ptd analysis of impedance structures based on an approximate dyadic diffraction coefficient equivalent currents ecs are derived for computing the scattering by a finite length impedance wedge of arbitrary angle the derived equivalent currents are implemented in a standard general purpose physical theory of diffraction ptd code and results are presented demonstrating the accuracy of the formulation for a number of impedance and dielectrically coated structures these include typical shapes such as plates finite length cones and cylinders which have been partially or fully coated the ptd implementation requires a dyadic physical optics diffraction coefficient which is presented in the appendix

comparison of lossy wedge diffraction coefficients with application to mixed path propagation loss prediction a heuristic lossy wedge diffraction coefficient is compared to the exact maliuzhinets diffraction coefficient for a 180 degrees wedge neglecting surface waves the comparison shows good agreement even for high incidence angles and greatly dissimilar wedge face materials this agreement provides justification for using the heuristic coefficient for the calculation of mixed path propagation loss since wedge angles are typically near 180 degrees for this application the heuristic coefficient has the advantages of being continuous at the reflection boundary and of being relatively simple to evaluate especially at angles other than 180 degrees

a utd solution for multiple rounded surfaces this paper describes the diffraction mechanism when a ray optical field is obstructed by a cascade of smooth convex obstructions a new formulation of the uniform theory of diffraction utd is presented that incorporates the slope diffraction term the solution follows a generic approach that can be applied to arbitrary placed convex structures it will be shown that the slope diffraction term provides the required accuracy in radio propagation predictions when more than one obstruction exists in the propagation path the proposed solution is then compared with measurements and other analytical models with excellent agreement it is also found that the presented slope utd solution seems to be superior in terms of computation time and complexity while achieving very accurate results in addition when the radius of the cylinder tends to be very small the solution approaches the utd solution for multiple knife edge predictions

double tip diffraction modeling 2 d numerical models versus high frequency asymptotics the subject of single and double diffraction phenomena has long been investigated by high frequency asymptotic techniques however integral or differential equation based numerical methods suffer from computational complexity due to electrically large geometries encountered in high frequency problems the main purpose of this paper is to present the finite element fem diffraction modeling of double tip structure and to compare its results with high frequency methods and other numerical models fem is made feasible for modeling of such an infinitely long structure by utilizing the locally conformal perfectly matched layer pml approach which enables the use of finite sized structure matlab codes are developed and various numerical examples are demonstrated in a comparative manner

algorithm for the computation of the generalized fresnel integral the generalized fresnel integral appears as a canonical function in the uniform ray field representation of several high frequency diffraction mechanisms such as double wedge or vertex diffraction here we propose an algorithm for its calculation which is valid both for real and imaginary arguments as required for treating the general case in the uniform geometrical theory of diffraction framework

a modified diffraction coefficient for imperfect conducting wedges and buildings with finite dimensions multiple building diffraction loss calculation is often based on the assumption that their dimensions are infinite in the direction perpendicular to the propagation path even though in reality this assumption is not valid in this paper a modified diffraction coefficient model for the calculation of a more realistic diffraction loss caused by a single building with finite dimensions is proposed the overall diffraction loss is calculated for a group of buildings in a row by using the proposed model in addition after carrying out the field measurements it is shown that for some particular cases where the transmitter and diffracting obstacles are not far away from each other i e microcells or picocells spherical wave modeling produces a better accuracy than the plane wave approximation it is also demonstrated that the theoretical results can further be improved by considering the higher order terms in the calculation of diffraction coefficients lastly excess path loss obtained for a row of buildings with spherical wave assumption is compared with our measurement data this comparison shows reasonable correspondence between our theoretical model and measurements therefore our proposed model can successfully be used for improving the theoretical methods for predicting path loss in urban environments where diffraction from a number of buildings has to be considered

rank dependency of group properties abstract the search for grand unified theories in physics has focussed attention on the need to know how the properties of lie groups depend explicitly on their rank some properties for some groups are essentially rank independent while for other groups they may be explicitly rank dependent the current status of such properties is reviewed specific examples relating to the tensor and spinor representations of the orthogonal and rotation groups are considered together with the n dependence of the group properties of the finite groups sn and an

reflections diffractions and surface waves for an interior impedance wedge of arbitrary angle the asymptotic impedance wedge solution for plane wave illumination at normal incidence is examined for interior wedge diffraction an efficient method for calculating the diffraction coefficient for arbitrary wedge angle is presented as previous calculations were very difficult except for three specific wedge angles for the uniform geometrical theory of diffraction utd expansion the asymptotic solution isolates the incident singly reflected multiply reflected diffracted surface wave and associated surface wave transition fields multiply reflected fields of any order are considered the multiply reflected fields from the exact solution arise as ratios of auxiliary maliuzhinets functions however by using properties of the maliuzhinets functions this representation can be reduced to products of reflection coefficients which are much more efficient for calculation a surface wave transition field is added to the surface wave boundaries computations are presented for interior wedge diffractions although the formulation is equally valid for both exterior and interior wedges with uniform but different impedances on each face for both soft and hard polarizations in addition the accuracy of the high frequency asymptotic expansion is examined for small diffraction distances by direct comparison of the exact and asymptotic solutions

symmetrised powers of rotation group representations the results given permit the unambiguous evaluation of all possible kronecker products of the irreducible representations tensor and spinor of on and son for n 2 nu and n 2 nu 1 a complete resolution of the second and third powers of the basic spinor representations of so2 nu and so2 nu 1 is given together with a prescription for analysing the fourth power of these representations detailed application is made to the enumeration of properties of so10 relevant to grand unified theories and sufficient information given to resolve the fourth power of any representation of so10

