simulating the kinematics of completely faceted surfaces we fully generalize a previously developed computational geometry tool 1 to perform large scale simulations of arbitrary 2 1d faceted surfaces z h x y the method is an explicit front tracking scheme that uses a compact three component facet edge junction storage mode because it naturally mirrors the intrinsic surface structure this scheme allows both rapid simulation of large ensembles and easy extraction of geometrical statistics to do so it must overcome the barrier of detecting and resolving a wide variety of topological changes that occur during surface evolution however while the variety of topological events is larger than in the case of 2d cellular networks it is still limited and our main result is a comprehensive algorithm performing these changes in the code

comparison of grid euler deconvolution with and without 2d constraints using a realistic 3d magnetic basement model we describe the application of a 2d constrained grid euler deconvolution method which is able to determine for each solution window whether the source structure is two dimensional three dimensional or poorly defined and to estimate the source location and depth in each solution window eigenvalues and eigenvectors are derived from the euler equations and compared to threshold levels a single eigenvalue below the given threshold and lying in the x y plane is shown to indicate a 2d source while the absence of such an eigenvalue indicates a 3d source geometry two small eigenvalues indicate the field in the window has no distinct source applying these criteria to each solution window allows us to generate a map of source geometry distribution we evaluate the effectiveness of 2d constrained grid euler deconvolution using synthetic magnetic data generated from a 3d basement model based on real topography from an area with surface exposed faulting this modeling strategy provides a complex nonidealized data set that compares euler depth estimates directly to the known basement surface depth our results indicate that noninteger structural indices can be the most appropriate choice for some data sets and the 2dconstrained grid euler method images magnetic basement structure more clearly and unambiguously than the conventional grid euler method

multiple solutions for a discrete boundary value problem involving the p laplacian multiple solutions for a discrete boundary value problem involving the p laplacian are established our approach is based on critical point theory

effects of low pass filtering on the calculated structural index from magnetic data euler and extended euler deconvolution applications use an assumed structural index si or calculate the si respectively for magnetic anomaly data within a specified window the structural index depends on the source type specifically the rate at which the field produced by the source decays we have examined the effects that the application of low pass filtering to magnetic data has on estimating the si using a simple low pass filter we derived the si for filtered field solutions directly over and away from a target based on the magnetic potential of a vertical dipole si 2 we validated this approach by applying extended euler deconvolution to synthetic and field examples in general filtered magnetic data will decrease the numerically determined si to a value lower than the theoretical one the slope and cutoff wavelength of the filter directly affect the estimated si solutions the results prove that one must take into account filtering for the application of euler deconvolution to locate dipole anomalies for unexploded ordnance detection

infinitely many solutions for a class of discrete non linear boundary value problems the existence of infinitely many solutions for a discrete non linear dirichlet problem involving the p laplacian under appropriate oscillating behaviours of the non linear term is established the approach is based on the critical point theory

approximation by γ convergence of a curvature depending functional in visual reconstruction where r2 is a bounded open set the image domain h1 denotes the one dimensional hausdorff measure and g l is the input image the functional has to be minimized over all closed sets c and all u c1 c the function u represents a denoised approximation of the input image g and c represents the set of boundaries of the segmentation the mumford and shah variational model can be extended to several visual reconstruction problems see march 29 computation of depth from stereo images shah 39 computation of optical flow nesi 34 shape from shading shah 40 the existence of minimizers of gms has been proved independently by dal maso morel and solimini 21 and de giorgi carriero and leaci 23 using the compactness and lower semicontinuity theorems of ambrosio 3 mumford and shah 33 studied the properties of minimizers u c of gms assuming that c is a finite union of simple c1 1 curves meeting and meeting each other only at their endpoints they proved that the vertices of c may only be i triple points where three curves meet with equal angles ii points on the boundary of where one curve meets perpendicularly iii crack tips where a curve ends and meets nothing in computer vision the constraints imposed on the segmentations highlighted by such results constitute a drawback of the variational model in particular corners and t junctions which are relevant features for pattern recognition are distorted since the length measure is not sensitive to corners and junctions to allow for such singularities in the segmentations it is necessary to consider curvature depending energies functionals

