gravimagnetic anomaly formulas for polyhedra of spatially linear media formulas for the gravity potential field and field gradient tensor are derived for a polyhedral target body of a spatially linear density medium the formulas also define the magnetic potential and field in the case of a medium of spatially linear magnetization this work generalizes existing solutions for the gravity field of a polyhedral target of linearly varying density the formulas are analyzed for singularities and for numerical error growth error growth with increasing target distance is found to be higher than in the corresponding uniform polyhedral case examination of the error sources reveals that some error reduction is possible on this basis new algorithms of improved error control over existing algorithms are proposed for the linear medium computational results confirm the expected improvement

on lp x norms the relation between a banach function space x and its subspaces formed by i those functions with absolutely continuous norm ii those with continuous norm and iii the closure of the set of

multiple solutions for a class of p x laplacian equations in involving the critical exponent in this paper we first establish a principle of concentration compactness in then based on this concentration compactness principle we study the existence of solutions for a class of p x lapl

new scheme for computing the magnetic field resulting from a uniformly magnetized arbitrary polyhedron the magnetic field at any point outside a uniformly magnetized polyhedron of arbitrary shape is obtained by adding the fields resulting from the effective free magnetic poles on each of the polygonal surfaces of the polyhedron for each polygonal surface the components of the field at the point of observation are expressed in terms of new line integrals around the edges of the polygon and the solid angle subtended by the polygon at the point of observation the line integrals are standard elementary forms this new approach makes the numerical evaluation of the magnetic fields for such models much simpler and faster than previously published methods

potential fields and their partial derivatives produced by a 2d homogeneous polygonal source a summary with some revisions we summarized and revised the present forward modeling methods for calculating the gravity and magnetic field components and their partial derivatives of a 2d homogeneous source with a polygonal cross section the responses of interest include the gravity field components and their first and second order partial derivatives and the magnetic field components and their first order partial derivatives the revised formulas consist of several basic quantities that are common in all the formulations a singularity appears when the observation point coincides with a polygon vertex this singularity is removable for the gravity formulas but not for the others the compact forms of the revised formulas make them easy to implement we compare the gravity and magnetic field components and their partial derivatives produced by a 2d prism whose polygonal cross section approximates a cylinder with the corresponding analytical fields and partial derivatives of the cylinder the perfect fittings presented by both data sets confirm the reliability of the updated formulas

on the gravitational attraction of a right rectangular prism with density varying with depth following a cubic polynomial juan garcia abdeslem 2005 geophysics 66 1793 1804 the author presents an interesting note that a cubic polynomial would be more appropriate than a quadratic polynomial to simulate the density depth variation of sedimentary rocks when the depth is confined to 7 km the author demonstrates this by simulating the density depth data from the green canyon offshore louisiana as presented by li 2001

on delay differential equations with impulses abstract sufficient conditions are obtained respectively for the asymptotic stability of the trivial solution of x t ax t j 1 bjx tj t tj t tj and for the existence of a nonoscillatory solution conditions are also obtained for all solutions to be oscillatory the asymptotic behaviour of an impulsively perturbed delay logistic equation is investigated as an extension to a nonlinear equation

a note on persistence and extinction of a randomized food limited logistic population model this paper addresses the issue of the asymptotic behavior for a non autonomous randomized food limited logistic population model several sufficient conditions are formulated and proved for p moment persistence and extinction of the population as well as in sense of almost sure results show that food limited assumption has an influence on the convergence rate of the solution to the equilibria for the deterministic and stochastic model some previously known results are improved numerical simulations are provided to support the results

dynamics of a discrete food limited population model with time delay abstract in this paper we consider a discrete food limited population model with time delay firstly the stability of the equilibrium of the system is investigated by analyzing the characteristic equation by choosing the time delay as a bifurcation parameter we prove that neimark sacker bifurcations occur when the delay passes a sequence of critical values then the explicit algorithm for determining the direction of the neimark sacker bifurcations and the stability of the bifurcating periodic solutions are derived finally some numerical simulations are given to verify the theoretical analysis

linear inversion of gravity data for 3 d density distributions we have developed an improved levenburg marquart technique to rapidly invert bouguer gravity data for a 3 d density distribution as a source of the observed field this technique is designed to replace tedious forward modeling with an automatic solver that determines density models constrained by geologic information supplied by the user where such information is not available objective models are generated the technique estimates the density distribution within the source volume using a least squares inverse solution that is obtained iteratively by singular value decomposition using orthogonal decomposition of matrices with sequential householder transformations the source volume is subdivided into a series of right rectangular prisms of specified size but of unknown density this discretization allows the construction of a system of linear equations relating the observed gravity field to the unknown density distribution convergence of the solution to the system is tightly controlled by a damping pa

on the singularities of the gravity field of a homogeneous polyhedral body the study of the gravity field of arbitrary polyhedral bodies of homogeneous density has provoked a series of publications over the last decades some of the researchers represented an arbitrary three dimensional body in terms of contours obtained by the intersection of horizontal planes with the body

