geometric implementation of hypercubic lattices with noninteger dimensionality by use of low lacunarity fractal lattices it is claimed that the abstract analytic continuation of hypercubic lattices to noninteger dimensionalities can be implemented explicitly by certain fractal lattices of low lacunarity these lattices are special examples of sierpinski carpets their being of low lacunarity means that they are arbitrarily close to being translationally invariant the claim is substantiated for the ising model in d 1 ensuremath epsilon dimensions and for resistor network models with 1ldl2

accurate projection methods for the incompressible navier stokes equations abstract this paper considers the accuracy of projection method approximations to the initial boundary value problem for the incompressible navier stokes equations the issue of how to correctly specify numerical boundary conditions for these methods has been outstanding since the birth of the second order methodology a decade and a half ago it has been observed that while the velocity can be reliably computed to second order accuracy in time and space the pressure is typically only first order accurate in the l norm this paper identifies the source of this problem in the interplay of the global pressure update formula with the numerical boundary conditions and presents an improved projection algorithm which is fully second order accurate as demonstrated by a normal mode analysis and numerical experiments in addition a numerical method based on a gauge variable formulation of the incompressible navier stokes equations which provides another option for obtaining fully second order convergence in both velocity and pressure is discussed the connection between the boundary conditions for projection methods and the gauge method is explained in detail

cohomological characterizations of projective spaces and hyperquadrics projective spaces and hyperquadrics are the simplest projective algebraic varieties and they can be characterized in many ways the aim of this paper is to provide a new characterization of them in terms of positivity properties of the tangent bundle theorem 1 1 the first result in this direction was mori s proof of the hartshorne conjecture in mor79 see also siu and yau sy80 that characterizes projective spaces as the only manifolds having ample tangent bundle then in wah83 wahl characterized projective spaces as the only manifolds whose tangent bundles contain ample invertible subsheaves interpolating mori s and wahl s results andreatta and wi niewski gave the following characterization

lacunarity and multifractal analysis of the large dla mass distribution abstract we show the methodology used to analyze fractal and mass multifractal properties of very large diffusion limited aggregation dla clusters with a maximum of 10 9 particles for 2d aggregates and 10 8 particles for 3d clusters to support our main result the scaling behavior obtained by our experimental results corresponds to the expected performance of monofractal objects in order to estimate lacunarity measures for large dla clusters we develop a variant of the gliding box algorithm which reduces the computer time needed to obtain experimental results we show how our mass multifractal data have a tendency to present monofractal behavior for the mass distribution of the cases presented in this paper in the limit of very large clusters lacunarity analysis shows provided we study small clusters mass distributions data which might be interpreted as two different values of fractal dimensions while the cluster grows however this effect tends to vanish when the cluster size increases further in such a way that monofractality is achieved the outcomes of this paper lead us to conclude that the previously reported mass multifractality behavior vicsek et al 1990 13 detected for dla clusters is a consequence of finite size effects and floating point precision limitations and not an intrinsic feature of the phenomena since the scaling behavior of our dla clusters space corresponds to monofractal objects being this situation remarkably noticeable in the limit of very large clusters

relationship of second order lacunarity hurst exponent brownian motion and pattern organization abstract second order lacunarity metrics are related to the hurst exponent h fractional brownian motion with h 0 5 is used as a benchmark for quantifying patterns local versus global correlation and implications for pattern organization are discussed

a fixed mesh method for incompressible flow structure systems with finite solid deformations a fixed mesh algorithm is proposed for simulating flow structure interactions such as those occurring in biological systems in which both the fluid and solid are incompressible and the solid deformations are large several of the well known difficulties in simulating such flow structure interactions are avoided by formulating a single set of equations of motion on a fixed eulerian mesh the solid s deformation is tracked to compute elastic stresses by an overlapping lagrangian mesh in this way the flow structure interaction is formulated as a distributed body force and singular surface force acting on an otherwise purely fluid system these forces which depend on the solid elastic stress distribution are computed on the lagrangian mesh by a standard finite element method and then transferred to the fixed eulerian mesh where the joint momentum and continuity equations are solved by a finite difference method the constitutive model for the solid can be quite general for the force transfer standard immersed boundary and immersed interface methods can be used and are demonstrated we have also developed and demonstrated a new projection method that unifies the transfer of the surface and body forces in a way that exactly conserves momentum the interface is still effectively sharp for this approach the spatial convergence of the method is observed to be between first and second order as in most immersed boundary methods for membrane flows the algorithm is demonstrated by the simulations of an advected elastic disk a flexible leaflet in an oscillating flow and a model of a swimming jellyfish

application of a fractional step method to incompressible navier stokes equations a numerical method for computing three dimensional time dependent incompressible flows is presented the method is based on a fractional step or time splitting scheme in conjunction with the approximate factorization technique it is shown that the use of velocity boundary conditions for the intermediate velocity field can lead to inconsistent numerical solutions appropriate boundary conditions for the intermediate velocity field are derived and tested numerical solutions for flows inside a driven cavity and over a backward facing step are presented and compared with experimental data and other numerical results

