on the limitations of the time domain impedance boundary condition limitations of the time domain leontovich impedance boundary condition are discussed for a canonical problem consisting of a planar interface with a conducting half space the exact analytical solution obtained using an extension of the cagniard dehoop technique is presented and compared with the approximate one found via the impedance boundary concept ranges of applicability of the latter are introduced and numerically tested on illustrative examples a proper understanding of these limitations may prove to be crucial in the emerging field of time domain integral equation formulations in planar stratified media

low dimensional real lie algebras all real lie algebras of dimension up to 8 that admit a nontrivial levi decomposition are found

on semi invariants and index for biparabolic seaweed algebras ii abstract in a continuation of i we study the semi invariant subalgebra sy q of s q for q a biparabolic subalgebra of a semisimple or of an affine lie algebra in most cases this is shown to be polynomial and we compute the degrees and weights of the generators we describe explicitly the canonical truncation q e of q for which sy q s q e q e let t denote the nilradical of q and q its derived algebra we show that the subalgebra s t q of s q q sy q is always polynomial and describe the degrees and weight of the generators finally a sum rule is given for the false degrees of s q q these are true degrees in most cases

windowed green function method for layered media scattering this paper introduces a new windowed green function wgf method for the numerical integral equation solution of problems of electromagnetic scattering by obstacles in the presence of dielectric or conducting half planes the wgf method which is based on the use of smooth windowing functions and integral kernels that can be expressed directly in terms of the free space green function does not require evaluation of expensive sommerfeld integrals the proposed approach is fast accurate flexible and easy to implement in particular straightforward modifications of existing accelerated or unaccelerated integral equation solvers suffice to incorporate the wgf capability the method relies on a certain integral equation posed on the union of the boundary of the obstacle and a small flat section of the interface between the penetrable media our analysis and numerical experiments demonstrate that both the near and far field errors resulting from the proposed approach decrease faster than any negative power of the window size in the examples considered in this paper the proposed method is up to thousands of times faster for a given accuracy than a corresponding method based on use of sommerfeld integrals

primitive localizations of an enveloping algebra abstract we prove a number of sufficient conditions for a localization of an enveloping algebra at a multiplicatively closed set of semi invariants to be a primitive ring in case of a central localization these conditions are also necessary

on symmetric invariants of centralisers in reductive lie algebras let g be a finite dimensional simple lie algebra of rank l over an algebraically closed field of characteristic 0 let e be a nilpotent element of g and let ge be the centraliser of e in g in this paper we study the algebra s ge ge of symmetric invariants of ge we prove that if g is of type a or c then s ge ge is always a graded polynomial algebra in l variables and we show that this continues to hold for some nilpotent elements in the lie algebras of other types in type a we prove that the invariant algebra s ge ge is freely generated by a regular sequence in s ge and describe the tangent cone at e to the nilpotent variety of g

exact image theory for the calculation of fields transmitted through a planar interface of two media exact image theory recently developed for the calculation of fields reflected from a planar interface between two homogenous media is extended for transmitted field calculation the image of a point source is found to be a line source located in complex space for proper convergence unlike for the reflection image the transmission image line is seen to be curved in complex space for efficient computation expressions are derived for the transmission image of the most general three dimensional electric and magnetic sources and for media with both epsilon and mu discontinuous across the interface the new theory is verified with several asymptotic tests and properties of the transmission image function are discussed as an application radiation from a resonant horizontal loop antenna into the ground is calculated and results are seen to agree with those given in the literature

time domain study of arbitrary dipole in planar geometry with discontinuity in permittivity and permeability transient fields arising from a general impulsive dipole above ground characterized by permittivity and permeability are studied a detailed analysis for arbitrary electric dipole excitation is given and the fields arising from the analog magnetic sources can be obtained by applying the duality transform the mathematical description of this problem is done using the hertz vector whose solution in the time domain is derived in a novel way using the exact image theory from this solution which involves a definite integral the actual fields are obtained through simple differentiations the efficiency of the method is demonstrated by solving 14 previously unknown closed form results for various field components in certain specific cases seven of these are obtained for electric dipole excitation and listed whereas the corresponding results in the case of magnetic excitation can be easily written by applying the duality transform

