the scaling limit of many two - dimensional critical lattice models and growth models encountered in statistical physics may be described in terms of fractals .the boundaries of the fortuin - kastelyn clusters in critical -state potts models and critical percolation models , for example , in this limit are known to be fractal , conformally invariant curves .stochastic loewner evolution ( also described as schramm - loewner evolution and abbreviated as sle ) is a rigorous mathematical tool for producing and studying stochastic conformally invariant curves in the plane ( see refs . for review ) .it was conjectured , therefore , that sle is a description of such statistical systems .in several special cases this conjecture has been rigorously proven .sle is based on the loewner equation taken from complex analysis .this equation contains a driving ( or forcing ) function , , which determines all the properties of sle .a few reasonable requirements constrain the choice of this function to a scaled brownian motion , which leaves only one free parameter .one of these requirements ( that the forcing function is continuous ) ensures that the curves produced by sle do not exhibit branching , thus leaving many systems , such as branching polymers , out of the picture . in this paper, we generalize sle by dropping the demand that the forcing stochastic function be continuous , but keeping the requirement of stationary and statistically - independent increments .this leads to a much broader class of forcing processes including , in particular , the so called lvy processes .this generalization might be a useful description of tree - like stochastic growth .standard sle has been studied in various geometries under the names of chordal , radial and other kinds of sle s . in this paperwe restrict ourselves to the chordal situation ( growth in the upper half plane ) .we believe that our generalized sle in a radial geometry may be relevant for description of the diffusion - limited aggregation , and plan to study it in a separate publication .  in the next section ,we define and describe the problem , in terms of the parameters which define the forcing .the forcing includes both a scaled brownian motion and a stable lvy process .the parameters are , which sets the normalization of the brownian term , which sets the normalization of the lvy term , and which determines its shape .this qualitative section ends with numerical simulations of the traces , the geometrical structures generated by the sle .the final section of the paper gives the analysis of the short - distance properties of the traces showing both analytically and numerically that the traces have qualitative change in behavior as and each pass though critical values , respectively at four and one .the transition at is quite analogous to a well known transition in the standard sle .the latter phase transition , at , is entirely new .this paper focuses upon the short - distance manifestations of this phase transitions in the chordal geometry .the accompanying large - scale features ( which may be different for different geometries ) will be the subject of a subsequent paper .we point out that most of the statements about the behavior of objects related to sle and its generalizations are probabilistic , and typically appended by qualifiers `` almost surely '' or `` with probability one '' in mathematical literature .we will not use these qualifiers explicitly assuming that they should be applied wherever it is necessary .some technical details of our calculations are given in the appendix .the so called chordal loewner evolution is described by a differential equation obeyed by a function . at each value of the `` time '' , this function is a conformal map from a subset of the upper half of the complex -plane ( which we refer to as the _ physical plane _ ) to the entire half plane ( _ mathematical plane _ ) .specifically we write here is the forcing , mentioned above .alternatively , the function obeys assuming that vanishes at .  for any point these equationsare valid up to the swallowing time , which is defined as the first time when or so that the right hand sides of eqs .( [ 11 ] ) and ( [ 11h ] ) become singular .the _ hull _ of the loewner evolution is defined as the set of all points which become singular ( are swallowed ) in this way up to time . at any given time , the s that newly enter the hull may represent a single point or a more complicated set , sometimes including a subset with a finite area .see figure [ fig1 ] .   for and zero elsewhere .left shows the situation just before touching ( ) ; right shows the situation just after ( ) .the trace is the thick dark line .the hull consists of that line plus the grey area .that area is added to the hull at .note that there is a continuum of points added to the hull at the time of touching , but only one of these is on the hull s boundary and is the point of the trace.,title="fig : " ]  for and zero elsewhere .left shows the situation just before touching ( ) ; right shows the situation just after ( ) .the trace is the thick dark line .the hull consists of that line plus the grey area .that area is added to the hull at .note that there is a continuum of points added to the hull at the time of touching , but only one of these is on the hull s boundary and is the point of the trace.,title="fig : " ]  another geometrical object