three positive solutions to a discrete focal boundary value problem abstract we are concerned with the discrete focal boundary value problem 3x t k f x t x a x t2 2x b 1 0 under various assumptions on f and the integers a t2 and b we prove the existence of three positive solutions of this boundary value problem to prove our results we use fixed point theorems concerning cones in a banach space

dependence on parameters for a discrete emden fowler equation abstract we investigate the dependence on parameters for the discrete boundary value problem connected with the emden fowler equation a variational method is used in order to obtain a general scheme allowing for investigating the dependence on parameters of discrete boundary value problems

existence of solutions for p laplacian discrete equations abstract this work is devoted to the study of the existence of at least one non zero solution to a problem involving the discrete p laplacian as a special case we derive an existence theorem for a second order discrete problem depending on a positive real parameter whose prototype is given by 2 u k 1 f k u k k z 1 t u 0 u t 1 0 our approach is based on variational methods in finite dimensional setting

multiple positive solutions of singular and nonsingular discrete problems via variational methods we employ the critical point theory to establish the existence of multiple solutions of some regular as well as singular discrete boundary value problems 2004 elsevier ltd all rights reserved msc 34b16 39a10

high precision potential field and gradient component transformations and derivative computations using cubic b splines potential field and gradient component transformations and derivative computations are necessary for many techniques of data enhancement direct interpretation and inversion we advance new unified formulas for fast interpolation differentiation and integration and propose flexible high precision algorithms to perform 3d and 2d potential field and gradient component transformations and derivative computations in the space domain using cubic b splines the spline based algorithms are applicable to uniform or nonuniform rectangular grids for the 3d case and to regular or irregular grids for the 2d case the fast fourier transform fft techniques require uniform grid spacing the spline based horizontal derivative computations can be done at any point in the computational domain whereas the fft methods use only grid points comparisons between spline and fft techniques through two gravity gradient examples and one magnetic example show that results computed with the spline technique agree better with t

degrees of homogeneity of potential fields and structural indices of euler deconvolution homogeneity is a well known property of the potential fields of simple point sources used infield inversion wefind that the analytical expressions of potential fields created by sources of complicated shape and constant or variable density or magnetization also show this property this is true if all variables of length dimension are involved in the test of homogeneity the coordinates of observation points and the source coordinates and sizes form an extended set of variables in relation to which the field expression is homogeneous in this case the principal definition of homogeneity applied to a potential field can be treated as an operator of a space transform of similarity the ratio between the transformed and original fields determines the value and sign of thedegreeofhomogeneityn thelattermaytakeonpositive zero ornegativevalues thedegreeofhomogeneitydepends onthetypeoffieldandontheassumedphysicalparameterof the field source and can be nonunique for a given field element we analyze the potential field of one singular point as thesimplestcaseofhomogeneity thus wededuceresultsfor the structural index n n in euler deconvolution the structural index can also be positive zero or negative but it has a unique value analytical considerations as well as numerical tests on the gravity contact model confirm the proposed physical interpretation of n and lead to an extended versionofeuler sdifferentialequationforpotentialfields

unbounded perturbations of nonlinear discrete periodic problem at resonance abstract in this paper we study the existence of solutions of nonlinear discrete boundary value problems 2 u t 1 k u t g t u t h t u 0 u t u 1 u t 1 where t 1 t h t r k is the k th eigenvalue of the linear problem 2 u t 1 u t 0 u 0 u t u 1 u t 1 g n r r satisfies some asymptotic nonuniform resonance conditions and g t u u 0 for u r

on the estimation of the structural index from low pass filtered magnetic data abstractthe estimation of the structural index and of the depth to the source is the main task of many popular methods used to analyze potential field data such as euler deconvolution however these estimates are unstable even in the presence of a weak amount of noise and euler deconvolution of noisy data leads to an underestimation of structural index and depth we have studied how the structural index and depth estimates are affected by applying low pass filtering to the data physically based low pass filters such as upward continuation and integration have been shown to be the best choice over a range of altitudes upward continuation or orders integration filters mainly because their outputs have a well defined physical meaning in contrast mathematical low pass filters require that the filter parameters be tuned carefully by means of several trial tests to produce optimally smoothed fields the c norm criterion is a reliable strategy to produce a stabilized vertical derivative and we disc