a multiplicity result for a nonlinear degenerate problem arising in the theory of electrorheological fluids the goal of this correction is to correct a mistake that appears in the proof of both lemma 3 9 and 3 10 in the article mih ilescu r dulescu 2006 1 the mistake in the proof of lemma 3 9 and 3 10 is mainly owing to the fact that we wrongly computed the expression of function g x t

existence and multiplicity of solutions for p x laplacian equations in rn abstract this paper investigates the existence and multiplicity of solutions for p x laplacian equations in r n

oscillation of solutions of second order mixed nonlinear differential equations under impulsive perturbations new oscillation criteria are obtained for second order forced mixed nonlinear impulsive differential equations of the form r t f a x q t f a x k 1nq k t f b k x e t t q ix q i a ix q i x q i b ix q i where f c s s c 1s and b 1 b 2 b m a b m 1 b n 0 if a 1 and the impulses are dropped then the results obtained by sun and wong y g sun j s w wong oscillation criteria for second order forced ordinary differential equations with mixed nonlinearities j math anal appl 334 2007 549 560 are recovered examples are given to illustrate the results

magnetic gradients produced by a 2d homogeneous polygonal source we use a set of analytic equations for calculating horizontal and vertical gradients of the magnetic anomaly produced by a 2d homogeneous magnetic source with a polygonal cross section the magnetization vector of the source is assumed to be constant and known and the anomalous field is assumed to be much smaller than the ambient geomagnetic field in magnitude the derivation begins from an expression of the magnetic potential then we evaluate clockwise integrations to establish the equations a comparison between this analytic approach and a numerical approach using the finite difference method to compute gradients indicates the reliability of the equations the proposed method is numerically efficient and does not present singularity on the edges of the polygon prism except at the vertices where the gradients approach to infinity

existence results for schrodinger p laplace equations involving critical growth in rn abstract we establish some existence results for schrodinger p laplace equations in r n with various potentials and critical growth of nonlinearity that may occur on some nonempty set although not necessarily the whole space r n the proofs are mainly based on concentration compactness principles in a suitable weighted variable exponent sobolev space and its imbeddings

analytic solution of the gravity anomaly of irregular 2d masses with density contrast varying as a 2d polynomial function the analytic solution of the gravity anomaly caused by a 2d irregular mass body with the density contrast varying as a polynomial function in the horizontal and vertical directions is extrapolated from a historical version in which the analytic solution for the gravity anomaly was given only at the origin of the coordinate system to any point for the density function in terms of variables relative to that origin to calculate the gravity anomaly at stations that are not at origins a coordinate transformation is performed in which case the polynomial density contrast function must also be expressed in the transformed coordinates or a transformed solution must be obtained these analytic solutions can be obtained at any station using 1 a solution transformation method in which the density function and boundary of a mass body are kept intact or 2 a coordinate transformation method in which polynomial coefficient and boundary of a mass body are transformed accordingly the issue of singularity and instability of the analytic methods has been related to case studies caution should be exercised in modeling or interpreting the gravity survey data using the analytic methods for large target distance to target size ratios outside the range of numerical stability compared with other published methods the analytic solution results agree very well with other numerical or seminumerical methods indicating the solution is correct and can be applied for any gravity anomaly calculation caused by an irregular 2d mass body with the density contrast approximated as a polynomial function of horizontal position and or vertical position when the observation is within the range of numerical stability

a compact embedding result for anisotropic sobolev spaces associated to a strip like domain and some applications abstract let m 1 and d 2 be integers and consider a strip like domain o r d where o r m is a bounded euclidean domain with smooth boundary furthermore let p o r d r be a uniformly continuous and cylindrically symmetric function we prove that the subspace of w 1 p x y o r d consisting of the cylindrically symmetric functions is compactly embedded into l o r d provided that m d p inf x y o r d p x y p sup x y o r d p x y as an application we study a neumann problem involving the p x y laplacian operator and an oscillating nonlinearity proving the existence of infinitely many cylindrically symmetric weak solutions our approach is based on variational and topological methods in addition to the principle of symmetric criticality

hopf bifurcations in a reaction diffusion population model with delay effect abstract a reaction diffusion population model with a general time delayed growth rate per capita is considered the growth rate per capita can be logistic or weak allee effect type from a careful analysis of the characteristic equation the stability of the positive steady state solution and the existence of forward hopf bifurcation from the positive steady state solution are obtained via the implicit function theorem where the time delay is used as the bifurcation parameter the general results are applied to a food limited population model with diffusion and delay effects as well as a weak allee effect population model

the principle of concentration compactness in lp x spaces and its application abstract in this paper we establish a principle of concentration compactness in l p x spaces and apply it to obtain the existence of solutions for p x laplacian equations with critical growth

on a stochastic logistic equation with impulsive perturbations a stochastic logistic model with impulsive perturbations is proposed and investigated first we give a new definition of a solution of an impulsive stochastic differential equation isde which is more convenient for use than the existing one using this definition we show that our model has a global and positive solution and obtain its explicit expression then we establish the sufficient conditions for extinction non persistence in the mean weak persistence persistence in the mean and stochastic permanence of the solution the critical value between weak persistence and extinction is obtained in addition the limit of the average in time of the sample path of the solution is estimated by two constants afterwards the lower growth rate and the upper growth rate of the solution are estimated finally sufficient conditions for global attractivity are established