considered in the sle literature is the growing tip of the hull .the point is picked from among the ( perhaps many ) points which newly enter the hull at time . among these , this point is unique for having the property of being on the boundary of the hull at the time , see figure [ fig1 ] .more formally , this point is defined by  where the limit is taken within .the union of the points up to a time is the trace of the evolution up to .the choice of the process defines the properties of the hull and the trace .in particular , if has a discontinuity ( jumps ) , branching or discontinuity occurs in the trace and the hull may become disconnected . for a sufficiently smooth ( a precise condition involves hlder continuity , see refs . ) the trace is a curve parametrized by .moreover , if the trace is a simple curve , that is a curve without double points and without touchings of the boundary , it then coincides with the hull .standard sle is the loewner evolution with scaled brownian forcing where is a standard brownian motion . in physical termsthis means that the `` time derivative '' is a white noise with intensity : .the increments of in distinct time intervals are stationary and statistically independent .these properties of ensure that the process is conformally invariant in the sense that the time evolution is consistent with composition of conformal maps .in particular , the ensemble of can be obtained by functional composition of with , each drawn from the correct ensemble .  if we want to generalize eq .( [ sleu ] ) but maintain the functional composition property , we may only discard the requirement of continuity of the forcing but keep its increments to be stationary and statistically independent .this leads to a broad class of stochastic processes , including the so called lvy processes , also called lvy flights in physics literature .the simplest of these are the so called stable lvy processes characterized by a single real parameter in the range .the process is composed of a succession of jumps of all sizes . in a time interval , jumps by an amount between and with the probability proportional to more precisely , the probability distribution function of is given by the fourier transform   the case of , formally corresponding to a brownian motion , is , in fact , quite different from .we find it interesting to combine a brownian motion with a stable lvy process as the input to our generalized version of the sle .thus we choose as our forcing the function every time a discontinuity of the driving force occurs the trace develops a branching point .with lvy flights , therefore , there are infinitely many branching points on all size - scales .the stable lvy processes are self - similar . for any positive number two processes , and , are `` equal in distribution '' , that is , statistically identical , which we denote as thus , a typical trajectory of a lvy flight looks like many clusters separated by long jumps . each cluster , if zoomed into , again looks like many clusters separated by long jumps , and so on .we choose the driving force in loewner evolution to be of the form ( [ sleul ] ) : if this process is scale - invariant .the run of the loewner evolution up to a time is statistically the same as the run up to a time , rescaled by a factor of .the properties of the growing hull do not change with time except for general size rescaling . in thisbrownian driving force is exceptional .the addition of lvy flights changes the situation : time becomes significant .consider a time rescaling , where is a positive number .we construct the rescaled conformal map it satisfies the initial condition and using the self - similarity property ( [ self - similarity ] ) we observe that obeys an equation of exactly the same form as eq .( [ 12 ] ) except that the value of is changed to the strength of the lvy process flows under time - rescaling , making sle not scale - invariant .( many calculations in the usual sle with pure brownian motion are based on scale - invariance so that they can not immediately be extended to the present case . )this situation can be phrased in the language of the renormalization group ( rg ) .the critical behavior of a system described by an sle with purely brownian forcing is determined by a fixed point of an rg .lvy flights make the system non - scale invariant , and their addition to the forcing can be thought of as a perturbation that is relevant at the fixed point .we do not know at present what such perturbations might be in terms of microscopic models of critical systems described by brownian sle .one possible candidate is a long range interaction of some kind .the flow of has a fixed point at zero , when we recover the result that sle with pure brownian motion is scale - invariant .for the ratio of and the effective changes with time . at time it is just at time the effective ratio of the two terms in the forcing is therefore , crossover behavior can be observed at a time at small times the evolution is dominated by the brownian motion , whereas at large times it is dominated by lvy flights .( if were very large or very small one would have to consider the independent effect of the three terms in the denominator of eq .