on groups of central type a finite group is said to be of central type if it possesses an irreducible complex character which takes the value zero on all noncentral elements equivalently the degree of this character is the square root of the index of the center in 1964 iwahori and matsumoto 15 conjectured that a group of central type must be solvable the paper of liebler and yellen 16 aims to prove this but there is a gap in their proof as explained below nevertheless they do correctly obtain substantial information about a minimal counterexample to the conjecture the aim of this paper is to fill the gap in the liebler yellen proof using as they do the classification of finite simple groups and to provide further information about which solvable groups can have central type we show for instance that every normal subgroup of the central factor group of a solvable group of central type has the property that its system normalizers have square index it is interesting to contrast this restrictive condition with the result of gagola that every solvable group is embeddable in the central factor group of a group of central type see 6 theorem 1 2 the arguments in 16 show that a minimal counterexample g to the iwahori matsumoto conjecture must have the following structure if k z is any minimal normal subgroup of g z where z z g the center then k is abelian k z is a 2 group s cg k is the maximal solvable normal subgroup of g and any minimal normal subgroup his of g s is the direct product of a number l of copies of gl 3 2 indeed the liebler yellen methods can be used to show that the sylow 2 subgroup of k considered as an h s module is a product o f copies of a gl 3 2 module of dimension 4 this module having a unique minimal submodule which is trivial and an irreducible factor module of dimension 3 the minimal submodules are amalgamated in the product

a general uniform double wedge diffraction coefficient the uniform theory of diffraction utd is extended to include double diffraction by an arbitrary configuration of two wedges a scalar double diffraction coefficient is defined which is uniform throughout space while reducing to the product of single diffraction coefficients outside overlapping transition regions the result is formulated in terms of double fresnel integrals which can be rapidly evaluated

the groups of order 256 abstract building on earlier work a new method for generating descriptions of p groups is developed the theory and implementation of this method are described and its application in determining the 56 092 groups of order 256 is outlined

an efficient ray tracing algorithm for multiple straight wedge diffraction we present an efficient algorithm for the tracing of multiply edge diffracted rays the algorithm assumes a given sequence of infinite edges and complete visibility among them the ray tracing problem is formulated as the minimization of the ray total path length since such a cost function is strictly convex except for coplanar edges in the plane wave far field regime then the problem admits a unique global minimum and allows the use of a modified newton search algorithm which exhibits a very high converging rate we also propose a convenient starting point to effectively initialize the minimization algorithm the proposed algorithm is tested by some numerical examples that show its efficiency and effectiveness

asymptotic behavior of some linear differential systems abstract consider the following chasing game in r amongbparticles whose trajectories are governed by the following differential equations fori s 1 2 b f i t j i q ij t f j t f i t wherefi t is the position at timetof the particleiandqij t which measures the instantaneous rate that particleiis attracted toward particlej is given byqij t pij t u i j forj i the rate function t is positive with limt t 0 u i j 0 is the cost function indicating the level of difficulty fromitojand the nonnegative numberspijfori jreflect the neighborhood structure of the particles particles are attracted to one another but such attraction becomes weaker as time goes on a sufficient condition on t is given for allfi t converging to a common finite limit

generalized young tableaux and kronecker products of so n representations abstract a new type of young tableaux with negative boxes is introduced the product of two such generalized young tableaux is defined and enables one to develop an algorithm for the reduction of the product of so n representations a general formula for such a reduction is presented for the case n 2 p 1

a hybrid asymptotic solution for the scattering by a pair of parallel perfectly conducting wedges the overlapping transition regions of the double diffraction by a pair of parallel wedge edges are considered for the hybrid case where the gap between the edges is small compared to the distances from the source and the observation point plane wave far field limit and the scatterer as a whole is large or infinite a closed form asymptotic solution for the scattered field continuous at all angles of incidence and scattering is constructed for this case the peculiar feature of this solution is a hybrid representation of the field singly diffracted by the first wedge a part of it is described by a nonuniform geometrical theory of diffraction gtd expression while the other part is described in terms of the uniform theory of diffraction utd the rest of the diffracted ray fields are described by nonuniform expressions with singularities mutually canceling on summation this solution is applied to the scattering by a perfectly conducting rectangular cylinder with appropriate geometrical parameters and agreement with moment method calculation is demonstrated

an improved impedance boundary algorithm for fourier split step solutions of the parabolic wave equation a new implementation of the previously published mixed fourier transform mft method for including impedance boundaries in split step parabolic equation solutions is described and demonstrated the new algorithm is formulated entirely in the discrete domain which results in extended applicability and increased computation speed a brief review of the original mft solution is followed by a detailed description of the discrete formulation the performance of the new algorithm is then demonstrated with a few examples which rely heavily on the accuracy of the impedance boundary these examples include 10 mhz surface wave propagation over smooth and rough sea surfaces and 10 ghz calculations utilizing an effective rough surface impedance

ray optical prediction of radio wave propagation characteristics in tunnel environments 1 theory a tunnel is modeled as congregates of walls with the wall being approximated by a uniform impedance surface the aim is to get a solution for a canonical problem of a wedge with uniform impedance surface the diffraction by a right angle wedge with different impedance boundary conditions at its two surfaces is first considered a functional transformation is used to simplify the boundary conditions the eigenfunction solutions for the transformed functions are replaced by integral representations which are then evaluated asymptotically by the modified pauli clemmow method of steepest descent the asymptotic solution is interpreted ray optically to obtain the diffraction coefficient for the uniform geometrical theory of diffraction utd the obtained diffraction coefficients are related directly to the keller diffraction coefficients in the uniform version the total field is continuous across the shadow of the geometrical optics fields

wiener hopf analysis of the em diffraction by an impedance discontinuity in a planar surface and by an impedance half plane the wiener hopf technique is used to solve two canonical problems the first problem considered is the electromagnetic em diffraction by a planar surface with an impedance discontinuity two part surface of an arbitrarily polarized plane wave obliquely incident to the axis of the two dimensional objects the second problem considers the em diffraction by a half plane with equal impedances on both sides the solutions obtained are cast in a matrix notation which is useful for diffraction problems the exact formal solutions are expressed in terms of integrals which can be asymptotically evaluated uniform asymptotic expressions are obtained where the presence of the geometrical optics go poles as well as the surface wave poles near the saddle point are fully taken into account several numerical examples are presented and it is shown that the solutions are continuous across the shadow boundaries of the go and surface wave fields

a heuristic utd slope diffraction coefficient for rough lossy wedges heuristic wedge diffraction coefficients for computing propagation path loss over finitely conducting earth are extended to include slope diffraction with the assumption that propagation of energy through the wedge is negligible the slope diffraction terms for the lossy wedge are obtained in an analogous manner as for the perfectly conducting case except that special care must be taken with the factor multiplying the incident field for grazing incidence results given show that the slope diffraction term produces continuous results that behave reasonably when compared with results for perfectly conducting wedges