euler deconvolution of gravity anomalies from thick contact fault structures with extended negative structural index the concept of extended euler homogeneity of potential fields is examined with respect to all variables of length dimension in their analytical expressions this reveals the possible existence of positive degrees of homogeneity or corresponding negative structural indices considered as extensions of thethompson s structural indices in euler deconvolution this approach is implemented for a contact gravity model represented by a 2d semi infinite slab with large thickness relative to its depth applying euler deconvolution on synthetic and field data indicates that the positive degree of homogeneity i e the extended negative structural index is the appropriate one for the inversion of gravity anomalies fromcontactstructures

nonlinear difference equations investigated via critical point methods abstract the multiplicity of solutions for nonlinear difference equations involving the p laplacian is investigated the approach is based on critical point theorems in the setting of finite dimensional banach spaces

existence of multiple solutions for second order discrete boundary value problems we give conditions on f involving pairs of discrete lower and discrete upper solutions which lead to the existence of at least three solutions of the discrete two point boundary value problem yk 1 2yk yk 1 f k yk vk 0 for k 1 n 1 y0 0 yn where f is continuous and vk yk yk 1 for k 1 n in the special case f k t p f t greater than or equal to 0 we give growth conditions on f and apply our general result to show the existence of three positive solutions we give an example showing this latter result is sharp our results extend those of avery and peterson and are in the spirit of our results for the continuous analogue c 2002 elsevier science ltd all rights reserved

unification of euler and werner deconvolution in three dimensions via the generalized hilbert transform the extended euler deconvolution algorithm is shown to be a generalization and unification of 2 d euler deconvolution and werner deconvolution after recasting the extended euler algorithm in a way that suggests a natural generalization to three dimensions we show that the 3 d extension can be realized using generalized hilbert transforms the resulting algorithm is both a generalization of extended euler deconvolution to three dimensions and a 3 d extension of werner deconvolution at a practical level the new algorithm helps stabilize the euler algorithm by providing at each point three equations rather than one we illustrate the algorithm by explicit calculation for the potential of a vertical magnetic dipole

tensor deconvolution a method to locate equivalent sources from full tensor gravity data we present a method dedicated to the interpretation of full tensor gravity gradiometry ftg data called tensor deconvolution it is especially designed to benefit from the simultaneous use of all the ftg components and of the gravity field in particular it uses tensor scalar invariants as a basis for source location the invariant expressions involve all of the independent components of the tensor this method is a tensor analog of euler deconvolution but has the following advantages compared to the conventional euler deconvolution method 1 it provides a solution at every observation point without the use of a sliding window 2 it determines the structural index automatically as a consequence the structural index follows the variations of the field morphology 3 it uses all components of the measured full gradient tensor and gravity field thus reducing errors caused by random noise it is based on scalar invariants that are by nature insensitive to the orientation of the measuring device we tested our method on both noise free and noise contaminated data these tests show that tensor solutions cluster in the vicinity of the center of causative bodies whereas euler solutions better outline their edges hence these methods should be combined for improved contouring and depth estimation in addition we use a clustering method to improve the selection of solutions which proves advantageous when data are noisy or when signals from close causative bodies interfere introduction

a γ convergence result for the two gradient theory of phase transitions the generalization to gradient vector fields of the classical double well singularly perturbed functionals i e u 1w u e 2 u 2 dx where w 0 if and only if a or b and a b is a rank 1 matrix is considered under suitable constitutive and growth hypotheses on w it is shown that i e converge to k h n 1 s u if u w 1 1 r d i u u bv a b otherwise where k is the constant interfacial energy per unit area

nontrivial solutions of boundary value problems of second order difference equations we employ the critical point theory to establish the existence of nontrivial solutions for some boundary value problems of second order difference equations

multiple source werner deconvolution a reformulation of the werner deconvolution algorithm using the analytic signal is extended to multiple source bodies the extended algorithm involves solving a linear least squares problem the coefficients so obtained determine a complex polynomial whose roots define the locations and depths of the body contacts the extended algorithm has been used to map the structure of the cobb offset zone of the juan de fuca ridge from aeromagnetic data both the top and bottom of the spreading center basalts can be delineated connections between the multiple source werner technique and compudepth are discussed