a generalized finite difference gfd ale scheme for incompressible flows around moving solid bodies on hybrid meshfree cartesian grids a scheme using the mesh free generalized finite differencing gfd on flows past moving bodies is proposed the aim is to devise a method to simulate flow past an immersed moving body that avoids the intensive remeshing of the computational domain and minimizes data interpolation associated with the established computational fluid methodologies as such procedures are time consuming and are a significant source of error in flow simulation in the present scheme the moving body is embedded and enveloped by a cloud of mesh free nodes which convects with the motion of the body against a background of cartesian nodes the generalized finite difference gfd method with weighted least squares wls approximation is used to discretize the two dimensional viscous incompressible navier stokes equations at the mesh free nodes while standard finite difference approximations are applied elsewhere the convecting motion of the mesh free nodes is treated by the arbitrary lagrangian eulerian ale formulation of the flow equations which are solved by a second order crank nicolson based projection method the proposed numerical scheme was tested on a number of problems including the decaying vortex flow external flows past moving bodies and body driven flows in enclosures

bose like oscillator in fractional dimensional space the system of a fractional dimensional bose like oscillator of one degree of freedom whose canonical variables satisfy the general wigner commutation relations is investigated momentum position uncertainty relations are obtained for states without definite parity the well known one dimensional heisenberg uncertainty principle is retained but for odd and even states momentum position uncertainty inequalities depending on the dimension of the space are obtained the motions of both the free particle and the harmonic oscillator in a fractional dimensional space are studied through the probability density function the existence of compression spread of the probability density for dimensions d 1 is shown fractional dimensional bose like operators are also deduced they together with the reflection operator form an r deformed heisenberg algebra with a deformation parameter depending on the dimension of the space

spectral dimension of a wire network by a rigorous analysis for diffusion on a wire network the spectral dimension of a fractal is shown to be independent of the local structure it is discussed that diffusion does not occur when the lattice spacing tends to zero on such a fractal that the spectral dimension is less than the hausdorff dimension if a free field on the fractal exhibits a certain long distance behaviour for a sierpinski carpet the spectral dimension is evaluated within bond moving approximation migdal kadanov renormalisation as a result the author obtains a value smaller than the hausdorff dimension

statistical analysis of digital images of periodic fibrous structures using generalized hurst exponent distributions abstract distinction of diverse two dimensional periodic structures can be based on a large number of methods and parameters while the quantitative description of differences between similar samples is usually difficult this article aims by the use of statistical random walk in a generalized q order dimensional space at introducing a methodology to qualify the networked structures on the basis of exemplary textile samples the presented results were obtained at 1 bit monochromatic maps obtained from optical microscopic pictures significant features of samples were represented by the obtained distributions of hurst exponents and shannon entropy calculations

phase transitions for polymers on fractal lattices abstract models of linear and branched polymers on finitely ramified fractal lattices are briefly reviewed their foremost interest is to provide exactly solvable systems with phase transitions at finite values of the fugacity or temperature this gives insight into the behaviour of polymers in inhomogeneous media their collapse transition and their adsorption at surfaces some unexpected results are also found such as the possible existence of essential singularities in generating functions or the non convergence of critical exponents towards their eucledean limit

space time finite element techniques for computation of fluid structure interactions we describe the space time finite element techniques we developed for computation of fluid structure interaction fsi problems among these techniques are the deforming spatial domain stabilized space time dsd sst formulation and its special version and the mesh update methods including the solid extension mesh moving technique semmt also among these techniques are the block iterative quasi direct and direct coupling methods for the solution of the fully discretized coupled fluid and structural mechanics equations we present some test computations for the mesh moving techniques described we also present numerical examples where the fluid is governed by the navier stokes equations of incompressible flows and the structure is governed by the membrane and cable equations overall we demonstrate that the techniques we have developed have increased the scope and accuracy of the methods used in computation of fsi problems 2005 elsevier b v all rights reserved

curvilinear immersed boundary method for simulating fluid structure interaction with complex 3d rigid bodies the sharp interface curvib approach of ge and sotiropoulos l ge f sotiropoulos a numerical method for solving the 3d unsteady incompressible navier stokes equations in curvilinear domains with complex immersed boundaries journal of computational physics 225 2007 1782 1809 is extended to simulate fluid structure interaction fsi problems involving complex 3d rigid bodies undergoing large structural displacements the fsi solver adopts the partitioned fsi solution approach and both loose and strong coupling strategies are implemented the interfaces between immersed bodies and the fluid are discretized with a lagrangian grid and tracked with an explicit front tracking approach an efficient ray tracing algorithm is developed to quickly identify the relationship between the background grid and the moving bodies numerical experiments are carried out for two fsi problems vortex induced vibration of elastically mounted cylinders and flow through a bileaflet mechanical heart valve at physiologic conditions for both cases the computed results are in excellent agreement with benchmark simulations and experimental measurements the numerical experiments suggest that both the properties of the structure mass geometry and the local flow conditions can play an important role in determining the stability of the fsi algorithm under certain conditions the fsi algorithm is unconditionally unstable even when strong coupling fsi is employed for such cases however combining the strong coupling iteration with under relaxation in conjunction with the aitken s acceleration technique is shown to effectively resolve the stability problems a theoretical analysis is presented to explain the findings of the numerical experiments it is shown that the ratio of the added mass to the mass of the structure as well as the sign of the local time rate of change of the force or moment imparted on the structure by the fluid determine the stability and convergence of the fsi algorithm the stabilizing role of under relaxation is also clarified and the upper bound of the under relaxation coefficient required for stability is derived