gravity anomalies of arbitrary 3d polyhedral bodies with horizontal and vertical mass contrasts up to cubic order abstracta new singularity free analytical formula has been developed for the gravity field of arbitrary 3d polyhedral mass bodies with horizontally and vertically varying density contrast using third order polynomial functions first the observation sites are moved to the origin of the coordinate system then the volume and surface integral theorems are invoked successively to transform the volume integrals into surface integrals over polygonal faces and into line integrals over the edges of the polyhedral mass bodies furthermore singularity free closed form solutions are derived for these line integrals over the edges thus the observation sites can be located inside on or outside the 3d distributions a synthetic prismatic mass body is adopted to verify the accuracy and singularity free property of our newly developed analytical expressions excellent agreements are obtained between our solutions and other published closed form solutions with relative errors in the order of 10 12 to 10 13 in a

efficient gravity inversion of basement relief using a versatile modeling algorithm abstractwe have developed a novel approach to compute in an efficient and versatile way the gravity anomaly produced by an arbitrary discrete 3d distribution of density contrast the method allows adjustable precision and is particularly suited for the interpretation of sedimentary basins because the gravity field decays with the square of the distance we use a discrete green s operator that may be made much smaller than the whole study area for irregularly positioned observations this discrete green s operator must be computed just at the first iteration and because each of its horizontal layers presents a center of symmetry only one eighth of its total elements need to be calculated furthermore for gridded data on a plane this operator develops translation symmetry for the whole region of interest and has to be computed just once for a single arbitrary observation position the gravity anomaly is obtained as the product of this small operator by any arbitrary density contrast distribution in

higher integrability for nonlinear elliptic equations with variable growth and discontinuous coefficients abstract in this paper based on the theory of variable exponent spaces we study the higher integrability for a class of nonlinear elliptic equations with variable growth and discontinuous coefficients under suitable assumptions we obtain a local gradient estimate in orlicz space for weak solution

oscillation properties of nonlinear impulsive delay differential equations and applications to population models comparison theorem and explicit sufficient conditions are obtained for oscillation and nonoscillation of solutions of nonlinear impulsive delay differential equations which can be utilized to population dynamic models our results in this paper generalize and improve several known results

gravitational attraction of a rectangular prism with depth dependent density the gravity effect produced by two and three dimensional bodies with nonuniform density contrast has been treated by several authors one of the first attempts in this direction made by cordell 1973 who developed a method to compute the gravity effect due to a two dimensional prism whose density decreases exponentially with depth a different approach was proposed by murthy and rao 1979 they extended the line integral method to obtain the gravity effect for bodies of arbitrary cross sections with density contrast varying linearly with depth chai and hinze 1988 have derived a wavenumber domain approach to compute the gravity effect due to a vertical prism whose density contrast varies exponentially with depth recently rao 1990 has developed a closed expression of the gravity field produced by an asymmetrical trapezoidal body whose density varies with depth following a quadratic polynomial

fourier forward modeling of vector and tensor gravity fields due to prismatic bodies with variable density contrast abstractprismatic bodies with variable density contrasts are very useful in modeling and inversion of gravity anomalies for sedimentary basins in which some simple mathematical functions offer better approximations for the density depth relationship of the basin fill than a constant density model we have explored fourier domain modeling of vector and tensor gravity anomalies due to 2d and 3d prismatic bodies following a wide class of variable density functions we placed the emphasis on general polynomial models because they provided a flexible way to approximate arbitrary density functions the recently proposed gauss fast fourier transform method was applied to improve the accuracy of inverse fourier transform the numerical performance of the presented method was examined by comparing with space domain analytical seminumerical or completely numerical solutions singularity and numerical stability of the algorithm were also analyzed with ranges of numerical stability for polynomial density models of

3 d inversion of gravity data we present two methods for inverting surface gravity data to recover a 3 d distribution of density contrast in the first method we transform the gravity data into pseudomagnetic data via poisson s relation and carry out the inversion using a 3 d magnetic inversion algorithm in the second we invert the gravity data directly to recover a minimum structure model in both approaches the earth is modeled by using a large number of rectangular cells of constant density and the final density distribution is obtained by minimizing a model objective function subject to fitting the observed data the model objective function has the flexibility to incorporate prior information and thus the constructed model not only fits the data but also agrees with additional geophysical and geological constraints we apply a depth weighting in the objective function to counteract the natural decay of the kernels so that the inversion yields depth information applications of the algorithms to synthetic and field data produce density models representative of true structures our results have shown that the inversion of gravity data with a properly designed objective function can yield geologically meaningful information

