amplitude effect of the free surface in elastic reverse time extrapolation we evaluate the physical validity of surface boundary conditions of the computational model in reverse time extrapolation of 3d three component 3 c elastic seismic data acquired at the earth s free surface by using mathematical derivations and numerical simulations reverse time extrapolation of elastic data assumes that only the incident p or s waves are reconstructed during extrapolation into the computational grid however superposition of the upgoing incident waves and the downgoing reflected and converted waves generated at the free surface also is recorded in data acquisition and is input into reverse time extrapolation in elastic reverse time extrapolation the computational model needs to have an absorbing top boundary when the 3d 3 c elastic data are inserted into the computational model during reverse time extrapolation the originally incident p or s wave is reconstructed in addition the free surface p to p reflected and p to s converted waves recombine to reconstruct a second in

direct vector field method to obtain angle domain common image gathers from isotropic acoustic and elastic reverse time migration we have developed an alternative new method to produce common image gathers in the incident angle domain by calculating wavenumbers directly from the p wave polarization rather than using the dominant wavenumber as the normal to the source wavefront in isotropic acoustic media the wave propagation direction can be directly calculated as the spatial gradient direction of the acoustic wavefield which is parallel to the wavenumber direction the normal to the wavefront instantaneous wavenumber obtained via a novel hilbert transform approach is used to calculate the local normal to the reflectors in the migrated image the local incident angle is produced as the difference between the propagation direction and the normal to the reflector by reordering the migrated images over all common source gathers with incident angle common image gathers are produced in the incident angle domain instantaneous wavenumber takes the place of the normal to the reflector in the migrated image p and s wave separ

controllability cost of conservative systems resolvent condition and transmutation abstract this article concerns the exact controllability of unitary groups on hilbert spaces with unbounded control operator it provides a necessary and sufficient condition not involving time which blends a resolvent estimate and an observability inequality by the transmutation of controls in some time l for the corresponding second order conservative system it is proved that the cost of controls in time t for the unitary group grows at most like exp l 2 t as t tends to 0 in the application to the cost of fast controls for the schrodinger equation l is the length of the longest ray of geometric optics which does not intersect the control region this article also provides observability resolvent estimates implying fast smoothing effect controllability at low cost and underscores that the controllability cost of a system is not changed by taking its tensor product with a conservative system

uniform stability of semilinear wave equations with arbitrary local memory effects versus frictional dampings abstract this paper is concerned with the mixed initial boundary value problem for semilinear wave equations with complementary frictional dampings and memory effects we successfully establish uniform exponential and polynomial decay rates for the solutions to this initial boundary value problem under much weak conditions concerning memory effects more specifically we obtain the exponential and polynomial decay rates after removing the fundamental condition that the memory effect region includes a part of the system boundary while the condition is a necessity in the previous literature moreover for the polynomial decay rates we only assume minimal conditions on the memory kernel function g without the usual assumption of g controlled by g

wave field separation in two dimensional anisotropic media until recently the term elastic usually implied two dimensional 2 d and isotropic in this limited context the divergence and curl operators have found wide use as wave separation operators for example mora 1987 used them in his inversion method to allow separate correlation of p and s arrivals although the separation is buried in the math and not obvious clayton 1981 used them explicitly in several modeling and inversion methods devaney and oristaglio 1986 used closely related operators to separate p and s arrivals in elastic vsp data

