on the existence and multiplicity of positive periodic solutions for first order vector differential equation abstract in this paper we consider the existence and multiplicity of positive periodic solutions for first order vector differential equation x t f t x t 0 a e t 0 under the periodic boundary value condition x 0 x here is a positive constant and f 0 r n r n is a caratheodory function some existence and multiplicity results of positive periodic solutions are derived by using a fixed point theorem in cones

remarks on the periodic boundary value problems for first order differential equations this paper is concerned with the existence of solutions and the monotone method of first order periodic boundary value problems when the lower and upper solutions and violate the boundary conditions 0 2 and 0 2 using the topological degree theory two existence theorems are established under weaker conditions than the one side lipschitz conditions an example is given which illustrates that pbvp may not have solutions between and without further restrictions to f t u the monotone method is also discussed with some new results

remarks on first and second order periodic boundary value problems on etudie les problemes aux valeurs limites periodiques d ordre 1 et 2 suivants u f t u u 0 u 2 et u f t u u 0 u 2 u 0 u 2 on obtient l existence de solutions extremales comme limites d iterees monotones

nonconvex duality in optimal control we derive a representation of the minimum cost of an optimal control problem in terms of the upper envelope of generalized semisolutions of the hamilton jacobi equation

the method of upper lower solutions and hyperbolic partial differential equations on presente l etude de la methode des solutions superieures et inferieures pour des equations differentielles hyperboliques on considere la theorie des inegalites differentielles hyperboliques d une facon generale et on demontre un resultat d existence

on the structure of the solution set for first order differential equations we study the structure of the solution set for the periodic boundary value problem for first order differential equations we present some existence results and give conditions under which the solution set is compact connected or acyclic if any of the assumptions is slightly modified then it is shown that the solution set may be chaotic

monotone method for second order delayed differential equations with boundary value conditions in this paper we investigate the existence of extremal solutions of boundary value problems for second order differential equations with delayed arguments we show that a lower and upper solution method has been applied successfully to such problems an one sided lipschitz condition with corresponding constants is important assumption in our research

the synthesis of universal feedback pursuit strategies in differential games we show how any generalized supersolution of the hamilton jacobi equation can be used to construct a feedback pursuit strategy which guarantees to any given tolerance a capture time not exceeding the solution s value if the supersolution is the value function then a near optimal pursuit strategy is obtained in this way an important feature of the construction is its universal nature i e the fact that the feedback law is uniformly effective on compact sets of initial conditions this implies in particular that the feedback construction is one that exploits nonoptimal behavior on the part of the evader

the method of lower and upper solutions for systems of boundary value problems the method of lower and upper solutions for second order differential system x f t x with dirichlet boundary condition x 0 x 1 0 is used to give sufficient conditions for the existence of at least one solution

ordinary differential equations with nonlinear boundary conditions of antiperiodic type abstract the quasilinearization method is successfully applied to differential problems with nonlinear boundary conditions of antiperiodic type given are sufficient conditions under which the monotone iterations converge quadratically to a unique solution of our problem the advantage of this approach lies in the fact that the approximate solutions are obtained iteratively as solutions of the above mentioned iterations

positive solution for system of nonlinear first order pbvps on time scales abstract in this paper we are concerned with the following system of nonlinear first order periodic boundary value problems on time scale t x i t f i t x 1 t x 2 t x n t 0 t 0 t x i 0 x i t i 1 2 n where f i 0 t 0 n r is continuous and there exists a constant m i 0 such that m i x i f i t x 1 x 2 x n 0 for x 1 x 2 x n 0 n t 0 t some existence criteria of positive solution are established by using a fixed point theorem for operators on cone

various lipschitz like properties for functions and sets i directional derivative and tangential characterizations in this work we introduce for extended real valued functions defined on a banach space x the concept of k directionally lipschitzian behavior where k is a bounded subset of x for different types of sets k e g zero singleton or compact the k directionally lipschitzian behavior recovers well known concepts in variational analysis locally lipschitzian directionally lipschitzian or compactly epi lipschitzian properties respectively characterizations of this notion are provided in terms of the lower dini subderivatives we also adapt the concept for sets and establish characterizations of the mentioned behavior in terms of the bouligand tangent cones the special case of convex functions and sets is also studied

