towards a physics on fractals differential vector calculus in three dimensional continuum with fractal metric abstract one way to deal with physical problems on nowhere differentiable fractals is the mapping of these problems into the corresponding problems for continuum with a proper fractal metric on this way different definitions of the fractal metric were suggested to account for the essential fractal features in this work we develop the metric differential vector calculus in a three dimensional continuum with a non euclidean metric the metric differential forms and laplacian are introduced fundamental identities for metric differential operators are established and integral theorems are proved by employing the metric version of the quaternionic analysis for the moisil teodoresco operator which has been introduced and partially developed in this paper the relations between the metric and conventional operators are revealed it should be emphasized that the metric vector calculus developed in this work provides a comprehensive mathematical formalism for the continuum with any suitable definition of fractal metric this offers a novel tool to study physics on fractals

gravity source moment inversion a versatile approach to characterize position and 3 d orientation of anomalous bodies we present a 3 d gravity interpretation method based on the inversion of source moments employing the series derived from the gravity anomaly expansion in multipoles and retaining moments up to second order only it presents the advantages of being essentially linear and allowing a straightforward computer implementation the method requires neither an explicit assumption about the source geometry nor a priori knowledge about the density contrast distribution which may even be nonuniform the method assumes implicitly 1 that the source fits in the interior of an imaginary sphere whose center coincides with the source center of mass and whose radius is smaller than the depth to the source center of mass relative to the measuring plane and 2 that the spatial density distribution presents three ortogonal planes of symmetry intersecting at the source center of mass the first assumption can be met by upward continuing the observed anomaly when both assumptions are met the method produces reliable and stable estimates of the total anomalous mass the coordinates of the center of mass the three principal axes directions of the anomalous body and the relative importance among the axes the method is particularly suited for interpreting compact isolated or disjoint but spatially correlated sources the method is sensitive to an incomplete removal of the regional field presumably superimposed on the residual anomaly of interest because the unremoved part of the regional field may strongly affect the estimates of the residual source moments

necessary and sufficient graphical conditions for formation control of unicycles the feasibility problem is studied of achieving a specified formation among a group of autonomous unicycles by local distributed control the directed graph defined by the information flow plays a key role it is proved that formation stabilization to a point is feasible if and only if the sensor digraph has a globally reachable node a similar result is given for formation stabilization to a line and to more general geometric arrangements

simultaneous multifrequency inversion of full waveform seismic data we present a simultaneous multifrequency inversion approach for seismic data interpretation this algorithm inverts all frequency data components simultaneously a data weighting scheme balances the contributions from different frequency data components so the inversion process does not become dominated by high frequency data components which produce a velocity image with many artifacts a gauss newton minimization approach achieves a high convergence rate and an accurate reconstructed velocity image by introducing a modified adjoint formulation we can calculate the jacobian matrix efficiently allowing the material properties in the perfectly matched layers pmls to be updated automatically during the inversion process this feature ensures the correct behavior of the inversion and implies that the algorithm is appropriate for realistic applications where a priori information of the background medium is unavailable two different regularization schemes an l2 norm and a weighted l2 norm function a

analysis of depth resolution in potential field inversion we study the inversion of potential fields and evaluate the degree of depth resolution achievable for a given problem to this end we introduce a powerful new tool the depth resolution plot drp the drp allows a theoretical study of how much the depth resolution in a potential field inversion is influenced by the way the problem is discretized and regularized the drp also allows a careful study of the influence of various kinds of ambiguities such as those from data errors or of a purely algebraic nature the achievable depth resolution is related to the given discretization regularization and data noise level we compute drp by means of singular value decomposition svd or its generalization gsvd depending on the particular regularization method chosen to illustrate the use of the drp we assume a source volume of specified depth and horizontal extent in which the solution is piecewise constant within a 3d grid of blocks we consider various linear regularization terms in a tikhonov damped least squares formulation some based on using higher order derivatives in the objective function drps are illustrated for both synthetic and real data our analysis shows that if the algebraic ambiguity is not too large and a suitable smoothing norm is used some depth resolution can be obtained without resorting to any subjective choice of depth weighting

