a three dimensional parabolic equation model that includes the effects of rough boundaries a three dimensional parabolic equation 3dpe model that handles wide propagation angles in depth narrow propagation angles in azimuth and rough boundaries is derived and solved numerically with the method of alternating directions a homogeneous boundary condition which is easily incorporated into the numerical solution of the 3dpe is applied at the ocean surface to approximate the effects of rough boundaries in terms of a reflection coefficient that depends on grazing angle calculations are presented to demonstrate the accuracy of the 3dpe and to demonstrate that horizontal coupling the effects due to the term involving azimuth derivatives can be important in shallow water previous work has focused on deep water applications the model is applied to problems involving rough boundaries and range dependence

an energy conserving parabolic equation for elastic media the accuracy of the parabolic equation pe method is improved for range dependent elastic media the energy conserving elastic pe is a generalization of the energy conserving acoustic pe a range dependent waveguide is approximated by a sequence of range independent regions a linear approximation is derived for the nonlinear condition corresponding to conservation of compressional energy flux at the vertical interfaces between range independent regions the rotated elastic pe is applied to generate reference solutions for demonstrating accuracy the energy conserving elastic pe is applied to solve problems involving the propagation of an interface wave along a sloping ocean bottom and a problem involving mode cutoff and coupling into shear wave beams in the ocean bottom the latter problem is analogous to a cutoff problem that was first solved with the acoustic pe f b jensen and w a kuperman j acoust soc am 67 1564 1566 1980

strong convergence theorems by hybrid methods for families of nonexpansive mappings in hilbert spaces abstract in this paper we prove a strong convergence theorem by the hybrid method for a family of nonexpansive mappings which generalizes nakajo and takahashi s theorems k nakajo w takahashi strong convergence theorems for nonexpansive mappings and nonexpansive semigroups j math anal appl 279 2003 372 379 simultaneously furthermore we obtain another strong convergence theorem for the family of nonexpansive mappings by a hybrid method which is different from nakajo and takahashi using this theorem we get some new results for a single nonexpansive mapping or a family of nonexpansive mappings in a hilbert space

strong convergence theorems for equilibrium problems and relatively nonexpansive mappings in banach spaces the purpose of the paper is to introduce modified halpern and ishikawa iteration for finding a common element of the set of fixed points of a relatively nonexpansive mapping and the set of solutions of an equilibrium problem in banach spaces we also consider two strong convergence theorems for relatively nonexpansive mappings with some proper restriction

approximating fixed points of nonexpansive mappings in a banach space by metric projections abstract in this paper a strong convergence theorem for nonexpansive mappings in a uniformly convex and smooth banach space is proved by using metric projections this theorem is different from the recent strong convergence theorem due to xu h k xu strong convergence of approximating fixed point sequences for nonexpansive mappings bull aust math soc 74 2006 143 151 which was established by generalized projections

a mapping approach for handling sloping interfaces a mapping approach for handling sloping interfaces in parabolic equation solutions is developed and tested at each range the medium is rigidly translated vertically so that a sloping interface becomes horizontal to simplify the approach the slope is assumed to be small and the extra terms that arise in the wave equation under the mapping are neglected the effects of these terms can be approximately accounted for by applying a leading order correction to the phase the mapping introduces variations in topography which are relatively easy to handle for the case of a pressure release boundary condition the accuracy of the approach is demonstrated for problems involving fluid sediments the approach should also be accurate for problems involving elastic sediments and should be useful for solving three dimensional problems involving variable topography

strong convergence theorems for a finite family of asymptotically nonexpansive mappings and semigroups strong convergence theorems are obtained for a finite family of asymptotically nonexpansive mappings and semigroups by the modified mann method

strong convergence to common fixed points of families of nonexpansive mappings abstract we introduce iteration schemes for families of nonexpansive mappings in hilbert spaces and prove that the iterates converge strongly to common fixed points of the mappings

strong convergence theorems for solving equilibrium problems and fixed point problems of ξ strict pseudo contraction mappings by two hybrid projection methods in this paper we introduce an iterative scheme by using the hybrid projection methods for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a x strict pseudo contraction mapping in hilbert spaces we obtain two strong convergence theorems under mild assumptions on parameters for the sequences generated by these processes the results presented in the paper extend and improve some recent results of marino and xu g marino h k xu weak and strong convergence theorems for strict pseudo contractions in hilbert spaces j math anal appl 329 2007 336 346 tada and takahashi a tada w takahashi weak and strong convergence theorems for a nonexpansive mapping and an equilibrium problem j optim theory appl 133 2007 359 370 and ceng et al l c ceng s al homidan q h ansari j c yao an iterative scheme for equilibrium problems and fixed point problems of strict pseudo contraction mappings j comput appl math 223 2009 967 974 and many others