influence of image resolution and evaluation algorithm on estimates of the lacunarity of porous media in recent years experience has demonstrated that the classical fractal dimensions are not sufficient to describe uniquely the interstitial geometry of porous media at least one additional index or dimension is necessary lacunarity a measure of the degree to which a data set is translationally invariant is a possible candidate unfortunately several approaches exist to evaluate it on the basis of binary images of the object under study and it is unclear to what extent the lacunarity estimates that these methods produce are dependent on the resolution of the images used in the present work the gliding box algorithm of allain and cloitre phys rev a 44 3552 1991 and two variants of the sandbox algorithm of chappard et al j pathol 195 515 2001 along with three additional algorithms are used to evaluate the lacunarity of images of a textbook fractal the sierpinski carpet of scanning electron micrographs of a thin section of a european soil and of light transmission photographs of a togolese soil the results suggest that lacunarity estimates as well as the ranking of the three tested systems according to their lacunarity are affected strongly by the algorithm used by the resolution of the images to which these algorithms are applied and at least for three of the algorithms producing scale dependent lacunarity estimates by the scale at which the images are observed depending on the conditions under which the estimation of the lacunarity is carried out lacunarity values range from 1 02 to 2 14 for the three systems tested and all three of the systems used can be viewed alternatively as the most or the least lacunar some of this indeterminacy and dependence on image resolution is alleviated in the averaged lacunarity estimates yielded by chappard et al s algorithm further research will be needed to determine if these lacunarity estimates allow an improved unique characterization of porous media

potts model on sierpinski carpets and their relation to hypercubic lattices of non integer dimensionality the authors study the critical frontiers and the high and low temperature asymptotic behavior of the potts model two site correlation function for several sierpinski carpets with fractal dimension lying between one and two they simulate the sierpinski carpets by means of appropriate hierarchical lattices obtaining from their basic cells the two graphs among others that arise from the anisotropic bond moving schemes they correspond to the asymptotic high and low temperature dominant terms for the hierarchical lattice two site correlation function exact result by taking the infinite limit size of the basic cell these asymptotic behaviors tends to those of the sierpinski carpets it is then possible to compare them with similar results obtained from analytical extension to d 2 of the hypercubic lattice their results indicate that these asymptotic behaviors do not always coincide suggesting that possibly they may not identify the sierpinski carpets exhibited by gefen et al 1984 as a correct implementation of the analytical extension of hypercubic lattices

on fano foliations abstract in this paper we address fano foliations on complex projective varieties these are foliations whose anti canonical class k is ample we focus our attention on a special class of fano foliations namely del pezzo foliations on complex projective manifolds we show that these foliations are algebraically integrable with one exceptional case when the ambient space is p n we also provide a classification of del pezzo foliations with mild singularities

phase transitions on fractals iii infinitely ramified lattices for pt ii see ibid vol 17 p 435 1984 in the first two papers of this series the authors considered self similar fractal lattices with a finite order of ramification r in the present paper they study physical models defined on a family of fractals with r infinity in order to characterise the geometry of these systems they need the connectivity q and the lacunarity l in addition to the fractal dimensionality d it is found that discrete symmetry spin models on these lattices undergo a phase transition at tc 0 an approximate renormalisation group scheme is constructed and used to find the dependence of tc and the critical exponents on the geometrical factors they also solve the problem of resistor networks on these fractals and discuss its consequences concerning spin models with continuous symmetry

decomposable multiphase entropic descriptor abstract to quantify the degree of spatial inhomogeneity for multiphase materials we adapt the entropic descriptor ed of a pillar model developed to greyscale images to uncover the contribution of each phase we introduce the suitable phase splitting of the adapted descriptor as a result each of the phase descriptors pds describes the spatial inhomogeneity attributed to each phase component obviously their sum equals the value of the overall spatial inhomogeneity we apply this approach to three phase synthetic patterns the black and grey components are aggregated or clustered while the white phase is the background one the examples show how the valuable microstuctural information related separately to each of the phases can be obtained at any integer length scale even dissimilar hidden statistical periodicities can be easily detected for chosen phases built up of compact regular clusters