2 d inversion of gravity data using sources laterally bounded by continuous surfaces and depth dependent density a description is given of numerical methods for 2 d gravity modeling and nonlinear inversion the forward model solution is suitable for calculating the gravity anomaly caused by a 2 d source body with depth dependent density that is laterally bounded by continuous surfaces and can easily accommodate different kinds of geologic structures the weighted and damped discrete nonlinear inverse method addressed here can invert both density and geometry of the source body both modeling and inversion methods are illustrated with several examples using synthetic and two field gravity data sets one over a sulfide ore body and other across a sedimentary basin a sensitivity analysis is carried out for the resulting solutions by means of the resolution covariance and correlation matrices providing insight into the capabilities and limitations of the inversion method the inversion of synthetic data provides meaningful results showing that the method is robust in the presence of noise its sensitivity analysis i

existence of solutions for systems arising in electromagnetism abstract in this paper we study the following p x curl systems u p x 2 u a x u p x 2 u f x u g x u u 0 in u p x 2 u n 0 u n 0 on where r 3 is a bounded simply connected domain with a c 1 1 boundary denoted by p 1 is a continuous function a l f g r 3 r 3 are caratheodory functions and are two parameters using variational arguments based on fountain theorem and dual fountain theorem we establish some existence and non existence results for solutions of this problem our main results generalize the results of xiang et al 2017 41 bahrouni and repovs 2018 9 and ge and lu 2019 22

gravity inversion using open reject and shape of anomaly fill criteria a gravity inversion method is developed by iteratively applying open reject and fill o r f criteria within a model space comprising a great many rectangular prisms each prism is assigned a density contrast the modeling procedure consists of filling some prisms while leaving others empty only one element is filled for each pass generally elements are added only to the periphery of the growing model models can be allowed to grow in any combination of directions or in all directions by application of a shape of anomaly fill criterion the model rapidly attains a form which yields gravity approximating the given gravity scaled down by some constant factor as the model continues to grow this scale factor approaches unity the method readily yields inverse models comprising several thousand individual prisms examples presented here give applications to 2 d problems the method is readily applicable to 212 d and 3 d problems as well overhanging elements are obtained by appropriate use of model

dynamics of high order bam neural networks with and without impulses the main aim of this paper is to study the existence and global exponential stability of periodic solution for high order bidirectional associative memory bam neural networks with and without impulses easily verifiable sufficient conditions are established the method is based on coincidence degree theory as well as a priori estimates and lyapunov functional it is shown that the convergence characteristics of periodic solution for the impulsive system are preserved by the corresponding nonimpulsive system with some restriction imposed on the impulse effect numerical simulation results are given to support the theoretical predictions

sobolev embedding theorems for spaces wk p x ω this paper gives a sobolev type embedding theorem for the generalized lebesgue sobolev space wk p x where is an open domain in rn n 2 with cone property and p x is a lipschitz continuous function defined on satisfying 1 p p nk the main result can be stated as follows for any measurable function q x x formula with p x q x p x np x n kp x there exists a continuous embedding from wk p x to lq x

picone s formula for linear non selfadjoint impulsive differential equations in this paper we derive a picone type formula for second order linear non selfadjoint impulsive differential equations having fixed moments of impulse actions and obtain a wirtinger type inequality a leighton type comparison theorem and a sturm picone comparison theorem for such equations moreover several oscillation criteria are also derived as applications

2 d gravity modeling with analytically defined geometry and quadratic polynomial density functions new methods for 2 d modeling of gravity anomaly data are developed following an approach that uses both analytic and numerical methods of integration the forward model solution developed here is suitable to calculate the gravity effect caused by a 2 d source body bounded either laterally or vertically by continuous functions in our models the density contrast is defined by a second order polynomial function of depth and distance along the profile we present several examples to show that our models are capable of accommodating a broad variety of geologic structures

global attractivity in a food limited population model with impulsive effects abstract sufficient conditions are obtained for the global attractivity of the positive equilibrium of the delay logistic equation with impulsive effect n t r t n t k n p t k c t n p t n k n k exp i k ln n k k p where r t c t are continuous functions p k are positive constants and k k 1 denotes the sequence of impulsive time which assumed to be strictly increasing some special cases are also considered

multiplicity of positive solutions for some quasilinear elliptic equation in rnwith critical sobolev exponent consider the equation pu g x u p 2u f x u p 2u 1 in rn where 1 0 be the principal eigenvalue of pu g x u p 2uinrn rng x u p 0 2 withu 1 0 the associated eigenfunction we prove i equation 1 has at least one positive solution if 0 1 ii suppose rn f x u 1 p 1such that 1 has at least one positive solution for 1 0 moreover ifp 2 there exists 0 1such that 1 has at least two positive solutions for 1 0

closed form formula of magnetic gradient tensor for a homogeneous polyhedral magnetic target a tetrahedral grid example abstracta closed form formula is developed for the full magnetic gradient tensor of a polyhedral body with a homogeneous magnetization vector it is based on the direct derivative technique on the closed form of the magnetic field these analytical expressions are implemented into an easy to use c package which simultaneously calculates the magnetic potential the magnetic field and the full magnetic gradient tensor for magnetic targets modern unstructured tetrahedral grids are adopted to represent the polyhedral body so that our code can deal with arbitrarily complicated magnetic targets a prismatic body is tested to verify the accuracies of our closed form formula excellent agreements are obtained between our closed form solutions and solutions of a prismatic magnetic body with differences up to machine precision a pipeline model is used to demonstrate its capability to deal with complicated magnetic targets this c code is freely available to the magnetic exploration community