a uniform double diffraction coefficient for a pair of wedges in arbitrary configuration an analytic closed form solution is presented for the double diffraction at a pair of arbitrarily placed wedges that is suitable to be used in a uniform theory of diffraction utd ray description framework here the particular assumption present in all the past literature that the two diffracting edges are coplanar is removed the doubly diffracted dd field at a pair of wedges is constructed via spectral synthesis such a procedure provides a double spectral integral representation of the dd field that is asymptotically evaluated by resorting to transition functions typical in double diffraction problems that can be expressed in terms of generalized fresnel integrals the final expression is arranged in the typical utd fashion and contains a uniform double diffraction dyadic coefficient the dd field behavior is analyzed in its transition regions to show how the dd field smoothly compensates for the total field discontinuities occurring at shadow boundaries where a singly diffracted field from a wedge is shadowed by the other wedge numerical examples are provided to verify such analysis and to test the accuracy of our utd ray description against method of moments full wave results

uniform asymptotic evaluation of surface integrals with polygonal integration domains in terms of utd transition functions the field scattered by a scattering body or by an aperture in the free space or in an unbounded homogenous medium can be described in terms of a double integral in this paper we show how a canonical integral on a polygonal domain with a constant amplitude function and a quadratic phase variation can be exactly expressed in terms of special functions namely fresnel integrals and generalized fresnel integrals this exact reduction represents a paradigm for deriving a new asymptotic evaluation for a more general integral this new asymptotic uniform integral evaluation is expressed in the format of the uniform geometrical theory of diffraction which is convenient for numerical computations

standard young tableaux and weight multiplicities of the classical lie groups by examining the branching rules for all irreducible representations of the classical groups u k su k so 2k 1 sp 2k and so 2k on restriction to u 1 u 1 u 1 standard young tableaux are specified for each of these groups it is shown that these tableaux determine the corresponding characters of the irreducible representations the rules for constructing these tableaux are derived and in this way the determination of weight multiplicities is reduced to a simple combinatorial exercise general formula for such weight multiplicities are given encompassing the most difficult case namely that of so 2k illustrative examples are provided including some yielding the explicit k dependence of weight multiplicities

global boundary flattening transforms for acoustic propagation under rough sea surfaces this paper introduces a conformal transform of an acoustic domain under a one dimensional rough sea surface onto a domain with a flat top this non perturbative transform can include many hundreds of wavelengths of the surface variation the resulting two dimensional flat topped domain allows direct application of any existing acoustic propagation model of the helmholtz or wave equation using transformed sound speeds such a transform model combination applies where the surface particle velocity is much slower than sound speed such that the boundary motion can be neglected once the acoustic field is computed the bijective one to one and onto mapping permits the field interpolation in terms of the original coordinates the bergstrom method for inverse riemann maps determines the transform by iterated solution of an integral equation for a surface matching term rough sea surface forward scatter test cases provide verification of the method using a particular parabolic equation model of the helmholt

oblique scattering by a pair of conducting half planes tm case an exact series solution for the oblique scattering by a pair of infinitely long parallel edges of perfectly conducting half planes is formulated for a tm plane wave using the mode matching technique mmt the scattered and guided fields are represented in terms of an infinite series of radial waveguide modes by applying the appropriate boundary conditions the coefficients of the scattered field are obtained the diffraction coefficient of double edges is subsequently derived from the scattered field

real argument incomplete hankel functions accurate and computationally efficient integral representations and their asymptotic approximants novel accurate computationally efficient integral representations of the real argument incomplete hankel functions of arbitrary order are presented leading to a straightforward numerical implementation these representations are shown to yield analytical approximants expressed through known special functions which are also accurate and valid for any arguments of the incomplete hankel functions through these representations the electromagnetic field distribution excited in planar and truncated cylindrical structures can be determined accurately and efficiently numerical results based on the exact and approximate representations are presented to demonstrate the effectiveness of the proposed integral representations in the analysis of the electromagnetic field distribution excited in complex structures

asymptotical behaviour of several interacting annealing processes we prove that the optimal convergence speed exponent for parallel annealing based on periodically interacting multiple searches with time periodr is always worse than for independent multiple searches whenever the cost function has only one global minimum our proofs will be based on large deviation estimates coming from the theory of generalized simulated annealing g s a

on the convergence rate of annealing processes for the class of inhomogeneous markov processes arising from simulated annealing it is shown that lim t to infty p x t i exp u i t t exists and is positive for each state i where t t is the temperature and u i is the energy level at i assuming that min u i 0 the method used is to consider the forward equations associated with such processes

terrain based propagation model for rural area an integral equation approach a terrain based propagation model for vertically polarized radio waves is described based on the field integral equation for a smooth surface the model is simplified from a 3d integral equation model to a one dimensional integral equation by assuming that the surface is magnetically perfectly conducting a soft surface with no transverse variations by assuming no back scattering the integral equation is turned into a simple integral the method is tested numerically with known solutions the integral equation model is also applied to actual terrain profiles at four frequencies 144 435 970 1900 mhz and the results are compared with results from the hata model and measurements the standard deviation std of the error is a growing function of frequency from 3 to 9 db

a note on double edge diffraction for parallel wedges an extended spectral theory of the diffraction approach is used in the investigation of double edge diffraction in the near field region of a parallel wedge geometry illuminated by a cylindrical wave a high frequency approximation can be formulated in terms of a canonical double stationary phase integral a uniform asymptotic reduction of the latter using the method of d s jones 1971 results in a compact representation involving the generalized fresnel integral with certain advantages in numerical applications over past representations

rank dependency of weight multiplicities for simple lie groups the determination of the rank dependence of the weight multiplicities for the groups o2k 1 sp2k and o2k is simplified using the properties of schur functions together with a simple weight space mapping the weight multiplicities are factorised into products of much simpler quantities whose rank dependence is readily computed the spinor weight multiplicities for so2k are shown to be simply related to those of o2k the weight multiplicities for the k part tensor representation of so2k are examined in terms of difference characters some specific results are tabulated for so2k

a uniform geometrical theory of diffraction for an imperfectly conducting half plane diffraction tensors are presented in the context of the uniform geometrical theory of diffraction utd for the high frequency scattering by an impedance half plane at normal and oblique skew incidence these are based on the exact wiener hopf solution and were derived according to the utd ansatz in addition unlike previous uniform diffraction coefficients the ones given here reduce to the known utd diffraction coefficients for the perfectly conducting case the coefficients are explicit and therefore appropriate for practical applications several scattering patterns are also presented and compared to a previous heuristic solution

