optimality conditions with state constraints for semilinear systems determined by operator valued measures abstract in this paper we develop the necessary conditions of optimality for a class of distributed parameter systems partial differential equations determined by operator valued measures and controlled by vector measures based on some recent results on existence of optimal controls from the space of vector measures we develop necessary conditions of optimality for a class of control problems the main results are the necessary conditions of optimality for problems without state constraints and those with state constraints also a conceptual algorithm along with a brief discussion of its convergence is presented

intertwining operators and unitary representations i abstract for a linear semisimple lie group with a compact cartan subgroup the authors obtain formulas for the action of intertwining operators on certain subspaces of standard induced representations these formulas provide explicit limitations on the pool of candidates for irreducible unitary representations since the only possible invariant inner product is given by such an intertwining operator

axisymmetric elastic waves in weakly coupled layered media of infinite radial extent abstract the propagation of axisymmetric waves in layered periodic elastic media of infinite radial extent is investigated hankel and laplace transforms are employed to convert radial and time dependence of displacement and stress within a layer to frequency and radial wavenumber continuity of stress and displacement is imposed at the interface between layers yielding transfer matrices the structure of propagation and attenuation zones pz s and az s of the system with infinite number of layers is studied when the ratios of shear and longitudinal mechanical impedances t l of the two layers are large the pz s of the layered system become narrow approaching certain limiting curves in the frequency wavenumber plane for large t l an asymptotic analysis is employed analytically to approximate the width of the pz s numerical computations of pz s are given which are in good agreement with the analytical predictions the spacing of the resonance points of a layered system with finite number of layers is found to depend mainly on the structure of the pz s of the corresponding system with an infinite number of layers

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

semismooth newton methods for optimal control of the wave equation with control constraints in this paper optimal control problems governed by the wave equation with control constraints are analyzed three types of control action are considered distributed control neumann boundary control and dirichlet control and proper functional analytic settings for them are discussed for treating inequality constraints semismooth newton methods are discussed and their convergence properties are investigated in the case of distributed and neumann control superlinear convergence is shown for dirichlet boundary control superlinear convergence is proved for a strongly damped wave equation for numerical realization a space time finite element discretization is discussed numerical examples illustrate the results

a minimal parameter adaptive notch filter with constrained poles and zeros a new algorithm is presented for adaptive notch filtering and parametric spectral estimation of multiple narrow band or sine wave signals in an additive broad band process the algorithm is of recursive prediction error rpe form and uses a special constrained model of infinite impulse response iir with a minimal number of parameters the convergent filter is characterized by highly narrow bandwidth and uniform notches of desired shape for sufficiently large data sets the variances of the sine wave frequency estimates are of the same order of magnitude as the cramer rao bound results from simulations illustrate the performance of the algorithm under a wide range of conditions

near field transient axisymmetric waves in layered structures effects of weak coupling axisymmetric stress wave transmission through the leading layers of layered structures of infinite radial but finite axial extent is numerically studied by employing two different computational approaches a technique based on the numerical inversion of double integral transformations dit and a finite element fe analysis considering the first approach careful selections of the limits of the numerical inversions and the sampling rates are required in order to overcome inherent numerical instabilities associated with exponential dichotomy this type of numerical instability is more evident in layered media with weak coupling in such systems direct multiplications of layer transfer matrices are avoided by employing a global scheme to assemble well conditioned global transfer matrices moreover the specific structure of the propagation and attenuation zones of the structure are taken into account for increasing the efficiency and effectiveness of the transfer matrix manipulations satisfactory agreement between the dit and fe numerical results is observed at least for early times close to the region of application of the external pressure the fe simulations suffer from the discretization of the applied load node to node oscillations and reflections from infinite elements silent boundaries using the aforementioned numerical techniques transient wave transmission in two layered systems one with weak and one with strong interlayer coupling is considered and the effects of weak coupling on the wave transmission is studied we show that at early times weak coupling results in stress localization in the region close to the applied pressure a result which can have potential application in the use of layered media as shock isolators 1997 by john wiley sons ltd

a note on minimizing the prediction error when the zeros are restricted to the unit circle it is shown that the problem of minimizing the prediction error under the constraint that the zeros of the predictor polynomial be on the unit circle can be solved by using a modification of the levinson algorithm consequences of this result are discussed

transient elastic waves in finite layered media one dimensional analysis transient waves induced by a trapezoidal pulse are studied for a class of finite periodic elastic media using a modal method for weakly coupled systems approximate relations are derived for peak compressive stress and group velocity peak stress behind the wave front is proportional to transmissibility the product of frequency interval of the first propagating zone and time interval of the equivalent rectangular pulse which conserves impulse along the stack stress is attenuated or magnified exponentially depending on transmissibility finally the effect on peak stress of disorder in the stack is investigated