2d and 3d elastic wavefield vector decomposition in the wavenumber domain for vti media a pragmatic decomposition of a vector wavefield into p and s waves is based on the helmholtz theory and the christoffel equation it is applicable to vti media when the plane wave polarization is continuous in the vicinity of a given wavenumber and is uniquely defined by that wavenumber except for the kiss singularities on the vti symmetry axis unlike divergence and curl which separate the wavefield into a scalar and a vector field the decomposed p and s wavefields are both vector fields with correct amplitude phase and physical units if the vector components of decomposed wavefields are added they reconstruct those of the original input wavefield wavefield propagation in any portions of a vti medium that have the same polarization distribution i e the same eigenvector in the wavenumber domain have the same decomposition operators and can be recon structed with a single 3d fourier transform for each operator e g one for p waves and one for s waves this applies to isotropic wavefields a

accurate simulations of pure quasi p waves in complex anisotropic media abstractreverse time migration rtm in complex anisotropic media requires calculation of the propagation of a single mode wave the quasi p wave this was conventionally realized by solving a 2 2 system of second order partial differential equations the implementation of this 2 2 system required at least twice the computational resources as compared with the acoustic wave equation the s waves an introduced auxiliary function in this system were treated as artifacts in the rtm furthermore the 2 2 system suffered numerical stability problems at the places in which abrupt changes of symmetric axis of anisotropy exist which brings more challenges to real data implementation on the other hand the alkhalifah s equation which governs the pure quasi p wave propagation was hard to solve because it was a pseudodifferential equation we proposed a pure quasi p wave equation that can be easily implemented within current imaging framework our new equation was obtained by decomposing the original pseudodif

the optimal form of selection principles for functions of a real variable abstract let t be a nonempty set of real numbers x a metric space with metric d and x t the set of all functions from t into x if f x t and n is a positive integer we set n f sup i 1 n d f b i f a i where the supremum is taken over all numbers a 1 a n b 1 b n from t such that a 1 b 1 a 2 b 2 a n b n the sequence n f n 1 is called the modulus of variation of f in the sense of chanturiya we prove the following pointwise selection principle if a sequence of functions f j j 1 x t is such that the closure in x of the set f j t j 1 is compact for each t t and lim n 1 n lim sup j n f j 0 then there exists a subsequence of f j j 1 which converges in x pointwise on t to a function f x t satisfying lim n n f n 0 we show that condition is optimal the best possible and that all known pointwise selection theorems follow from this result including helly s theorem also we establish several variants of the above theorem for the almost everywhere convergence and weak pointwise convergence when x is a reflexive separable banach space

maps of several variables of finite total variation ii e helly type pointwise selection principles abstract given a a 1 a n b b 1 b n r n with a b componentwise and a map f from the rectangle i a b a 1 b 1 a n b n into a metric semigroup m m d denote by tv f i a b the hildebrandt leonov total variation of f on i a b which has been recently studied in v v chistyakov yu v tretyachenko maps of several variables of finite total variation i j math anal appl 2010 submitted for publication the following helly type pointwise selection principle is proved if a sequence f j j n of maps from i a b into m is such that the closure in m of the set f j x j n is compact for each x i a b and c sup j n tv f j i a b is finite then there exists a subsequence of f j j n which converges pointwise on i a b to a map f such that tv f i a b c a variant of this result is established concerning the weak pointwise convergence when values of maps lie in a reflexive banach space m with separable dual m

modular metric spaces ii application to superposition operators abstract applying the theory of modular metric spaces developed in the first part of this paper v v chistyakov 2009 1 we define a metric semigroup and an abstract convex cone of functions of finite generalized variation in the approach of schramm m schramm functions of bounded variation and riemann stieltjes integration trans amer math soc 287 1 1985 49 63 which are significantly larger as compared to the spaces of bounded variation in the sense of jordan wiener young and waterman we present a complete description of generators of lipschitz continuous bounded and some other classes of superposition nemytskii operators mapping in these semigroups and cones which extends recent results by matkowski and mi j matkowski j mi on a characterization of lipschitzian operators of substitution in the space bv a b math nachr 117 1984 155 159 maligranda and orlicz l maligranda w orlicz on some properties of functions of generalized variation monatsh math 104 1987 53 65 zawadzka g zawadzka on lipschitzian operators of substitution in the space of set valued functions of bounded variation rad mat 6 1990 279 293 and chistyakov v v chistyakov mappings of generalized variation and composition operators j math sci new york 110 2 2002 2455 2466 v v chistyakov lipschitzian nemytskii operators in the cones of mappings of bounded wiener variation folia math 11 1 2004 15 39