cooling schedules for optimal annealing a monte carlo optimization technique called simulated annealing is a descent algorithm modified by random ascent moves in order to escape local minima which are not global minima the level of randomization is determined by a control parameter t called temperature which tends to zero according to a deterministic cooling schedule we give a simple necessary and sufficient condition on the cooling schedule for the algorithm state to converge in probability to the set of globally minimum cost states in the special case that the cooling schedule has parametric form tt c log1 t the condition for convergence is that c be greater than or equal to the depth suitably defined of the deepest local minimum which is not a global minimum state

an improved heuristic utd solution for multiple edge transition zone diffraction this paper describes a new heuristic approach for multiple edge diffraction modeling based on the uniform theory of diffraction utd it incorporates the slope diffraction terms which can be realized as a second order diffraction process vital for accurate predictions when more than one edge exists the predictions of this approach are compared with the vogler 1982 solution and were found to be in very good agreement for many different geometrical arrangements with low computational complexity the solution is also extended to wedge diffraction

recursive two way parabolic equation approach for modeling terrain effects in tropospheric propagation the fourier split step method is a one way marching type algorithm to efficiently solve the parabolic equation for modeling electromagnetic propagation in troposphere the main drawback of this method is that it characterizes only forward propagating waves and neglects backward propagating waves which become important especially in the presence of irregular surfaces although ground reflecting boundaries are inherently incorporated into the split step algorithm irregular surfaces such as sharp edges introduce a formidable challenge in this paper a recursive two way split step algorithm is presented to model both forward and backward propagation in the presence of multiple knife edges the algorithm starts marching in the forward direction until the wave reaches a knife edge the wave arriving at the knife edge is partially reflected by imposing the boundary conditions at the edge and is propagated in the backward direction by reversing the paraxial direction in the parabolic equation in other words the wave is split into two components and the components travel in their corresponding directions the reflected wave is added to the forward wave in each range step to obtain the total wave the wave splitting is performed each time a wave is incident on one of the knife edges this procedure is repeated until convergence is achieved inside the entire domain

explicit evaluation of coupling coefficients for the most degenerate representations of so n explicit results for particular coupling coefficients for the most degenerate representations of so n are given the isoscalar factors which allow a recursive calculation of arbitrary coupling coefficients from those of so 3 are derived 6j and 9j symbols for most degenerate representations are also briefly discussed

an efficient algorithm for the computation of the utd t transition function we present an efficient algorithm for the numerical calculation of the t canonical transition function which is encountered in the uniform geometrical theory of diffraction utd such a transition function allows the uniform ray field description of various high frequency diffraction mechanisms such as double wedge or vertex diffraction the proposed algorithm is valid for both real and imaginary arguments as required to deal with the general case in utd applications

a utd triple diffraction coefficient for straight wedges in arbitrary configuration a new utd solution is presented for the description of the ray triply diffracted by straight wedges in arbitrary configuration illuminated by a spherical wavefront field the proposed utd coefficient permits to uniformly describe the field at any aspect in particular including the case of overlapping transition regions where the subsequent application of utd single wedge diffraction coefficient fails the triple diffracted ray field is expressed in an analytical form providing physical insight into the triple diffraction mechanism and a new effective engineering tool within the utd framework as required in a modern ray based code

a hybrid finite difference surface green s function method for computing transmission losses in an inhomogeneous atmosphere over irregular terrain a standard parabolic equation spe is used to approximate the helmholtz equation for electromagnetic propagation in an inhomogeneous atmosphere an implicit finite difference ifd scheme to solve the spe is applied between the irregularly shaped ground and an altitude z z sub h below which all inhomogeneities of the medium are assumed localized the boundary condition at z z sub h is obtained by matching the ifd solution to a surface green s function sgf solution within the uniform region above z z sub h for ground slopes above about 1 degrees the ifd implementation of the impedance boundary condition at the ground is shown to maintain the validity of the approximation only for vertically polarized waves predictions using this hybrid finite difference fd sgf method agree well with results obtained using other computational methods

perfectly matched absorbing layers for the paraxial equations a new absorbing boundary technique for the paraxial wave equations is proposed and analyzed numerical results show the efficiency of the method

stochastic relaxation gibbs distributions and the bayesian restoration of images we make an analogy between images and statistical mechanics systems pixel gray levels and the presence and orientation of edges are viewed as states of atoms or molecules in a lattice like physical system the assignment of an energy function in the physical system determines its gibbs distribution because of the gibbs distribution markov random field mrf equivalence this assignment also determines an mrf image model the energy function is a more convenient and natural mechanism for embodying picture attributes than are the local characteristics of the mrf for a range of degradation mechanisms including blurring nonlinear deformations and multiplicative or additive noise the posterior distribution is an mrf with a structure akin to the image model by the analogy the posterior distribution defines another imaginary physical system gradual temperature reduction in the physical system isolates low energy states annealing or what is the same thing the most probable states under the gibbs distribution the analogous operation under the posterior distribution yields the maximum a posteriori map estimate of the image given the degraded observations the result is a highly parallel relaxation algorithm for map estimation we establish convergence properties of the algorithm and we experiment with some simple pictures for which good restorations are obtained at low signal to noise ratios

efficient algorithm for the evaluation of the physical optics scattering by nurbs surfaces with relatively general boundary condition an adaptive integration algorithm is presented for the computation of the physical optics po electric and magnetic field scattered by electrically large objects modeled by non uniform rational b splines nurbs the algorithm is the customization of a more general purpose result that has been recently published by using a unique formulation both impenetrable e g impedance surfaces coated conductors as well as transparent thin sheet materials e g thin dielectric panels or frequency selective surfaces are treated via their fresnel reflection and transmission coefficients the po radiation integral is evaluated over the nurbs parametric domain since most of the computer aided geometric design cagd tools are based on nurbs the proposed algorithm allows a straightforward electromagnetic analysis of the structures by exploiting the standard available geometrical description with no need of generating new geometrical models furthermore the proposed adaptive sampling requires a number of integration points that is found to be drastically smaller than that resulting from standard nyquist based sampling integration algorithms such reduction of the sampling points is achieved by resorting to high frequency technique concepts and allows a significant reduction of the cpu computational burden therefore the algorithm is efficient and particulary suitable for the electromagnetic characterization of real life electrically large objects