( [ 12 ] ) , but for of order one this is the only crossover that need be considered . ) there is a similar crossover in space at a spatial scale , with the brownian motion dominating the small scale events .we approximate the integration of the sle by introducing a discretized version , where each realization of is replaced by a piecewise constant function with jumps appropriately distributed .let , , be a sequence of real stochastic variables .we define a realization of the discretized driving force to be constant in the interval  ] .being continuous , the process can exit $ ] either through ( with probability ) , or through ( with probability ) .clearly , is either , or . then dynkin s formula becomes and allows us to find , the probability for to hit before hitting .now if we take the limits it becomes the probability for to hit in a finite time , which is the probability for the point on the physical plane to be swallowed by the hull : in general , the order of limits matters here .if it is reversed , is the probability for to come arbitrarily close to 0 , that is , for the hull to come arbitrarily close to the boundary.  the generator for the process driven by the brownian motion ( [ sleu ] ) is  a zero mode of this operator is substituting this into eqs .( [ 40 ] , [ 50 ] ) we find that the answer is independent of , thus the probability for the hull to touch the boundary for the order of limits is important and we find that note that in order to determine the phases , only the behavior of a zero mode of at zero and at infinity is necessary .examples of traces of standard sle are shown in figures [ pick2 ] and [ pick6 ] .touching is visible for the and apparently does not occur for .the case of the generalized sle is different from the brownian case : now has discontinuities .therefore , at the exit time the value of does not have to be or : the process can jump and overshoot the boundary of the interval . to avoid this difficulty , we define to be the exit time from the set ,\ ] ] andalso restrict ourselves to zero modes that are even functions of .the process can exit by either hitting ( no overshooting possible ) , or by taking a value beyond or , in which case the overshooting is possible .but we consider the limit .the behavior of zero modes , considered below , is simple at infinity : they either diverge or are constant . with these changescomplete , we return to eqs .( [ 40 ] , [ 50 ] ) .the generator for the process defined by eqs . ( [ 11h ] , [ sleul ] ) is ( see ) , where is the same as in eq .( [ a_b ] ) , and this operator contains the principal value integral , which ensures convergence at small for .the meaning of this term is most transparent in the fourier space , where it is equivalent to the multiplication by .once again , we need to find a zero mode , obeying .these zero modes are studied in detail in the appendix . herewe summarize the results .first of all , we find by direct calculation that the function is a zero mode of . comparing this with eq .( [ f_b ] ) , we conclude that if , or along this line on the phase diagram we have an explicit expression for a zero mode of the full operator .secondly , for arbitrary and we can transform the zero mode equation , , into fourier space : - 2k\tilde{f}(k ) = 0,\ ] ] where is the fourier transform of a solution of this equation is apart from a multiplicative constant and a possible delta function at , this solution is unique .its asymptotic behavior is note that brownian motion determines the behavior at large ( small distances in -space ) , whereas lvy flights determine the behavior at small ( large distances in -space ) .analysis of the asymptotic behavior of the inverse fourier transform of eq .( [ nuinu ] ) gives the following : strictly speaking , we are able to obtain these asymptotics only for .the trouble is that the function that behaves as with for small would lead to a divergent integral in eq .( [ 101 ] ) , and so can not be a zero mode of .nevertheless , on physical grounds we believe that there is nothing special happening at , and that our results for phases that we state next , apply to all values .  ) .the dashed line is the hyperbola given by eq .( [ line ] ) in the text . along this linean explicit expression for the zero mode of the generator is known , see eq .( [ f_l ] ) . ]  the use of the asymtotic behavior ( [ 1004 ] ) in the formulas ( [ 40 ] , [ 50 ] ) gives the phases of sle driven by the noise ( [ sleul ] ) :  * * isolated trees * , : , .an example of the trace is shown in figure [ pick2m07 ] .the lvy flights have broad distribution .long jumps occur often .the trace can jump far from the existing trees and then a new isolated tree starts growing .the chances of returning to one of the old trees are small , so the old trees stop growing .we hope to make this point quantitatively in a future publication . ** a forest of trees * , : , .an example of the trace is shown in figure [ pick2m13 ] .the lvy flights rarely produce very big jumps , so that the trace starts a new tree close to the old one , or does not leave the old tree at all .the hull looks like trees growing close to each other with their branches almost touching .