance faces oblique incidence of a plane electromagnetic wave this paper studies diffraction of an obliquely incident arbitrarily polarized plane electromagnetic wave by an anisotropic impedance wedge with an opening angle 2 spl phi between 0 and 2 spl pi and presents a closed form exact solution to a class of impedance wedge faces and the related uniform asymptotic solution uas on use of a unitary similarity transform the boundary conditions on the wedge faces is brought into a form which makes the exactly soluble class of impedance faces evident the exact solution is found with help of the sommerfeld malyuzhinets 1896 1958 technique a generalized malyuzhinets function spl chi sub spl phi and the so called s integrals a standard procedure yields therefrom the uas the exact solution agrees with known analytical results in special cases and the numerical results of uas are confirmed by that of parabolic equation method pem

mode conversion corner singularities and matrix wiener hopf factorization in diffraction theory the diffraction of acoustic electromagnetic and elastic waves by an edge is a subject of continuing mathematical and physical importance the physical motivation for research is the vast number of technological and geophysical applications of the fundamental models of linear wave propagation in continuous media while the mathematical drive is to understand and devise new analytical and numerical techniques for their quantification this paper is concerned with certain physical aspects of diffraction which have proven to give the most troublesome of mathematical difficulties these are mode conversion that is when the incident wave is scattered among different modes supported by the propagating medium and edge or corner singularities when the diffracting obstacle creates a stress singularity it is shown by way of a particular example that the exact solution of diffraction problems involving these phenomena will in general give rise to a vector wiener hopf equation with a matrix kernel whose factorization is non commutative the main result of this paper is a new procedure motivated by a precise treatment of both the corner singularity and mode conversion for determining the factors in this case

two dimensional diffraction by a wedge with impedance boundary conditions the two dimensional problem of diffraction by a wedge with impedance boundary conditions on its faces is explicitly solved in a form that admits effective numerical simulation by simple perfectly scalable algorithms with unlimited capability for parallel processing the solution is represented as a superposition of the geometric field that is completely determined by elementary ray analysis and of the waves diffracted by the tip of the wedge the diffracted field is explicitly represented as a mathematical expectation of a specified functional on trajectories of a random motion determined by the configurations of the problem and by the boundary conditions the numerical results confirm the efficiency of this approach

stability analysis of matrix wiener hopf factorization of daniele khrapkov class and reliable approximate factorization this paper presents new stability results for matrix wiener hopf factorization the first part of the paper examines conditions for stability of wiener hopf factorization in the daniele khrapkov class the second part of the paper concerns the class of matrix functions which can be exactly or approximately reduced to the factorization of the daniele khrapkov matrices the results of the paper are demonstrated by numerical examples with partial indices 1 1 0 0 and 1 1

mixed boundary value problems of the type of sommerfeld s half plane problem on etudie des modifications du probleme de demi plan de sommerfeld gouverne par deux equations elliptiques propres dans des demi espaces complementaires de r 3 et on pose des conditions aux limites et de transmission sur les deux demi plans qui ensemble forment la frontiere commune de

generalised factorisation for a class of jones form matrix functions a systematic approach is proposed for the generalised factorisation of certain non rational n n matrix functions the first main result consists in a transformation of a meromorphic into a generalised factorisation by algebraic means it closes a gap between the classical wiener hopf procedure and the operator theoretic method of generalised factorisation secondly as examples we consider certain matrix functions of jones form or of n part form which are equivalent to each other in a sense the factorisation procedure is complete and explicit based only on the factorisation of scalar functions of rational matrix functions and upon linear algebra applications in elastodynamic diffraction theory are treated in detail and in a most effective way

