on idempotent affine mappings let v be a vector space and end v the semigroup of endomorphisms of v an affine mapping of v is a map a v v given by x a x a where a belongs to end v and a is some element of v let v be the semigroup of affine mappings of v let e denote the non injective idempotents of end v and let denote the idempotents of v in this paper is determined in terms of e when end v consists of all endomorphisms of v and when end v only contains the continuous endomorphisms in which case we restrict v to being an inner product space

classification of semisimple algebraic monoids let x be a semisimple algebraic monoid with unit group g associated with e is its polyhedral root system x c where x x t is the character group of the maximal torus t c g c x t is the set of roots and c x t is the character monoid of t c e zariski closure the correspondence e x c is a complete and discriminating invariant of the semisimple monoid e extending the well known classification of semisimple groups in establishing this result monoids with preassigned root data are first constructed from linear representations of g that done we then show that any other semisimple monoid must be isomorphic to one of those constructed to do this we devise an extension principle based on a monoid analogue of the big cell construction of algebraic group theory thls ultimately yields the desired conclusions consider the classification problem for semisimple algebraic monoids over the algebraically closed field k what sort of problem is this first the definition a semisimple algebraic monoid is an irreducible affine algebraic variety e defined over k together with an associative morphlsm m e x e e and a two sided unit 1 e e for m we assume further that e has a 0 the unit group g which is always linear algebraic and dense in e is reductive e g gl k dim zg 1and e is a normal variety the problem then presents us with two familiar objects let t c g be a maximal torus then z t c e zariski closure is an affine torus embedding and g c e is a reductive algebraic group torus embeddings have been introduced by demazure in 6 in his study of cremona groups and are classified numerically using rational polyhedral cones 14 on the other hand reductive groups have been studied at least in principle since the nineteenth century their classification in the modern sense being achieved largely by chevalley 4 that numerical classification uses the now familiar root systems l l chapter 31 of killing that were introduced by him 15 in his penetrating formulation of the classification e cartan s of semisimple lie algebras thus in the classification of semisimple monoids we are compelled to consider the root system x x t t of g and the polyhedral cone c x z c x t of t c z the two objects are canonically related by the weyl group action on x which leaves c stable received by the editors november 16 1984 1980 mathematics subject classification primary 14m99 secondary 20m99

free vibration analysis of a uniform cantilever beam with point masses by an analytical and numerical combined method abstract the natural frequencies and mode shapes of a uniform cantilever beam carrying any number of concentrated masses were determined by using an analytical and numerical combined method anc method the eigenvalue equation was first derived analytically by using the expansion theorem and then the eigenvalues and eigenvectors were calculated numerically in comparison with the general finite element method fem the anc method has the advantage that there is no necessity to derive the property matrices of each beam element and then to develop the overall ones of the entire beam in comparison with the transfer matrix method tmm the anc method requires no tedious matrix multiplication so that some computer time may be saved the anc method is also better than the pure analytical closed form method since instead of a few special cases a uniform beam carrying any number of point masses of various magnitudes or distributions along the beam length can be easily solved besides by using the anc method there is no difficulty in taking more modes in the mode superposition equation and hence more higher mode natural frequencies of better accuracy may be obtained

monoids on groups with bn pairs the theory of linear algebraic monoids has been developed in the last few years by l renner and the author 21 361 as a generalization of the theory of linear algebraic groups 16 40 in applications of the theory of linear algebraic groups one is naturally led to consider groups with bnpairs or tits systems 12 4 9 42 431 this class of groups includes in particular the groups of lie type so that the abstract properties of tits systems are useful in studying the conjugacy classes and representations of groups of lie type see 4 9 our interest is in monoids so let m be a monoid such that the group of units g has a bn pair in general g has very little to do with the rest of m in fact for a given g m g could be an arbitrary semigroup so we introduce three axioms which intimately connect the idempotent structure of m with the structure of g we then say that m is a monoid on g the prime example is a connected regular linear algebraic monoid with zero but there are other natural examples for instance renner 37 is developing a theory of monoids of lie type along the lines of steinberg 41 for groups another example arises from the work of smith 39 and cabanes 3 from their work one can associate with certain irreducible modular representations 4 g gl v a set of idempotents es end v then the monoid m generated by 4 g and e is a monoid on 4 g in the examples we have worked out and very likely in general in the situation of linear algebraic monoids there is a natural type map from the finite lattice of conjugacy classes of idempotents g the lattice of f classes into the power set of the dynkin diagram of g the idea of the type map is due to lex renner and is exploited by renner and the author 31 to show in particular that the system of idempotents can be constructed from the type map in the present situation of monoids on g there is again a type map i which characterizes the system of idempotents 139 0021 8693 89 3 00