time domain analysis of the sommerfeld vmd problem based on the exact image theory a novel time domain solution of the sommerfeld half space problem is constructed through the exact image theory the radiation from a vertical magnetic dipole vmd above the planar interface of two lossless and dispersionless dielectric media is analyzed the reflection field is given as a laplace transform when the refraction index of the lower medium is greater the converse case of refraction indexes is also studied and the reflection field is written as originating from an image source located in homogeneous space the present theory is seen to lead to a numerically efficient result which is demonstrated with examples

the frobenius semiradical of a lie algebra univ limburg dept math b 3590 diepenbeek belgium ooms ai univ limburg dept math univ campus b 3590 diepenbeek belgium alfons ooms luc ac be

an algebraic slice in the coadjoint space of the borel and the coxeter element abstract let g be a complex simple lie algebra and b a borel subalgebra the algebra y of polynomial semi invariants on the dual b of b is a polynomial algebra on rank g generators grothendieck and dieudonne 1965 1967 16 the analogy with the semisimple case suggests there exists an algebraic slice to coadjoint action that is an affine translate y v of a vector subspace of b such that the restriction map induces an isomorphism of y onto the algebra r y v of regular functions on y v this holds in type a and even extends to all biparabolic subalgebras joseph 2007 20 but the construction fails in general even with respect to the borel moreover already in type c 2 no algebraic slice exists very surprisingly the exception of type c 2 is itself an exception indeed an algebraic slice for the coadjoint action of the borel subalgebra is constructed for all simple lie algebras except those of types b 2 m c n and f 4 outside type a the slice obtained meets an open dense subset of regular orbits even though the special point y of the slice is not itself regular this explains the failure of our previous construction

exact image theory for the sommerfeld half space problem part iii general formulation the general sommerfeld problem with both epsilon and mu discontinuous and a source consisting of arbitrarily oriented electric and or magnetic dipoles at the same location is considered in terms of image theory the problem is formulated with electric and magnetic fields instead of potential quantities resulting in a vector transmission line interpretation of the fourier transformed problem the image sources are seen to be located in complex space expressable in terms of a certain basic image current function which was encountered in part ii of this paper on the vertical electric dipole problem the horizontal electric magnetic dipole image is solved and found to consist of both vertical and horizontal current components the image concept is generalized to the most general three dimensional sources as a check the well known reflection coefficient method is obtained as the far field approximation of the present theory

identification of linear stochastic systems via second and fourth order cumulant matching the identification problem for time invariant single input single output linear stochastic systems driven by non gaussian white noise is considered the system is not restricted to be minimum phase and it is allowed to contain all pass components a least squares criterion that involves matching the second and the fourth order cumulant functions of the noisy observations is proposed knowledge of the probability distribution of the driving noise is not required an order determination criterion that is a modification of the akaike information criterion is also proposed strong consistency of the proposed estimator is proved under certain sufficient conditions simulation results are presented to illustrate the method

exact image method for impedance computation of antennas above the ground the principle of exact images recently developed by two of the present authors for the sommerfeld problem is here applied for the classical impedance problem of antennas above the ground in the present method the effect of the ground is taken into account with an image source which is located in complex space leading to a modified green function for the antenna current integral equation the image of the antenna can be effectively determined through new expressions evaluated for the image current function the stationary functional for the antenna impedance is extended to take into account the effect of the ground as examples of the theory impedance expressions for small dipoles and half wave dipoles are formulated with numerical results calculated and compared with previous results a simple method applicable for small microcomputers is suggested for calculating the impedance of wire antennas with finite thickness from that of infinitesimally thin antenna if the height of the antenna from the ground is not too small also a practical method is suggested for the determination of ground parameters from horizontal lambda 2 dipole minimum resistance and height measurements