* * a forest of bushes * , : , .an example of the trace is shown in figure [ pick6m13 ] .the brownian motion , which controls the small scales , is stronger now .the trees become thicker and different branches touch each other . * * isolated bushes * , : and are equal and finite , that is , the probability for a fixed point on the real axis to be swallowed by the hull at a finite time is a function of the position of this point .this function is defined by any of the equations ( [ 40 ] , [ 50 ] ) ( the order of limits is unimportant in this phase ) , where is the inverse fourier transform of ( [ nuinu ] ) . is a symmetric function decaying from to with the asymptotic behavior : an example of the trace is shown in figure [ pick6m07 ] . since varies between 0 and 1 , parts of the real axis will not be swallowed , leaving gaps among trees .  as long as , the first three phases do not depend on at all .this is an expected property , because unlike and , flows under the renormalization ( [ resc ] ) and and are constants . a phase diagram of the generalized sle is shown in fig .[ pic - phasediagram ] .   .there is a direct correspondence between the areas swallowed by the trace on the physical plane and the process on the mathematical plane . ]figures [ pick6 ] , [ pick6m07 ] and [ pick6m13 ] seem to show that the trace swallows whole areas and touches itself . in the numerical approximation where the forcing consists of discontinuous steps the trace does nt exactly touch itself, however it comes closer to touching as and whole areas thus appear to be swallowed .we can then try to determine the probabilities for swallowing by setting a small - distance cut - off and looking for limiting behavior as the cut - off goes to zero .  in order to measure consider the real stochastic process on the mathematical plane defined by eq .( [ 11h ] ) .a point on the axis is considered to be swallowed at time if for some .our method for determining is shown in figure [ pic - onaxis - cor ] : for one particular realization of the noise we calculate for on the real axis .we use the same realization to draw the trace on the physical plane . from the tracewe can infer which points of the axis have been surrounded by the curve and thus have been swallowed . in the figurethe boundaries of a swallowed area on the real axis are marked by the arrows ; the dashed line corresponds to the cut off , we see that all points at which appear to be swallowed in the physical plane .we repeat for several noise realizations to determine the value of the cut off accurately .the existence of the cut - off means we will be unable to distinguish between and , so that the case , ( the forest of trees ) will be similar to , ( the forest of bushes ) . for all cases we have taken .  ) .main - large behavior : probability of a point x on the real axis not to be swallowed until time ( , 1000000 steps , , , statistics for 7000 independent noise realizations ) . inset - small behavior : probability ( , 1000000 steps , , statistics for 3000 independent noise realizations ) . in both casesthe dotted line is the theoretical prediction ( eq . ( [ 1004 ] ) ) which is followed accurately by the numerics for four decades of .deviations for small are due to the finite time step of the noise realizations ; the data points approach the theoretical prediction as we approach the scaling limit ( not shown ) ]  figure [ pic - onaxis - bl-1 ] shows the results of the probability distribution of a point on the real axis being swallowed by the time for a case with , ( isolated bushes ) .the numerical results closely follow the theoretical prediction of eq .( [ asymp ] ) .  in general , we reproduce the theoretical form of rather well in this phase , but less well in the other phases .this is not surprising , it is the effect of finite time in our calculation . in the phases with forests ,points are surrounded by a process which includes multiple returns to the same region of . in the simulations ,not enough time is allowed for this multiple return to work itself out .furthermore , our use of the cut - off means we are unable to distinguish between and , so that the method makes the forest of trees look similar to the forest of bushes .thus , we shall have to put together a more refined method of analysis and do much longer runs to make this distinction .the generalization of stochastic loewner evolution , suggested in this paper , has many new features . the hull acquires branching and the growth becomes a tree - like stochastic process .the phases of such hull are very different from the usual sle .although the driving force of sle is still self - similar , the growing hull is not , the force of the lvy flights increases with time , being a relevant parameter under size rescaling .the non - stationary nature of the growth considered in this paper is related to a technical difficulty in its study .many properties of the standard sle are obtained using the so - called random time change which , in the case of the brownian forcing , is consistent with the form of the loewner equation ( [ 11 ] ) . in the case of lvyprocesses a similar random time change exists , but changes the nature of the loewner equation ( [ 12 ] ) and , therefore , can not be used to extract useful information .  in terms of the geometry of the generalized sle driven by lvy processes, the non - stationarity means that many such properties as , for example , the fractal dimension of the trace can not be globally defined .instead , they may possibly be defined locally at each stage of the evolution .the question of how one can characterize the geometry of the generalized sle will be addressed in a future publication .this research was supported in part by nsf mrsec program under dmr-0213745 .ig is an alfred p. sloan research fellow , and was also supported by an award from research corporation and the nsf career award under dmr-0448820 .we wish to acknowledge many helpful discussions with paul wiegmann , eldad bettelheim , and seung yeop lee .ig also acknowledges useful communications with john cardy and nigel goldenfeld .in this appendix we study zero modes of the operator , where    a zero mode of is given in eq .