exact renormalization group approach to critical phenomena in an ising system with long range interactions on the sierpinski gasket lattice abstract an ising spin system with two point long range interactions on the sierpinski gasket lattice is considered the interactions are assigned along edges of all equilateral triangles being part of the self similar structure in addition the long range couplings i e all couplings besides nearest neighbor interactions k are assumed to be independent of the distance i e q1 q l 1 2 where q1 denotes an interaction between the two spins on the distance 2 a with a being a lattice constant and q a positive parameter using an exact renormalization group method based upon the decimation transformation the system is shown to display a line of nontrivial fixed points k k q each point of this line corresponds to a phase transition of a rather high order the critical behavior of the system near every nontrivial fixed point is claimed to be characterized by two dimensions viz the fractal dimension and the so called critical dimension which reflects some topological properties of the system the critical dimension is defined through considering the decay of an effective two point correlation function and depends on q

quantum field theory models on fractal spacetime continuing the analysis of a previous paper the present work applies rigorous renormalization group methods to the hierarchical models to establish the existence of field theories with non gaussian ultraviolet renormalization group fixed points in 4 e dimensions

splitting method for the combined formulation of the fluid particle problem abstract in this paper a splitting method for the direct numerical simulation of solid liquid mixtures is presented where a symmetric pressure equation is newly proposed through numerical experiment it is found that the newly proposed splitting method works well with a matrix free formulation for some benchmark problems avoiding an erroneous pressure field which appears when using the conventional pressure equation of a splitting method when deriving a typical pressure equation of a splitting method the motion of a solid particle has to be approximated by the intermediate velocity instead of treating it as unknowns since it is necessary as a boundary condition therefore the motion of a solid particle is treated in such an explicit way that a particle moves by the known form drag pressure drag that is calculated from the pressure equation in the previous step from the numerical experiment it was shown that this method gives an erroneous pressure field even for a very small time step size as the particle velocity increases in this paper coupling the unknowns of particle velocities in the pressure equation is proposed where the resulting matrix is reduced to the symmetric one by applying the projector of the combined formulation it has been tested over some benchmark problems and gives reasonable pressure fields

on the enforcement of the zero mean pressure condition in the spectral element approximation of the stokes problem this paper considers the spectral element approximation of the stokes problem and the conditioning of the resulting discrete problem the well posedness of the variational formulation of the stokes problem and in particular the uniqueness of the pressure has been demonstrated when the subspace of square integrable functions having vanishing mean is chosen for the pressure space in the discrete setting a conforming subspace is chosen for the discrete pressure space and the analysis of the discrete problem is based on this choice however this space is not used in practical implementations of finite or spectral element methods for the stokes problem in this paper it is shown how the zero volume condition on the pressure may be accommodated within the trial space by modifying the continuity equation in a consistent manner the dependence of the condition number of the uzawa operator and the accuracy of the spectral element approximation on the weighting factor is investigated

lacunarity of the zero crossings of gaussian processes a lacunarity analysis of the zero crossings derived from gaussian stochastic processes with oscillatory autocorrelation functions is evaluated and reveals distinct multiscaling signatures depending on the smoothness and degree of anticorrelation of the process these bear qualitative similarities and quantitative distinctions from an oscillatory deterministic signal and a poisson random process both possessing the same mean interval size between crossings at very small and large scales compared with the correlation length of the random processes the lacunarity is similar to the poisson but exhibits significant departures from poisson behavior if there is a zero frequency component to the process s power spectrum a comparison of exact results with the gliding box technique that is frequently used to determine lacunarity demonstrates its inherent bias

a singular value decomposition svd based generalized finite difference gfd method for close interaction moving boundary flow problems in this paper we present a study on a singular value decomposition svd based generalized finite difference gfd method and a nodal selection scheme for moving body boundary flow problems formulated on a hybrid cartesian cum meshfree grid system the present study shows that the svd based method is more robust and accurate than the conventional least squares based gfd scheme a nodal selection scheme is also introduced to overcome the problem of numerical instability associated with the clustering of computational nodes such nodal clustering occurs dynamically when moving bodies or boundaries approach within close proximity of each other resulting in the overlap of their meshfree grids the nodal scheme is applied to close interaction flow problems as exemplified by the squeezing action of a circular cylinder through a very narrow slot and the close proximity bypass interaction of two oscillating circular cylinders

a three dimensional hybrid meshfree cartesian scheme for fluid body interaction a numerical method based on a hybrid meshfree cartesian grid is developed for solving three dimensional fluid solid interaction fsi problems involving solid bodies undergoing large motion the body is discretized and enveloped by a cloud of meshfree nodes the motion of the body is tracked by convecting the meshfree nodes against a background of cartesian grid points spatial discretization of second order accuracy is accomplished by the combination of a generalized finite difference gfd method and conventional finite difference fd method which are applied to the meshfree and cartesian nodes respectively error minimization in gfd is carried out by singular value decomposition the discretized equations are integrated in time via a second order fractional step projection method a time implicit iterative procedure is employed to compute the new evolving position of the immersed bodies together with the dynamically coupled solution of the flow field the present method is applied on problems of free falling spheres and tori in quiescent flow and freely rotating spheres in simple shear flow good agreement with published results shows the ability of the present hybrid meshfree cartesian grid scheme to achieve good accuracy an application of the method to the self induced propulsion of a deforming fish like swimmer further demonstrates the capability and potential of the present approach for solving complex fsi problems in 3d copyright 2011 john wiley sons ltd