singularity removal a refinement of resistivity modeling techniques modeling techniques commonly exhibit errors of 3 to 10 percent or more in the calculation of apparent resistivities over earth models for which analytic solutions are easily available a singularity occurs in the solution of any elliptic partial differential equation for which the forcing function is not smooth the inability to adequately represent in discrete space a discontinuous function in this case the delta function describing the introduction of current at a point commonly results in numerical error near the source of a modeled singularity inspection of an integrated finite difference method for modeling the dc resistivity geophysical technique indicates much of the error encountered is of singular origin a procedure is herein detailed by which the singularity is mathematically removed from the modeling process and reintroduced as a last step thus preventing it from contributing to the numerical error using this procedure the average error in apparent resistivity values for a model of a polar dipole traverse over a nonconducting sphere is reduced by 40 percent for a dipole dipole traverse of a two layer model the error decreases by 75 percent and in the case of a wenner profile of a model of a vertically faulted earth the average error is diminished by 90 percent

high precision fourier forward modeling of potential fields we analyzed the numerical forward methods in the fourier domain for potential fields existing fourier domain forward methods applied the standard fast fourier transform fft algorithm to inverse transform a conjugate symmetrical spectrum into a real field it had significant speed advantages over space domain forward methods but suffered from problems including aliasing imposed periodicity and edge effect usually grid expansion was needed to reduce these errors which was equivalent to the numerical evaluation of the oscillatory fourier integral using the trapezoidal rule with smaller steps we tested a high precision fourier domain forward method based on a combined use of shift sampling technique and gaussian quadrature theory the trapezoidal rule applied by the standard fft algorithm to evaluate the continuous fourier transform was modified by introducing a shift parameter by choosing optimum values of as gaussian quadrature nodes we developed a gauss fft method for fourier forward modeling of potential fields no grid expansion was needed the sources can be set near the boundary of the fields or even go beyond the boundary the gauss fft method converged to the space domain solution much faster than the standard fft method with grid expansion forward modeling results almost identical to space domain ones can be obtained in less time numerical examples of both simple and complex 2d and 3d source forward modeling revealed the reliability and adaptability of the method

a gamma convergence approach to the critical sobolev embedding in variable exponent spaces abstract in this paper we study the critical sobolev embeddings w 1 p l p for variable exponent sobolev spaces from the point of view of the convergence more precisely we determine the limit of subcritical approximation of the best constant associated with this embedding as an application we provide a sufficient condition for the existence of extremals for the best constant

gravity inversion of a discontinuous relief stabilized by weighted smoothness constraints on depth we present a new stable gravity inversion method applied to the mapping of an interface separating two homogeneous media in contrast with previous similar methods it does not impose an overall smoothness on the estimated interface to stabilize the solution the density contrast between the media is assumed to be known the interpretation model for the upper medium consists of rectangular juxtaposed prisms whose thicknesses represent the depths to the interface and are the parameters to be estimated the true interface is assumed to be flat everywhere except at faults to incorporate this attribute into the estimated relief we developed an iterative process in which three kinds of constraints are imposed on parameters 1 proximity between values of adjacent parameters 2 lower and upper bounds to parameters and 3 proximity between the values of parameters and fixed numerical values starting with an initial solution which presents an overall smooth relief the method enhances initially estimated

bifurcation results for a class of p x laplacian equations abstract in this paper we study the following p x laplacian equation div u p x 2 u u p x 2 u f x u x u x 0 x where r n is bounded 0 under suitable assumptions we obtain the existence of global branches of solutions for the above problem via the subsolution supersolution method

comparison of gravimetric formulas for uniform polyhedra the gravitational anomaly of a polyhedral target of uniform density is amenable to analytical solution and has received wide attention we analyze a number of apparently dissimilar solutions for consistency and comment on their suitability for numeric computation consistency is determined by evaluating a method over a special contour for which all methods must agree numerical performance is assessed through operation counts and algorithm classification into vertex line or surface type each type having a known accuracy bound for a given floating point precision we verify performance predictions through model calculations in particular we demonstrate the effectiveness of revised vertex and line algorithms and of a new analytical surface method our scheme of analysis and classification has general application it provides criteria by which existing and novel solutions may be examined for consistency and numerical performance

perron s theorem for linear impulsive differential equations with distributed delay in this paper it is shown that under a perron condition trivial solution of linear impulsive differential equation with distributed delay is uniformly asymptotically stable

environmental periodicity and time delays in a food limited population model abstract sufficient conditions are obtained for the existence of a globally attracting positive periodic solution of the food limited population system modelled by the equation n t r t k t n t m k t c t r t n t m where m is a nonnegative integer and k r c are continuous positive periodic functions of period