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

adjoint optimization of one dimensional hyperbolic equations with constrained periodic boundary conditions abstract this paper presents a continuous adjoint based optimization theory for a general closed loop transport hyperbolic model controlled via a periodic boundary control to minimize a cost functional the periodic boundary control is subject to a nonlinear differential equation constraint thus resulting in a coupling between the hyperbolic equation and the ordinary differential equation variational principles are used to derive the pontryagin s minimum principle for optimality that results in a dual adjoint system a numerical optimization method is implemented using the adjoint based second order gradient method to solve for the optimal trajectory of the control numerical methods for solving the hyperbolic equation using an explicit scheme wave splitting method and for solving the adjoint equation using an implicit scheme and a quasi steady state method are described

quantum generalized harish chandra isomorphisms abstract we give a different proof of generalized harish chandra isomorphisms proven by khoroshkin nazarov and vinberg 9 and joseph 6 7 as well as of their analogues in the quantum case

unitary representations induced from maximal parabolic subgroups abstract it is known that the problem of classifying the irreducible unitary representations of a linear connected semisimple lie group g comes down to deciding which langlands quotients j man v are infinitesimally unitary here man is any cuspidal parabolic subgroup is any discrete series or nondegenerate limit of discrete series representation of m and v is any complex valued linear functional on the lie algebra of a satisfying certain positivity and symmetry properties the authors determine which langlands quotients are infinitesimally unitary under the conditions that g is simple that dim a 1 and that g is neither split f 4 nor split g 2

waves in periodically layered media a comparison of two theories two distinct asymptotic theories for wave propagation in one dimensional inhomogeneous media are compared in their common domain of validity one theory due to santosa and symes applies to long wavelength propagation in periodic media with arbitrary contrast in material properties the o doherty anstey theory on the other hand is explicitly intended to describe time dependent wave propagation in media that are finely layered but characterized by relatively small reflectivity the two theories are compared in detail in the doubly asymptotic limit of low frequency wave propagation in periodic media with small contrasts the equivalence is demonstrated by deriving the asymptotic limit of the dispersion curve of the fundamental bloch wave according to each theory the analysis for the o doherty anstey theory sheds some new light on its strengths and limitations particularly in periodic media it is shown that it correctly predicts the leading order dispersion curve of the first branch for frequencies of

indefinite intertwining operators for a wide class of linear connected semisimple lie groups one obtains formulas limiting the langlands parameters of irreducible unitary representations obtained from maximal parabolic subgroups the formulas relate unitarity to the number of roots satisfying certain conditions some evidence is presented that the formulas are sharp the results confirm aspects of conjectures that relate unitary parameters to cohomological induction

optimal boundary control of hyperbolic equations with pointwise state constraints in this paper we consider dynamic optimization problems for hyperbolic systems with boundary controls and pointwise state constraints in contrast to parabolic dynamics such systems have not been sufficiently studied in the literature the reason is the lack of regularity in the case of hyperbolic dynamics we present necessary optimality conditions for both neumann and dirichlet boundary control problems and discuss differences and relationships between them

neumann boundary control of hyperbolic equations with pointwise state constraints we consider optimal control problems for hyperbolic equations with controls in neumann boundary conditions with pointwise constraints on the control and state functions focusing on the multidimensional wave equation with a nonlinear term we derive new necessary optimality conditions in the form of a pointwise pontryagin maximum principle for the state constrained problem under consideration our approach is based on modern methods of variational analysis that allow us to obtain refined necessary optimality conditions with no convexity assumptions on integrands in the minimizing cost functional

dispersion of stress waves in one dimensional semi infinite weakly coupled layared systems an analytical study of the dispersion of transient stress waves in the first layer of a weakly coupled semi infinite bi layered system is carried out the analysis is based on approximate fourier transform inversions and makes use of the fact that the weakly coupled system possesses small propagation zones pzs in the frequency domain low and high frequency asymptotic approximations to the transient waves are computed taking into account frequency components of the transfer function in the first and second pzs respectively the derived analytic expressions are superpositions of non oscillating terms and convolution integrals with decaying oscillatory kernels it is shown that depending on the frequency and the amplitude of the convolution kernels the dispersed waves overshoot or undershoot the applied impulsive excitation this result is of significant practical importance in the design of layered systems as stress attenuators