dilute ising model on fractal lattices the authors study dilute ising models on a family of infinitely ramified exact fractals whose fractal dimension varies from 1 to 2 they calculate the phase diagram and critical exponents as a function of the fractal dimensionality within migdal s approximation 1975

an immersed interface method for simulating the interaction of a fluid with moving boundaries in the immersed interface method boundaries are represented as singular force in the navier stokes equations which enters a numerical scheme as jump conditions recently we systematically derived all the necessary spatial and temporal jump conditions for simulating incompressible viscous flows subject to moving boundaries in 3d with second order spatial and temporal accuracy near the boundaries sheng xu z jane wang systematic derivation of jump conditions for the immersed interface method in three dimensional flow simulation siam j sci comput 2006 in press in this paper we implement the immersed interface method to incorporate these jump conditions in a 2d numerical scheme we study the accuracy efficiency and robustness of our method by simulating taylor couette flow flow induced by a relaxing balloon flow past single and multiple cylinders and flow around a flapping wing our results show that 1 our code has second order accuracy in the infinity norm for both the velocity and the pressure 2 the addition of an object introduces relatively insignificant computational cost 3 the method is equally effective in computing flow subject to boundaries with prescribed force or boundaries with prescribed motion

neuromorphometric characterization with shape functionals this work presents a procedure to extract morphological information from neuronal cells based on the variation of shape functionals as the cell geometry undergoes a dilation through a wide interval of spatial scales the targeted shapes are ensuremath alpha and ensuremath beta cat retinal ganglion cells which are characterized by different ranges of dendritic field diameter image functionals are expected to act as descriptors of the shape gathering relevant geometric and topological features of the complex cell form we present a comparative study of classification performance of additive shape descriptors namely minkowski functionals and the nonadditive multiscale fractal we found that the proposed measures perform efficiently the task of identifying the two main classes ensuremath alpha and ensuremath beta based solely on scale invariant information while also providing intraclass morphological assessment

self avoiding walks on self similar structures finite versus infinite ramification self avoiding walks saws generated by exact enumeration techniques are studied on the sierpinski carpet d 2 and on the sierpinski sponge d 3 also called sierpinski square lattices a detailed comparison of the results for saws on these infinitely ramified fractals to saws on finitely ramified sierpinski gaskets sierpinski triangular lattices on regular lattices and on the incipient percolation cluster is done providing insight into the behaviour of saws on ordered and disordered structures the saws on sierpinski square lattices are found to display a kind of intermediate behaviour sharing aspects of both saws on ordered and on fractal structures as a consequence a des cloizeaux relation does not seem to hold for this structure as opposed to its validity for saws on regular lattices on sierpinski triangular lattices and on the incipient percolation cluster

a 3d immersed interface method for fluid solid interaction abstract in immersed interface methods solids in a fluid are represented by singular forces in the navier stokes equations and flow jump conditions induced by the singular forces directly enter into numerical schemes this paper focuses on the implementation of an immersed interface method for simulating fluid solid interaction in 3d the method employs the mac scheme for the spatial discretization the rk4 scheme for the time integration and an fft based poisson solver for the pressure poisson equation a fluid solid interface is tracked by lagrangian markers intersections of the interface with mac grid lines identify finite difference stencils on which jump contributions to finite difference schemes are needed to find the intersections and to interpolate jump conditions from the lagrangian markers to the intersections parametric triangulation of the interface is used the velocity of the lagrangian markers is interpolated directly from surrounding mac grid nodes with interpolation schemes accounting for jump conditions numerical examples demonstrate that 1 the method has near second order accuracy in the infinity norm for velocity and the accuracy for pressure is between first and second order 2 the method conserves the volume enclosed by a no penetration boundary and 3 the method can efficiently handle multiple moving solids with ease

duality on planar fractal and hierarchical lattices duality constructions are presented for the classes of planar fractal and hierarchical lattices interesting results include the interpretation of dual sierpinski gasket and carpets as fractal domain models that the constructions take a lattice l with intrinsic dimension d and connectivity q to a dual l with d d q and q 1 q and a novel but trivial dual for the one dimensional lattice

critical properties of ising models with long range interactions on sierpinski gasket lattices in higher euclidean dimensions abstract ising systems with two spin long range interactions are studied on sierpinski gasket lattices generalized to euclidean dimensions d 3 4 5 6 for each lattice the long range interactions are assigned to edges that are part of the appropriate self similar structure in consequence these systems are infinitely ramified and exhibit critical behavior at finite temperatures an analysis of critical properties of the systems in particular the calculation of critical exponents is carried out by using an exact renormalization group method it is shown that there exists some similarity between critical properties of the long range interacting fractal models under study here and critical properties of short range interacting ising models defined on abstract regular lattices interpolated to noninteger dimensionalities