convergence theorems of common fixed points for a family of lipschitz quasi pseudocontractions abstract in an infinite dimensional hilbert space the normal mann s iteration has only weak convergence in general even for nonexpansive mappings the purpose of this paper is to modify the normal mann s iteration to have strong convergence for a family of lipschitz quasi pseudocontractions in the framework of hilbert spaces our results improve and extend the corresponding ones announced by many others

applications and time domain solution of higher order parabolic equations in underwater acoustics a higher order parabolic equation and the corresponding higher order time domain parabolic equation are derived from a pade series and solved numerically these models provide accurate solutions for problems involving very wide angle propagation e g propagation in the nearfield or over a hard ocean bottom propagation in domains in which sound speed variations are large e g propagation in deep water deep within the ocean bottom in high speed ocean bottoms and possibly of different wave types and propagation out to very long ranges the possibility of modeling elastic wave propagation with a similar approach is considered

a higher order parabolic equation for wave propagation in an ocean overlying an elastic bottom a higher order elastic parabolic equation hepe is derived for wave propagation in depth dependent and weakly range dependent fluid solid media galerkin s method is used to discretize the depth operators in the hepe within layers in which depth variations in the lame constants and density are continuous discontinuities in material properties are handled with centered differences for the interface conditions between layers the numerical solution of the hepe also involves the method of alternating directions and crank nicolson integration the hepe is applied to underwater wave propagation problems involving a water column over an elastic bottom including a weakly range dependent problem the accuracy of the hepe is demonstrated with benchmark calculations

coupling scattering from the sea surface to a one way marching propagation model via conformal mapping validation propagation models in underwater acoustics usually incorporate sea surface scattering effects in an ad hoc manner which in most cases requires making severe approximations in particular to include in a coherent manner in a marching acoustic propagation model the scattering that occurs at a rough sea surface poses a serious problem dozier j acoust soc am 75 1415 1432 1984 introduced a rigorous approach in the framework of the split step parabolic equation model which used a sequence of conformal mappings to flatten segments of the sea surface locally each conformal mapping preserved the elliptic form of the wave equation in each transformed space the parabolic approximation is made and the solution advanced one range step the method has the attractive feature of handling surface roughness within a propagation model in a mathematically consistent manner including refraction and multiple surface interactions when and where they occur in this work the technique developed by dozier is impleme

two parabolic equations for propagation in layered poro elastic media parabolic equation methods for fluid and elastic media are extended to layered poro elastic media including some shallow water sediments a previous parabolic equation solution for one model of range independent poro elastic media collins et al j acoust soc am 98 1645 1656 1995 does not produce accurate solutions for environments with multiple poro elastic layers first a dependent variable formulation for parabolic equations used with elastic media is generalized to layered poro elastic media an improvement in accuracy is obtained using a second dependent variable formulation that conserves dependent variables across interfaces between horizontally stratified layers furthermore this formulation expresses conditions at interfaces using no depth derivatives higher than first order this feature should aid in treating range dependence because convenient matching across interfaces is possible with discretized derivatives of first order in contrast to second order

higher order pade approximations for accurate and stable elastic parabolic equations with application to interface wave propagation higher order pade approximations are applied to derive accurate and stable parabolic equations for sound propagation in oceans bounded below by an elastic bottom or bounded above by ice cover accuracy is achieved by placing constraints on the derivatives of the pade approximations at the point corresponding to the reference wave number stability is achieved by requiring that the pade approximations map part of the lower left quadrant of the complex plane into the upper half of the complex plane elastic parabolic equations based on these pade series can handle problems involving compressional shear and interface waves very wide propagation angles and large depth variations and weak range variations in the seismoacoustic parameters a finite difference spectral solution is developed for generating reference solutions and starting fields the rotated elastic parabolic equation is used to investigate the accuracy of the elastic parabolic equation for range dependent problems

a single scattering correction for large contrasts in elastic layers the single scattering solution is implemented in a formulation that makes it possible to accurately handle solid solid interfaces with the parabolic equation method problems involving large contrasts across sloping stratigraphy can be handled by subdividing a vertical interface into a series of two or more scattering problems the approach can handle complex layering and is applicable to a large class of seismic problems the solution of the scattering problem is based on an iteration formula which has improved convergence in the new formulation and the transverse operator of the parabolic wave equation which is implemented efficiently in terms of banded matrices accurate solutions can often be obtained by using only one iteration

