an accurate and systematic surface wave pole location method for multilayered media finding the surface wave modes in multilayered media is a problem that is of great importance in many full wave numerical electromagnetic solvers in this communication an efficient and systematic method is proposed to locate all of the surface wave poles for general multilayered structures it is observed that the characteristic equations can be cast into a function composed of eri kz i and ti tan kz i di where the ti occur for the substrates that have finite thicknesses the pole location method is carried out in the complex kzm plane where kzm represents the wave number along the z axis for the substrate with the maximum permittivity two sets of critical points are determined in the kzm plane according to the parameters of the structure since there is at most one pole between two consecutive critical points these critical points establish proper limits for the zero search algorithm muller s method can thus be applied to find the pole between the two consecutive critical points if it exists the proposed method is validated by means of several numerical examples

applications of the ashtekar gravity to four dimensional hyperkahler geometry and yang mills instantons the ashtekar jacobson smolin mason newman equations are used to construct the hyperkahler metrics on four dimensional manifolds these equations are closely related to anti self dual yang mills equations of the infinite dimensional gauge lie algebras of all volume preserving vector fields several examples of hyperkahler metrics are presented through the reductions of anti self dual connections for any gauge group anti self dual connections on hyperkahler manifolds are constructed using the solutions of both nahm and laplace equations

second order pdes in four dimensions with half flat conformal structure we study second order partial differential equations pdes in four dimensions for which the conformal structure defined by the characteristic variety of the equation is half flat self dual or ant

efficient determination of the poles and residues of spectral domain multilayered green s functions that are relevant in far field calculations in this work the total least squares algorithm tlsa is applied to the determination of the proper and improper poles of spectral domain multilayered green s functions that are closer to the branch point and to the determination of the residues at these poles the introduction of an adequate transformation in the spectral domain permits that the tlsa provides accurate values of the poles and residues regardless of the proximity of the poles to the branch point it is shown that the poles and residues supplied by the tlsa can be used to write the far field of the spatial domain green s functions in terms of closed form expressions that are reliable in a wide variety of scenarios

partner symmetries and non invariant solutions of four dimensional heavenly equations we extend our method of partner symmetries to the hyperbolic complex monge ampere equation and the second heavenly equation of plebanski we show the existence of partner symmetries and derive the relations between them for certain simple choices of partner symmetries the resulting differential constraints together with the original heavenly equations are transformed to systems of linear equations by an appropriate legendre transformation the solutions of these linear equations are generically non invariant as a consequence we obtain explicitly new classes of heavenly metrics without killing vectors

symmetry reduction for nonlinear relativistically invariant equations symmetry reduction is studied for the relativistically invariant scalar partial differential equation h u u 2 u 0 in n 1 dimensional minkowski space m n 1 the introduction of k symmetry variables 1 k as invariants of a subgroup g of the poincare group p n 1 having generic orbits of codimension k n in m n 1 reduces the equation to a pde in k variables all codimension 1 symmetry variables in m n 1 n arbitrary reducing the equation studied to an ode are found as well as all codimension 2 and 3 variables for the low dimensional cases n 2 3 the type of equation studied includes many cases of physical interest in particular nonlinear klein gordon equations such as the sine gordon equation and hamilton jacobi equations

complex reflection groups braid groups hecke algebras presentations a la coxeter are given for all irreducible nite com plex re ection groups they provide presentations for the corresponding generalized braid groups for all but six cases which allow us to generalize some of the known properties of nite coxeter groups and their associated braid groups such as the computation of the center of the braid group and the construction of deformations of the nite group algebra hecke algebras we introduce monodromy representations of the braid groups which factorize through the hecke algebras extending results of cherednik opdam kohno and others summary introduction complex re ection groups and their presentations a background from complex re ection groups b presentations braid groups and their diagrams a generalities about hyperplane complements b generalities about the braid groups c the braid diagrams proofs of the main theorems for the braid groups b de e r a notation and prerequisites b computation of b de e r and of its center for d c computation of b e e r and of its center hecke algebras a background from di erential equations and monodromy b a family of monodromy representations of the braid group c hecke algebras diagrams and tables appendix generators of the monodromy around a divisor appendix tables to mathematics subject classi cation primary g we thank jean michel peter orlik pierre vogel for useful conversations and the isaac newton institute for its hospitality while the last version of this manuscript was written up the second named author gratefully acknowledges nancial support by the fondation alexander von humboldt for his stays in paris michel brou e gunter malle raphael rouquier