critical behavior of the kinetic ising model on a fractal lattice the authors study the critical behaviour static and dynamic of the ising model defined on a fractal sierpinski carpet using monte carlo calculations estimates for various critical exponents are obtained from a finite size scaling analysis the results are compared with the available predictions for the dynamical critical exponent z they find the dynamic exponent z near 2 2 to be lower than predicted

potts model on a fractal lattice the authors consider the dilute potts model on the sierpinski carpet the model has a phase transition at a finite temperature and both critical and tricritical behaviour are observed for qqc where q is the number of potts states they use migdal kadanoff type recursion relations to obtain the value of qc and of critical and tricritical exponents and investigate their dependence on various geometrical characteristics of the fractal

the immersed interface method for elliptic equations with discontinuous coefficients and singular sources the authors develop finite difference methods for elliptic equations of the form nabla cdot beta x nabla u x kappa x u x f x in a region omega in one or two space dimension

a suggested generalization for the lacunarity index abstract nowadays lacunarity is a well accepted concept to complement fractality studies of complex sets lacunarity features are used to assess the largeness of gaps or holes of complex signals and images as it stands the lacunarity analysis used is an index averaging the behavior of small and large size structures in a given set the aim of this work is to propose a generalized lacunarity concept on the basis of generalized moments in this form the lacunarity index depending on a discrimination parameter is oriented to quantify lacunarity associated with objects of different size textbook and real images are used to illustrate the ability of the generalized lacunarity index for displaying interesting features of real complex sets

noninteger dimensional hyper euclidean lattices on sierpinski carpets random walks on sierpinski carpets are studied with bond moving renormalisation using the results the author constructs fractal lattices whose fractal dimension of random walk is two such fractal lattices are called hyper euclidean lattices it turns out that hyper euclidean lattices have a connectivity which is one smaller than the fractal dimension

deterministic models of fractal and multifractal growth abstract four deterministic models are presented in order to get more insight into the geometry and multifractal behavior associated with fractal growth phenomena these models allow for exact treatment and are used to demonstrate such properties as directed self affinity and self similarity and multifractal growth probability and mass distribution it is pointed out that in some cases the various fractal dimension definitions may lead to inconsistent results

on computational issues of immersed finite element methods the objective of this paper is to provide a review of recent finite element formulations for immersed methods in these finite element formulations independent lagrangian solid meshes are introduced to move on top of a background eulerian fluid mesh this key feature allows the handling without excessive fluid mesh adaptation multiple deformable solids immersed in viscous fluid in particular pros and cons of both explicit and implicit approaches are illustrated along with subtle differences between incompressible and slightly compressible models

quantum field theory models on fractal spacetime i introduction and overview deux types de modeles de champs scalaires sont consideres l un a son propagateur donne par un temps propre ou une representation chaleur noyau l autre a un propagateur hierarchique la version du reseau fractal du propagateur a temps propre est une reflexion positive

a suggested lacunarity expression for sierpinski carpets a suggested approximate expression for lacunarity is proposed some numerical investigations show that the old expression fails in describing the homogeneity of elimination for some sierpinski carpets

exact renormalization group treatment of the piecewise directed random walks on fractals abstract an exact evaluation of the critical exponent v associated with the mean squared end to end distance of the piecewise directed random walks is presented in the case of the complete family of the sierpinski gasket type of fractals

self referred approach to lacunarity this paper describes an approach to lacunarity which adopts the pattern under analysis as the reference for the sliding window procedure the superiority of such a scheme with respect to more traditional methodologies especially when dealing with finite size objects is established and illustrated through applications to diffusion limited aggregation pattern characterization it is also shown that given the enhanced accuracy and sensitivity of this scheme the shape of the window becomes an important parameter with advantage for circular windows

lacunarity and universality a new expression for the lacunarity of sierpinski carpets is proposed it turns out that the universality can be described with the new expression the fractal dimension and the connectivity

scale dependent lacunarity and fractal ising models as an alternate way of measuring the lacunarity of sierpinski carpets the authors have computed the site to site fluctuation of the quantity m n namely the total number of lattice sites within the chemical distance of n in long scales of n the resultant lacunarity agrees qualitatively with the results obtained in the past using the covering method in short scales of n however the lacunarity shows a different trend among fractals it varies monotonically with the ising correlation length exponent a trend which was sought before but not found with the previous method this applies only for the carpets of the first kind

exact results for an ising system with long range interactions on the sierpinski gasket lattice thermodynamic properties of an ising model with two point long range interactions on the sierpinski gasket lattice are studied using an exact recursive methodfor calculating the partition function the long range interactions are introduced in a self similar manner reflecting the structure of the lattice and are assumed to be of the form ql q 2 l 1 where q 0 0 are parameters and l 1 2 indicates that ql couples pairs of spins on the distance 2l measured in units of the lattice spacing in the case of constant long range interactions 0 exact analytic expression for the critical temperature and the critical exponent v are derived it is also shown that the recursive method can be applied to analyze the problem of the existence of phase transitions at finite temperatures in the case of 0 i e when the long range coupling decrease with the distance