windowed green function method for the helmholtz equation in the presence of multiply layered media this paper presents a new methodology for the solution of problems of two and three dimensional acoustic scattering and in particular two dimensional electromagnetic scattering by obstacles and defects in the presence of an arbitrary number of penetrable layers relying on the use of certain slow rise windowing functions the proposed windowed green function approach efficiently evaluates oscillatory integrals over unbounded domains with high accuracy without recourse to the highly expensive sommerfeld integrals that have typically been used to account for the effect of underlying planar multilayer structures the proposed methodology whose theoretical basis was presented in the recent contribution bruno et al 2016 siam j appl math 76 1871 1898 doi 10 1137 15m1033782 is fast accurate flexible and easy to implement our numerical experiments demonstrate that the numerical errors resulting from the proposed approach decrease faster than any negative power of the window size in a number of examples considered in this paper the proposed method is up to thousands of times faster for a given accuracy than corresponding methods based on the use of sommerfeld integrals

strongly consistent identification algorithms and noise insensitive mse criteria windowed cumulant projections of nongaussian linear processes yield autocorrelation estimators which are immune to additive gaussian noise of unknown covariance by establishing strong consistency of these estimators strongly consistent and noise insensitive recursive algorithms are developed for parameter estimation these computationally attractive schemes are shown to be optimal with respect to a modified mean square error mse criterion which implicitly exploits the high signal to noise ratio domain of cumulant statistics the novel mse objective function is expressed in terms of the noisy process but it is shown to be a scalar multiple of the standard mse criterion as if the latter was computed in the absence of noise simulations illustrate the performance of the proposed algorithms and compare them with the conventional algorithms

a numerical methodology for efficient evaluation of 2d sommerfeld integrals in the dielectric half space problem the analysis of 2d scattering in the presence of a dielectric half space by integral equation formulations involves repeated evaluation of sommerfeld integrals deformation of the contour to the steepest descent path results in a well behaved integrand that can be readily integrated a well known drawback of this method is that an analytical expression for the path is available only for evaluation of the reflected fields but not for the evaluation of the transmitted fields a simple scheme for numerical determination of the steepest descent path valid for both cases is presented the computational cost of the numerical determination is comparable to that of evaluating the analytical expression for the steepest descent path for reflected fields when necessary contributions from branch cut integrals and a second saddle point are taken into account certain ranges of the input parameters which result in integrands that vary rapidly in the neighborhood of the saddle point require special treatment alternative paths and specialized gaussian quadrature rules for these cases are also proposed an implementation of the proposed numerically determined steepest descent path nd sdp method is freely available for download

on the coadjoint representation of z2 contractions of reductive lie algebras abstract we study the coadjoint representation of contractions of reductive lie algebras associated with symmetric decompositions let g g 0 g 1 be a symmetric decomposition of a reductive lie algebra g then the semi direct product of g 0 and the g 0 module g 1 is a contraction of g we conjecture that these contractions have many properties in common with reductive lie algebras in particular it is proved that in many cases the algebra of invariants is polynomial we also discuss the so called codim 2 property for coadjoint representations and its relationship with the structure of algebra of invariants

efficient calculation of the fields of a dipole radiating above an impedance surface in this paper the classic problem of field computation for an infinitesimal dipole radiating above an impedance half space is revisited the expressions for the traditional solution consist of integrals of the sommerfeld type that cannot be evaluated in closed form and due to their highly oscillatory nature are difficult to evaluate numerically a method known as exact image theory which has previously been applied to vertical electric and magnetic dipoles is used to derive explicit expressions for dipoles of arbitrary orientation above impedance surfaces starting from the spectral representation of the field the reflection coefficients are cast in the form of exact laplace transforms and then by changing the order of integrations field expressions in terms of rapidly converging integrals are obtained these expressions are exact and valid for any arbitrary source alignment or observation position it is shown that the formulation for a horizontal dipole contains an image in the conjugate complex plane resulting in a diverging exponential term not previously addressed in the literature it is shown through further mathematical manipulations that the diverging term is a contribution of the mirror image which can be extracted comparison of numerical results from exact image theory and the original sommerfeld type expressions shows good agreement as well as a speedup in computation time of many orders of magnitude which depends on the distance between the transmitter and the receiver this formulation can effectively replace the approximate asymptotic expressions used for predicting wave propagation over a smooth planar ground having different regions of validity the exact image formulation is also of practical use in evaluation of the green s function for various applications in scattering problems where approximate solutions are not sufficient