( [ f_b ] ) . to find zero modes of ,let us consider the integral latexmath:[\[i_{\alpha,\beta}(x ) = \dashint \frac{|y + x|^\beta -  converges at large if , at small for any and any ( because of the principal value ) , and at for .thus , we assume .  by rescaling reduce the integral ( [ ialphabetax ] ) ( for nonzero ) to the last integral here can be evaluated .first , we split the integration interval into three intervals and change variables in each appropriately : the integrals here can be found in ref .some of them contain hypergeometric functions that have to be transformed further by applying the kummer s connection formulas ( see sec .2.9 . in chapter 2 of ref . ) . combining all the results , we obtain finally , in the limit , the last term here vanishes , and we get note that this expression vanishes identically if , thus showing that is a zero mode of .let us now consider the limit of . specifically , we note that the last integral can be obtained form eq .( [ ialphabeta ] ) by carefully taking the limit .expanding gamma functions we get finally , using properties of the gamma and digamma functions , we get this expression vanishes at showing that is a zero mode of for .thus , we have shown that is a zero mode of .next we note that if or . along this line on the phase diagram we have an explicit expression for a zero mode of the full operator , which behaves for small and large values of exactly as claimed in the text of the paper .however , this is restricted to the range of ( because must be positive ) .this is , actually , more generally true .indeed , the integral with converges at near only for .thus , our claims in the paper about the phases can be justified strictly speaking only for .   after the fourier transform the zero mode equation becomes a differential equation - 2k\tilde{f}(k ) = 0,\ ] ] let us consider . introducing the new function we rewrite the differential equation as this is immediately integrated to give the case is treated similarly , and we get in the end an even function ( as expected ) : apart from a multiplicative constant and a possible delta function at , this solution is unique .  performing the inverse fourier transform, we obtain a representation of our zero mode : this integral diverges at the lower limit for , but this is a spurious divergence that is easy to amend .namely , we can formally subtract a ( possibly infinite ) constant from replacing it by this integral converges at the lower limit for all values , which is all we need .situation with convergence at the upper limit is trickier . by chartier s criterion ( see ref . ) , the integral ( [ ift1 ] ) converges at the upper limit only if .this is the same condition that was obtained in the end of the previous subsection where the fourier transform was not used at all .so , this seems to be a more problematic issue .we also note that after the subtraction performed in eq .( [ subtraction ] ) , the convergence at the upper limit worsens .now the integral is convergent only for .but this can be ignored .we simply perform the integration for and then analytically continue to smaller values of .convergence also improves when we take derivative in either of eqs .( [ ift1 ] , [ subtraction ] ) with respect to the strength of the lvy term .we can then in principle obtain integrating in .another comment is in order here .if we tried to obtain the zero mode for the brownian sle case using the fourier transform , we would get instead of eq .( [ ift1 ] ) .the last integral has the same behavior at infinity as the one in eq .( [ ift1 ] ) .but the zero mode of given in eq .( [ f_b ] ) exist for all values of .this is another reason for us to believe that the value is not special , and that our results for the phases of the generalized sle apply for all values of .consider now the integral where we ignored the overall constant coming from pulling a power out of the integral , as well as rescaling of and .this integral can be evaluated explicitly through hypergeometric functions for such values of that where are mutually prime , see ref . , eq .. the mathematica software can also do the integral with these values of .the simplest of these corresponds to , that is , .the corresponding integral is   for all values of given by eq .( [ alpha ] ) the structure of the answers is similar .they are given in terms of hypergeometric functions .the asymptotic analysis of these expressions can be performed with the help of ref . , and in all cases gives the result given in eq .( [ 1004 ] ) .99  b. nienhuis in _ phase transitions and critical phenomena _ , vol . 11 , c. domb and j. l. lebowitz , eds . , academic press , 1987 .b. duplantier , phys .lett . * 84 * , 1363 ( 2000 ; _ conformal fractal geometry and boundary quantum gravity _ , in _ fractal geometry and applications : a jubilee of benoit mandelbrot , part 2 _ , 365 , proc .pure math ., 72 , part 2 , providence , ri : american mathematical society , 2004 ; arxiv : math - ph/0303034 .