a scaled mapping parabolic equation for sloping range dependent environments parabolic equation solutions use various techniques for approximating range dependent interfaces one is a mapping approach m d collins et al j acoust soc am 107 1937 1942 2000 where at each range the domain is vertically translated so that sloping bathymetry becomes horizontal and range dependence is transferred to the upper surface in this paper a scaled mapping is suggested where the domain is vertically distorted so that both the bathymetry and upper surface are horizontal accuracy is demonstrated for problems involving fluid sediments generalizations of the approach should be useful for environments with layer thicknesses that vary with range

the flattened surface parabolic equation a simple surface flattening transformation is applied to the problem of underwater acoustic scattering from an irregular sea surface the flattened surface elliptic wave equation is factored to obtain a parabolic equation that has the same high angle capability as recent parabolic equation models the flattened surface parabolic equation contains an additional surface slope term but includes forward scattering from the irregular sea surface the transformation creates additional volume irregularities and irregularities in the water sediment interface that the range and depth mesh need to resolve the surface flattening transformation is easy to implement in a parabolic equation model it compares very well with results obtained using boundary integral equations an example is given of scattering at 400 hz from a sea surface with wave heights of 3 m or nearly a wavelength

t class algorithms for pseudocontractions and κ strict pseudocontractions in hilbert spaces in this paper we study iterative algorithms for finding a common element of the set of fixed points of strict pseudocontractions or finding a solution of a variational inequality problem for a monotone lipschitz continuous mapping the last problem being related to finding fixed points of pseudocontractions these algorithms were already studied in 1 and 9 but our aim here is to provide the links between these know algorithms and the general framework of t class algorithms studied in 3

the stabilized self starter a stability problem is eliminated from the self starter which is a parabolic equation technique for generating initial conditions the stability problem is associated with evanescent modes that correspond to nearly vertical propagation it causes the solution to break down at certain frequencies for problems involving relatively low attenuation the error is restricted to relatively short ranges unless the medium is strongly range dependent near the source the stability problem is eliminated by placing the pole of the smoothing operator in the self starter away from the eigenvalues of the depth operator

generalization of the rotated parabolic equation to variable slopes the rotated parabolic equation j acoust soc am 87 1035 1037 1990 is generalized to problems involving ocean sediment interfaces of variable slope the approach is based on approximating a variable slope in terms of a series of constant slope regions the original rotated parabolic equation algorithm is used to march the field through each region an interpolation extrapolation approach is used to generate a starting field at the beginning of each region beyond the one containing the source for the elastic case a series of operators is applied to rotate the dependent variable vector along with the coordinate system the variable rotated parabolic equation should provide accurate solutions to a large class of range dependent seismo acoustics problems for the fluid case the accuracy of the approach is confirmed through comparisons with reference solutions for the elastic case variable rotated parabolic equation solutions are compared with energy conserving and mapping solutions

nonuniform depth grids in parabolic equation solutions the parabolic wave equation is solved using a finite difference solution in depth that involves a nonuniform grid the depth operator is discretized using galerkin s method with asymmetric hat functions examples are presented to illustrate that this approach can be used to improve efficiency for problems in ocean acoustics and seismo acoustics for shallow water problems accuracy is sensitive to the precise placement of the ocean bottom interface this issue is often addressed with the inefficient approach of using a fine grid spacing over all depth efficiency may be improved by using a relatively coarse grid with nonuniform sampling to precisely position the interface efficiency may also be improved by reducing the sampling in the sediment and in an absorbing layer that is used to truncate the computational domain nonuniform sampling may also be used to improve the implementation of a single scattering approximation for sloping fluid solid interfaces

single scattering parabolic equation solutions for elastic media propagation including rayleigh waves the parabolic equation method with a single scattering correction allows for accurate modeling of range dependent environments in elastic layered media for problems with large contrasts accuracy and efficiency are gained by subdividing vertical interfaces into a series of two or more single scattering problems this approach generates several computational parameters such as the number of interface slices an iteration convergence parameter and the number of iterations n for convergence using a narrow angle approximation the choices of n 1 and 2 give accurate solutions analogous results from the narrow angle approximation extend to environments with larger variations when slices are used as needed at vertical interfaces the approach is applied to a generic ocean waveguide that includes the generation of a rayleigh interface wave results are presented in both frequency and time domains

a single scattering correction for the seismo acoustic parabolic equation an efficient single scattering correction that does not require iterations is derived and tested for the seismo acoustic parabolic equation the approach is applicable to problems involving gradual range dependence in a waveguide with fluid and solid layers including the key case of a sloping fluid solid interface the single scattering correction is asymptotically equivalent to a special case of a single scattering correction for problems that only have solid layers kusel et al j acoust soc am 121 808 813 2007 the single scattering correction has a simple interpretation conservation of interface conditions in an average sense that facilitated its generalization to problems involving fluid layers promising results are obtained for problems in which the ocean bottom interface has a small slope