knapp wallach szego integrals and generalized principal series representations the parabolic rank one case abstract to each discrete series representation of a connected semisimple lie group g with finite center a g equivariant embedding into a generalized principal series representation is given this representation is induced from specified parameters on a maximal parabolic subgroup of g and the mapping is defined by an integral formula analogous to the szego integral introduced by knapp and wallach for a minimal parabolic subgroup in a limiting case embeddings of limits of discrete series representations are obtained and used to exhibit a reducibility theorem

self inversive hilbert space operator polynomials with spectrum on the unit circle abstract hilbert space operator polynomials with self inversive structure are studied if the inner numerical radius of an associated polynomial is greater than or equal to one then the spectrum lies on the unit circle and consists of normal approximate characteristic values

adapted pairs in type a and regular nilpotent elements abstract let g be a simple lie algebra over an algebraically closed field k of characteristic zero and g its adjoint group a biparabolic subalgebra q of g is the intersection of two parabolic subalgebras whose sum is g the algebra sy q of semi invariants on q of a proper biparabolic subalgebra q of g is polynomial in most cases in particular when g is simple of type a or c on the other hand q admits a canonical truncation q such that sy q sy q y q where y q denotes the algebra of invariant functions on q an adapted pair for q is a pair h q q such that is regular in q and a d h in joseph 2008 9 adapted pairs for every truncated biparabolic subalgebra q of a simple lie algebra g of type a were constructed and then provide weierstrass sections for y q in q these weierstrass sections are linear subvarieties v of q such that the restriction map induces an algebra isomorphism of y q onto the algebra of regular functions on v the main result of the present work is to show that for each of the adapted pairs h constructed in joseph 2008 9 one can express not quite uniquely as the image of a regular nilpotent element y of g under the restriction map g q this is a significant extension of joseph and fauquant millet 2011 12 which obtains this result in the rather special case of a truncated biparabolic of index one observe that y must be a g translate of the standard regular nilpotent element defined in terms of the already chosen set of simple roots consequently one may attach to y a unique element of the weyl group w of g ultimately one can then hope to be able to describe adapted pairs in general that is not only for g of type a through the weyl group

the poles of symmetric linear prediction models lie on the unit circle symmetric linear prediction models have their poles on the unit circle a simple proof of this result is presented

quasicrystals and icosians a family of quasicrystals of dimensions 1 2 3 4 governed by the root lattice e8 is constructed the use of the icosian ring found in the quaternions with coefficients in q square root 5 allows simultaneous interpretation of the construction both in physical space and as a result of the standard cut and projection method in double dimension icosians are seen to provide a natural co ordination scheme for these quasicrystals nested sequences of quasicrystals form systems whose symmetries are all derivable from inflational and reflective symmetries directly related to the arithmetic of the icosians the use of coxeter diagrams clarifies the amazing relationship of e8 and quasicrystal symmetries and leads to the fundamental chain e8 contains implies d6 contains implies a4 contains implies a1 a1 that underlies five fold symmetry in quasicrystals decomposition of quasicrystals into concentric shells and a counting formula for the cardinalities of these shells is discussed

pontryagin s principle for state constrained control problems governed by parabolic equations with unbounded controls this paper deals with optimal control problems governed by semilinear parabolic equations with pointwise state constraints and unbounded controls under some strong stability assumption we obtain necessary optimality conditions in the form of a pontryagin s minimum principle in qualified form a pontryagin s principle in nonqualified form is also proved without any stability condition

sufficient optimality conditions for dirichlet boundary control of wave equations we study optimal control problems for wave equations focusing on the multidimensional wave equation with control functions in the dirichlet boundary conditions under pointwise control and we admit state by assuming weak hypotheses constraints