understanding imaging methods for potential field data several noniterative imaging methods for potential field data have been proposed that provide an estimate of the 3d magnetization density distribution within the subsurface or that produce images of quantities related or proportional to such distributions they have been derived in various ways using generalized linear inversion wiener filtering wavelet and depth from extreme points dexp transformations crosscorrelation and migration we demonstrated that the resulting images from each of these approaches are equivalent to an upward continuation of the data weighted by a possibly depth dependent function source distributions or related quantities imaged by all of these methods are smeared diffuse versions of the true distributions but owing to the stability of upward continuation resolution may be substantially increased by coupling derivative and upward continuation operators these imaging techniques appeared most effective in the case of isolated compact and depth limited sources beca

signed consensus problems on networks of agents with fixed and switching topologies abstractthis paper deals with signed consensus problems for networks of agents in the presence of both fixed and switching topologies by specifying signs for the agents they are divided into two groups and are enabled to reach agreement on a consensus value which is the same in modulus for both groups but not in sign using nearest neighbour interaction rules we propose the distributed protocols and address their exponential convergence problem it is shown that the quasi strong connectivity of fixed networks or joint quasi strong connectivity of switching networks can provide a necessary and sufficient guarantee for all agents to achieve signed consensus exponentially fast in particular the signed consensus results can include as special cases those of bipartite consensus in signed networks with fixed and switching topologies numerical simulations are also provided to illustrate the exponential convergence performance of the proposed signed consensus protocols

a computationally efficient tool for assessing the depth resolution in large scale potential field inversion in potential field inversion careful management of singular value decomposition components is crucial for obtaining information about the source distribution with respect to depth in principle the depth resolution plot provides a convenient visual tool for this analysis but its computational complexity has hitherto prevented application to large scale problems to analyze depth resolution in such problems we developed a variant approxdrp which is based on an iterative algorithm and therefore suited for large scale problems because we avoid matrix factorizations and the associated demands on memory and computing time we used the approxdrp to study retrievable depth resolution in inversion of the gravity field of the neapolitan volcanic area our main contribution is the combined use of the lanczos bidiagonalization algorithm established in the scientific computing community and the depth resolution plot defined in the geoscience community

scaled consensus a notion of scaled consensus is defined wherein network components states approach dictated ratios in the asymptote a linear network dynamics that achieves scaled consensus for a prescribed interaction graph is introduced and an equivalence with the class of negative singular m matrices is explored a few further characterizations of the scaled synchronization process are given and finally a modification that allows tracking on the stable manifold is presented

applying compactness constraints to differential traveltime tomography tomographic imaging problems are typically ill posed and often require the use of regularization techniques to guarantee a stable solution minimization of a weighted norm of model length is one commonly used secondary constraint tikhonov methods exploit low order differential operators to select for solutions that are small flat or smooth in one or more dimensions this class of regularizing functionals may not always be appropriate particularly in cases where the anomaly being imaged is generated by a nonsmooth spatial process time lapse imaging of flow induced velocity anomalies is one such case flow features are often characterized by spatial compactness or connectivity by performing inversions on differenced arrival time data the properties of the time lapse feature can be directly constrained we develop a differential traveltime tomography algorithm whichselects for compact solutions i e models with a minimum area of support through application of model space iteratively reweighted leas

compact gravity inversion we present a new criterion for the inversion of gravity data the principle employed is to minimize the volume of the causative body which is equivalent to maximizing its compactness the anomalous density distribution is obtained using an iterative technique which is numerically stable and rapidly convergent the principle can also be adapted to include modeling of gravity anomalies by single density sources the advantage of this approach is that desirable geologic characteristics are automatically incorporated into the model with a minimum of subjective judgments on the part of the interpreter the treatment of noise in the data fits naturally into the formulation of the inversion procedure the method is illustrated by the inversion of noise free and noisy data generated from a two dimensional model consisting of a regular array of identical rectangular blocks whose densities can be individually specified in every case the algorithm successfully recovers the correct density distribution from the dat