propagation modeling over terrain using the parabolic wave equation we address the numerical solution of the parabolic wave equation over terrain using the fourier split step approach the method referred to as a shift map generalizes that of beilis and tappert 1979 who introduced a coordinate transformation technique to flatten the boundary this technique is extended to a wide angle form allowing larger propagation angles with respect to the horizon a new impedance boundary condition is derived for electromagnetic waves incident on a finitely conducting surface that enables solution of the parabolic wave pwe using the previously developed mixed fourier transform it is also shown by example that in many cases of interest the boundary may be approximated by discrete piecewise linear segments without affecting the field solution a more accurate shift map solution of the pwe for a piecewise linear boundary is therefore developed for modeling propagation over digitally sampled terrain data the shift map solution is applied to various surface types including ramps wedges curved obstacles and actual terrain where possible comparisons are made between the numerical solution and an exact analytical form the examples demonstrate that the shift map performs well for surface slopes as large as 10 15 spl deg and discontinuous slope changes on the order of 15 20 spl deg to accommodate a larger range of slopes it is suggested that the most viable solution for general terrain modeling is a hybrid of the shift map with the well known terrain masking knife edge diffraction approximation

multiple time domain diffraction for uwb signals the time domain td solution of the two dimensional multiple diffraction case is investigated the proposed td solution is based on the representation of the inverse fourier transform of the corresponding frequency domain fd solution in closed form as it is given by the uniform theory of diffraction utd and it incorporates the td representation of the higher order diffraction coefficients an algorithm to predict the td diffracted field after an arbitrary number of objects is also presented in the proposed algorithm different types of objects along the propagation path can be applied as well such as absorbing knife edges and metallic or nonperfectly conducting wedges the comparison between the td solution and the numerical inverse fast fourier transform of the fd solution proves the validity of the proposed solution

construction of kac moody superalgebras as minimal graded lie superalgebras and weight multiplicities for kac moody superalgebras we construct a kac moody superalgebra l as the minimal graded lie superalgebra with local part v ge v where g is a smaller lie superalgebra inside l v is an irreducible highest weight g module and v is the contragredient of v we show that the weight multiplicities of irreducible highest weight modules over kac moody superalgebras of finite type and affine type more precisely kac moody superalgebras of type b 0 r b 1 0 r a 4 2r 0 a 2 2r 1 0 and c 2 r 1 are given by polynomials in the rank r the degree of these weight multiplicity polynomials are less than or equal to the depth of weights

a scheme for automatic computation of fock type integrals the evaluation of fock type integrals i e integrals that are formally the fourier transform of a slowly varying factor comprising a quotient of terms containing airy functions and their derivatives is developed in a computationally efficient way the integrals are evaluated by use of a simple extension of a fourier quadrature method that is related to the discrete fourier transform the resulting quadrature is applicable to both surface field and reflected field fock integrals for a lossy impedance surface the computation is believed to be efficient enough to allow the use of these functions in field computations codes where the fock integrals must be computed on an as needed basis

a uniform gtd formulation for the diffraction by a wedge with impedance faces a uniform high frequency solution is presented for the diffraction by a wedge with impedance faces illuminated by a plane wave perpendicularly incident on its edge arbitrary uniform isotropic impedance boundary conditions may be imposed on the faces of the wedge and both the transverse electric te and transverse magnetic tm cases are considered this solution is formulated in terms of a diffraction coefficient which has the same structure as that of the uniform geometrical theory of diffraction utd for a perfectly conducting wedge its extension to the present case is achieved by introducing suitable multiplying factors which have been derived from an asymptotic evaluation of the exact solution given by maliuzhinets when the field point is located on the surface near the edge a more accurate asymptotic evaluation is employed to obtain a high frequency expression for the diffracted field which is suitable for several specific applications the formulation described in this paper may provide a useful rigorous basis to search for a more numerically efficient but yet accurate approximation

greene approximation wide angle parabolic equation for radio propagation this paper presents the greene second order parabolic equation pe solution to model radio propagation over irregular terrain this solution is implemented by finite difference and shift map technique the second order finite difference approach performs well for slopes up to about 15 and discontinuous slope changes up to about 30 which is better than can be achieved using the split step fourier approach the results prove that the greene pe solution has greater propagation angles than does the claerbout pe solution thus the greene second order solution can certainly give better results for large angle propagation over complicated terrain boundaries besides we treat the important problem of accuracy of the different approximations of the pe based propagation models and derive the general pe solutions with respect to the terrain slope for the different approximations it is useful to have collected in one paper the most important pe approximations for radio propagation

a uniform asymptotic solution for the diffraction by a right angled dielectric wedge we propose a uniform asymptotic solution for the field diffracted by a lossless right angled dielectric wedge illuminated by a plane wave at normal incidence the diffraction problem is solved by splitting the observation domain in the inner region of the wedge and the surrounding free space the scattered electric field in each region is assumed to be originated by a set of equivalent electric and magnetic surface currents involved in the well known radiation integral such currents are localized on the interior and exterior faces of the wedge and expressed in terms of the corresponding geometrical optics field useful analytical manipulations and asymptotic evaluations of the resulting integrals allow one to obtain the diffraction coefficients in terms of the fresnel s reflection and transmission coefficients of the structure and the transition function of the uniform geometrical theory of diffraction the related diffracted field compensates the discontinuities of the geometrical optics field and gives total field levels in good agreement with finite difference time domain results

on camina groups of prime power order this paper is devoted to the class of finite p groups with the property that every nontrivial coset of the commutator subgroup consists of conjuw x gate elements this class introduced by camina c has been studied by w x w x w x w x macdonald mann and scoppola mc1 mc2 m ms and also in the w x w x case of class 2 by beisiegel and verardi be v under the name of semiextraspecial p groups our main result here shows that the nilpotency class of such a group is bounded by three on the way we will also obtain a number of structural properties of these groups results from the papers of mcdonald mann and scoppola mentioned above will be stated without proof w x it should be noted that the sketch of proof given for lemma 2 1 in mc2 does not appear to be conclusive not much of the literature seems to be w x affected by this possible gap and those few results in ms that are affected are in fact superseded by our results here the best result that we w x can prove in the direction of lemma 2 1 in mc2 is lemma 5 below