oscillation and stability of linear impulsive delay differential equations the main results of this paper are that the oscillation and the stability of a linear impulsive delay differential equation are equivalent respectively to the oscillation and the stability of a corresponding linear delay differential equation without impulses the results of this paper can be used to improve and derive some of the known results in the literature

an analytical expression for the gravity field of a polyhedral body with linearly varying density over the past 24 years the gravity fields of homogeneous polyhedra have been studied extensively paul 1974 barnett 1976 okabe 1979 golizdra 1981 strakhov et al 1986a 1986b gotze and lahmeyer 1988 pohanka 1988 ivan 1990 pohanka 1990 holstein and ketteridge 1996 this intensive scrutiny is well deserved because polyhedra provide a framework in which complex body shapes can be modeled parsimoniously and because the analytic expressions for the gravity fields of uniform polyhedra involve only elementary transcendental functions available as part of the run time library for most computational programming languages

lyapunov type inequalities for second order forced mixed nonlinear impulsive differential equations in this paper we present some new lyapunov and hartman type inequalities for second order forced impulsive differential equations with mixed nonlinearities x t p t x t 1 x t q t x t 1 x t f t t i x t p i x t 1 x t q i x t 1 x t f i t i where p q f are real valued functions pi qi fi are real sequences and 0 1 2 no sign restrictions are imposed on the potential functions p q and the forcing term f and the sequences pi qi fi the inequalities obtained generalize and complement the existing results for the special cases of this equation in the literature

stability switches and hopf bifurcations of an isolated population model with delay dependent parameters the dynamical behaviors of an isolated population model involving delay dependent parameters are investigated it is shown that the positive equilibrium switches from being stable to unstable and then back to stable as the delay increases and the hopf bifurcation occurs finite times between the two critical values of stability changes which can be analytically determined moreover the bifurcating periodic solutions are expressed analytically in an approximate form by the perturbation approach and floquet technique the direction and stability of the bifurcating periodic solutions are also determined finally the validity of the results is shown by the consistency with the numerical simulations

conjugate complex variables method for the computation of gravity anomalies using conjugate complex variables a generalized method is presented to derive formulas to calculate first and higher order derivatives of the gravity potential due to selected mass models double integrals in the computation of gravity gradient anomalies are transformed into complex contour integrals analytical expressions for higher order derivatives of the gravitational potential in arbitrary directions due to two dimensional 2 d polygonal mass models are derived the method is extended to 2 d polygonal bodies whose density contrasts vary with depth and horizontal distance and can be generalized to deal with 2 d bodies of any shape the vertical gravity field and its first derivatives due to a homogeneous radially symmetric body are also computed using conjugate complex variables derivation of gravity and gravity gradient formulas generally is greatly simplified by the use of complex variables

gravimagnetic similarity in anomaly formulas for uniform polyhedra gravitational and magnetic anomalies of an arbitrary target body are linked through poisson s differential relation for a uniform polyhedral target poisson s relation reduces to an algebraic link between gravity and magnetic anomaly formulas the derivation is given in tensor form it identifies for each target facet edge a vector function in terms of which the gravitational and magnetic potential and field anomaly formulas are similarly expressed as appropriately weighted linear combinations this similarity unifies the theory of uniform polyhedral anomalies it benefits analysis and construction of software that naturally embraces all anomalies in a single code the analysis is exemplified by a discussion of singularities and by the adaptation of three gravity field algorithms to the remaining gravitational and magnetic cases while retaining the respective computational advantages of the former gravity field algorithms

inversion modeling of gravity with prismatic mass bodies a combined method for forward and inverse modeling of gravity data is presented based on the fourier transform of poisson s equation the forward modeling is suitable for observation points above within and below causative masses with any prescribed density distribution the inversion is linearized in the spatial domain by superimposing numerous prismatic bodies each having constant but different density and fixed geometry our inversion algorithm adopts a sampling window to reduce memory storage and computations testing with synthetic and field data demonstrates that a successful inversion can be obtained from crudely estimated a priori density distributions and uncertainties lateral variations in density are well resolved but depth resolution often requires better constrained a priori information under various a priori conditions our modeling indicates that sediment density tends to vary exponentially with depth in the san jacinto basin southern california

a versatile solution for the gravity anomaly of 3d prism meshed bodies with depth dependent density contrast abstractwe have developed a generalized solution for computing the gravity anomalies of 3d irregular mass bodies with complicated density contrast variation the 3d irregular shaped bodies can be approximated flexibly by a collection of finite juxtaposed right rectangular prisms the complicated density contrast variation of each prism can be well represented by a depth dependent polynomial function a novel analytic solution of gravity anomalies due to a right rectangular prism with an arbitrary order of polynomial density contrast function of depth is then derived the solution is singularity free in the upper half space over the prism and its singularity in the lower half space containing the prism is resolved by assigning their limit values to the singular terms the numerical stability of the solution is also evaluated through numerical tests hence the solution can be used to compute the gravity anomalies of 3d irregular bodies with variable density contrasts without singularities when computation