modular lie algebras and the gelfand kirillov conjecture let mathfrak g be a finite dimensional simple lie algebra over an algebraically closed field mathbb k of characteristic 0 let mathfrak g mathbb z be a chevalley form of mathfrak g and mathfrak g bbbk mathfrak g mathbb z otimes mathbb z bbbk where bbbk is the algebraic closure of mathbb f p let g bbbk be a simple simply connected algebraic bbbk group with operatorname lie g bbbk mathfrak g bbbk in this paper we apply recent results of rudolf tange on the fraction field of the centre of the universal enveloping algebra u mathfrak g bbbk to show that if the gelfand kirillov conjecture from 1966 holds for mathfrak g then for all p 0 the field of rational functions bbbk mathfrak g bbbk is purely transcendental over its subfield bbbk mathfrak g bbbk g bbbk very recently it was proved by colliot thelene kunyavski popov and reichstein that the field of rational functions mathbb k mathfrak g is not purely transcendental over its subfield mathbb k mathfrak g mathfrak g if mathfrak g is of type b n n 3 d n n 4 e6 e7 e8 or f4 we prove a modular version of this result valid for p 0 and use it to show that in characteristic 0 the gelfand kirillov conjecture fails for the simple lie algebras of the above types in other words if mathfrak g is of type b n n 3 d n n 4 e6 e7 e8 or f4 then the lie field of mathfrak g is more complicated than expected

the polynomiality of the poisson center and semi center of a lie algebra and dixmier s fourth problem abstract let g be a finite dimensional lie algebra over an algebraically closed field k of characteristic zero we provide necessary and also some sufficient conditions in order for its poisson center and semi center to be polynomial algebras over k this occurs for instance if g is quadratic of index 2 with g g g and also if g is nilpotent of index at most 2 the converse holds for filiform lie algebras of type l n q n r n and w n we show how dixmier s fourth problem for an algebraic lie algebra g can be reduced to that of its canonical truncation g moreover dixmier s statement holds for all lie algebras of dimension at most eight the nonsolvable ones among them possess a polynomial poisson center and semi center

application of the modified image method to inversion of synthetic active electromagnetic bathymetric data a modified image method is used to invert synthetic active electromagnetic data generated from shallow ocean models by means of the exact sommerfeld theory the modified image method offers approximate solutions to the sommerfeld problem the response of a layered structure of ohmic conductors to an active electromagnetic source the image plane for the active electromagnetic source is assumed to be at a complex depth given by q m delta 1 exp ipi 4 rad 2 where delta 1 is the skin depth of the first layer and q m is the multilayer correction factor two ad hoc correction factors have been introduced into the modified image method they bring the image field h i of the single image at a complex depth into very good agreement with the secondary field h s of the exact sommerfeld theory of a two layer bathymetric model we demonstrate that these correction factors can be calculated from synthetic data this allows for an accurate algebraic and fast inversion of active electromagnetic data in terms of the parameters of a two layer model the conductivities of the sea and sea bottom and the depth and ohmic skin depth of the sea the beta factor which takes into account the departure of the primary active electromagnetic field from a plane wave form at the air sea interface is incorporated into the complex two layer correction function q m a rescaling factor f brings h i into good agreement with h s in the depth regime where the beta correction factor is ineffective i e where the first layer depth is greater than two skin depths