numerical study of diffraction and slope diffraction at anisotropic impedance wedges by the method of parabolic equation space waves the method of parabolic equation pe has been successfully applied to the numerical determination of diffraction slope diffraction and multiple diffraction coefficients of scalar impedance wedges illuminated by a line source as a continuation this paper studies for the first time to the authors knowledge another important canonical problem for the uniform geometrical theory of diffraction utd namely diffraction and slope diffraction of an incident cylindrical wave at wedges with anisotropic impedance surfaces by using the same method for the diffracted fields the exact helmholtz equation is asymptotically approximated by the corresponding parabolic one it is proved that the sufficient conditions for the unique solution of the helmholtz equation also guarantee the uniqueness of the solution of the parabolic one the latter is then efficiently solved by using crank nicholson finite difference fd scheme due to the lack of exact solutions the pe results were compared to uniform asymptotic theory of diffraction uat ones for weak anisotropy and in this case very good agreement has been achieved the diffraction and slope diffraction behavior dependent upon the measure of the weakness of the anisotropy has been demonstrated by several examples

differential calculus for some p norms of the fundamental matrix with applications abstract for the fundamental matrix t e a t of a complex n n matrix a the differential properties of the mapping t t p at every point t t 0 r 0 t r t 0 are investigated where p is the matrix operator norm associated with the vector norm p in c n or r n as the case may be for p 1 2 moreover formulae for the first two right derivatives d k t p k 1 2 are calculated and applied to determine the best upper bounds on t p in certain classes of bounds these results cannot be obtained by the methods used so far the systematic use of the differential calculus for norms as done here for the first time could lead to major advances also in other branches of mathematics and of other sciences notably in engineering for example in the simulation of dynamic problems with excitation

definite triples of hermitian matrices and matrix polynomials let a b and c be three n n nonzero hermitian matrices the triple a b c is called defnite if the convex hull of the joint numerical range f a b c x ax x bx x cx r3 x cn x x 1 does not contain 0 0 0 if the triple a b c is nondefinite then the numerical ranges of the matrix polynomials q a v3 b v2 c v1 v3 v2 v1 0 and l a 2 b ic 1 2 1 0 coincide with the whole complex plane providing no information as a consequence it is of particular interest to characterize a definite triple a b c and find the distance between 0 0 0 and the boundary f a b c the distance between a nondefinite triple a b c and the nearest definite triples with specified properties is also investigated moreover applications of definite triples on matrix polynomials of special interest are presented

a novel technique for the solution of second order difference equations the class of second order difference equations that arise in the diffraction of a plane wave at skew incidence on an impedance wedge is investigated for a typical equation which was solved by the introduction of elliptic integrals it is shown that the solution can be obtained in a very trivial manner from a convolution type integral equation and the results are in complete agreement with those previously found

on the vibration suppression property and monotonicity behavior of a special weighted norm for dynamical systems x ax x t 0 x 0 in an earlier paper the author formed a positive definite matrix r as the sum of positive semi definite matrices that are eigenmatrices of a matrix eigenproblem associated with the lyapunov matrix equation this positive definite matrix r was then used to define the weighted norm email protected r in c n which led to new two sided bounds on the solution of the initial value problem email protected ax x t 0 x 0 in any vector norm email protected in the present paper the quantity x t r itself is analyzed showing that the solution x t in the weighted norm x t r suppresses the vibration behavior inherent to x t this new sight at the norm x t r may be of considerable practical use in technical dynamical systems as a direct important application in engineering the quantity x t r may be used as a measure to assess the damping behavior of the studied free dynamical system since the vibratory part is absent moreover it is shown that x t r is monotonically decreasing for sufficiently large t if matrix a is asymptotically stable thus as a further application a two sided estimate of the form c 0 d email protected x t r email protected t r

wiener hopf factorization for a group of exponentials of nilpotent matrices abstract a complete study of the generalized factorization for a group of 2 2 matrix functions of the form g i n where c r i denotes the 2 2 identity matrix and n represents a rational nilpotent matrix function is presented a closely related class involving the same matrix n is also studied the canonical and non canonical factorizations are considered and explicit formulas are obtained for the partial indices and the factors in such factorizations it is shown in particular that only one of the columns in the factors needs to be determined as a solution to a homogeneous linear riemann hilbert problem the other column being expressed in terms of the first necessary and sufficient conditions for existence of a canonical factorization within the same class are established as well as explicit formulas for the factors in this case