stability of multiagent systems with time dependent communication links we study a simple but compelling model of network of agents interacting via time dependent communication links the model finds application in a variety of fields including synchronization swarming and distributed decision making in the model each agent updates his current state based upon the current information received from neighboring agents necessary and or sufficient conditions for the convergence of the individual agents states to a common value are presented thereby extending recent results reported in the literature the stability analysis is based upon a blend of graph theoretic and system theoretic tools with the notion of convexity playing a central role the analysis is integrated within a formal framework of set valued lyapunov theory which may be of independent interest among others it is observed that more communication does not necessarily lead to faster convergence and may eventually even lead to a loss of convergence even for the simple models discussed in the present paper

three dimensional gravity inversion using graph theory to delineate the skeleton of homogeneous sources abstractwe developed a new strategy based on graph theory concepts to invert gravity data using an ensemble of simple point masses our method consisted of a genetic algorithm with elitism to generate a set of possible solutions each estimate was associated to a graph to solve the minimum spanning tree mst problem to produce unique and stable estimates we restricted the position of the point masses by minimizing the statistical variance of the distances of an mst jointly with the data misfit function during the iterations of the genetic algorithm hence the 3d spatial distribution of the point masses identified the skeleton of homogeneous gravity sources in addition our method also gave an estimation of the anomalous mass of the source so together with the anomalous mass the skeleton could aid other 3d methods with promising geometric a priori parameters several tests with different values of regularizing parameter were made to bespeak this new regularizing strategy the inversion results ap

fractal solids product measures and fractional wave equations this paper builds on the recently begun extension of continuum thermomechanics to fractal porous media that are specified by a mass or spatial fractal dimension d a surface fractal dimension d and a resolution length scale r the focus is on pre fractal media i e those with lower and upper cut offs through a theory based on a dimensional regularization in which d is also the order of fractional integrals employed to state global balance laws in effect the governing equations are cast in forms involving conventional integer order integrals while the local forms are expressed through partial differential equations with derivatives of integer order but containing coefficients involving d d and r this procedure allows a specification of a geometry configuration of continua by fractal metric coefficients on which the continuum mechanics is subsequently constructed while all the derived relations depend explicitly on d d and r upon setting d 3 and d 2 they reduce to conventional forms of governing equations for continuous media with euclidean geometries whereas the original formulation was based on a riesz measure and thus more suited to isotropic media the new model is based on a product measure making it capable of grasping local fractal anisotropy finally the one two and three dimensional wave equations are developed showing that the continuum mechanics approach is consistent with that obtained via variational energy principles

harmonic and biharmonic biases in potential field inversion abstractwe found that minimum l2 norm and smoothness constrained continuous solutions of the linear inverse problem of potential field data are harmonic and biharmonic respectively in the case of a discrete distribution the minimum l2 norm and smoothness constrained solutions become biased toward being harmonic or biharmonic respectively as a result the estimated discrete distribution of density or magnetization contrast tends to be smooth and to satisfy the maximum principle which forces the solution maxima and minima to lie on any boundary of the discretized region the above findings were illustrated with 2d numerical examples the harmonic or biharmonic bias is brought forth when the strengths of the minimum l2 norm or the smoothness constraint are enhanced relative to all other constraints by approximating the continuous case a large number of discretizing cells relative to the number of independent observations and or by using a regularizing parameter we discovered that by inspecting th

scaled consensus problems on switching networks this note aims to solve a class of consensus problems on switching networks of agents in the sense that all agents reach agreement upon a consensus value but with different time varying scales that is multi agent networks achieve the so called scaled consensus necessary and sufficient conditions are established to guarantee the exponential convergence of agents under the action of distributed protocols which are conducted by nearest neighbor interaction rules simulation tests are performed to illustrate the scaled consensus performances for agents in the presence of time varying scales and switching topologies