markov chains with exponentially small transition probabilities first exit problem from a general domain i the reversible case we consider general ergodic aperiodic markov chains with finite state space whose transition probabilities between pairs of different communicating states are exponentially small in a large parameter we extend previous results by m i freidlin and a d wentzell fw on the first exit problem from a general domainq in the present paper we analyze the case ofreversible markov chains the general case will be studied in a forthcoming paper we probe in a purely probabilistic way and without using the fw graphical technique some results on the first exit problem from a general domainq containing many attractors in particular we analyze the properties of special domains calledcycles and by using the new concept oftemporal entropy we obtain new results leading to a complete description of the typical tube of trajectories during the first excursion outsideq

peruse a numerical treatment of rough surface scattering for the parabolic wave equation scattering of underwater acoustic signals from real ocean surfaces often does not fit into any of the classical theoretical approaches to the problem thus the need for a numerical approach is clear a novel method is presented that uses a sequence of conformal mappings to locally flatten successive segments of the surface which is assumed piecewise linear and frozen in time each conformal mapping preserves the elliptic form of the reduced wave equation so that in each transformed space the parabolic approximation can be made and the solution advanced one range step using the split step fourier algorithm this peruse pe rough surface numerical method is validated by exhibiting classically predicted bragg angle peaks for single scatter of a plane wave from a sinusoidal surface for multiple rough surface scattering in surface ducts peruse results are analyzed by resolution in depth into the virtual modes of labianca j acoust soc am 53 1137 1147 1973 the resulting modal intensities are then

kronecker product of sp 2n representations using generalised young tableaux using generalised young tableaux the authors obtain an explicit formula for the reduction of the kronecker product of irreducible representations of the symplectic groups this extends a previous work devoted to the case of orthogonal groups

calculation of higher order diffracted fields for multiple edge transition zone diffraction a well known problem in conventional uniform theory of diffraction utd is multiple edge transition zone diffraction here higher order diffracted fields have to be used in order to improve the result and these fields are added by means of a series unfortunately this series will converge slowly for many edges which means that a good enough result might require a large number of terms however by summing up the series properly the computation time can be kept reasonable which will be shown for knife edges the summation technique is though valid for wedges as well

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

an alternative direction decomposition scheme and error analysis for parabolic equation model alternate direction implicit adi method has been widely used to analyze the tunnel propagation or electromagnetic scattering problem for the finite difference method in this paper a novel alternate direction decomposition add scheme is proposed to simplify the split step fourier transform ssft algorithm in parabolic equation pe model the spatial matrix of electric field is decoupled into two subarrays by add ssft algorithm which is used to decrease the computational burden for 3 d pe problems in a large scale environment after the integration order exchanged also the phase error of this approximation is analyzed to show the algorithm performance along with several numerical results including the tunnel cases with traditional modal theory and they are provided to validate and calibrate the pe model and demonstrate the efficiency and accuracy of the proposed algorithm moreover the scattering computation is investigated and compared with multilevel fast multipole method to show that add ssft is appropriate for solving the scattering problems for the electrical large target

polynomial approximation of maliuzhinets function approximate formulas for the maliuzhinets functions which give the solutions for the wedge diffraction with face impedance are derived using the chebyshev polynomials for various wedge angles the coefficients of the polynomials are listed in a table although the formulas are derived for real variables it is found that they can be used for the restricted range of complex variables since the validity of the formulas depends on both real and imaginary part when applied to complex variables the equi error contour of the formulas is given for the special case where the exact solution is available an alternative but empirical formula for maliuzhinets functions is also obtained which can be applied to arbitrary wedge angles

utd diffraction coefficients for higher order wedge diffracted fields an asymptotic expansion of a multiple integral for knife edge diffraction is derived the expansion expresses the field in terms of single edge diffraction by considering this expression a similar expansion for higher order utd diffracted fields is proposed by means of a set of transition functions the result removes some of the shortcomings of the original set out of the utd when the incident field is not a ray optical field

diffraction by lossy dielectric wedges using both heuristic utd formulations and fdtd an improvement of the uniform theory of diffraction utd coefficient for the case of a lossy dielectric wedge when a transmitted ray exists is presented we elaborated two new terms that are added to the classical utd diffraction coefficient so that we obtain continuity of the total field this new utd formulation is compared to a numerical method based on finite difference time domain fdtd we outline the adaptation of the fdtd grid calculation which was necessary to isolate only one edge diffraction and to treat two dimensional 2 d structures with two infinite sides this comparison allows one to conclude that the new diffraction coefficient is relevant for the case of a lossy dielectric wedge then we present a comparison between two different versions of the utd diffraction coefficient based on single or multiple reflection in the case of a dielectric slab thus we can conclude to the significance of the multipaths for modeling dielectric structures finally we analyze the results obtained with two consecutive wedge vertices in order to show that the slope diffraction related to the doubly diffracted field allows one to predict the field behind the structure when the transmitted field does not exist

high frequency green s function for planar periodic phased arrays with skewed grid and polygonal contour the derivation of an uniform high frequency asymptotic representation of the array green s function agf for finite planar periodic phased arrays with arbitrary polygonal contour and skewed grid is presented this result generalizes those obtained in a series of recent papers in the case of rectangular phased arrays with rectangular grid for the treatment of the high frequency phenomena the actual finite array is rigorously decomposed in terms of canonical constituents i e infinite semi infinite and sectoral arrays the final result is a representation of the agf in terms of spatially truncated floquet waves fws and fw induced diffracted fields arising from the edges and vertices of the polygonal rim of the array consequently the number of field contributions necessary to reconstruct the total field is independent of the number of elements of the array leading to a very efficient algorithm a series of numerical results is provided to demonstrate the effectiveness of the high frequency representation