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

exact analytical solutions for elastodynamic impact abstract we consider the one dimensional impact problem in which a semi infinite flyer collides with and adheres to the front face of a stationary target plate of finite thickness with the back face of the target bonded to another semi infinite medium all three bodies are assumed to be linear elastic and homogeneous our interest is in explicit expressions for the stress and velocity in the target at all times after impact the analysis of this problem is simplified by reducing it to an initial boundary value problem for the target only which is solved by combining the d alembert solution of the wave equation with the laplace transform method an appropriate impact boundary condition is required on the front face of the target in the literature this is usually taken as a prescribed step in stress or velocity but the correct boundary condition involves a linear combination of the unknown stress and velocity at the impact face our solutions are expressed in an apparently new compact form involving the floor or greatest integer function the results are amenable to asymptotic analysis in particular solutions for stress free or rigid back faces follow easily as limiting cases of the backing impedance and the long time asymptotes of stress and velocity in the target are seen to be independent of the target s elastic properties all of our results are corroborated by derivation of exact discrete solutions from recursive equations for the impact problems

a pontryagin maximum principle for infinite dimensional problems a basic idea of the classical approach for obtaining necessary optimality conditions in optimal control is to construct suitable needle like control variations we use this idea to prove the main result of the present paper a pontryagin maximum principle for infinite dimensional optimal control problems with pointwise terminal constraints in arbitrary banach state space by refining the classical variational technique we are able to replace the differentiability of the norm of the state space guaranteed by the strict convexity of its dual norm which is assumed in the known results by a separation argument we also drop another key assumption which is common in the existing literature on infinite dimensional control problems that the set of variations in the state space of the state trajectory s endpoint resulting from the control variations be finite codimensional instead we require only that it has nonempty interior in its closed affine hull as an application of the abstract result we present an illustrative example an optimal control problem for an age structured system with pointwise terminal state constraints

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

a dispersive effective medium for wave propagation in periodic composites the problem of wave propagation through a periodic medium is considered it is assumed that the ratio between the cell size and the shortest wavelength of the initial disturbance is small within this regime an effective medium model of the propagation phenomena is obtained using the bloch expansion the effective medium obtained is shown to be dispersive it is known that a dominant feature of the transient wave train in a periodic medium is that of dispersion it is shown that our model recovers homogenization formally when the cell size approaches zero explicit formula for the constants appearing in the effective medium equation is derived for the case of one dimension the accuracy of the resulting approximation is assessed in numerical simulations

pontryagin s principle for state constrained boundary control problems of semilinear parabolic equations this paper deals with state constrained optimal control problems governed by semilinear parabolic equations we establish a minimum principle of pontryagin s type to deal with the state constraints we introduce a penalty problem by using ekeland s principle the key tool for the proof is the use of a special kind of spike perturbations distributed in the domain where the controls are defined conditions for normality of optimality conditions are given

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

matrix based higher order spectral analysis for three wave coupling processes composite linear and quadratic systems produce three wave coupling when stimulated by random phase input sinusoids due to the nonlinearity of the system the output frequencies are arithmetically related to the input using third order cumulant statistics and their associated bispectrum techniques are devised based on phase insensitive matrix structures for detection and frequency estimation of coupling frequencies the separation of the third order cumulant series into symmetric and skew symmetric portions allows one to exploit their characteristic eigendecompositions for frequency estimation after symmetrization biphases can be easily extracted as coefficients of the cumulant sequence using a generalized eigenvector representation one can relate symmetric and skew symmetric bases by a subspace rotation algorithm biphases can be estimated directly from generalized eigenvalues of the matrix pencil formed by symmetric and skew symmetric matrices the dimensionality of the matrices can be reduced through the use of cumulant projections that yield a slice of the bispectrum the radon transform procedure is related to bispectral processing through an isotropic radial slice volterra filter the compact third order kronecker product matrix formulation and algorithms for coupling frequency estimation can also be converted for use in biphase estimation simulations showing the performance of the above procedures are also presented for both synthetic and biomedical time series

an exact solution for the one dimensional elastic wave equation in layered media a new approach to the direct scattering problem in one dimensional inhomogeneous bodies is presented a general formulation considering density wave speed and cross section variation is introduced and a compact form for the wave equation in the characteristic plane with a generalized acoustical impedance profile as a parameter is established with the aid of a nonsingular variable transformation the wave propagation pattern in layered or discretized continuous media is then examined and an exact algebraic solution for the reflected wave is obtained the resulting formula is adequate to an arbitrary input pulse furnishing a quick computation of all multiple reflection contributions to the final echo

on the liouville integrability of the periodic kostant toda flow on matrix loops of level k in this work we consider the periodic kostant toda flow on matrix loops in mathfrak s mathfrak l n mathbb c of level k which correspond to periodic infinite band matrices with period n with lower bandwidth equal to k and fixed upper bandwidth equal to 1 with 1 s on the first superdiagonal we show that the coadjoint orbits through the submanifold of such matrix loops can be identified with those of a finite dimensional lie group which appears in the form of a semi direct product we then characterize the generic coadjoint orbits and obtain an explicit global cross section for such orbits we also establish the liouville integrability of the periodic kostant toda flow on such orbits via the construction of action angle variables