degree of convergence of modified averaged iterations for fixed points problems and operator equations abstract let h be a real hilbert space we propose a modification for averaged mappings to approximate the unique fixed point of a mapping t h h such that t is boundedly lipschitzian and t is monotone we not only prove strong convergence theorems but also determine the degree of convergence using this result an iteration process is given for finding the unique solution of the equation a x f where a h h is strongly monotone and boundedly lipschitzian

strong convergence of the cq method for fixed point iteration processes strong convergence theorems are obtained for the cq method for an ishikawa iteration process a contractive type iteration process for nonexpansive mappings and the proximal point algorithm for maximal monotone operators in hilbert spaces

the problem of energy conservation in one way models it is shown that the standard stair step representation of a sloping bottom may result in significant prediction errors in fact current parabolic equation implementations are not energy conserving the problem is shown to derive from the approximate treatment of the interface conditions at vertical boundaries along the stair steps several improved interface conditions are proposed

strong convergence theorems for nonexpansive mappings and nonexpansive semigroups in this paper we show strong convergence theorems for nonexpansive mappings and nonexpansive semigroups in hilbert spaces by the hybrid method in the mathematical programming

strong convergence theorem of an iterative method for variational inequalities and fixed point problems in hilbert spaces in this paper we introduce a new iteration method based on the hybrid method in mathematical programming and the descent like method for finding a common element of the set of solutions for a variational inequality and the set of fixed points for a nonexpansive mapping in hilbert spaces our method modifies and improves some methods in literature

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

strong convergence theorems for a countable family of quasi lipschitzian mappings and its applications abstract we use the hybrid method in mathematical programming to obtain strong convergence to common fixed points of a countable family of quasi lipschitzian mappings as a consequence several convergence theorems for quasi nonexpansive mappings and asymptotically strict pseudo contractions are deduced we also establish strong convergence of iterative sequences for finding a common element of the set of fixed point the set of solutions of an equilibrium problem and the set of solutions of a variational inequality with an appropriate setting we obtain the corresponding results due to tada takahashi and nakajo shimoji takahashi

injectivity of quasi projective modules projectivity of quasi injective modules and projective cover of injective modules abstract in k r fuller on indecomposable injectives over artinian rings pacific j math 29 1969 115 135 theorem 3 1 k r fuller gave necessary and sufficient conditions for projective left modules to be injective over a left artinian ring in y baba and k oshiro on a theorem of fuller preprint we studied this theorem in the present paper first we generalize the theorem by giving necessary and sufficient conditions for quasi projective modules to be injective and ones for quasi injective modules to be projective in k r fuller on indecomposable injectives over artinian rings pacific j math 29 1969 115 135 theorem 2 4 fuller also gave a natural one to one correspondence between the homogeneous components of the k th upper resp lower loewy factor of an injective right r module e and the k th lower resp upper loewy factor of fr whenever rf is the projective cover of the socle of e where f is a primitive idempotent in a right artinian ring r second we shall give a complete correspondence between simple submodules of the 1st upper loewy factor of e and the 1st lower loewy factor of rf

strong convergence theorems for an infinite family of quasi nonexpansive mappings and generalized equilibrium problems and variational inequality problems in this paper we introduce a new iterative scheme for finding a common element of the set of common fixed points of an infinite family of uniformly continuous quasi nonexpansive mappings and the set of solutions to a generalized equilibrium problem and the set of solutions to a variational inequality problem in a real hilbert space we then prove strong convergence of the scheme to a common element of the three mentioned sets our results extend important recent results

the coupled mode parabolic equation the coupled mode parabolic equation pe is a generalization of the adiabatic mode pe that includes mode coupling terms it is practical to apply this approach to large scale problems involving coupling of energy between both modes and azimuths the solution is expressed in terms of the normal modes and mode coefficients which satisfy coupled horizontal wave equations the coupled mode pe may be solved efficiently with the splitting method the first step is equivalent to solving the adiabatic mode pe over one range step the second step involves the integration of the coupling term the coupling mode pe solution conserves energy which is an important aspect of a range dependent propagation model the derivation of the coupled mode pe which involves completing the square of an operator is related to the derivation of an adiabatic mode pe that accounts for ambient flow examples are presented to illustrate the accuracy of the coupled mode pe