superposition operators and functions of bounded p variation ii we continue the study of the superposition operator tf g f g on the space bvp1 i of primitives of real valued functions of bounded p variation on an interval i we give first a characterization of the functions f such that tf takes bvp1 i to itself then we characterize the functions f for which tf is continuous uniformly continuous and differentiable as a mapping of bvp1 i to itself respectively by exploiting the peetre s imbedding theorem and the fubini property we derive partial results on continuity of tf in besov spaces bp qs rn for a smoothness parameter s satisfying 0 s 1 1 p

a pointwise selection principle for metric semigroup valued functions abstract let t r x d be an additive commutative semigroup with metric d satisfying d x z y z d x y for all x y z x and x t the set of all functions from t into x if n n and f g x t we set n f g t sup i 1 n d f t i g s i g t i f s i where the supremum is taken over all numbers s 1 s n t 1 t n from t such that s 1 t 1 s 2 t 2 s n t n we prove the following pointwise selection theorem if a sequence of functions f j j n x t is such that the closure in x of the set f j t j n is compact for each t t and lim n 1 n lim n sup j k n j k n f j f k t 0 then it contains a subsequence which converges pointwise on t we show by examples that this result is sharp and present two of its variants

regular caratheodory type selectors under no convexity assumptions we prove the existence of caratheodory type selectors that is measurable in the first variable and having certain regularity properties like lipschitz continuity absolute continuity or bounded variation in the second variable for multifunctions mapping the product of a measurable space and an interval into compact subsets of a metric space or metric semigroup

scalar reverse time depth migration of prestack elastic seismic data reflected p to p and p to s converted seismic waves in a two component elastic common source gather generated with a p wave source in a two dimensional model can be imaged by two independent scalar reverse time depth migrations the inputs to migration are pure p and s waves that are extracted by divergence and curl calculations during shallow extrapolation of the elastic data recorded at the earth s surface for both p to p and p to s converted reflected waves the imaging time at each point is the p wave traveltime from the source to that point the extracted p wave is reverse time extrapolated and imaged with a p velocity model using a finite difference solution of the scalar wave equation the extracted s wave is reverse time extrapolated and imaged similarly but with an s velocity model converted s wave data requires a polarity correction prior to migration to ensure constructive interference between data from adjacent sources synthetic examples show that the algorithm gives satisfactory resul

maps of several variables of finite total variation i mixed differences and the total variation given two points a a1 an and b b1 bn from rn with a b componentwise and a map f from the rectangle iab a1 b1 an bn into a metric semigroup m m d we study properties of the total variation tv f iab of f on iab introduced by the first author in v v chistyakov a selection principle for mappings of bounded variation of several variables in real analysis exchange 27th summer symposium opava czech republic 2003 pp 217 222 such as the additivity generalized triangle inequality and sequential lower semicontinuity this extends the classical properties of c jordan s total variation n 1 and the corresponding properties of the total variation in the sense of hildebrandt t h hildebrandt introduction to the theory of integration academic press 1963 n 2 and leonov a s leonov on the total variation for functions of several variables and a multidimensional analog of helly s selection principle math notes 63 1998 61 71 n n for real valued functions of n variables

discontinuous local semiflows for kurzweil equations leading to lasalle s invariance principle for differential systems with impulses at variable times abstract in this paper we consider an initial value problem for a class of generalized odes also known as kurzweil equations and we prove the existence of a local semidynamical system there under certain perturbation conditions we also show that this class of generalized odes admits a discontinuous semiflow which we shall refer to as an impulsive semidynamical system as a consequence we obtain lasalle s invariance principle for such a class of generalized odes due to the importance of lasalle s invariance principle in studying stability of differential systems we include an application to autonomous ordinary differential systems with impulse action at variable times