bounds for the fractal dimension of space the authors discuss the possibility that space has to be described as being of non integer dimension they elaborate on some of the problems associated with this idea and investigate the chance to observe effects due to such a fractional dimension two experiments show rather stringent bounds for the deviation of the space dimension from the value three these are the perihelion shift of planetary motion and the lamb shift in hydrogen

a hybrid cartesian immersed boundary method for simulating flows with 3d geometrically complex moving bodies a numerical method is developed for solving the 3d unsteady incompressible navier stokes equations in cartesian domains containing immersed boundaries of arbitrary geometrical complexity moving with prescribed kinematics the governing equations are discretized on a hybrid staggered non staggered grid layout using second order accurate finite difference formulas the discrete equations are integrated in time via a second order accurate dual time stepping artificial compressibility iteration scheme unstructured triangular meshes are employed to discretize complex immersed boundaries the nodes of the surface mesh constitute a set of lagrangian control points used to track the motion of the flexible body at every instant in time the influence of the body on the flow is accounted for by applying boundary conditions at cartesian grid nodes located in the exterior but in the immediate vicinity of the body by reconstructing the solution along the local normal to the body surface grid convergence tests are carried out for the flow induced by an oscillating sphere in a cubic cavity which show that the method is second order accurate the method is validated by applying it to calculate flow in a cartesian domain containing a rigid sphere rotating at constant angular velocity as well as flow induced by a flapping wing the ability of the method to simulate flows in domains with arbitrarily complex moving bodies is demonstrated by applying to simulate flow past an undulating fish like body and flow past an anatomically realistic planktonic copepod performing an escape like maneuver euver

self similar and translationally invariant structures in a bond diluted ising model a bond diluted ising model is used to simulate a process from a cantor set to a translationally invariant lattice the free energy function shows that there is no singularity which seem to imply that no transition exists in this process

on the fractal distribution of primes and prime indexed primes by the binary image analysis abstract in this paper the distribution of primes and prime indexed primes pips is studied by mapping primes into a binary image which visualizes the distribution of primes these images show that the distribution of primes and pips is similar to a cantor dust moreover the self similarity with respect to the order of pips already proven in batchko 2014 can be seen as an invariance of the binary images the index of primes plays the same role of the scale for fractals so that with respect to the index the distribution of prime indexed primes is characterized by the self similarity alike any other fractal in particular in order to single out the scale dependence the pips fractal distribution will be evaluated by limiting to two parameters fractal dimension and lacunarity that are usually used to measure the fractal nature because of the invariance of the corresponding binary plots the fractal dimension and lacunarity of primes distribution are invariant with respect to the index of pips

critical behaviour of the long range interaction model on a fractal lattice a particular class of fractal lattices is considered where the ising model with power law decaying interactions can be well approximated to a hierarchical type model applying the exact recurrence relations the critical behaviour is calculated and its universality with respect to the fractal dimensionality d is examined

a high order meshless method with compact support abstract in spite of their advantages in handling complex geometries meshless methods have been restricted very much to applications at low spatial order typically of the second order this is because the conventional meshless methods would require a very large supporting set of nodes when high order interpolants are used and this could pose problems for neighborhood and nodal selection in complex domains in this work a new meshless method is proposed for obtaining high order approximation on a highly compact nodal set whereas conventional meshless methods utilize the lowest order solution data to reconstruct the solution and its derivatives in the neighborhood of the reference node the present method increases the usage of available information at the nodes by interpolating over nodal set of vector data the utilization of the increased degrees of freedom and corresponding high order information at nodes results in a significant reduction in the number of support nodes needed for solution reconstruction high order accuracy of up to the 6th order is demonstrated for problems in linear advection diffusion and poisson equation the scheme is also applied to incompressible viscous fluid flows governed by the nonlinear system of navier stokes equations in all cases nine support nodes including the reference node are used for data interpolation and spatial reconstruction

remarks on duality lattices and the sierpinski gasket duality constructions for the sierpinski gasket and other fractals are studied coxeter s definition 1969 of a dual lattice is used to prove the existence of two kinds of dual lattice for the sierpinski gasket

irreducible components of the space of holomorphic foliations of degree two in cp n n 3 in this paper we will prove that the space of holomorphic fo liations of codimension 1 and degree 2 in cp n n 3 has six irreducible components

the immersed boundary method a projection approach a new formulation of the immersed boundary method with a structure algebraically identical to the traditional fractional step method is presented for incompressible flow over bodies with prescribed surface motion like previous methods a boundary force is applied at the immersed surface to satisfy the no slip constraint this extra constraint can be added to the incompressible navier stokes equations by introducing regularization and interpolation operators the current method gives prominence to the role of the boundary force acting as a lagrange multiplier to satisfy the no slip condition this role is analogous to the effect of pressure on the momentum equation to satisfy the divergence free constraint the current immersed boundary method removes slip and non divergence free components of the velocity field through a projection the boundary force is determined implicitly without any constitutive relations allowing the present formulation to use larger cfl numbers compared to some past methods symmetry and positive definiteness of the system are preserved such that the conjugate gradient method can be used to solve for the flow field examples show that the current formulation achieves second order temporal accuracy and better than first order spatial accuracy in l 2 norms for one and two dimensional test problems results from two dimensional simulations of flows over stationary and moving cylinders are in good agreement with those from previous experimental and numerical studies