a split step pade solution for the parabolic equation method a split step pade solution is derived for the parabolic equation pe method higher order pade approximations are used to reduce both numerical errors and asymptotic errors e g phase errors due to wide angle propagation this approach is approximately two orders of magnitude faster than solutions based on pade approximations that account for asymptotic errors but not numerical errors in contrast to the split step fourier solution which achieves similar efficiency for some problems the split step pade solution is valid for problems involving very wide propagation angles large depth variations in the properties of the waveguide and elastic ocean bottoms the split step pade solution is practical for global scale problems

higher order paraxial wave equation approximations in heterogenous media a new family of paraxial wave equation approximations is derived these approximations are of higher order accuracy than the parabolic approximation and they can be applied to the same computationa

implicit iterative algorithms for asymptotically nonexpansive mappings in the intermediate sense and lipschitz continuous monotone mappings in this paper we introduce some implicit iterative algorithms for finding a common element of the set of fixed points of an asymptotically nonexpansive mapping in the intermediate sense and the set of solutions of the variational inequality problem for a monotone lipschitz continuous mapping these implicit iterative algorithms are based on two well known methods extragradient and approximate proximal methods we obtain some weak convergence theorems for these implicit iterative algorithms based on these theorems we also construct some implicit iterative processes for finding a common fixed point of two mappings such that one of these two mappings is taken from the more general class of lipschitz pseudocontractive mappings and the other mapping is asymptotically nonexpansive

modeling rayleigh and stoneley waves and other interface and boundary effects with the parabolic equation an improved approach for handling boundaries interfaces and continuous depth dependence with the elastic parabolic equation is derived and benchmarked the approach is applied to model the propagation of rayleigh and stoneley waves depending on the choice of dependent variables the operator in the elastic wave equation may not factor or the treatment of interfaces may be difficult these problems are resolved by using a formulation in terms of the vertical displacement and the range derivative of the horizontal displacement these quantities are continuous across horizontal interfaces which permits the use of galerkin s method to discretize in depth this implementation extends the capability of the elastic parabolic equation to handle arbitrary depth dependence and should lead to improvements for range dependent problems

an approximation to the three dimensional parabolic equation method for acoustic propagation we have reported previously ralph n baer j acoust soc am 69 70 75 1981 on an algorithm based on the parabolic approximation to the reduced wave equation for the propagation of sound in three dimensions in the ocean we present here a simpler algorithm solve n two dimensional problems and combine the results to form an approximate three dimensional solution analytic and numerical results show that this n 2d approach is an excellent approximation to the original algorithm for realistic ocean environments even those where fronts and eddies are present provided redirection of energy in azimuth due to boundary interaction is not important we compare the two computer models based on these algorithms for several test cases by considering how they distribute energy spatially and by simulating the performance of a hypothetical horizontal array of hydrophones placed in the calculated complex valued acoustic fields in an example of propagation through a large and strong ocean eddy the n 2d metho

a self starter for the parabolic equation method an efficient method is developed for generating accurate starting fields for the parabolic equation pe method for both fluid and solid media the self starter which is constructed by solving a one dimensional boundary value problem bvp involving the pe operator is more efficient than the normal mode starter which requires the solution of a large number of similar bvps the self starter depends on the depth dependent properties of the medium and satisfies all interface and boundary conditions since the self starter is based on higher order parabolic approximations it is accurate for problems involving wide propagation angles large depth variations in the properties of the medium low frequencies interface waves and the continuous spectrum

a higher order energy conserving parabolic equqation for range dependent ocean depth sound speed and density outgoing solutions of the wave equation including parabolic equation pe and normal mode solutions are usually formulated so that pressure is continuous with range for range dependent problems the accuracy of normal mode solutions has been improved by conserving energy rather than maintaining continuity of pressure porter et al the problem of energy conservation in one way equations j acoust soc am 89 1058 1067 1991 this approach is applied to derive a higher order energy conserving pe that provides improved accuracy for problems involving large ocean bottom slopes and large range and depth variations in sound speed and density a special numerical approach and complex pade coefficients are applied to suppress gibbs oscillations the back propagated half space field an improved pe starter is applied to handle wide propagation angles reference solutions generated with a complex ray model and with the rotated pe are used for comparison

asymptotically strict pseudocontractive mappings in the intermediate sense abstract it is proved that the modified mann iteration process x n 1 1 n x n n t n x n n n where n is a sequence in 0 1 with n 1 for some 0 1 converges weakly to a fixed point of an asymptotically strict pseudocontractive mapping t in the intermediate sense which is not necessarily lipschitzian we also develop cq method for this modified mann iteration process which generates a strongly convergent sequence