a fast butterfly algorithm for generalized radon transforms abstractgeneralized radon transforms such as the hyperbolic radon transform cannot be implemented as efficiently in the frequency domain as convolutions thus limiting their use in seismic data processing we have devised a fast butterfly algorithm for the hyperbolic radon transform the basic idea is to reformulate the transform as an oscillatory integral operator and to construct a blockwise low rank approximation of the kernel function the overall structure follows the fourier integral operator butterfly algorithm for 2d data the algorithm runs in complexity o n2 log n where n depends on the maximum frequency and offset in the data set and the range of parameters intercept time and slowness in the model space from a series of studies we found that this algorithm can be significantly more efficient than the conventional time domain integration

estimation of surface wave group velocity using slant stack in the generalized s transform domain abstractwe have developed the slant stacking approach in the generalized s transform gst domain of seismic data for the estimation of the group velocity of the multimodal surface wave we used two versions of the gst the constant scale gst uses a constant scaling factor in the gaussian window to control the spectral time frequency localization we found that a smaller scaling factor should be chosen for the low frequency surface wave whereas for higher frequencies a larger scaling factor should be chosen therefore the transform exhibits a trade off in the group velocity resolution of low frequencies with small values of the scaling factor and the group velocity resolution of high frequencies with large values of the scaling factor the modified s transform mst another version of the gst used in this study enhanced the time frequency resolution by projecting the frequency into a linear frequency function in the gaussian window this property allowed us to estimate the group velocity for a broad

periodic n dimensional first order coupled systems and periodic schizophrenia phenomena abstract this paper introduces some existence and location results for periodic solutions of first order coupled nonlinear systems of n ordinary differential equations the arguments used are based on the arzela ascoli theorem and schauder s fixed point theorem combined with the upper and lower solution technique to illustrate the applicability of these problems an application to a mathematical model of the thyroid pituitary interaction and their homeostatic mechanism is included for specific cases of periodic events such as relapsing catatonia

waveform relaxation methods for periodic differential functional systems the convergence of waveform relaxation methods applied to a periodic differential functional system is studied the continuous dependence is also discussed

a generalized upper and lower solutions method and multiplicity results for nonlinear first order ordinary differential equations abstract we introduce a generalized upper and lower solutions method for the solvability of first order ordinary differential equations u t t u t u 0 u 1 in order to cover the case when the function satisfies caratheodory conditions this method is then applied to get multiplicity results when the nonlinearity interacts with the real eigenvalue of the linearized problem our proofs are based on differential inequalities and classical leray schauder degree

a generalization of the method of upper and lower solutions for discontinuous first order problems with nonlinear boundary conditions in this paper we prove the existence of extremal solutions for a discontinuous bvp between a lower and an upper solution which may be discontinuous in a finite set of points furthermore we present a monotone method with which we can explicitly construct a pair of sequences that converge to the extremal solutions despite the discontinuities of the right hand side of the equation

suppression of multiple reflections using the radon transform multiple suppression using a variant of the radon transform is discussed this transform differs from the classical radon transform in that the integration surfaces are hyperbolic rather than planar this specific hyperbolic surface is equivalent to parabolae in terms of computational expense but more accurately distinguishes multiples from primary reflections the forward transform separates seismic arrivals by their differences in traveltime moveout multiples can be suppressed by an inverse transform of only part of the data examples show that multiples are effectively attenuated in prestack and stacked seismograms

three parameter radon transform based on shifted hyperbolas abstractthe focusing power of the conventional hyperbolic radon transform decreases for long offset seismic data due to the nonhyperbolic behavior of moveout curves at far offsets furthermore conventional radon transforms are ineffective for processing data sets containing events of different shapes the shifted hyperbola is a flexible three parameter zero offset traveltime slowness and focusing depth function which is capable of generating linear and hyperbolic shapes and improves the accuracy of the seismic traveltime approximation at far offsets radon transform based on shifted hyperbolas thus improves the focus of seismic events in the transform domain we have developed a new method for effective decomposition of seismic data by using such three parameter radon transform a very fast algorithm is constructed for high resolution calculations of the new radon transform using the recently proposed generalized fourier slice theorem gfst the gfst establishes an analytic expression between the f