toroidal varieties and the weak factorization theorem we develop the theory of stratified toroidal varieties which gives together with the theory of birational cobordisms 73 a proof of the weak factorization conjecture for birational maps in characteristic zero a birational map between complete nonsingular varieties over an algebraically closed field k of characteristic zero is a composite of blow ups and blow downs with smooth centers

a unified mathematical framework and an adaptive numerical method for fluid structure interaction with rigid deforming and elastic bodies many problems of interest in biological fluid mechanics involve interactions between fluids and solids that require the coupled solution of momentum equations for both the fluid and the solid in this work we develop a mathematical framework and an adaptive numerical method for such fluid structure interaction fsi problems in which the structure may be rigid deforming or elastic we employ an immersed boundary ib formulation of the problem that permits us to avoid body conforming discretizations and to use fast cartesian grid solvers rigidity and deformational kinematic constraints are imposed using a formulation based on distributed lagrange multipliers and a conventional ib method is used to describe the elasticity of the immersed body we use cartesian grid adaptive mesh refinement amr to discretize the equations of motion and thereby obtain a solution methodology that efficiently captures thin boundary layers at fluid solid interfaces as well as flow structures shed from such interfaces this adaptive methodology is validated for several benchmark problems in two and three spatial dimensions in addition we use this scheme to simulate free swimming including the maneuvering of a two dimensional model eel and a three dimensional model of the weakly electric black ghost knifefish

potts model on sierpinski carpets migdal kadanoff bond moving renormalisation is used to study the q state potts model on sierpinski carpets a general approximate recursion relation including q as a parameter is given the property of bond interchanging invariance is found and used in deriving the recursion relation fixed points and critical exponents for some carpets are presented marginal fixed points instead of unstable ones are found several typical flow diagrams are also shown

a front tracking method for the computations of multiphase flow direct numerical simulations of multiphase flows using a front tracking method are presented the method is based on writing one set of governing equations for the whole computational domain and treating the different phases as one fluid with variable material properties interfacial terms are accounted for by adding the appropriate sources as functions at the boundary separating the phases the unsteady navier stokes equations are solved by a conventional finite volume method on a fixed structured grid and the interface or front is tracked explicitly by connected marker points interfacial source terms such as surface tension are computed on the front and transferred to the fixed grid advection of fluid properties such as density is done by following the motion of the front the method has been implemented for fully three dimensional flows as well as for two dimensional and axisymmetric ones first the method is described for the flow of two or more isothermal phases the representation of the moving interface and its dynamic restructuring as well as the transfer of information between the moving front and the fixed grid are discussed applications and extensions of the method to homogeneous bubbly flows atomization flows with variable surface tension solidification and boiling are then presented

on codimension 1 del pezzo foliations on varieties with mild singularities in this paper we extend to the singular setting the theory of fano foliations developed in our previous paper araujo and druel adv math 238 70 118 2013 a mathbb q fano foliation on a complex projective variety x is a foliation fancyscript f subsetneq t x whose anti canonical class k fancyscript f is an ample mathbb q cartier divisor in the spirit of kobayashi ochiai theorem we prove that under some conditions the index i fancyscript f of a mathbb q fano foliation is bounded by the rank r of fancyscript f and classify the cases in which i fancyscript f r next we consider mathbb q fano foliations fancyscript f for which i fancyscript f r 1 these are called del pezzo foliations we classify codimension 1 del pezzo foliations on mildly singular varieties

phase transitions on fractals i quasi linear lattices magnetic spin models and resistor networks are studied on certain self similar fractal lattices which are described as quasi linear because they share a significant property of the line finite portions can be isolated from the rest by removal of two points sites in all cases there is no long range order at finite temperature the transition at zero temperature has a discontinuity in the magnetisation and the associated magnetic exponent is equal to the fractal dimensionality d when the lattice reduces to a non branching curve the thermal exponent v 1 y is equal to d when the lattice is a branching curve y is related respectively to the dimensionality of the single channel segments of the curve for the ising model or to the exponent describing the resistivity for models with continuous spin symmetry

magnetization of a potts ferromagnet on a sierpinski carpet the authors study the inhomogeneous magnetization behaviour of a q state potts ferromagnet on a sierpinski carpet as a function of the temperature using a real space renormalization group procedure two qualitatively different regions on the carpet are distinguished a the interior of the carpet and b the set of internal walls which border the eliminated areas the spontaneous mean magnetization curves for these regions are obtained with the corresponding critical exponents beta and beta w for different values of q they find that the magnetization of region b goes to zero at the critical temperature always below the one of region a