on commutative polarizations let l be a finite dimensional lie algebra over a field k of characteristic zero and let u l be its enveloping algebra with quotient division ring d l let p be a commutative lie subalgebra of l in o2 the necessary and sufficient condition on p was given in order for d p to be a maximal commutative subfield of d l in particular this condition is satisfied if p is a commutative polarization cp with respect to any regular f l and the converse holds if l is ad algebraic the purpose of this paper is to study lie algebras admitting these cp s and to demonstrate their widespread occurrence first we have the following characterisation if l is completely solvable p is a cp of l if and only if there exists a descending chain of lie subalgebras

the gelfand kirillov conjecture for semi direct products of lie algebras let g be an n dimensional lie algebra over a field k of characteristic zero and let w be a g module such that dimw n sufficient conditions are given in order for the semi direct product g w to satisfy the gelfand kirillov conjecture this implies that this conjecture holds for an important class of frobenius lie algebras special attention is devoted to the case where g sl 2 k

slices for biparabolic coadjoint actions in type a abstract let q be a truncated biparabolic subalgebra of a simple lie algebra in previous work it was shown that the invariant subalgebra y q of the algebra r q of regular functions on q is polynomial in most cases this includes all cases if g s l n and so provides 2 2 n 1 mostly non isomorphic lie algebras q for which y q is polynomial in general the structure of the generators and even their number have no known simple expression one may recall that the set of companion matrices forms a slice to the regular co adjoint orbits in g g s l n the present work is a quite remarkable and far reaching generalisation of this companion slice precisely one gives an element y q and a subspace v q such that restriction of functions gives an algebra isomorphism of y q onto r y v the construction is a combinatorial procedure based on the data specifying q unlike the semisimple case there can be many equivalence classes of such pairs as a consequence it is found that the nilfibre of the geometric quotient map may have many irreducible components containing a regular element namely the closure of the co adjoint orbit defined by such y q an example shows it may also have a component with no regular elements

exact closed form expression of a sommerfeld integral for the impedance plane problem a sommerfeld integral for an impedance half plane is one of the classical problems in electromagnetic theory in this paper the integral is evaluated into two series representations which are expressed in terms of the exponential integral and the lommel function respectively it is shown that the series expansions absolutely converge for any parameters such as distance or surface impedance then based on the lommel function expansion an exact closed form expression of the integral is formulated this expression is written in terms of incomplete weber integrals which are directly related to incomplete cylindrical functions such as incomplete lipschitz hankel integrals additionally a complete asymptotic series of the integral is obtained based on the exponential integral expansion with conventional methods such as steepest descent method it is cumbersome to derive the divergent series the validity of all the formulations derived in this paper is demonstrated through a comparison with a numerical integration of the integral for various situations

the gelfand kirillov conjecture for lie algebras of dimension at most eight univ reims dept math f 51687 reims 2 france univ limburg dept math b 3590 diepenbeek belgium alev j univ reims dept math bp 1039 f 51687 reims 2 france

a fast finite difference method for propagation predictions over irregular inhomogeneous terrain propagation of electromagnetic waves over irregular inhomogeneous terrain is solved by a finite difference scheme the method is fast and requires considerably less memory than the integral equation methods the method requires a storage space of order o n and an execution time of order o n sup 2 fields generated by a te sub 2 line source are represented in an integral form in terms of the field over a flat constant impedance plane the incident field and the field scattered by the terrain irregularities and inhomogeneities accurate expressions are provided for the incident field and the green s function whose evaluation is otherwise accomplished by the rather time consuming sommerfeld s integrals measured equation of invariance is used to terminate the computational domain the sparse matrix generated by the method is inverted by the ricatti transform numerical results are presented for the ground wave as well as for the sky wave comparison is made for known geometries to establish the validity and limitations of the method

a class of counterexamples to the gel fand kirillov conjecture limburgs univ centrum dept wni b 3500 diepenbeek belgium alev j univ reims ufr sci dept math moulin housse bp 347 f 51062 reims france