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

extremal solutions of periodic boundary value problems for first order integro differential equations of mixed type this paper investigates the maximal and minimal solutions of periodic boundary value problems for first order nonlinear integro differential equations of mixed type by establishing a comparison result and using the monotone iterative technique

mean value inequalities in hilbert space we establish a new mean value theorem applicable to lower semicontinuous functions on hilbert space a novel feature of the result is its multidirectionality it compares the value of a function at a point to its values on a set we then discuss some refinements and consequences of the theorem including applications to calculus flow invariance and generalized solutions to partial differential equations rtsumt on etablit un nouveau theoreme de la valeur moyenne qui s applique aux fonctions semicontinues inferieurement sur un espace de hilbert on deduit plusieurs consequences du resultat portant par exemple sur les fonctions monotones et sur les solutions generalisees des equations aux derivees partielles

monotone method for second order periodic boundary value problems in recent years there has been an extensive study of the existence of periodic solutions l s 11 14 151 in 8 111 the existence of solutions of first and second order pbvp period boundary value problems has been obtained by a novel approach of combining the classical method of lower and upper solutions and the method of alternative problems lyapunovschmidt method which provide conditions that are easily verifiable and which covers previous known results of other authors as a constructive method for obtaining extremal solutions of initial and boundary value problems the monotone iterative procedure has been employed by several researchers 5 7 11 13 151 the objective of this paper is to employ this useful technique for second order pbvp to obtain the minimal and maximal solutions as limits of monotone iterates our method can be used to study semilinear parabolic initial boundary value problems and other problems at resonance

a smooth variational principle with applications to subdifferentiability and to differentiability of convex functions we show that typically lower semicontinuous functions on a banach space densely inherit lower subderivatives of the same degree of smoothness as the norm in particular every continuous convex function on a space with a g teaux weak hadamard frechet smooth renorm is densely g teaux weak hadamard frechet differentiable our technique relies on a more powerful analogue of ekeland s variational principle in which the function is perturbed by a quadratic like function this smooth variational principle has very broad applicability in problems of nonsmooth analysis

monotone method for a system of second order periodic boundary value problems the existence of coupled maximal and minimal quasi solutions of a system of second order periodic boundary value problems is discussed when the right hand side possesses a mixed monotone property

nonlinear second order periodic boundary value problems abstract in this paper we study a periodic boundary value problem pbvp for brevity of second order it is well known that if and are upper and lower solutions respectively with then there exists a solution u such that u when nothing was known the main purpose of this paper is to consider this last case also we study the structure of the set of solutions of the pbvp in the sector when and we apply the results to the case

quadratic approximation of solutions for differential equations with nonlinear boundary conditions the quasilinearization method is used for nonlinear ordinary differential equations with nonlinear boundary conditions given are sufficient conditions when corresponding monotone sequences converge to the unique solution and this convergence is quadratic

high resolution velocity gathers and offset space reconstruction we present a high resolution procedure to reconstruct common midpoint cmp gathers first we describe the forward and inverse transformations between offset and velocity space then we formulate an underdetermined linear inverse problem in which the target is the artifacts free aperture compensated velocity gather we show that a sparse inversion leads to a solution that resembles the infinite aperture velocity gather the latter is the velocity gather that should have been estimated with a simple conjugate operator designed from an infinite aperture seismic array this high resolution velocity gather is then used to reconstruct the offset space the algorithm is formally derived using two basic principles first we use the principle of maximum entropy to translate prior information about the unknown parameters into a probabilistic framework in other words to assign a probability density function to our model second we apply bayes s rule to relate the a priori probability density function pdf with the pdf corresponding to the experimental uncertainties likelihood function to construct the a posteriori distribution of the unknown parameters finally the model is evaluated by maximizing the a posteriori distribution when the problem is correctly regularized the algorithm converges to a solution characterized by different degrees of sparseness depending on the required resolution the solutions exhibit minimum entropy when the entropy is measured in terms of burg s definition we emphasize two crucial differences in our approach with the familiar burg method of maximum entropy spectral analysis first burg s entropy is minimized rather than maximized which is equivalent to inferring as much as possible about the model from the data second our approach uses the data as constraints in contrast with the classic maximum entropy spectral analysis approach where the autocorrelation function is the constraint this implies that we recover not only amplitude information but also phase information which serves to extrapolate the data outside the original aperture of the array the tradeoff is controlled by a single parameter that under asymptotic conditions reduces the method to a damped least squares solution finally the high resolution or aperture compensated velocity gather is used to extrapolate near and far offset traces