toward a three dimensional automatic interpretation of potential field data via generalized hilbert transforms fundamental relations the paper extends to three dimensions 3 d the two dimensional 2 d hilbert transform relations between potential field components for the 3 d case it is shown that the hilbert transform is composed of two parts with one part acting on the x component and one part on the y component as for the previously developed 2 d case it is shown that in 3 d the vertical and horizontal derivatives are the hilbert transforms of each other the 2 d cauchy riemann relations between a potential function and its hilbert transform are generalized for the 3 d case finally the previously developed concept of analytic signal in 2 d can be extended to 3 d as a first step toward the development of an automatic interpretation technique for potential field data

combining spatial components and hilbert transforms to interpret ground time domain electromagnetic data abstractwe have developed a method for displaying or imaging data from a ground time domain electromagnetic system and for extracting the geometric parameters of a small conductor the parameters are determined directly from the data using combinations of the spatial components of the secondary fields and their hilbert transforms the position of the target coincides with the peaks of the energy envelope ee or the t component of the response here the ee is the square root of the sum of the squares of the three spatial components and their hilbert transforms whereas the t component response is an analogous quantity that excludes the hilbert transform terms studies on synthetic models indicate that the t component response is sharper than the ee in most possible target orientations once the position of a body has been determined using the peak of the t component response the dip of the target can be quantified using the ratio of the full width at half magnitude fwhm of the t component response and

towards a variational theory of phase transitions involving curvature an anisotropic area functional is often used as a model for the free energy of a crystal surface for models of faceting the anisotropy is typically such that the functional becomes nonconvex and then it may be appropriate to regularize it with an additional term involving curvature when the weight of the curvature term tends to 0 this gives rise to a singular perturbation problem the structure of this problem is comparable to the theory of phase transitions studied first by modica and mortola their ideas are also use ful in this context but they have to be combined with adequate geometric tools in particular a variant of the theory of curvature varifolds intro duced by hutchinson is used in this paper this allows an analysis of the asymptotic behaviour of the energy functionals

interface evolution in three dimensions with curvature dependent energy and surface diffusion interface controlled evolution phase transitions epitaxial growth of elastic films when the interfacial energy is a nonconvex function of orientation the anisotropic curvature flow equation becomes backward parabolic to overcome the instability thus generated a regularization of the equation that governs the evolution of the interface is needed in this paper we develop a regularized theory of curvature flow in three dimensions that incorporates surface diffusion and bulk surface interactions the theory is based on a superficial mass balance configurational forces and couples consistent with superficial force and moment balances a mechanical version of the second law that includes via the configurational moments work that accompanies changes in the curvature of the interface a constitutive theory whose main ingredient is a positive definite isotropic quadratic dependence of the interfacial energy on the curvature tensor two special cases are investigated i the interface is a boundary between bulk phases or grains and ii the interface separates an elastic thin film bonded to a rigid substrate from a vapor phase whose sole action is the deposition of atoms on the surface

the structural index in gravity and magnetic interpretation errors uses and abuses abstractthe structural index si is based on the concept of euler homogeneity a description of scaling behavior it has found wide use in potential field depth estimation and is a constant integer for simple sources with single singularities points lines thin bed faults sheet edges infinite contacts for these cases the si is identical to the index of a simple power law field fall off with distance the simple euler formulation is only strictly correct for such simple sources and integer si values the widespread use of the simple euler method on more complex structures using fractional si values is likely to produce misleading results because the si is no longer a constant for any given source we examine a recently published example that used an arbitrary si to estimate depth to the base of the crust for africa and produced misleading results extension to more complex sources such as tabular bodies or thick steps requires one of several more generalized approaches which recognize all variabl

three positive fixed points of nonlinear operators on ordered banach spaces we generalize the fixed point theorem of leggett williams which is a theorem giving conditions that imply the existence of three fixed points of an operator defined on a cone in a banach space we then show how to apply our theorem to prove the existence of three positive solutions to a second order discrete boundary value problem 2001 elsevier science ltd all rights reserved

a note on the multiplicity of solutions for a second order difference equation with a parameter abstract we undertake the question of the multiplicity of solutions to the discrete emden fowler bvp by correcting some results contained in 2