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

recent advances of stretched gaussian distribution underlying hausdorff fractal distance and its applications in fitting stretched gaussian noise abstract this paper summarizes the latest advances of the third author s research group on stretched gaussian distribution underlying the hausdorff fractal theory and its applications in fitting stretched gaussian noise firstly the hausdorff fractal metrics are introduced as an extension of non euclidean distance based on the fractal scaling the hausdorff derivative is derived which can describe the problems of interest to construct non integer differential equations secondly we introduce the hausdorff derivatives in tackling partial differential equations and the fundamental solution of hausdorff derivative diffusion equation is stretched gaussian distribution thirdly we analyze the feasibility of the least square method for stretched gaussian noise which obeys stretched gaussian distribution the least square method is inapplicable when the noise level is larger than 5 finally by using the hausdorff fractal distance we introduce the stretched least square method which improves the traditional least square method to fit stretched gaussian noise

inversion and uncertainty estimation of gravity data using simulated annealing an application over lake vostok east antarctica interpretation of gravity data warrants uncertainty estimation because of its inherent nonuniqueness although the uncertainties in model parameters cannot be completely reduced they can aid in the meaningful interpretation of results here we have employed a simulated annealing sa based technique in the inversion of gravity data to derive multilayered earth models consisting of two and three dimensional bodies in our approach we assume that the density contrast is known and we solve for the coordinates or shapes of the causative bodies resulting in a nonlinear inverse problem we attempt to sample the model space extensively so as to estimate several equally likely models we then use all the models sampled by sa to construct an approximate marginal posterior probability density function ppd in model space and several orders of moments the correlation matrix clearly shows the interdependence of different model parameters and the corresponding trade offs such correlation plots are used to stud

3d magnetic data space inversion with sparseness constraints i have developed an inversion approach that determines a 3d susceptibility distribution that produces a given magnetic anomaly the subsurface model consists of a 3d equally spaced array of dipoles the inversion incorporates a model norm that enforces sparseness and depth weighting of the solution sparseness is imposed by using the cauchy norm on model parameters the inverse problem is posed in the data space leading to a linear system of equations with dimensions based on the number of data n this contrasts with the standard least squares solution derived through operations within the m dimensional model space m being the number of model parameters hence the data space method combined with a conjugate gradient algorithm leads to computational efficiency by dealing with an n n system versus an m m one where n m tests on synthetic data show that sparse inversion produces a much more focused solution compared with a standard model space least squares inversion the inversion of aeromagnet

analysis of gravity gradiometer inverse problems using optimal design measures tensor components provide related but independent measures of the gravity field in commercial systems some of the components are measured directly while others are calculated with five independent tensor components available for interpretation the question of which components to use arises using ideas from optimal survey design we investigated the information content provided by each of the tensor components and combinations thereof the main tool used is the singular value decomposition of the design matrix resulting from posing the 3d inversion problem of determining densities from gravity gradiometer data optimal design measures suggest that at larger measurement source distances all components and combinations provide similar levels of information concerning the source density distribution simply adding more components to the inverted data set does not provide any advantages this changes at smaller measurement source distances where multicomponent combinations provide stronger information con