3 d finite difference modeling of elastic waves in borehole environments full wavefield sonic logs may be simulated for arbitrarily complicated three dimensional 3 d borehole environments using 3 d elastic finite differences to ensure reliability through large contrasts in poisson s ratio across mud casing lithology contacts a staggered grid finite difference formulation is used cylindrical symmetry is not assumed so the responses of features such as asymmetrical washouts and dipping structure are easily obtained when features are asymmetrical seismic responses vary significantly at various points around the hole circumference at any depth even for simple hole geometries observed responses are complicated because of coupling between waves inside and outside the hole observations are also sensitive to the source receiver separation output formats include fixed time snapshots of displacement divergence and curl and seismograms for the center and edge of a borehole this allows detailed arrival identification and interpretation to our knowledge this is the most

time reversal focusing of the initial state for kirchhoff plate consider a kirchhoff plate partial t 2 u delta 2 u partial t 2 delta u 0 in omega times 0 t with boundary data u delta u 0 on partial omega times 0 t and unknown initial data u cdot 0 u 0 and partial t u cdot 0 u 1 in omega we study an inverse problem of determining u 0 u 1 from an interior observation u omega times 0 t here omega is a bounded domain omega a nonempty open subset of omega and t 0 a suitable time duration by means of an iterative time reversal technique we derive an asymptotic formula of reconstructing u 0 u 1 approximately with a logarithmical convergence rate for smooth initial data the convergence becomes uniform and exponential when omega omega t satisfies the geometric control condition introduced by bardos lebeau and rauch

3d ps wave imaging with elastic reverse time migration abstractone of the important features of elastic reverse time migration is that it is a depth domain prestack migration method based on a vectorial wavefield pp and ps wave images can be easily generated by implementing an imaging condition to pure wave modes separated with divergence and curl operators in isotropic media in the 3d case however the curl calculation will generate a 3c vector s wave which makes it impossible to generate a scalar ps image with the crosscorrelation imaging condition therefore we first analyzed the polarity distribution of a 3d s wave and concluded that the separated s wave keeps perpendicular to a plane determined by the particle motion direction and the propagation direction of the s wave this specific plane is the raypath plane when the s wave is excited by an incident p wave the normal direction of the raypath plane can be calculated by computing the vector product of the propagation directions of source and receiver wavefields then we used this normal directio

phase correction in separating p and s waves in elastic data two dimensional elastic data containing reflected p waves and converted s wave generated by a p source may be separated using dilatation and rotation calculation sun 1999 the algorithm is a combination of elastic full wavefield extrapolation sun and mcmechan 1986 chang and mcmechan 1987 1994 and wave type separation using dilatation divergence and rotation curl calculations dellinger and etgen 1990 it includes 1 downward extrapolating the multicomponent elastic data in an elastic velocity model using the elastic wave equation 2 calculating the dilatation to represent pure p waves and calculating the rotation to represent pure s waves at some depth and 3 upward extrapolating the dilatation in a p velocity model and upward extrapolating the rotation in an s velocity model using the acoustic wave equation for each

the control transmutation method and the cost of fast controls in this paper the null controllability in any positive time t of the first order equation 1 dot x t e i theta ax t bu t theta pi 2 fixed is deduced from the null controllability in some positive time l of the second order equation 2 ddot z t az t bv t the differential equations 1 and 2 are set in a banach space b is an admissible unbounded control operator and a is a generator of cosine operator function the control transmutation method makes explicit the input function u of 1 in terms of the input function v of 2 u t int mathbb r k t s v s ds where the compactly supported kernel k depends only on t and l this method proves roughly that the norm of a u steering the system 1 from an initial state x 0 x 0 to the final state x t 0 grows at most like x 0 exp alpha l 2 t as the control time t tends to zero the rate alpha is characterized independently by a one dimensional controllability problem in applications to the cost of fast controls for the heat equation l is roughly the length of the longest ray of geometric optics which does not intersect the control region