sheets and hearts of prime ideals in enveloping algebras of semisimple lie algebras abstract consider the enveloping algebra u g of a complex semisimple lie algebra g the heart of a prime ideal i of u g is the center of the total ring of fractions of u g i this is an extension field of the field of fractions of the center of u g i let d be the degree of this field extension an old problem of j dixmier asked whether d 1 a recent paper of the second author r rentschler a negative answer to the problem of dixmier on hearts of prime quotients of enveloping algebras preprint 2004 gave a negative answer by an example in sl 4 the present paper provides many more examples involving the so called sheets of primitive ideals introduced and studied by a joseph and the first author in w borho a joseph sheets and topology of primitive spectra for semisimple lie algebras j algebra 244 2001 76 167 a sheet corresponds to a prime ideal i which has a heart of degree d the main result of this paper is that d equals the covering degree of the sheet as introduced in w borho a joseph sheets and topology of primitive spectra for semisimple lie algebras j algebra 244 2001 76 167 8 7

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 stability of periodic orbits of high dimensional autonomous hamiltonian systems we study the stability of periodic orbits of autonomous hamiltonian systems with n 1 degrees of freedom or equivalently of 2n dimensional symplectic maps with n 1 we classify the different stability types introducing a new terminology which is perfectly suited for systems with many degrees of freedom since it clearly reflects the configuration of the eigenvalues of the corresponding monodromy matrix on the complex plane the different stability types correspond to different regions of the n dimensional parameter space s defined by the coefficients of the characteristic polynomial of the monodromy matrix all the possible direct transitions between different stability types are classified and the corresponding transition hypersurface in s is determined the dimension of the transition hypersurface is an indicator of how probable to happen is the corresponding transition as an application of the general results we consider the well known cases of hamiltonian systems with two and three degrees of freedom we also describe in detail the different stability regions in the three dimensional parameter space s of a hamiltonian system with four degrees of freedom or equivalently of a six dimensional symplectic map 2001 elsevier science b v all rights reserved

dirichlet boundary control problems for parabolic equations with quadratic cost analyticity and riccati s feedback synthesis riccati type feedback synthesis of optimal controls for dirichlet boundary parabolic equations is considered the functional cost penalizes the l2 energy over 0 t of state and control action u and also final state y t at t t this latter fact makes the functional cost discontinuous on the space of admissible controls l2 0t after overcoming some technical difficulties related to the above mentioned discontinuity we prove that the optimal control uo can be written in the desired feedback form uo t cp t yo t for all 0 t t

pontryagin s principle for local solutions of control problems with mixed control state constraints this paper deals with optimal control problems of semilinear parabolic equations with pointwise state constraints and coupled integral state control constraints we obtain necessary optimality conditions in the form of a pontryagin s minimum principle for local solutions in the sense of l p p leq infty

z transform methods for the optimal design of one dimensional layered elastic media in this work we develop a finite trigonometric series representation for the stress in a multilayered goupillaud type elastic strip subjected to transient heaviside loading on one end while the other end is held fixed this representation is achieved by means of the z transform method and involves the so called base angles generally different layered designs could share the same set of base angles and the more layers the design has the more base angles are expected necessary conditions for the base angles and design parameters for any given design are described as a result of the stress representation we are able to identify optimal layered designs which provide the smallest stress amplitude for two and three layered designs for which the coefficients of the stress representation are easy to find the optimization results are achieved using a custom made discrete optimization technique applied in a p velo and g a gazonas int j solids structures 40 2003 pp 6417 6428 for other laye

asymptotic impact behavior of goupillaud type layered elastic media abstract in this paper we use the method of riemann invariants to develop a system of recurrence relations for the one dimensional elastic impact problem in which a semi infinite flyer collides with and adheres to a stationary goupillaud type layered elastic target plate of finite thickness that is bonded to a semi infinite half space we derive explicit expressions for the long time asymptotes of stress and particle velocity that are independent of the layered target s elastic properties this result is verified by numerical simulations of impact into layered targets with periodic or randomly distributed elastic properties and holds when the half space backing the target is replaced by either free or fixed boundary conditions our discrete recurrence relation solutions for impact into layered media generalize our recent analytical results concerning impact into homogeneous single layer targets as well as prior discrete recurrence relation solutions for stress wave propagation in goupillaud type layered elastic media