explicit 3d depth migration with a constrained operator numerical anisotropy is one of the main problems in the design of explicit 3d depth extrapolation operators this paper introduces a new method based on constraining the number of independent coefficients for the full 3d extrapolation operator the extrapolation operator is divided into two regions the coefficients for the inner part of the extrapolation operator are treated the same as the full 3d extrapolation operator the coefficients for the outer part of the extrapolation operator are constrained to be constant as a function of azimuth for a given radius this strategy reduces the number of floating point operations because for each extrapolation step the number of complex multiplications are reduced and replaced by complex additions the numerical workload of this alternative scheme is comparable to the hale mcclellan scheme impulse responses are compared with finite difference solutions for the two way acoustic wave equation it is demonstrated that the numerical anisotropy for the proposed sc

representation of seismic sources in the one way wave equations one way extrapolation of downgoing and upgoing acoustic waves plays an essential role in the current practice of seismic migration berkhout 1985 stolt and benson 1986 claerbout 1985 gardner 1985 generally one way wave equations are derived for the source free situation sources are then represented as boundary conditions for the one way extrapolation problem this approach is valid provided the source representation is done with utmost care for instance it is not correct to represent a monopole source by a spatial delta function and to use this as input data for a standard one way extrapolation scheme this yields an erroneous directivity pattern as illustrated below

true amplitude one way wave equation migration in the mixed domain one way wave equation migration is a powerful imaging tool for locating accurately reflectors in complex geologic structures however the classical formulation of one way wave equations does not provide accurate amplitudes for the reflectors when dynamic information is required after migration such as studies for amplitude variation with angle or when the correct amplitudes of the reflectors in the zero offset images are needed some modifications to the one way wave equations are required the new equations which are called true amplitude one way wave equations provide amplitudes that are equivalent to those provided by the leading order of the ray theoretical approximation through the modification of the transverse laplacian operator with dependence of lateral velocity variations the introduction of a new term associated with the amplitudes and the modification of the source representation in a smoothly varying vertical medium the extrapolation of the wavefields with the true amplitude one way

a novel approach to determine the frequency equations of combined dynamical systems abstract a novel approach is presented which can be used to reduce the generalized eigenvalue problem associated with the free vibration of a linear elastic structure carrying lumped elements atsdistinct locations usingncomponent modes in the assumed modes method the free vibration of such a combined dynamical system is governed by the solution of a generalized eigenvalue problem of ordern n whose stiffness and mass matrices consist of diagonal matrices modified by a total ofsrank one matrices wherescorresponds to the number of attachment points this generalized eigenvalue problem can be manipulated such that the natural frequencies governing free vibration can be calculated instead by solving a much smaller characteristic determinant of orders s interestingly enough this smaller and simpler characteristic determinant can also be obtained by using the lagrange multipliers formalism in conjunction with lagrange s equations

new full wave phase shift approach to solve the helmholtz acoustic wave equation for modeling we have developed a method of solving the helmholtz equation based on a new way to generalize the one way wave equation impose correct boundary conditions and eliminate exponentially growing evanescent waves the full two way nature of the helmholtz equation is included but the equation is converted into a pseudo one way form in the framework of a generalized phase shift structure consisting of two coupled first order partial differential equations for wave propagation with depth a new algorithm based on the particular structure of the coupling between p z and 2p z2 is introduced to treat this problem by an explicit approach more precisely in a depth marching strategy the wave operator is decomposed into the sum of two matrices the first one is a propagator in a reference velocity medium whereas the second one is a perturbation term which takes into account the vertical and lateral variation of the velocity the initial conditions are generated by solving the lippmann schwinger integral e