a pointwise selection principle for maps of several variables via the total joint variation a b s t r a c t given a rectangle in the real euclidean n dimensional space and two maps f and g defined on it and taking values in a metric semigroup we introduce the notion of the total joint variation tv f g of these maps this extends similar notions considered by hildebrandt 1963 17 leonov 1998 18 chistyakov 2003 2005 5 8 and the authors 2010 we prove the following irregular pointwise selection principle in terms of the total joint variation if a sequence of maps fj 1 from the rectangle into a metric semigroup is pointwise precompact and lim supj k tv fj fk is finite then it admits a pointwise convergent subsequence whose limit may be a highly irregular e g everywhere discontinuous map this result generalizes some recent pointwise selection principles for real functions and maps of several real variables

separating p and s waves in prestack 3d elastic seismograms using divergence and curl the reflected p and s waves in a prestack 3d three component elastic seismic section can be separated by taking the divergence and curl during finite difference extrapolation the elastic seismic data are downward extrapolated from the receiver locations into a homogeneous elastic computational model using the 3d elastic wave equation during downward extrapolation divergence a scalar and curl a three component vector of the wavefield are computed and recorded independently at a fixed depth as a one component seismogram and a three component seismogram respectively the p and s velocities in the elastic computational model are then split into two independent models the divergence seismogram containing p waves only is then upward extrapolated using the scalar wave equation through the p velocity model to the original receiver locations at the surface to obtain the separated p waves the x component y component and z component seismograms of the curl containing s waves only are upward ex

3 d elastic prestack reverse time depth migration by combining and extending previous algorithms for 2 d prestack elastic migration and 3 d prestack acoustic migration a full 3 d elastic prestack depth migration algorithm is developed reverse time extrapolation of the recorded data is by 3 d elastic finite differences computation of the image time for each point in the 3 d volume is by 3 d acoustic finite differences the algorithm operates on three component vector wavefield common source data and produces three component vector reflectivity distributions converted p to s reflections are automatically imaged with the primary p wave reflections there are no dip restrictions as the full wave equation is used the algorithm is illustrated by application to synthetic data from three models a flat reflector a dipping truncated wedge overlying a flat reflector and the classical french double dome and fault model

amplitude balancing in separating p and s waves in 2d and 3d elastic seismic data the reflected p and s waves in elastic displacement component data recorded at the earth s surface are separated by reverse time downward extrapolation of the data in an elastic computational model followed by calculations to give divergence dilatation and curl rotation at a selected reference depth the surface data are then reconstructed by separate forward time upward scalar extrapolations from the reference depth of the magnitude of the divergence and curl wavefields and extraction of the separated p and s waves respectively at the top of the models a p wave amplitude will change by a factor that is inversely proportional to the p velocity when it is transformed from displacement to divergence and an s wave amplitude will change by a factor that is inversely proportional to the s velocity when it is transformed from displacement to curl consequently the ratio of the p to the s wave amplitude the p s amplitude ratio in the form of divergence and curl postseparation is different

prestack scalar reverse time depth migration of 3d elastic seismic data using two independent 3d scalar reverse time depth migrations we migrate the reflected p and s waves in a prestack 3d three component 3 c elastic seismic data volume generated with a p wave source in a 3d model and recorded at the top of the model reflected p and s waves are extracted by divergence a scalar and curl a 3 c vector calculations respectively during shallow downward extrapolation of the elastic seismic data the imaging time for the migrations of both the reflected p and p s converted waves at each point is the one way p wave traveltime from the source to that point the divergence the extracted p waves is reverse time extrapolated using a finite difference solution of the 3d scalar wave equation in a 3d p velocity modeland is imaged to obtain the migrated p image the curl the extracted s waves is first converted into a scalar s wavefield by taking the curl s absolute value as the absolute value of the scalar s wavefield and assigning a positive sign if the curl is counterc