the monotone method for first order problems with linear and nonlinear boundary conditions in this paper we develop the monotone method for the first order problem u t f t u t for a e t a b when f is a caratheodory function and u w 1 1 a b we consider the nonlinear boundary conditions l u a u b 0 with l c r 2 r nondecreasing in x or nonincreasing in y and the linear boundary conditions a 0u 0 b 0u t l 0 with a 0 b 0 and l 0 r we prove the existence of solutions and the validity of the monotone method if there exists a lower solution a and an upper solution b with either a b for the linear conditions we obtain eight new concepts of lower and upper solutions which generalize previous known cases

existence of closed solutions for a polynomial first order differential equation we find new criteria for the existence of closed solutions in a first order polynomial differential equation which contains the abel equation as a particular case such results are applied to the problem of the existence of limit cycles in planar polynomial vector fields

existence of periodic solutions for first order differential equations in this paper we study the periodic boundary value problem for first order differential equations by combining techniques of the theory of differential inequalities namely the method of upper and lower solutions and the alternative method for nonlinear problems at resonance the results obtained are in terms of the behavior of the nonlinear part at infinity

existence of solutions of differential equations with nonlinear multipoint boundary conditions abstract we establish sufficient conditions for the existence of solutions unique or extremal of differential equations with nonlinear multipoint boundary conditions using the monotone iterative technique

iterative approximation of limit cycles for a class of abel equations this article considers the analytical approximation of limit cycles that may appear in abel equations written in the normal form the procedure uses an iterative approach that takes advantage of the contraction mapping theorem thus the obtained sequence exhibits uniform convergence to the target periodic solution the effectiveness of the technique is illustrated through the approximation of an unstable limit cycle that appears in an abel equation arising in a tracking control problem that affects an elementary second order bilinear power converter

existence and approximation of solutions for discontinuous functional differential equations in this paper existence of solutions of initial value problems for discontinuous functional differential equations is investigated firstly by applying the method of upper and lower solutions which may be discontinuous some existence results of extremal solutions are obtained furthermore we also develop a monotone iterative technique for obtaining extremal solutions which are obtained as limits of monotone sequences

direct mapping of seismic data to the domain of intercept time and ray parameter a plane wave decomposition marine seismic data recorded as a function of source receiver offset and traveltime are mapped directly to the domain of intercept or vertical delay time and horizontal ray parameter this is a plane wave decomposition based on beam forming of wide aperture seismic array data to determine automatically the loci of coherent seismic reflection and refraction events in this computation semblance in addition to the required slowness or horizontal ray parameter stack is found for linear x t trajectories across subarrays subsequently semblance is used to derive a windowing filter that is applied to the slowness stack to determine the points of stationary phase and eliminate aliasing the resulting filtered slowness stacks for multiple subarrays can then be linearly transformed and combined according to ray parameter range and time the resulting function of intercept time and horizontal ray parameter offers significant computational and interpretational advantages for the case of horizontal homogeneou

generalized quasilinearization and first order periodic boundary value problem the method of quasilinearization was extended and generalized to include functions which are neither convex nor concave this is known as generalized quasilinearization the generalized method yields both sided bounds as in the monotone method but the rate of convergence is quadratic as in the quasilinearization method in this paper we extend the generalization method to a first order periodic boundary value problem

monotone iterative techniques of v lakshmikantham for a boundary value problem for systems of impulsive differential difference equations by means of the monotone iterative techniques of v lakshmikantham a couple of minimal and maximal quasisolutions of a two point boundary value problem for a system of impulsive differential difference equations are constructed

positive solutions for first order periodic boundary value problem this paper discusses the existence and multiplicity of positive solutions for the following first order periodic boundary value problemx t f t x t 0 0

periodic solutions of nonlinear boundary value problems one of the well known techniques in the theory of nonlinear boundary value problems bvp is the method of differential inequalities or the method of upper and lower solutions the method of alternative problems a global variant of the lyapunov schmidt method has been used in the study of problems at resonance the investigation of periodic bvp s forms an important subclass of problems at resonance our aim is to combine the two approaches to discuss nonlinear problems at resonance we restrict ourselves in this paper to the discussion of periodic boundary value problems in section 1 we shall indicate the method of upper and lower solutions the results for systems are given in a general way so as to include known results and also offer new directions section 2 deals with the abstract existence results at resonance in the desired framework in section 3 we consider the scalar periodic bvp s by combining the two techniques namely alternative methods and the method of differential inequalities