twistor theory of hyper kahler metrics with hidden symmetries we review the hierarchy for the hyper kahler equations and define a notion of symmetry for solutions of this hierarchy a four dimensional hyper kahler metric admits a hidden symmetry if it embeds into a hierarchy with a symmetry it is shown that a hyper kahler metric admits a hidden symmetry if it admits a certain killing spinor we show that if the hidden symmetry is tri holomorphic then this is equivalent to requiring symmetry along a higher time and the hidden symmetry determines a twistor group action as introduced by bielawski twistor quotients of hyper kahler manifolds world scientific river edge nj 2001 this leads to a construction for the solution to the hierarchy in terms of linear equations and variants of the generalized legendre transform for the hyper kahler metric itself given by ivanov and rocek commun math phys 182 291 1996 we show that the ale spaces are examples of hyper kahler metrics admitting three tri holomorphic killing spinors these metrics are in this sense

some integrable hierarchies in 2 1 dimensions and their twistor description the twistor space construction due to mason and sparling for solutions of the nonlinear schrodinger and korteweg devries hierarchies is generalized resulting in families of integrable models in 2 1 dimensions soliton solutions to one such hierarchy associated with the gauge group su 2 are constructed by adapting the methods used to construct instanton and monopole solutions these integrable models have the feature that their solutions depend on a number of arbitrary functions as well as arbitrary constants in contrast to the closely related davey stewartson hierachy

symmetries of the self dual einstein equations i the infinite dimensional symmetry group and its low dimensional subgroups this is the first of two papers in which the authors give a complete classification of symmetry reduced solutions of plebanski s potential equation for self dual einstein spaces in this first part the infinite pseudogroup of symmetries of plebanski s equation is described and the conjugacy classes of all local subgroups of dimensions one two and three over both the real and complex numbers are classified then in the second paper this classification is used to obtain all symmetry reduced solutions

formula for the reflection length of elements in the group g m p n we deduce a closed formula for the reflection length functions on the reflection group g m p n with p m and p m the formula is shown separately in three cases p 1 p m and p 2 m 1 see the proof of theorems 2 1 3 1 and 4 4

hyper kahler hierarchies and their twistor theory a twistor construction of the hierarchy associated with the hyper kahler equations on a metric the anti self dual einstein vacuum equations asdve in four dimensions is given the recursion operator r is constructed and used to build an infinite dimensional symmetry algebra and in particular higher flows for the hyper kahler equations it is shown that r acts on the twistor data by multiplication with a rational function the structures are illustrated by the example of the sparling tod eguchi hansen solution

sdiff 2 and uniqueness of the plebanski equation the group of area preserving diffeomorphisms showed importance in the problems of self dual gravity and integrability theory we discuss how representations of this infinite dimensional lie group can arise in mathematical physics from pure local considerations then using lie algebra extensions and cohomology we derive the second plebanski equation and its geometry we do not use kahler or other additional structures but obtain the equation solely from the geometry of area preserving transformations group we conclude that the plebanski equation is lie remarkable

a new method for locating the poles of green s functions in a lossless or lossy multilayered medium a new method is presented to locate the poles of the green s functions for a general multilayered medium whether lossy or lossless the problem associated with the pole extraction is reduced to solve the contour integrals in the complex plane that are represented in terms of the spectral domain transmission coefficients with the help of cauchy s theorem the proposed method can accurately and rapidly find all surface wave poles with a few contour integrals the numerical examples are performed to show the efficiency of the method