total variation regularization for depth to basement estimate part 2 physicogeologic meaning and comparisons with previous inversion methods we applied the mathematical basis of the total variation tv regularization to analyze the physicogeologic meaning of the tv method and compared it with previous gravity inversion methods weighted smoothness and entropic regularization to estimate discontinuous basements in the second part we analyze the physicogeologic meaning of the tv method and compare it with previous gravity inversion methods weighted smoothness and entropic regularization to estimate discontinuous basements presenting a mathematical review of these methods we show that minimizing the tv stabilizing function favors discontinuous solutions because a smooth solution to honor the data must oscillate and the presence of these oscillations increases the value of the tv stabilizing function these three methods are applied to synthetic data produced by a simulated 2d graben bordered by step faults tv regularization and weighted smoothness are also applied to the real anomaly of steptoe valley nevada u s a in all applications the three methods perform similarly tv regularization however has the advantage compared with weighted smoothness of requiring no a priori information about the maximum depth of the basin as compared with entropic regularization tv regularization is much simpler to use because it requires in general the tuning of just one regularization parameter

3 d magnetic inversion with data compression and image focusing we develop a method of 3 d magnetic anomaly inversion based on traditional tikhonov regularization theory we use a minimum support stabilizing functional to generate a sharp focused inverse image an iterative inversion process is constructed in the space of weighted model parameters that accelerates the convergence and robustness of the method the weighting functions are selected based on sensitivity analysis to speed up the computations and to decrease the size of memory required we use a compression technique based on cubic interpolation our method is designed for inversion of total magnetic anomalies assuming the anomalous field is caused by induced magnetization only the method is applied to synthetic data for typical models of magnetic anomalies and is tested on real airborne data provided by exxonmobil upstream research company

on the consensus and bipartite consensus in high order multi agent dynamical systems with antagonistic interactions abstract the aim of this paper is to address consensus and bipartite consensus for a group of homogeneous agents under the assumption that their mutual interactions can be described by a weighted signed connected and structurally balanced communication graph this amounts to assuming that the agents can be split into two antagonistic groups such that interactions between agents belonging to the same group are cooperative and hence represented by nonnegative weights while interactions between agents belonging to opposite groups are antagonistic and hence represented by nonpositive weights in this framework bipartite consensus can always be reached under the stabilizability assumption on the state space model describing the dynamics of each agent on the other hand nontrivial standard consensus may be achieved only under very demanding requirements both on the laplacian associated with the communication graph and on the agents description in particular consensus may be achieved only if there is a sort of equilibrium between the two groups both in terms of cardinality and in terms of the weights of the conflicting interactions amongst agents

practical applications of uniqueness theorems in gravimetry part i constructing sound interpretation methods to obtain a unique and stable solution to the gravity inverse problem a priori information reflecting geological attributes of the gravity source must be used mathematical conditions to obtain stable solutions are established in tikhonov s regularization method where the a priori information is introduced via a stabilizing functional which may be suitably designed to incorporate some relevant geological information however there is no unifying approach establishing general uniqueness conditions for a gravity inverse problem rather there are many theorems usually establishing just abstract mathematical conditions and making it difficult to devise the type of geological information needed to guarantee a unique solution in part i of these companion papers we show that translating the mathematical uniqueness conditions into geological constraints is an important step not only in establishing the type of geological setting where a particular method may be applied but also in designing new gravity inv

an approach to solving mechanics problems for materials with multiscale self similar microstructure this article proposes an efficient method for solving mechanics boundary value problems formulated for domains with multiscale self similar microstructure in particular composite materials for which one of the phases has a fractal like structure with scale cut offs are considered the boundary value problems are solved using a finite element procedure with enriched shape functions that incorporate information about the geometric complexity the use of these shape functions makes possible the definition of a unique parametrically defined model from which the solution for configurations with an arbitrary number of scales can be derived the proposed method is primarily useful for structures with a large number of self similar scales for which using the usual finite element method would be too expensive in order to exemplify the method a 2d composite with fractal microstructure is considered and several boundary value problems are solved