a monotone method for the equation uxyz f comparison theorems for the nonlinear boundary value problem u x y z f x y z u u x u y u z u x y u y z u z x with the boundary conditions u o y z a y z u x 0 z b z x u x y 0 c x y a y 0 c 0 y a 0 z b z 0 b 0 x c x 0 are established under certain conditions then it is shown that there exist sequences of lower and upper solutions which converge uniformly and monotonically to minimal and maximal solutions respectively

periodic boundary value problem for first order differential equations with impulses at variable times in this paper we show the validity of the method of upper and lower solutions to obtain an existence result for a periodic boundary value problem of first order impulsive differential equations at variable times

upper and lower solutions with jumps abstract we study a periodic boundary value problem for a first order differential equation from a new point of view which permits us to obtain an existence result involving upper and lower solutions with jump discontinuities we present some applications of our result with illustrative examples

dualization of the euler and hamiltonian inclusions optimal control emerged as a distinct eld of research in the late 1950s with the derivation of the pontryagin maximum principle a fundamental set of necessary conditions on minimizing arcs which takes account of di erential constraints the pontryagin maximum principle directly generates the euler equation and other classical necessary conditions from the calculus of variations when specialised to problems without dynamic constraints however deeper connections between necessary conditions in optimal control and the calculus of variations were only subsequently revealed when clarke developed new analytic machinery now referred to as nonsmooth analysis a term introduced by clarke himself for extending classical necessary conditions to apply to the generalized

comparison results for parabolic differential equations at resonance the objective of this paper is to develop systematically general comparison techniques for semilinear parabolic equations with periodic and homogeneous neumann boundary conditions these results are expected to play a significant role in the existence theory for parabolic boundary value problems at resonance

comparison results for first and second order boundary value problems at resonance various types of comparison results for first and second order periodic boundary value problems are developed it is hoped that these comparison results play an important role in the existence theory of boundary value problems at resonance

proximal analysis and the minimal time function under general hypotheses on the target set s and the dynamics of the system we show that the minimal time function ts cdot is a proximal solution to the hamilton jacobi equation uniqueness results are obtained with two different kinds of boundary conditions a new propagation result is proven and as an application we give necessary and sufficient conditions for ts cdot to be lipschitz continuous near s a petrov type modulus condition is also shown to be sufficient for continuity of ts cdot near s

existence of solutions of boundary value problems for differential equations with delayed arguments we study the existence and approximation of solution of boundary value problems for differential equations with delayed arguments sufficient conditions are established for the existence of a unique solution or extremal ones of the given problem a monotone iterative technique is applied

existence of periodic solutions for fully nonlinear first order differential equations in this paper we establish some existence results for periodic solutions of fully nonlinear first order scalar differential equation f t x x 0 the approach here uses the viscosity solutions method developed in recent decades and the classical upper lower solutions method

existence of solutions for some nonlinear problems with p laplacian like operators abstract the purpose of this paper is to obtain some existence results of solutions for the nonlinear boundary value problems with p laplacian like operators

extremal solutions of measure differential equations abstract we investigate the existence of the least and greatest solutions to measure differential equations as well as the relation between the extremal solutions and lower or upper solutions along the way we prove a fairly general local existence theorem and an analogue of peano s uniqueness theorem for measure differential equations finally we show that the general theory is applicable in the study of differential equations with impulses or dynamic equations on time scales

approximation of solutions for nonlinear periodic boundary value problems with discontinuous upper and lower solutions we prove that it is possible to approximate the solutions of a nonlinear periodic boundary value problem via monotone sequences starting at piecewise continuous upper and lower solutions in fact we present here two different iterative methods explicit expression of each iteration is obtained by solving some linear initial value problems

rapid convergence of the iterative technique for first order initial value problems in this paper we study the convergence of the approximate solutions of the first order problem u t f t w t u o u o we prove that if there exists kf u k k 1 and it is a continuous function then it is possible to construct two sequences of approximate solutions converging to a solution with rate of convergence of order k