on the application of euler deconvolution to the analytic signal standard euler deconvolution is applied to potential field functions that are homogeneous and harmonic homogeneity is necessary to satisfy the euler deconvolution equation itself whereas harmonicity is required to compute the vertical derivative from data collected on a horizontal plane according to potential field theory the analytic signal modulus of a potential field is a homogeneous function but is not a harmonic function hence the vertical derivative of the analytic signal is incorrect when computed by the usual techniques for harmonic functions and so also is the consequent euler deconvolution we show that the resulting errors primarily affect the structural index and that the estimated values are always notably lower than the correct ones the consequences of this error in the structural index are equally important whether the structural index is given as input as in standard euler deconvolution or represents an unknown to be solved for the analysis of a case history confirms serious errors in the estimation of structural index if the vertical derivative of the analytic signal is computed as for harmonic functions we suggest computing the first vertical derivative of the analytic signal modulus taking into account its nonharmonicity by using a simple finite difference algorithm when the vertical derivative of the analytic signal is computed by finite differences the depth to source and the structural index consistent with known source parameters are in fact obtained

grid euler deconvolution with constraints for 2d structures the conventional formulation of 3d euler deconvolution assumes that the observed field in each euler window varies in all directions where the source is 2d this assumption leads to the production of poorly constrained solutions if the source is 2d the problem leads to a rank deficient normal equations matrix having an eigenvector associated with a zero eigenvalue this vector lies in the horizontal plane and is pointing along the strike direction thus allowing for the identification of a 2d structure and its strike finding a pseudoinverse via eigenvector expansion allows accurate source location and the strike information allows the automatic implementation of profile based techniques like extended euler deconvolution to gridded data thus allowing for the first time the estimation of strikes dips and susceptibilities from grids using an automatic process we present a grid based version of euler deconvolution that has the ability to define within an euler operating window whether the source is 2

positive solutions and nonlinear eigenvalue problems for third order difference equations first solutions that lie in a cone intersected with an annular type region are obtained for the third order difference equation 3u t a t f u t 0 2 t t 2 satisfying the boundary conditions u 0 u 1 u t 3 0 in the cases that f is either superlinear or sublinear the second part is devoted to determining values of for which there exist positive solutions of 3u t a t f u t 0 2 t t 2 satisfying the boundary conditions u 0 u 1 u t 3 0 when both limx 0 f x x and limx f x x exist as positive real numbers

steerable wavelet frames based on the riesz transform we consider an extension of the 1 d concept of analytical wavelet to n d which is by construction compatible with rotations this extension called a monogenic wavelet yields a decomposition of the wavelet coefficients into amplitude phase and phase direction the monogenic wavelet is based on the hypercomplex monogenic signal which is defined using riesz transforms and perfectly isotropic wavelets frames employing the new concept of clifford frames we can show that the monogenic wavelet generates a wavelet frame furthermore this approach yields wavelet frames that are steerable with respect to direction applications to descreening and contrast enhancement illustrate the versatility of this approach to image analysis and reconstruction

existence of multiple solutions for a second order difference equation with a parameter applying the critical point theorem we establish existence and multiple solutions for a second order difference boundary value problem and show the explicit intervals of l such that the equation has at least 2n distinct solutions

existence of solutions for some discrete boundary value problems with a parameter in this paper the existence and multiplicity results of solutions are obtained for the discrete nonlinear two point boundary value problem bvp email protected 2u k 1 email protected k u k email protected z 1 t u 0 email protected t where t is a positive integer z 1 t 1 2 t d is the forward difference operator defined by du k u k 1 u k and f z 1 t xr r is continuous email protected r is a parameter by using the critical point theory and morse theory we obtain that the above bvp has solutions for l being in some different intervals

eigenvalue and spectral intervals for a nonlinear algebraic system abstract this paper studies the nonlinear algebraic system of the form x af x where 0 a is a positive n n square matrix x x 1 x 2 x n t f x f x 1 f x 2 f x n t the nondecreasing continuous function f is defined on 0 and f u 0 for u 0 the system covers many problems that arise in applications such as difference equations boundary value problems and dynamical networks let f 0 lim u 0 cf u u and f lim u f u u we classify the system according to six pairs of possible values of f 0 and f with the assumption that both f 0 and f exist including for each case r 0 is divided into intervals for the value of intervals that corresponds to existence multiplicity and nonexistence of positive solutions of the system respectively we then obtain intervals that contain only the eigenvalues of the nonlinear operator t af also the resolvent and the spectrum of t are determined two examples are given to show the applications of the results