elastic least squares reverse time migration via linearized elastic full waveform inversion with pseudo hessian preconditioning abstracttime domain elastic least squares reverse time migration lsrtm is formulated as a linearized elastic full waveform inversion problem the elastic born approximation and elastic reverse time migration rtm operators are derived from the time domain continuous adjoint state method our approach defines p and s wave impedance perturbations as unknown elastic images our algorithm is obtained using continuous functional analysis in which the problem is discretized at the final stage optimize before discretize approach the discretized numerical versions of the elastic born operator and its adjoint elastic rtm operator pass the dot product test the conjugate gradient least squares method is used to solve the least squares migration quadratic optimization problem in other words the hessian operator for elastic lsrtm is implicitly inverted via a matrix free algorithm that only requires the action of forward and adjoint operators on vectors the diagonal of the pseudo hessian operator is used t

inverse spectral problem for analytic domains i balian bloch trace formula this is the first in a series of papers z3 z4 on inverse spectral resonance problems for analytic plane domains in this paper we present a rigorous version of the balian bloch trace formula bb1 bb2 it is an asymptotic formula for the trace tr1 r k i log k of the regularized resolvent of the dirichlet or neumann laplacian of as k with 0 when the support of open image in new window contains the length l of precisely one periodic reflecting ray then the asymptotic expansion of tr1 r k i log k is essentially the same as the wave trace expansion at the raison d etre for this approach is that it leads to relatively simple explicit formulae for wave invariants for example we give the first formulae for wave invariants of bouncing ball orbits of plane domains the details will appear in z3 although we only present details in dimension 2 the methods and results extend with few modifications to all dimensions

multicomponent kirchhoff migration this paper presents a method for elastic and viscoelastic imaging of multicomponent seismic data the method is based on claerbout s survey sinking concept and the visco elastic kirchhoff integral for the displacement field assuming a multishot and multireceiver experiment the migration process is formulated as a wavefield reconstruction problem using the visco elastic kirchhoff integral twice first the receiver coordinates are downward continued second the source coordinates are downward continued the multicomponent seismic data are treated as a vector wavefield in which the data measurements may be displacement velocity or traction pressure the theoretical formulation is based on the viscoelastic hooke s law and newton s equation of motion as the physical model for seismic wave propagation it is valid for linear viscoelastic media with any anisotropic symmetry when the lowest order ray approximation is introduced the migration equation takes a form similar to conventional kirchhoff migra

exponential stability for the wave model with localized memory in a past history framework abstract in this paper we discuss the asymptotic stability as well as the well posedness of the damped wave equation posed on a bounded domain of r n n 2 x u t t u 0 g s div a x u t s d s b x u t 0 subject to a locally distributed viscoelastic effect driven by a nonnegative function a x and supplemented with a frictional damping b x 0 acting on a region a of where a 0 in a assuming that x is constant considering that the well known geometric control condition t 0 holds and supposing that the relaxation function g is bounded by a function that decays exponentially to zero we prove that the solutions to the corresponding partial viscoelastic model decay exponentially to zero even in the absence of the frictional dissipative effect in addition in some suitable cases where the material density x is not constant it is also possible to remove the frictional damping term b x u t that is the localized viscoelastic damping is strong enough to assure that the system is exponentially stable the semi linear case is also considered

controllability and observability of a heat equation with hyperbolic memory kernel abstract the exact controllability and observability for a heat equation with hyperbolic memory kernel in anisotropic and nonhomogeneous media are considered due to the appearance of such a kind of memory the speed of propagation for solutions to the heat equation is finite and the corresponding controllability property has a certain nature similar to hyperbolic equations and is significantly different from that of the usual parabolic equations by means of carleman estimate we establish a positive controllability and observability result under some geometric condition on the other hand by a careful construction of highly concentrated approximate solutions to hyperbolic equations with memory we present a negative controllability and observability result when the geometric condition is not satisfied