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

convergence speed in distributed consensus and averaging we study the convergence speed of distributed iterative algorithms for the consensus and averaging problems with emphasis on the latter we first consider the case of a fixed communication topology we show that a simple adaptation of a consensus algorithm leads to an averaging algorithm we prove lower bounds on the worst case convergence time for various classes of linear time invariant distributed consensus methods and provide an algorithm that essentially matches those lower bounds we then consider the case of a time varying topology and provide a polynomial time averaging algorithm

consensus seeking in multiagent systems under dynamically changing interaction topologies this note considers the problem of information consensus among multiple agents in the presence of limited and unreliable information exchange with dynamically changing interaction topologies both discrete and continuous update schemes are proposed for information consensus this note shows that information consensus under dynamically changing interaction topologies can be achieved asymptotically if the union of the directed interaction graphs have a spanning tree frequently enough as the system evolves

reaching a consensus in a dynamically changing environment a graphical approach this paper presents new graph theoretic results appropriate for the analysis of a variety of consensus problems cast in dynamically changing environments the concepts of rooted strongly rooted and neighbor shared are defined and conditions are derived for compositions of sequences of directed graphs to be of these types the graph of a stochastic matrix is defined and it is shown that under certain conditions the graph of a sarymsakov matrix and a rooted graph are one and the same as an illustration of the use of the concepts developed in this paper graph theoretic conditions are obtained which address the convergence question for the leaderless version of the widely studied vicsek consensus problem

gravity and magnetic inversion with minimization of a specific functional the nonuniqueness of gravity or magnetic data inversion is well known in order to remove ambiguity some authors have sought solutions minimizing a functional describing geometrical or physical properties last and kubik 1983 in particular developed a method explaining the observed anomaly by structures of minimum volume in this method the domain where anomalous sources are searched is divided into elementary prisms of a constant density or susceptibility contrast each elementary contrast is allowed to vary individually thus a contrast distribution is computed the search for this kind of solution leads in general to geologically more appropriate bodies but exceptions do occur in this paper the technique is broadened to include the search for solutions minimizing the moment of inertia with respect to the center of gravity or with respect to a given dip line passing through it the resulting structures are both deeper and more compact precisely as is required in specific cases theoretical and

transcale average consensus of directed multi vehicle networks with fixed and switching topologies abstractthis paper contributes to solving a class of so called transcale average consensus problems for directed networks of vehicles with single integrator dynamics it generalises the idea of average consensus by taking into account the measurement scale of every vehicle s information state using the nearest neighbour interaction rules distributed algorithms are presented which are shown with the ability to achieve the transcale average consensus for directed networks associated with balanced graphs such consensus results are applicable to networks in the presence of both fixed topology and switching topologies which are also illustrated via simulation tests moreover improved algorithms and results of consensus are given for directed networks associated with unbalanced graphs

structural inversion of gravity data using linear programming structural inversion of gravity data deriving robust images of the subsurface by delineating lithotype boundaries using density anomalies is an important goal in a range of exploration settings e g ore bodies salt flanks application of conventional inversion techniques in such cases using l2 norms and regularization produces smooth results and is thus suboptimal we investigate an l1 norm based approach which yields structural images without the need for explicit regularization the density distribution of the subsurface is modeled with a uniform grid of cells the density of each cell is inverted by minimizing the l1 norm of the data misfit using linear programming lp while satisfying a priori density constraints the estimate of the noise level in a given data set is used to qualitatively determine an appropriate parameterization the 2 5d and 3d synthetic tests adequately reconstruct the structure of the test models the quality of the inversion depends upon a good prior estimation of

consensus problems on networks with antagonistic interactions in a consensus protocol an agreement among agents is achieved thanks to the collaborative efforts of all agents expresses by a communication graph with nonnegative weights the question we ask in this paper is the following is it possible to achieve a form of agreement also in presence of antagonistic interactions modeled as negative weights on the communication graph the answer to this question is affirmative on signed networks all agents can converge to a consensus value which is the same for all agents except for the sign necessary and sufficient conditions are obtained to describe cases in which this is possible these conditions have strong analogies with the theory of monotone systems linear and nonlinear laplacian feedback designs are proposed