exact image theory for the sommerfeld half space problem part ii vertical electrical dipole applying the laplace transform the exact distributed image current function is obtained for the classical sommerfeld half space problem with vertical magnetic current source the resulting field integral is well behaved when the image current is situated in complex space unlike previous approximate images the present theory is valid for any distance height of the source frequency and half space parameters it is demonstrated that the present image theory reduces to the well known dipole image at complex depth for large dielectric parameters of the half space also the reflection coefficient method is obtained as a farfield approximation calculation of fields through exact image integration is seen to be simple and accurate and require modest computer capacity and time in an appendix some properties of the multivalued green s function arising from a dipole source in complex space are also studied

application of the fast fourier transform to the computation of the sommerfeld integral for a vertical electric dipole above a half space the fast fourier transform fft is used in conjunction with analytical techniques to obtain an approximate expression for the sommerfeld integral associated with a vertical electric dipole above a half space the resulting expression which is an analytic function of the source and observation locations can be used in lieu of a numerical integration of the sommerfeld integral it can be used in either the near zone or the far zone and it can even be applied for observation angles close to grazing the accuracy of the expansion directly depends on the number of terms used in the fft results obtained using the expansion are compared with those obtained both from a direct numerical integration of the sommerfeld integral and the reflection coefficient method

on certain maximal subfields in the quotient division ring of an enveloping algebra univ limburg dept math b 3590 diepenbeek belgium ooms ai univ limburg dept math b 3590 diepenbeek belgium

computing invariants and semi invariants by means of frobenius lie algebras abstract let u be the enveloping algebra of a finite dimensional lie algebra over a field k of characteristic zero z u its center and sz u its semi center a sufficient condition is given in order for sz u to be a polynomial algebra over k surprisingly this condition holds for many lie algebras especially among those for which the radical is nilpotent in which case sz u z u in particular it allows the explicit description of z u for more than half of all complex indecomposable nilpotent lie algebras of dimension at most 7

the poisson center and polynomial maximal poisson commutative subalgebras especially for nilpotent lie algebras of dimension at most seven abstract let g be a finite dimensional lie algebra over an algebraically closed field k of characteristic zero we collect some general results on the poisson center of s g including some simple criteria regarding its polynomiality and also on certain poisson commutative subalgebras of s g these facts are then used to complete our previous work on the subject ooms 2009 o5 5 i e to give an explicit description for the poisson center of all indecomposable nilpotent lie algebras of dimension at most seven among other things we also provide a polynomial maximal poisson commutative subalgebra of s g enjoying additional properties as a by product we show that a conjecture by milovanov is valid in this situation these results easily carry over to the enveloping algebra u g

symmetric invariant non degenerate bilinear form on a lie algebra the killing form is a powerful tool in the study of semi simple lie algebras unfortunately it is non degenerate only in the semi simple case therefore one has to look more generally for a form which has all the useful properties of the killing form bilinearity symmetry invariance nondegeneracy such a form will be a scalar multiple of the killing form in the central simple case but in the general case it may not even exist just take the solvable lie algebra of dimension 2 in this work we study what we call regular quadratic lie algebra g cp i e lie algebra g equipped with a symmetric invariant non degenerate bilinear form cp if d is a derivation of g belonging to the orthogonal lie algebra o cp then following the ideas of v g kac we define a regular quadratic lie algebra of dimension 2 dim g denoted g pi which we call the extension of g cp by d proposition 2 3 the interesting point of this extension is the fact that any indecomposable regular quadratic lie algebra with nontrivial center is of the form g pi proposition 2 9 which allows an inductive classification moreover a regular quadratic solvable lie algebra has a non zero center lemma 2 8 therefore the solvable case is completely covered proposition 2 11 classified all extensions of a given g cp