necessary conditions in optimal control a new approach we present necessary conditions of optimality for a general control problem formulated in terms of a differential inclusion these conditions unify and substantially extend previous results in the literature and also incorporate a stratified feature of a novel nature when specialized to the calculus of variations the results yield necessary conditions and regularity theorems that go significantly beyond the previous standard they also give rise to new and stronger maximum principles of pontryagin type the principal focus here is the development of structural criteria on the problem which guarantee a priori that the necessary conditions apply to any local minimizer

existence results for discontinuous functional differential systems in this paper we study solvability of first order discontinuous functional differential systems equipped with discontinuous functional initial conditions a concrete example is presented and solved to illustrate the obtained results

links between directional derivatives through multidirectional mean value inequalities we prove in the general setting of lower semicontinuous functions on banach spaces the relation between the rockafellar directional derivative and the mixed lower limit of the lower dini derivatives as a byproduct we derive the famous inclusions of tangent cones of closed sets in banach spaces the results are established using as principal tool multidirectional mean value inequalities aussel et al siam j optim 9 3 690 706 1999

nonsmooth constrained optimization and multidirectional mean value inequalities we establish a general fermat rule for the problem of minimizing a lower semicontinuous function on a convex subset of a banach space our basic tool is a constrained variational principle derived from the smooth variational principle of borwein and preiss specializing the fermat rule to the case when the convex set is a drop we obtain a multidirectional rolle type inequality from which in turn we deduce a multidirectional mean value inequality in the line of clarke and ledyaev we follow the abstract approach of our previous paper trans amer math soc 347 1995 pp 4147 4161 thus covering all standard situations met in applications while stressing the links between the results and the few key properties that are needed

a proximal characterization of the reachable set we show that the graph of the reachable set of a control system given by a differential inclusion is uniquely characterized by a hamilton jacobi equation involving proximal normals

path dependent hamilton jacobi equations in infinite dimensions abstract we propose notions of minimax and viscosity solutions for a class of fully nonlinear path dependent pdes with nonlinear monotone and coercive operators on hilbert space our main result is well posedness existence uniqueness and stability for minimax solutions a particular novelty is a suitable combination of minimax and viscosity solution techniques in the proof of the comparison principle one of the main difficulties the lack of compactness in infinite dimensional hilbert spaces is circumvented by working with suitable compact subsets of our path space as an application our theory makes it possible to employ the dynamic programming approach to study optimal control problems for a fairly general class of delay evolution equations in the variational framework furthermore differential games associated to such evolution equations can be investigated following the krasovski subbotin approach similarly as in finite dimensions

existence of solutions to a global eikonal equation abstract we derive a geometric necessary and sufficient condition for the existence of solutions to a global eikonal equation we also study the existence of a minimal solution to this equation and its relation with the well known minimal time function

simultaneous source separation using a robust radon transform abstractwe adopted the robust radon transform to eliminate erratic incoherent noise that arises in common receiver gathers when simultaneous source data are acquired the proposed robust radon transform was posed as an inverse problem using an l1 misfit that is not sensitive to erratic noise the latter permitted us to design radon algorithms that are capable of eliminating incoherent noise in common receiver gathers we also compared nonrobust and robust radon transforms that are implemented via a quadratic l2 or a sparse l1 penalty term in the cost function the results demonstrated the importance of incorporating a robust misfit functional in the radon transform to cope with simultaneous source interferences synthetic and real data examples proved that the robust radon transform produces more accurate data estimates than least squares and sparse radon transforms

deconvolutive radon transform abstractthe radon transform rt plays an important role in seismic data processing for its ability to focus seismic events in the transform domain the band limited nature of seismic events due to the blurring effects of the source wavelet however causes a decrease in the temporal resolution of the transform we have developed the deconvolutive rt decrt as a generalization of conventional rt and to increase the temporal resolution unlike the conventional counterpart the new basis functions can take an arbitrary shape in the time direction this method is thus proposed to adaptively infer the temporal wave shape from the input data while finding a sparse representation of it the new transform significantly improves the sparsity and thus the temporal resolution of the resulting seismic data the applicability of the hyperbolic decrt in seismic data processing is demonstrated for random noise attenuation primary and multiple separation high quality stacking and automatic velocity model building t