clarification of issues on the closed form green s functions in stratified media the closed form green s functions cfgf derived for the vector and scalar potentials in planar multilayer media have been revisited to clarify some issues and misunderstandings on the derivation of these green s functions in addition the range of validity of these green s functions is assessed with and without explicit evaluation of the surface wave contributions as it is well known the derivation of the cfgf begins with the approximation of the spectral domain green s functions by complex exponentials and continues with applying the sommerfeld identity to cast these approximated spectral domain green s functions into the space domain in closed forms questions and misunderstandings of this derivation which have mainly originated from the approximation process of the spectral domain green s functions in terms of complex exponentials can be categorized and discussed under the topics of 1 branch point contributions 2 surface wave pole contributions and 3 the accuracy of the obtained cfgf when these issues are clarified the region of validity of the cfgf so obtained may be defined better therefore in this paper these issues will be addressed first and then their origins and possible remedies will be provided with solid analysis and numerical demonstrations

an efficient pole extraction technique for the computation of green s functions in stratified media using a sine transformation an efficient and automatic algorithm for the extraction of surface wave poles of the multilayered spectral domain green s function is presented with the sine transformation we have managed to locate any pole no matter how close to the branch point singularity lies something that has not been feasible with techniques proposed till now the aforementioned transformation maps the krho plane to the w plane where the branch point is removed thus enabling the application of a fast numerical contour integration technique proposed which is inaccurate when dealing with poles very close to the branch point

the virasoro and kac moody symmetries for the principal chiral model for the principal chiral model two hierarchies of symmetries and the related virasoro and kac moody algebras are derived by using a recursion operator

self dual gravity is completely integrable we discover a multi hamiltonian structure of a complex monge ampere equation cma set in a real first order 2 component form therefore by magri s theorem this is a completely integrable system in four real dimensions we start with lagrangian and hamiltonian densities and obtain a symplectic form and the hamiltonian operator that determines the dirac bracket we have calculated all point symmetries of the 2 component cma system and hamiltonians of the symmetry flows we have found two new real recursion operators for symmetries which commute with the operator of a symmetry condition on solutions of the cma system these operators form two lax pairs for the 2 component system the recursion operators applied to the first hamiltonian operator generate infinitely many real hamiltonian structures we show how to construct an infinite hierarchy of higher commuting flows together with the corresponding infinite chain of their hamiltonians

bi differential calculi and integrable models the existence of an infinite set of conserved currents in completely integrable classical models including chiral and toda models as well as the kp and self dual yang mills equations is traced back to a simple construction of an infinite chain of closed respectively covariantly constant 1 forms in a gauged bi differential calculus the latter consists of a differential algebra on which two differential maps act in a gauged bi differential calculus these maps are extended to flat covariant derivatives

symplectic structures their backlund transformations and hereditary symmetries abstract it is shown that compatible symplectic structures lead in a natural way to hereditary symmetries we recall that a hereditary symmetry is an operator valued function which immediately yields a hierarchy of evolution equations each having infinitely many commuting symmetries all generated by this hereditary symmetry furthermore this hereditary symmetry usually describes completely the soliton structure and the conservation laws of these equations this result then provide us with a method for constructing hereditary symmetries and hence exactly solvable evolution equations in addition we show how symplectic structures transform under backlund transformations this leads to a method for generating a whole class of symplectic structures from a given one several examples and applications are given illustrating the above results also the connection of our results with those of gelfand and dikii and of magri is briefly pointed out

a connection between the einstein and yang mills equations it is our purpose here to show an unusual relationship between the einstein equations and the yang mills equations we give a correspondence between solutions of the self dual einstein vacuum equations and the self dual yang mills equations with a special choice of gauge group the extension of the argument to the full yang mills equations yields einstein s unifield equations we try to incorporate the full einstein vacuum equations but the approach is incomplete we first consider yang mills theory for an arbitrary lie algebra with the condition that the connection 1 form and curvature are constant on minkowski space this leads to a set of algebraic equations on the connection components we then specialize the lie algebra to be the infinite dimensional lie algebra of a group of diffeomorphisms of some manifold the algebraic equations then become differential equations for four vector fields on the manifold on which the diffeomorphisms act in the self dual case if we choose the connection components from the lie algebra of the volume preserving 4 dimensional diffeomorphism group the resulting equations are the same as those obtained by ashtekar jacobsen and smolin in their remarkable simplification of the self dual einstein vacuum equations an alternative derivation of the same equations begins with the self dual yang mills connection now depending only on the time then choosing the lie algebra as that of the volume preserving 3 dimensional diffeomorphisms when the reduced full yang mills equations are used in the same context we get einstein s equations for his unified theory based on absolute parallelism to incorporate the full einsteinvacuum equations we use as the lie group the semi direct product of the diffeomorphism group of a 4 dimensional manifold with the group of frame rotations of anso 1 3 bundle over the 4 manifold this last approach however yields equations more general than the vacuum equations