on the semi center of a universal enveloping algebra throughout this paper l will be a nonzero finite dimensional lie algebra over a field k of characteristic zero let u l be the universal enveloping algebra of l with center z u l d l the division ring of quotients of u l with center z d l for each 2 e l we denote by d l the set of those ued l such that ad x u n x u for all xe l its elements are called the semi invariants of d l relative to 1 clearly d l l c al a for all i p e l the sum of the d l is direct and is a subalgebra sz d l of d l which is called the semi center of d l we put u l u l n d l the sum sz u l of the u l is called the semi center of u l the semi center sz s l of the symmetric algebra s l of l can be defined in a similar fashion the semi center is a commutative subalgebra of u l which never reduces to k 7 and which is stable under all automorphisms of u l 16 from the fact that it is a unique factorization domain in the algebraically closed case 131 we are able to deduce that in the general case sz u l is contained in a factorial subalgebra of u l theorem 1 2 in particular any nonzero semi invariant of u l can be factored in a unique way as the product of irreducible semiinvariants proposition 1 5 this result proves to be extremely useful throughout the paper we also obtain some characterizations of the semiinvariants of d l proposition 1 8 let a l resp a l be the set of all z e l such that u l 0 resp d l o a l is a semi group which is not always finitely generated 9 the group n l which is isomorphic to kharchenko s group of x inner automorphisms of u l 16 turns out to be the additive subgroup of l generated by a l corollary 1 9 next the centralizer of sz u l in u l coincides with u l where l is the intersection of ker 2 ac l corollary 1 16 l 324 0021 8693 85 3 00

a degree inequality for lie algebras with a regular poisson semi center for lie algebras whose poisson semi center is a polynomial ring we give a bound for the sum of the degrees of the generating semi invariants this bound was previously known in many special cases

on the jordan kernel of a universal enveloping algebra abstract first it is shown that the jordan kernel jk d l of the division ring of quotients d l of the universal enveloping algebra u l of a finite dimensional lie algebra l is isomorphic to the group ring of the free abelian group of weights of l in d l over a weyl algebra a m z where z is a polynomial ring over the center z d l of d l in particular it is a maximal order in its division ring of quotients and its center is a unique factorization domain these two properties are then investigated for the jordan kernel jk u l of u l therefore the centralizer c u l of the characteristic ideal l of l in u l is studied since it coincides with jk u l in most cases in particular we compute its gelfand kirillov dimension and we derive necessary and sufficient conditions in order for c u l to coincide with z u l finally we sharpen the previous results in the case that l is a frobenius lie algebra

complete closed form expression of dyadic green s function and its far and near field approximations for an impedance half plane a closed form expression of the dyadic green s function is formulated for an impedance half plane which is written in terms of the incomplete cylindrical function of poisson form due to the branch cut of the logarithm function that is required to calculate the input argument of the incomplete cylindrical function the closed form representation consists of two formulations since the closed form expression contains a singularity at 0 the small argument expansion of the expression is also derived to rigorously characterize the behavior of the function at 0 the previously reported complete asymptotic expansion for the sommerfeld integral for an impedance half plane is not accurate for practically important cases such as near earth propagation and or when the surface is highly conductive hence in this paper a new complete asymptotic series of the sommerfeld integral are derived for the case that the existing asymptotic series is not accurate the two asymptotic series not only allow efficient numerical computation but also provide more accurate results for virtually all propagation scenarios based on the two asymptotic series the complete asymptotic series of the dyadic green s function is derived all derived formulations are numerically verified and their accuracies are investigated

locating all the modes of green s function for a three layered medium based on the path tracking algorithm through a mathematical analysis on the relationship between the characteristic equations of the microstrip geometry and a configuration with a three layered medium a method based on the path tracking algorithm is proposed for precisely locating all the modes for a three layered medium with moderate thickness including surface wave modes leaky wave modes and improper real modes with this method the difficulty caused by branch points and cuts in the contour integral method could be avoided this method can be applied not only in the discrete complex image method dcim but also in the steepest descent path sdp method the proposed method can be extended to more complicated configurations