a multiplicity result for p x laplacian problem in rn abstract in this paper we discuss the solutions of the following p x laplacian problem div u p x 2 u u p x 2 u f x u x r n where 1 p p x p n based on the theory of the variable exponent sobolev spaces w 1 p x r n we get that there exist at least two non trivial weak solutions

practical applications of uniqueness theorems in gravimetry part ii pragmatic incorporation of concrete geologic information we illustrate the importance of establishing solution uniqueness through mathematical restrictions reflecting a source attribute we also illustrate the validity and utility of a guideline derived in an accompanying paper for constructing sound gravity inversion methods for the class of sources presenting either homogeneous or depth independent density distributions the two part guideline is 1 to introduce a priori information favoring uniqueness either by assuming that a nonnull density distribution depending only on x and y is confined to the interior of a horizontal slab with known position or by limiting the class of possible solutions to homogeneous simply connected polygons or polyhedra with known density displaying no fancy shapes and no curling apophyses at their borders and 2 to introduce information favoring solution stability by estimating only the features of the source which may be resolved by the data following the guideline we apply different methods to gravity data using interp

higher integrability for nonlinear elliptic equations with variable growth abstract in this paper based on the theory of variable exponent spaces we study the regularity for a class of nonlinear elliptic equation with p x laplacian under suitable assumptions we obtain a local gradient estimate in the orlicz space for weak solution

multiple solutions for noncooperative p x laplacian equations in rn involving the critical exponent abstract in this paper we study the multiplicity of solutions for a class of noncooperative p x laplacian equations in r n involving the critical exponent by using the limit index theory due to li y q li a limit index theory and its application nonlinear anal 25 1995 1371 1389 and concentration compactness principle we obtain a sequence of radially symmetric solutions

3d vector gravity potential and line integrals for the gravity anomaly of a rectangular prism with 3d variable density contrast three dimensional rectangular prisms are building blocks for calculating gravity anomalies from irregular 3d mass bodies with spatially variable density contrasts a 3d vector gravity potential is defined for a 3d rectangular prism with density contrast varying in depth and horizontally the vertical component of the gravity anomaly equals the flux of the 3d vector gravity potential through the enclosed surface of the prism thus the 3d integral for the gravity anomaly is reduced to a 2d surface integral in turn a 2d vector gravity potential is defined the vertical component of the gravity anomaly equals the net circulation of the 2d vector gravity potential along the enclosed contour bounding the surfaces of the prism the 3d integral for the gravity anomaly is reduced to 1d line integrals further analytical or numerical solutions can then be obtained from the line integrals depending on the forms of the density contrast functions if an analytical solution cannot be obtained the line integral method is semianalytical requiring numerical quadratures to be carried out at the final stages singularity and discontinuity exist in the algorithm and the method of exclusive infinitesimal sphere or circle is effective to remove them then the vector potential line integral method can calculate the gravity anomaly resulting from a rectangular prism with density contrast varying simply in one direction and sophisticatedly in three directions the advantage of the method is that the constraint to the form of the density contrast is greatly reduced and the numerical calculation for the gravity anomaly is fast

elastic and piezoelectric fields due to polyhedral inclusions abstract we derive explicit closed form expressions describing elastic and piezoelectric deformations due to polyhedral inclusions in uniform half space and bi materials our analysis is based on the linear elasticity theory and green s function method the method involves evaluation of volume and surface integrals of harmonic and bi harmonic potentials in case of polyhedra such integrals are expressed through algebraic functions our results generalize numerous studies on this subject and they allow to obtain fully analytical solutions for a number of physical and engineering problems in the limiting case of an infinite space our relations have an essentially more compact form than relations obtained by other authors we present solutions to classical mindlin and cherruti problems we describe the elastic relaxation of a misfitting polygonal quantum dot in bi materials assuming isotropic and vertically isotropic properties it is explained how to analyze non hydrostatic and non uniform inclusions we also study piezoelectric fields induced by inclusions in materials with cubic and hexagonal lattices among other results we have found that a cubic inclusion in an isotropic material reproduces fields of quantum dots in gaas 0 0 1 and gaas 1 1 1 depending on the orientation of the cube this suggests that one can qualitatively model crystals with different lattices by choosing an appropriate inclusion shape

2d vector gravity potential and line integrals for the gravity anomaly caused by a 2d mass of depth dependent density contrast using line integrals lis used to calculate the gravity anomaly caused by a 2d mass of complicated geometry and spatially variable density contrast is a computationally efficientalgorithm thatreducesthecalculationfromtwodimensions to one dimension this work has developed a mechanism for defining lis systematically for different types of density functions two dimensional vector gravity potential is defined as a vector the net circulation of which along the closedcontourboundinga2dmass equalsthegravityanomaly caused by the 2d mass two representative types of lis aredefined anliwithanarctangentkernelforanydepth dependent density contrast function which has been studied historically and an li with a simple algebraic kernel for any integrabledensity contrastfunction thepresentworkoffers 1avectorial basedderivationofformulasthatdonotsuffer from the arbitrary sign conventions found in some historical approaches and2asimplealgebraickernelinlineintegrals as an alternative to the historical arctangent kernel with the possibilityofextensiontomoregeneralcases theconceptof 2dvectorgravitypotentialprovidesausefultoolfordefining lis systematically for any mass density function helping us understand how dimensions can be reduced in a calculating gravity anomaly especially when the density contrast varies with space lis have been tested in case studies the maximumdifferencesincalculatedgravityanomaliesbydifferent lis for the case studies were between 5 9310