solution of a second order difference equation using the bilinear relations of riemann a recently proposed technique to solve a class of second order functional difference equations arising in electromagnetic diffraction theory is further investigated by applying it to a case of intermediate complexity the proposed approach is conceptually simple and relies on first obtaining well defined branched solutions to a pair of associated first order difference equations the construction of these branched expressions leads to an equation system whose solution requires relationships akin to riemann s bilinear relations for differentials of the first and third kinds their derivation necessitates the application of cauchy s theorem on riemann surfaces of in this particular instance genera one and three branch free solutions of the second order difference equation are then obtained by taking appropriate linear combinations of the branched solutions of the first order equations analysis and computation demonstrate that the resulting expressions have the desired analytical properties and recover known solutions in the appropriate limit

a factorisation procedure for two by two matrix functions on the circle with two rationally independent entries the aim of this paper is the explicit canonical or standard factorisation of matrix functions with wiener algebra elements the present approach covers all regular 2 2 matrices where two entries are arbitrary and the remaining two are linear combinations of the former with rational coefficient functions it is based on the knowledge of how to factorise scalar functions and rational matrix functions in general one also needs the approximation of any scalar wiener algebra function with a rational function however this can be easily circumvented in many applications by intuitive manipulations with rational matrix functions

a utd solution for the scattering by a wedge with anisotropic impedance faces skew incidence case asymptotic expressions for the fields scattered by an anisotropic impedance wedge at oblique incidence are derived in the context of the uniform geometrical theory of diffraction utd they are obtained by resorting to a perturbative approach considering the normal incidence case as the imperturbed configuration we observe that the limits of applicability of this approximate analytical solution extend far beyond those of standard perturbative approaches allowing us to account for deviations from the normal incidence case of 20 spl deg to 30 spl deg

solution of the matrix eigenvalue problem va a v µv with applications to the study of free linear dynamical systems the new idea is to study the stability behavior of the solution x x t of the initial value problem email protected ax t t 0 x t 0 x 0 with email protected c n x n in a weighted semi norm email protected r where r is taken as an appropriate solution of the matrix eigenvalue problemra a email protected rather than as the solution of the algebraic lyapunov matrix equationra a r s with given positive semi definite matrix s substantially better results are obtained by the new method for example if a is diagonalizable and all eigenvalues l i a have negative real parts i e email protected i a 0 t which is much more than the old result which only states that lim t x t 0 especially the semi norms email protected r i have a decoupling and filter effect on x t further new two sided bounds depending on x 0 for the asymptotic behavior can be derived the best constants in the bounds are obtained by the differential calculus of norms applications are made to free linear dynamical systems and computations underpin the theoretical findings

line source and point source scattering of acoustic waves by the junction of transmissive and soft hard half planes abstract firstly the analysis of a buyukaksoy g cinar a h serbest scattering of plane waves by the junction of transmissive and soft hard half planes zamp 55 2004 483 499 for the scattering of plane waves by the junction of transmissive and soft hard half planes is extended to the case of a line source the introduction of the line source changes the incident field and the method of solution requires a careful analysis in calculating the scattered field the graphical results are presented using mathematica we observe that the graphs of the plane wave situation a buyukaksoy g cinar a h serbest scattering of plane waves by the junction of transmissive and soft hard half planes zamp 55 2004 483 499 can be recovered by shifting the line source to a large distance subsequently the problem is further extended to the case of scattering due to a point source using the results obtained for a line source excitation the introduction of a point source three dimensions involves another variable which then requires the calculation of an additional integral appearing in the inverse transform

the wiener hopf solution of the isotropic penetrable wedge problem diffraction and total field the diffraction of an incident plane wave by an isotropic penetrable wedge is studied using generalized wiener hopf equations and the solution is obtained using analytical and numerical analytical approaches that reduce the wiener hopf factorization to fredholm integral equations of second kind mathematical aspects are described in a unified and consistent theory for angular region problems the formulation is presented in the general case of skew incidence and several numerical tests at normal incidence are reported to validate the new technique the solutions consider engineering applications in terms of gtd utd diffraction coefficients and total fields