escape function conditions for the observation control and stabilization of the wave equation for the linear wave equation with time invariant coefficients on a connected compact riemannian manifold omega g with c 3 boundary the geodesics condition of bardos lebeau and rauch siam j control optim 30 1992 pp 1024 1065 is characterized in terms of escape functions which are some lyapunov functions on the phase space s bar omega the unit sphere cotangent bundle differentiable escape functions yield a sufficient condition which is slightly less sharp but does not refer to geodesics the escape function condition yields a straightforward geometric proof that the geodesics condition holds in the situations where first order differential multiplier methods apply using microlocal control results it allows us to generalize some control results that were obtained by multiplier methods to variable coefficients and lower order terms it also allows us to prove in some class of simple situations e g in mathbb r 2 with constant coefficients that no first order differential multiplier method can reach the optimal control time or control regions

uniform decay rate estimates for schrodinger and plate equations with nonlinear locally distributed damping on a compact n dimensional riemannian manifold m g we establish uniform decay rate estimates for the linear schrodinger and plate equations subject to an internal nonlinear damping locally distributed on the manifold our approach can be also employed for other equations provided that inverse inequality for the linear model occurs in the particular case of the wave equation where the well known geometric control condition gcc is equivalent to the observability inequality our method generalizes the results due to cavalcanti et al 2010 2009 9 10 regarding the optimal choice of dissipative regions

the application of high order differencing to the scalar wave equation the second order central difference is often used to approximate the derivatives of the wave equation it is demonstrated that gains in computational efficiency can be made by using high order approximations for these derivatives a one dimensional model is used to illustrate the relative accuracy of o t2 x2 o t2 x4 o t4 x4 ando t4 x10 central difference schemes for comparison o t2 ando t4 pseudospectral schemes are used as an additional measure of performance the results indicate that o t4 x10 differencing can achieve similar accuracy as the o t4 spectral scheme for practical illustration a two dimensional form of the o t4 x10 algorithm is used to compute the exploding reflector response of a salt dome model and compared with a fine grid o t2 x4 result transmissive sponge like boundary conditions are also examined and shown to be effective

stability for semilinear hyperbolic coupled system with frictional and viscoelastic localized damping abstract in this paper we prove stability results for a semilinear hyperbolic coupled system subject to a viscoelastic localized damping acting in the first equation and a frictional localized one acting in the second equation of the system we divide the proof into two parts in the first part the equations are posed in a homogeneous medium with the damping acting in a boundary neighbourhood in the second part the equations posed in an inhomogeneous medium with the damping acting in a collar of the boundary and in an appropriate mesh in the interior due to the assumptions on the nonlinear function involving the frictional damping general decay rates are obtained to prove the results we used the tools of the microlocal analysis theory

asymptotic stability of the wave equation on compact manifolds and locally distributed damping a sharp result let m g be a n dimensional n geqq 2 compact riemannian manifold with boundary where g denotes a riemannian metric of class c this paper is concerned with the study of the wave equation on m g with locally distributed damping described by left begin array l u tt delta bf g u a x g u t 0 quad hbox on thinspace m times left 0 infty right u 0 hbox on partial m times left 0 infty right end array right where m represents the boundary of m and a x g ut is the damping term the main goal of the present manuscript is to generalize our previous result in cavalcanti et al trans ams 361 9 4561 4580 2009 treating the conjecture in a more general setting and extending the result for n dimensional compact riemannian manifolds m g with boundary in two ways i by reducing arbitrarily the region m ast subset m where the dissipative effect lies this gives us a totally sharp result with respect to the boundary measure and interior measure where the damping is effective ii by controlling the existence of subsets on the manifold that can be left without any dissipative mechanism namely a precise part of radially symmetric subsets an analogous result holds for compact riemannian manifolds without boundary