phase transition and landscape statistics of the number partitioning problem the phase transition in the number partitioning problem npp i e the transition from a region in the space of control parameters in which almost all instances have many solutions to a region in which almost all instances have no solution is investigated by examining the energy landscape of this classic optimization problem this is achieved by coding the information about the minimum energy paths connecting pairs of minima into a tree structure termed a barrier tree the leaves and internal nodes of which represent respectively the minima and the lowest energy saddles connecting those minima here we apply several measures of shape balance and symmetry as well as of branch lengths barrier heights to the barrier trees that result from the landscape of the npp aiming at identifying traces of the easy hard transition we find that it is not possible to tell the easy regime from the hard one by visual inspection of the trees or by measuring the barrier heights only the difficulty measure given by the maximum value of the ratio between the barrier height and the energy surplus of local minima succeeded in detecting traces of the phase transition in the tree in addition we show that the barrier trees associated with the npp are very similar to random trees contrasting dramatically with trees associated with the p spin glass and random energy models we also examine critically a recent conjecture on the equivalence between the npp and a truncated random energy model

a slope utd solution for a cascade of multishaped canonical objects the case of multiple diffraction from a cascade of two dimensional multi shaped canonical objects such as wedges knife edges and cylinders is investigated a heuristic slope uniform theory of diffraction utd solution is presented that can give accurate and uniform field predictions at any area point and for any kind of environment the solution is then compared against measurements taken in an anechoic chamber and a very good agreement is observed furthermore an algorithm for implementing the proposed utd solution to rural area field predictions is also presented based on a novel algorithm for optimum fit of canonical shapes to the terrain irregularities comparisons between the model and measurements taken in real terrain profile show a very good agreement and uniform field predictions the importance of the choice of the used canonical object to terrain modeling is also discussed since different diffraction patterns are observed for different combination of the canonical objects

electromagnetic scattering from electrically large coated flat and curved strips entire domain galerkin formulation efficient numerical solutions are presented for electromagnetic scattering for classes of electrically large coated perfectly conducting strips which are flat or curved the formulation is based on the solution of a coupled system of electric and magnetic field integral equations using the method of moments mm entire domain galerkin representations for the currents are used on the surface of the coating and at the coating conductor interface the resulting symmetric matrix equation is well conditioned and admits rapid accurate solutions numerical results are presented for various coating thicknesses strip widths and curvatures for the transverse electric te and transverse magnetic tm cases the convergence of the galerkin solution is examined as a function of these parameters the effect of the edge approximation on the choice of expansion functions is discussed the numerical results are compared with experimental measurements

wide angle shift map pe for a piecewise linear terrain a finite difference approach a wide angle parabolic wave equation solution is presented using shift map and finite difference techniques the corresponding split step fourier solution is well known the solution using finite difference technique where the standard parabolic wave equation is modified into the so called claerbout equation allowing propagation angles up to 45deg from the paraxial direction is also well known here we present an extension to that solution in which the shift map technique is incorporated into the finite difference scheme allowing a varying terrain to be considered the result is a solution that corresponds to the well known split step solution which is believed to perform well for terrain slopes up to 10deg 15deg and discontinuous slope changes on the order of 15deg 20deg this solution is a first order one with respect to the terrain slope however when using the finite difference technique it is also possible to find a second order solution with respect to the terrain slope this new solution performs well for slopes up to about 15deg and discontinuous slope changes up to about 30deg which is an improvement

simple expressions for a function occurring in diffraction theory the maliuzhinets integral function arises in connection with diffraction by a half plane two simple expressions are derived which when used in conjunction with known identities serve to approximate the function to a high degree of accuracy throughout the entire complex plane

analytic expressions for a function occurring in diffraction theory simple expressions are derived for the maliuzhinets integral function which arises in connection with diffraction by an impedance wedge the given expressions allow the computation of the function with remarkable accuracy throughout the entire complex plane for any exterior wedge angle

diffraction by a thin dielectric half plane a uniform diffraction coefficient is derived for a thin dielectric half plane illuminated by an h polarized incident plane wave this is accomplished by modeling the half plane using a pair of resistive and modified conductive sheets and then solving the resulting integral equations via the wiener hopf technique in contrast to a recently published solution the one presented here satisfies reciprocity and the numerical results which are included confirm its accuracy

asymptotic high frequency green s function for a large rectangular planar periodic phased array of dipoles with weakly tapered excitation in two dimensions this paper deals with the derivation and physical interpretation of a uniform high frequency representation of the green s function for a large planar rectangular phased array of dipoles with weakly varying excitation thereby our earlier published results valid for equiamplitude excitation and those for tapered illumination in one dimension are extended to tapering along both dimensions including dipole amplitudes tending to zero at the edges as previously the field obtained by direct summation over the contributions from the individual radiators is restructured into a double spectral integral whose high frequency asymptotic reduction yields a series of propagating and evanescent floquet waves fws together with corresponding fw modulated diffracted fields which arise from fw scattering at the array edges and vertexes to accommodate the weak amplitude tapering new generalized periodicity modulated edge and vertex slope transition functions are introduced accompanied by a systematic procedure for their numerical evaluation special attention is given to the analysis and physical interpretation of the complex vertex diffracted ray fields a sample calculation is included to demonstrate the accuracy of the asymptotic algorithm the resulting array green s function forms the basic building block for the full wave analysis of planar weakly amplitude tapered phased array antennas and for the description of electromagnetic radiation and scattering from weakly amplitude tapered rectangular periodic structures

a hybrid spectral spatial method to evaluate the active green s function of large planar rectangular arrays a combined asymptotic numerical algorithm the article illustrates a formulation to evaluate the array green s function agf of large finite planar phased array for observation points on the array plane possibly close to the array contour the procedure is based on the agf representation in terms of a double spectral integral whose integration paths are properly deformed to have an exponential attenuation of the integrand the diffraction integral is evaluated numerically for point close to the array edges while an asymptotic treatment is proposed far from the edges this latter comprises higher order contributions thanks to the convergence properties the final algorithm is numerically accurate stable and more efficient with respect to the individual element summation for large arrays it also constitutes the basic step for the efficient evaluation of the agf of a multilayer environment

a limit theorem for a class of inhomogeneous markov processes pour un processus de markov xh irreductible et aperiodique avec un espace d etats fini et avec un taux de transition q ij t x t devient homogene dans le cas q ij t q ij et admet une loi limite qui est independante de x o le but de cet article est de demontrer qu un resultat similaire peut etre obtenu dans le cas de processus de markov non homogenes avec un certain type de taux de transition