gravity inversion of an interface above which the density contrast varies exponentially with depth mapping of an interface above which the density contrast varies exponentially with depth as is common at the basement surface of sedimentary basins is efficiently achieved by a theoretically precise gravity method which can be applied to either profile data or twodimensional data the contrast in mass above the interface is modeled by an array of vertical rectangular prisms with density contrasts varying exponentially with depth gravity anomalies due to the prisms are calculated in the wavenumber domain and then converted to the space domain the precision of the inverse numerical fourier transform in this procedure is significantly increased by a shift sampling technique based on the discrete fourier deviation equation depth to the interface is determined by iterative adjustment of the vertical extent of the prisms in accordance with observed gravity anomaly data the basement surface of the los angeles basin california calculated by this method closely duplicates the published configuration based

compact imbedding theorems with symmetry of strauss lions type for the space w1 p x ω abstract in this paper we extend some compact imbedding theorems of strauss lions type to the space w 1 p x when the domain has some symmetric properties and p x satisfies some conditions

existence and multiplicity of solutions for a p x laplacian equation with critical growth abstract in this work we study the existence and multiplicity of weak solutions for a class of problems involving the p x laplacian operator in a bounded domain where the nonlinearity has a critical growth the main tool used is the variational method combined with the genus theory for even functionals

nonlinear elliptic equations with variable exponent old and new abstract in this survey paper by using variational methods we are concerned with the qualitative analysis of solutions to nonlinear elliptic problems of the type div a x u u q x 2 u in u 0 on where is a bounded or an exterior domain of r n and q is a continuous positive function the results presented in this paper extend several contributions concerning the lane emden equation and we focus on new phenomena which are due to the presence of variable exponents

on existence of a globally attractive periodic solution of impulsive delay logarithmic population model in this paper it is shown that a logarithmic population model which is governed by impulsive delay differential equation has a globally attractive periodic solution

the gravity anomaly of two dimensional sources with continuous density distribution and bounded by continuous surfaces the gravity anomaly of a complicated two dimensional 2 d source having arbitrary surfaces and density varying in either horizontal or vertical direction is calculated using a combination of closed form solutions and numerical integration the surfaces and density can be defined by continuous or piecewise continuous two dimensional functions in the integration interval for example the anomalies for intrusions or folded sedimentary units having an arbitrary density in the horizontal direction and a polynomial density distribution in the vertical direction can be calculated using surfaces represented by functions of the horizontal dimension when modeling dipping layered intrusions or sedimentary beds the surfaces are represented by functions of the vertical dimension in which case the density can be an arbitrary function of depth and a polynome function of horizontal coordinate the accuracy of the method is defined by the user the method gives simple and general equations to calculate anomalies of c

gravimetric analysis of uniform polyhedra analytical formulas for the gravity anomaly of a uniform polyhedral body are subject to numerical error that increases with distance from the target while the anomaly decreases this leads to a limited range of target distances in which the formulas are operational beyond which the calculations are dominated by rounding error we analyze the sources of error and propose a combination of numerical and analytical procedures that exhibit advantages over existing methods namely 1 errors that diminish with distance 2 enhanced operating range and 3 algorithmic simplicity the latter is achieved by avoiding the need to transform coordinates and the need to discriminate between projected observation points that lie inside on or outside a target facet boundary our error analysis is verified in computations based on a published code and on a code implementing our methods the former requires a numerical precision of one part in 1016 double precision in problems of geophysical interest whereas our c

existence of solutions for p x laplacian problems on a bounded domain in this paper we study the following p x laplacian problem div a x vd uv p x 2 d u b x v uv p x 2 u f x u x e u u 0 on p u where 1 p 1 l p x l p 2 n u s r n is a bounded domain and applying the mountain pass theorem we obtain the existence of solutions in w 0 1 p x for the p x laplacian problems in the superlinear and sublinear cases 2004 elsevier inc all rights reserved

new method for fast computation of gravity and magnetic anomalies from arbitrary polyhedra we show that at any point the gravity field from a solid body bounded by plane surfaces and having uniform density can be computed as a field from a fictitious distribution of surface mass density on the same body the surface mass density at every surface element is equal to the product of the volume density of the body and the scalar product of 1 the unit outward vector normal to that surface element and 2 the position vector of the surface element with respect to the point of observation accordingly the contribution to the gravity field from any plane surface of the body vanishes if the observation point lies in the plane of that surface as a result we can compute the gravity field everywhere including points inside on the surface on an edge or at a corner of the body where more than two surfaces meet this new result lets us compute the gravity field using exactly the same simple procedure as for the magnetic field of a uniformly magnetized object computed from an equivalent surface distr