congruence classes of presentations for the complex reflection groups g m p n we give an explicit description in terms of rooted graphs for representatives of all the congruence classes of presentations or r c p for brevity for the imprimitive complex reflection group g m p n also we show that s ps forms a presentation of g m p n where s is any generating reflection set of g m p n of minimally possible cardinality and ps is the set of all the basic relations on s

a self dual yang mills hierarchy and its reductions to integrable systems in 1 1 and 2 1 dimensions the self dual yang mills equations play a central role in the study of integrable systems in this paper we develop a formalism for deriving a four dimensional integrable hierarchy of commuting nonlinear flows containing the self dual yang mills flow as the first member we show that upon appropriate reduction and suitable choice of gauge group it produces virtually all well known hierarchies of soliton equations in 1 1 and 2 1 dimensions and can be considered as a universal integrable hierarchy prototypical examples of reductions to classical soliton equations are presented and related issues such as recursion operators symmetries and conservation laws are discussed

the symmetry structure of the anti self dual einstein hierarchy an important example of a multi dimensional integrable system is the anti self dual einstein equations by studying the symmetries of these equations a recursion operator is found and the associated hierarchy constructed owing to the properties of the recursion operator one may construct a hierarchy of symmetries and find the algebra generated by them in addition the lax pair for this hierarchy is constructed

on differential equations characterized by their lie point symmetries we study the geometry of differential equations determined uniquely by their point symmetries that we call lie remarkable we determine necessary and sufficient conditions for a differential equation to be lie remarkable furthermore we see how in some cases lie remarkability is related to the existence of invariant solutions we apply our results to minimal submanifold equations and to monge ampere equations in two independent variables of various orders

an infinite hierarchy of conservation laws and nonlinear superposition principles for self dual einstein spaces self dual einstein spaces are shown to admit an infinite hierarchy of conservation laws and this hierarchy is then used to derive a formal version of penrose s twistor construction the set of formal holomorphic bundles of fiber dimension 2 over the riemann sphere p1 is shown to form a formal infinite group which is used to derive nonlinear superposition principles as an example of our methods a new self dual einstein space is obtained as the result of a collision of complex pp waves traveling in opposite directions

non abelian infinite algebra of generalized symmetries for the sdiff 2 toda equation we determine the non abelian algebra of generalized symmetries for the sdiff 2 toda equation a pde for a single function of three independent variables the solutions of which determine self dual vacuum solutions of the einstein field equations this algebra is a realization of two copies of the abstract algebra sdiff 2 along with an additional pair of elements that have derivation like properties on both of the copies it contains as a subalgebra the doubly infinite abelian algebra equivalent to the infinite hierarchy of higher flows found by takasaki and takebe an infinite prolongation of the jet bundle for the original pde to include all the variables allowed in their hierarchy is required for the presentation of this generalization because these symmetries have non zero commutators they generate a recursion relation allowing the generation and description of the entire algebra

certain imprimitive reflection groups and their generic versions the present paper is concerned with the connection between the imprimitive reflection groups g m m n m n and the affine weyl group a n 1 we show that a n 1 is a generic version of the groups g m m n m n we introduce some new presentations of these groups which are shown to have some group theoretic advantages then we define the hecke algebras of these groups and of their braid versions each in two ways according to two presentations finally we give a new description for the affine root system of a n 1 such that the action of a n 1 on is compatible with that of g m m n on its root system in some sense