closure properties for integral problems driven by regulated functions via convergence results abstract in this paper we give necessary and sufficient conditions for the convergence of kurzweil stieltjes integrals with respect to regulated functions using the notion of asymptotical equiintegrability one thus generalizes several well known convergence theorems as applications we provide existence and closure results for integral problems driven by regulated functions both in single and set valued cases in the particular setting of bounded variation functions driving the equations we get features of the solution set of measure integrals problems

suppressing the unwanted reflections of the full wave equation in many instances in exploration geophysics we are interested in the so called one way wave equation this equation allows the wave fields to propagate in the positive depth direction but not in the reverse z direction some modeling and migration methods such as the f k method stolt 1978 and the phase shift method gazdag 1978 produce in a natural way the one way wave equation the main advantage of the one way wave equation is that it does not give rise to multiples or interlayer reverberations and enables the observation of primary events only

a selection principle for mappings of bounded variation e helly s selection principle states that an infinite bounded family of real functions on the closed interval which is bounded in variation contains a pointwise convergent sequence whose limit is a function of bounded variation we extend this theorem to metric space valued mappings of bounded variation then we apply the extended helly selection principle to obtain the existence of regular selections of non convex set valued mappings any set valued mapping from an interval of the real line into nonempty compact subsets of a metric space which is of bounded variation with respect to the hausdorff metric admits a selection of bounded variation also we show that a compact valued set valued mapping which is lipschitzian absolutely continuous or of bounded riesz variation admits a selection which is lipschitzian absolutely continuous or of bounded riesz variation respectively

asymmetric variations of multifunctions with application to functional inclusions abstract under certain initial conditions we prove the existence of set valued selectors of univariate compact valued multifunctions of bounded jordan variation when the notion of variation is defined taking into account only the pompeiu asymmetric excess between compact sets from the target metric space for this we study subtle properties of the directional variations we show by examples that all assumptions in the main existence result are essential as an application we establish the existence of set valued solutions x t of bounded variation to the functional inclusion of the form x t f t x t satisfying the initial condition x t 0 x 0

depth filtering of first breaks and ground roll much of the energy removed in a typical standard seismic data preprocessing sequence corresponds to approximately horizontal propagation in very near surface structure such energy includes direct first break compressional waves direct shear waves and ground roll these may be effectively separated from deeper reflections by downward continuation of the recorded common source wavefield by reverse time extrapolation to a depth beneath which these waves propagate the subhorizontally traveling waves get left behind in the shallow part of the model subsequent upward continuation of the wavefield reconstructs the original surface recorded wavefield with the subhorizontally traveling waves removed only a very simple even constant velocity distribution is required for the wavefield extrapolation no net distortion is produced since both downward and upward continuations are performed using the same velocity model

selections of bounded variation under the excess restrictions let x be a metric space with metric d c x denote the family of all nonempty compact subsets of x and given f g c x let e f g supx finfy gd x y be the hausdorff excess of f over g the excess variation of a multifunction f a b c x which generalizes the ordinary variation v of single valued functions is defined by v f a b sup i 1me f ti 1 f ti where the supremum is taken over all partitions ti i 0m of the interval a b the main result of the paper is the following selection theorem if f a b c x v f a b t0 a b and x0 f t0 then there exists a single valued function f a b x of bounded variation such that f t f t for all t a b f t0 x0 v f a t0 v f a t0 and v f t0 b v f t0 b we exhibit examples showing that the conclusions in this theorem are sharp and that it produces new selections of bounded variation as compared with v v chistyakov selections of bounded variation j appl anal 10 1 2004 1 82 in contrast to this a multifunction f satisfying e f s f t c t s for some constant c 0 and all s t a b with s t lipschitz continuity with respect to e admits a lipschitz selection with a lipschitz constant not exceeding c if t0 a and may have only discontinuous selections of bounded variation if a t0 b the same situation holds for continuous selections of f a b c x when it is excess continuous in the sense that e f s f t 0 as s t 0 for all t a b and e f t f s 0 as s t 0 for all t a b simultaneously