a curvilinear coordinate based split step parabolic equation method for propagation predictions over terrain propagation of radiowaves over irregular terrain and in an inhomogeneous atmosphere is solved by the parabolic equation method using the split step fourier algorithm on a terrain conformal mesh a piecewise continuous coordinate system is generated by the specification of 1 the terrain profile shape at discrete points and 2 an upper height the resulting mesh is conformal to the terrain at the lower boundary and gradually flattens off at the maximum height in addition to preserving the number of points on any vertical line between the terrain and the maximum height from one range step to another the coordinate transformation used in the paper produces a correction term in the refractive index whose gradient diminishes with height as a result the sampling requirements over steep terrain are relaxed when compared to the beilis tappert transformation formulation and results are given both for the horizontal and vertical polarizations

the weyl groups and weight multiplicities of the exceptional lie groups the weyl group wg of each exceptional simple lie group g is described in detail its structure is defined in terms of its coset decomposition with respect to the weyl group wh of a classical semi simple lie group h embedded naturally in g the concepts of g dominance and g equivalence are defined and used to determine from the character formula of weyl the branching rule associated with the restriction of group elements from g to h the weyl group wg is used further to impose constraints on both the branching multiplicities for g to h and the weight multiplicities of g these constraints are used to evaluate the weight multiplicities of f4 e6 e7 and e8 together with the branching multiplicities for e8 to so 16

generalization of utd double diffraction to the case of impenetrable wedges with relatively general boundary conditions in the framework of the uniform geometrical theory of diffraction utd a novel high frequency solution is provided which allows the description of the rays doubly diffracted by a pair of impenetrable straight wedges with relatively general boundary conditions under a spherical wave illumination such a utd solution is heuristically derived by imposing the continuity of the total field at the shadow boundaries of the first order i e geometrical optics utd wedge single diffracted ray field utd solution it is the extension to the case of double diffraction of the first order solutions proposed by luebbers and by burnside and burgener it allows a uniform description of the field at any observation point including those aspects where transition regions overlap and the cascading of utd single wedge diffraction coefficients fails the proposed solution does not consider surface wave effects and for the special case of two perfectly conducting wedges it reduces to the formulation present in the literature of which it can be considered a generalization indeed it requires the same special functions involved in utd double diffraction by two perfectly conducting wedges the double diffracted ray field is expressed in analytical closed form providing a deeper understanding of the physics involved in double diffraction and a new powerful tool to be employed in any modern ray based code

a novel utd type diffraction coefficient for a straight edge in a curved screen a novel uniform theory of diffraction utd type diffraction coefficient for a straight edge in a cylindrically curved screen valid for both convex and concave side wave incidence is obtained via uniform asymptotic theory of diffraction uat formulations the proposed curved screen utd type coefficient csutd allows uniform description of the fields at all aspect angles including the grazing incidence case where utd coefficient based predictions deviate sharply from available measurements when it is applied to the problem of monostatic radar cross section rcs of cylindrically curved plate

the numerical steepest descent path method for calculating physical optics integrals on smooth conducting quadratic surfaces in this paper we use the numerical steepest descent path nsdp method to analyze the highly oscillatory physical optics po integral on smooth conducting parabolic surfaces including both monostatic and bistatic cases quadratic variations of the amplitude and phase functions are used to approximate the integrand of po integral then the surface po integral is reduced into several highly oscillatory line integrals by invoking the nsdp method these highly oscillatory po line integrals are defined on the corresponding nsdps furthermore the critical point contributions for the po integral are exactly extracted and represented based on the nsdps the proposed nsdp method for calculating the po integral on the smooth conducting surfaces is frequency independent and error controllable compared with the traditional asymptotic expansion approach the nsdp method significantly improves the po integral accuracy by around two digits when the working wave frequencies are not extremely large numerical results are given to validate the nsdp method

groups with two extreme character degrees and their normal subgroups we study the finite groups g for which the set cd g of irreducible complex character degrees consists of the two most extreme possible values that is 1 and g z g 1 2 we are easily reduced to finite p groups for which we derive the following group theoretical characterization they are the p groups such that ig z g l is a square and whose only normal subgroups are those containing g or contained in z g by analogy we also deal with pgroups such that g z g i p2n 1 is not a square and we prove that cd g 1 pn if and only if a similar property holds for any n g either g n or nz g z g p the proof of these results requires a detailed analysis of the structure of the p groups with any of the conditions above on normal subgroups which is interesting for its own sake it is especially remarkable that these groups have small nilpotency class and that if the nilpotency class is greater than 2 then the index of the centre is small and in some cases we may even bound the order of g

noise attenuation by a hard wedge shaped barrier this paper is concerned with the problem of sound screening by a wedge like barrier the sound source is assumed to be point like and the receiver is located in the shadow of the source sound field so that according to geometrical optics only the field diffracted by the edge of the barrier is considered first for the hard wedge in space three models are used for calculating the amplitude of the edge diffracted field these are the uniform theory of diffraction utd the hadden pierce model both in the frequency domain and the biot tolstoy theory of diffraction which is a time domain formulation it is first shown that even at relatively low frequencies the frequency domain models perform quite satisfactorily as compared to the exact time domain theory hence and due to its relative simplicity the utd is proposed as an accurate calculation scheme for solving problems with edge diffraction by hard wedges it is also proved from theoretical calculations that the amplitude of the edge diffracted field increases for an increasing angle of the wedge and consequently the hard half plane gives the lowest field amplitude in the shadow zone some applications are then considered for evaluating the performance of a barrier on a flat ground either completely hard or with mixed homogeneous boundary conditions an improvement of the scheme for calculating the sound field in the all hard case is achieved through considering the multiple diffraction in this case only to the second order between the top of the wedge barrier and its base the results show that for usually occurring situations increasing the angle of the hard wedge barrier affects negatively its efficiency through diminishing its insertion loss these conclusions are also supported by the results of some experimental measurements conducted at a scale model level

markov chains with rare transitions and simulated annealing we consider approximate stationary markov chains in which the entries of the one step transition probability matrix are known to be of different orders of magnitude and whose structure that is the orders of magnitude of the transition probabilities does not change with time for such markov chains we present a method for generating order of magnitude estimates for the t step transition probabilities for any t we then notice that algorithms of the simulated annealing type may be represented by markov chains which are approximately stationary over fairly long time intervals using our results we obtain a characterization of the convergent cooling schedules for the most general class of algorithms of the simulated annealing type