a new approach for determining the natural frequencies and mode shapes of a uniform beam carrying any number of sprung masses abstract in theory one may obtain five equations from each attaching point of a spring mass system and two boundary equations from each end of the uniform beam hence for a uniform beam carrying n spring mass systems simultaneous equations of the form b c 0 will be obtained the solutions of b 0 where represents a determinant give the natural frequencies of the constrained beam and the substitution of each corresponding values of c j j 1 4 into the eigenfunction will define the associated mode shapes while the foregoing theory is simple the lengthy explicit mathematical expressions become impractical if the total number of spring mass systems is larger than two for this reason it was applied to do the free vibration analysis of a uniform beam carrying one spring mass system only in the existing literature the purpose of this paper is to present a numerical technique to apply the foregoing theory to obtain the exact solutions for the lowest several natural frequencies and mode shapes of a uniform beam carrying any number of spring mass systems with various boundary conditions to this end each integration constant c vi and each mode displacement z v v 1 n i 1 4 at the attaching point and the two ends of the beam are considered as nodal displacements of a finite beam element and are assigned an appropriate degree of freedom dof hence each associated coefficient matrix will be equivalent to the stiffness matrix of a beam element and the conventional numerical assembly technique for the finite element method fem may be used to determine the overall coefficient matrix b therefore the eigenvalue equation b c 0 is easily obtained

improved eigenvalues for combined dynamical systems using a modified finite element discretization scheme new approaches are presented to discretize an arbitrarily supported linear structure carrying various lumped attachments specifically the exact eigendata i e the exact natural frequencies and mode shapes of the linear structure without the lumped attachments are first used to modify its finite element mass and stiffness matrix so that the eigensolutions of the discretized system coincide with the exact modes of vibration this is achieved by identifying a set of minimum changes in the finite element system matrices and enforcing certain constraint conditions once the updated matrices for the linear structure are found the finite element assembling technique is then used to include the lumped attachments by adding their parameters to the appropriate elements in the modified mass and stiffness matrices numerical experiments show that for the same number of elements the proposed scheme returns higher natural frequencies that are substantially more accurate than those given by the finite element model alternatively the proposed discretization scheme allows one to efficiently and accurately determine the higher natural frequencies of a combined system without increasing the number of elements in the finite element model

reductive monoids are von neumann regular let k be an algebraically closed field an a e algebraic monoid e is an affme algebraic variety defined over k together with an associative morphism m e x e e and a two sided unit 1 e e for m in categorical terminology an afftne algebraic monoid is a representable functor from the category of afftne varieties to the category of monoids algebraic group theory is concerned mainly with reductive groups and their representations whereas the theory of monoids is concerned largely with von neumann regular monoids the main result of this paper provides a fundamental link between these two objects

smoothing operators for waveform tomographic imaging the analytic kernel in the space time domain for the frechet derivative of acoustic waveform data with respect to changes in the slowness model is given by the born approximation solution to the integral equation of waveform scattering preconditioning operators in the solution of this forward problem which may incorporate a priori information and approximate solutions are smoothing operators in the imaging problem the first iteration of a nonlinear inversion for the slowness model some preconditioning operators are determined for solutions to the parabolic wave equation and then used to create new sensitivity functions that retain appropriate characteristics of the true frechet kernel in forward calculations the new sensitivity functions define near source near receiver and far field kernels as well as kernels that exhibit an amplitude decay off the ray yielding ray perpendicular sensitivity that scales with the fresnel zone size a sample calculation from a synthetic cross well imaging experimen

classification of semisimple rank one monoids consider the classification problem for irreducible normal algebraic monoids with unit group g we obtain complete results for the groups s12 k x k g12 k and pgi2 k x k if g is one of these groups let 4 g denote the set of isomorphy types of normal algebraic monoids with zero element and unit group g our main result establishes a canonical one to one correspondence g q where q is the set of positive rational numbers the classification is achieved in two steps first we construct a class of monoids from linear representations of g that done we show that any other e must already be one of those constructed to do this we devise an extension principle analogous to the big cell construction of algebraic group theory this yields a birational comparison morphism qp er e for some r e q which is ultimately an isomor phism because the monoid er e g is regular