existence and multiplicity of solutions for nonlinear second order difference boundary value problems in the present paper the following nonlinear second order difference boundary value problems d p n 1 du n 1 q n u n f n u n 0 n z 1 n u 0 u n p 0 du 0 p n du n are studied by using the critical theory and three new multiple results of solutions are obtained where z 1 n 1 2 n f z 1 n xr r is a continuous function in the second variable p q z 0 n r and p n 0

structure and rigidity of functions in bv2loc r2 with gradients taking only 3 values consider a function u bv2loc r2 such that u takes values in a fixed set of 3 vectors almost everywhere this condition implies that u is piecewise affine away from a closed set of vanishing 1 dimensional hausdorff measure furthermore there is some rigidity in the sense that away from the exceptional set small perturbations of u will result only in controllable changes of the structure

multiridge analysis of potential fields geometric method and reduced euler deconvolution a new method based on 3d multiridge analysis of potential fields assumes a 3d subset in the harmonic region and studies the behavior of potential field ridges which are built by joining extreme points of the analyzed field computed at different altitudes three types of ridges are formed by searching for the zeros of the first horizontal and first vertical derivatives of the potential field types i and ii respectively and the zeros of the potential field itself type iii this method uses a redundant set of ridges called a multiridge set to determine source type and location for homogeneous potential fields generated by simple sources all of the ridges are straight lines converging to the source position this method analyzes the multiridges by using a geometric criterion to find the source position at the intersection of the multiridge set and by solving the three reduced euler equations associated with ridge types i ii and iii the reduced euler type i and ii equations are used to obtain the structural index and the vertical and horizontal source positions equation type iii estimates the horizontal and vertical source positions tests on synthetic as well as the bishop model field yield good results even with noise corrupted data results obtained using magnetic data collected over the wreck of a military ship in the tyrrhenian sea successfully determine its vertical and horizontal positions and the structural index

magsounddst 3d automatic inversion of magnetic and gravity data based on the differential similarity transform we present an automatic procedure magnetic and gravity sounding differential similarity transformmagsounddst for inversion of regular or irregular magnetic and gravitygrid data measured on even or uneven surfaces it solves for horizontal position depth and structural index of simple sources and is independent of a linear background in addition it estimates the shape of sources consisting of several singular points and lines the method uses the property of the differential similarity transform dst of a magnetic or a gravity anomaly to become zero or linear at all observation points when the central point of similarity of the transform which we refer to as the probing point coincides with a source s singular point it uses a measured anomalous field and its calculated or measured gradiometry first order derivatives the method is independent of the magnetization vector direction in the magnetic data case and does not require reduction to the pole transformed data as input with magsounddst we provide an important alternative interpretation technique to the euler deconvolution procedures combining a moving window method whereby the solutions are linked to singular points of causative bodies with an approach in which the solutions are linked to the real sources the procedure involves calculating a 3d function that evaluates the linearity of the dst for different integer or noninteger structural indices using a moving window we sound the subsurface along a vertical line under each window center then we combine the 3d results for different structural indices and present them in three easy tointerpret maps avoiding the need for clustering techniques we deduce only one solution for location and type of simple sources which is a major advantage over euler deconvolution application to different cases of synthetic and real data shows the method s applicability to various types of magnetic and gravity field investigations

generalized hilbert transforms of the effect of single magnetic sources abstractthe generalized hilbert transforms of potential fields particularly magnetic fields provide a useful resource for improving interpretation even though x and y hilbert transforms of a potential field on a plane can be approximately computed from the whole observed field they are locally independent of the observations when working on reduced areas of the plane and therefore on those areas hilbert transforms can be combined with them to better constrain and stabilize local inversions extended euler deconvolution based on euler s homogeneity equation makes extensive use of this principle we investigated closed form expressions for evaluating in the space domain the generalized x and y hilbert transforms of magnetic fields generated by single point sources given in the form of dipoles monopoles and newtonian potential type sources apart from providing a way to calculate hilbert transforms via equivalent sources this approach adding little computer effort can be applied to solve loc