on the eigenfrequencies of a cantilever beam with attached tip mass and a spring mass system this paper deals with the derivation of the frequency equation of a special combined dynamic system it consists of a clamped free bernoulli euler beam with a tip mass where a spring mass system is attached to it the derivation of the frequency equation is essentially carried out by means of the lagrange multipliers method frequency equations of some simpler systems are also obtained from the general expression by using limiting processes these equations are then numerically solved for various combinations of physical parameters the comparison with the results from the solution of the corresponding bernoulli euler frequency equations shows that the frequency equations established are very accurate

green s functions for uniform timoshenko beams abstract the vibration and control of distributed parameter systems is a topic of ongoing interest the availability of closed form expressions for the green s functions or equivalently transfer functions of individual elements facilitates the analysis of these systems in this paper a concise formulation is given for the green s functions of uniform timoshenko beams it is shown that green s functions for uniform euler bernoulli beams both with and without constant axial loads can be expressed in the same form

theory of true amplitude one way wave equations and true amplitude common shot migration one way wave operators are powerful tools for forward modeling and migration here we describe a recently developed true amplitude implementation of modified one way operators and present some numerical examples by true amplitude one way forward modeling we mean that the solutions are dynamically correct as well as kinematically correct that is ray theory applied to these equations yields the upward and downward traveling eikonal equations of the full wave equation and the amplitude satisfies the transport equation of the full wave equation the solutions of these equations are used in the standard wave equation migration imaging condition the boundary data for the downgoing wave is also modified from the one used in the classic theory because the latter data is not consistent with a point source for the full wave equation when the full wave form solutions are replaced by their ray theoretic approximations the imaging formula reduces to the common shot kirchhoff inversion formula in this sense

conjugacy classes in algebraic monoids let m be a connected linear algebraic monoid with zero and a reductive group of units g the following theorem is established theorem there exist affne subsets ml mk of m reductive groups g1 gk with antiautomorphisms surjective morphisms 0t mi gi such that 1 every element of m is conjugate to an element of some ms and 2 two elements a b in mi are conjugate in m if and only if there exists x e gi such that x0 a x si b as a consequence it is shown that m is a union of its inverse submonoids introduction the objects of study in this paper are connected linear algebraic monoids m with zero this means by definition that the underlying set of m is an irreducible affine variety and that the product map is a morphism i e a polynomial map we will further assume that the group of units g of m is reductive this means 1 3 that the unipotent radical of g is trivial then by 6 10 m is unit regular i e m e m g where e e m e e mle2 e in this paper we study the conjugacy classes of m an initial study was made by the author 8 where the general problem was reduced to nilpotent elements the approach here is quite different yielding a more complete answer to be precise we show that there exist affine subsets m1 mk of m reductive groups g1 gk with antiautomorphisms surjective morphisms si mi gi i 1 k such that 1 every element of m is conjugate to an element of some mi and 2 if a b e mi then a is conjugate to b in m if and only if there exists x e gi such that xsi a x si b as an application of this result we show that m is a union of its inverse submonoids an inverse semigroup is a semigroup s with the property that for each a e s there exists a unique a e s such that aaa a and aaa a see 2 finally in 3 we use our main results to briefly analyze the conjugacy classes of nilpotent elements 1 preliminaries throughout this paper z will denote the set of all positive integers and k an algebraically closed field let g be a connected linear algebraic group defined over k the radical r g is the maximal closed connected normal solvable subgroup of g and the unipotent radical ru g is the group of unipotent elements of r g we will assume throughout that g is a reductive group i e ru g 1 then r g c c g the center of g moreover g r g go where go g g is a semisimple group i e r go 1 also 3 theorem 27 5 go is a product of the simple closed normal subgroups of g in particular we have the following received by the editors november 3 1986 1980 mathematics subject classification 1985 revision primary 20g99 20m10 c 1987 american mathematical society 0002 9947 87 1 00 25 per page