Abstract: standard stochastic loewner evolution ( sle ) is driven by a continuous brownian motion , which then produces a continuous fractal trace . if jumps are added to the driving function , the trace branches . we consider a generalized sle driven by a superposition of a brownian motion and a stable lvy process . the situation is defined by the usual sle parameter , , as well as which defines the shape of the stable lvy distribution . the resulting behavior is characterized by two descriptors : , the probability that the trace self - intersects , and , the probability that it will approach arbitrarily close to doing so . using dynkin s formula , these descriptors are shown to change qualitatively and singularly at critical values of and . it is reasonable to call such changes `` phase transitions '' . these transitions occur as passes through four ( a well - known result ) and as passes through one ( a new result ) . numerical simulations are then used to explore the associated touching and near - touching events .   = 10000


the peridynamic theory , originally devised by silling , is a nonlocal extension of classical continuum mechanics , which is based on partial differential equations . since partial derivativesdo not exist on crack surfaces and other singularities , the classical equations of continuum mechanics can not be applied directly when such features are present .in contrast , the peridynamic balance of linear momentum is formulated as an integral equation , which remains valid in the presence of material discontinuities .therefore , the peridynamic theory can be applied directly to modelling both bulk _ and _ interface properties , using the same mathematical model .additionally , peridynamics is readily implemented in a meshless formulation , which facilitates the simulation of large deformations when compared to the traditional mesh - based finite - element method used for simulating solid mechanics in the classical continuum theory .  with these desirable features ,peridynamics has received considerable attention by researchers interested in numerically describing fundamental crack growth and failure effects in brittle materials . however ,the scope of the original peridynamic formulation included only so - called _ bond - based _models which were limited to a fixed poisson ratio for linear isotropic materials and could not describe true plastic yielding .further development of the theory led to _ state - based _models , which can , in principle , describe all classic material behaviour . in particular, the state - based theory offers a route to approximating the classical deformation gradient , which can be used to obtain a classical stress tensor .the stress tensor can then be converted into nodal forces using a peridynamic integral equation. this promising situation , i.e. , the ability to use classical material models with a method that remains valid at material discontinuities , has prompted a number of studies where plastic yielding , damage , and failure was simulated using this new meshless method .however , there exists a demand for studies which compare peridynamics to other meshless methods .while the theoretical correspondence of peridynamics with classical elasticity theory has been established , no information is available on the accuracy of the discretized peridynamic expression suitable for computer implementation .relevant questions that need to be addressed include , but are not limit to , whether linear fields can be exactly reproduced by the discretized theory , and what the the order of convergence is when peridynamic solutions are compared against exact results .little is known about how common problems encountered with other meshless methods , e.g. , the tensile instability or the rank deficiency problem affect peridynamics . as an exception to this general observation , bessa _ etal _ have published a study which demonstrates the equivalence of state - based peridynamics and the reproducing kernel particle method ( rkpm ) , if nodal integration is used . however , no published studies are available to the best of this author s knowledge , which compare peridynamics to other meshless methods .the goal of our paper is to elucidate on some properties of peridynamics with respect to other meshless methods . in particular, we establish that the discrete equations of the peridynamic formulation using classical material models is identical to a very well - known meshless technique : smooth particle hydrodynamics ( sph ) in the total - lagrangian formulation .this equivalence facilitates understanding of peridynamics using the large body of literature already published for other meshless methods , see e.g. and .the key observations of our work address two issues : ( i ) discretizations of peridynamics directly arrive at correct equations which conserve linear and angular momentum .these features can only be obtained in sph by assuming _ad hoc _ corrections such as explicict symmetrization .( ii ) all of the problems that apply to collocation - type particle methods also apply to peridynamics if this theory is discretized using nodal integration .the remainder of this paper is organized as follows : we begin by deriving the fundamental expressions of the sph approximation including the most important corrections for this method , which allow it to be used with the minimal level of accuracy required for solid mechanics simulations .then , the essential peridynamic expressions required for simulating classical material models are derived .building on this foundation , the equivalence of sph and this particular variant of peridynamics is shown . finally , the implications of this observation are discussed and peridynamics ( with classical material models )is characterized using the established terminology encountered in the sph literature .smooth particle hydrodynamics was originally devised as a lagrangian particle method with the smoothing kernel moving with the particle , thus redefining the interaction neighbourhood for every new position the particle attains . in this sense , the kernel of the original sph formulation has eulerian character , as other particles move through the interaction neighbourhood .the tensile instability encountered in sph , where particles clump together under negative pressure conditions , has been found to be caused by the eulerian kernel functions .consequently , total lagrangian formulations were developed , which use a constant reference configuration for defining the interaction neighborhood of the particles .see typically , the initial , undeformed configuration is taken for this purpose . in the following , this concept and the associated nomenclatureis briefly explained with the limited scope of obtaining sph expressions that are to be compared with the peridynamic expressions . for a more detailed derivation, the reader is referred to the works cited above .   in the total lagrangian formulation , conservation and constitutive equationsare expressed in terms of the material coordinates , which are taken to be the coordinates of the initial , undeformed reference configuration . a mapping between the current coordinates , and the reference coordinates describes the body motion at time : here , are the current , deformed coordinates and the reference ( lagrangian ) coordinates . the displacement is given by the conservation equations for mass , impulse , and energy in the total lagrangian formulation are given by where and are the current and initial jacobian determinants . is the current mass density and is the initial mass density , is the nominal stress tensor ( the transpose of the first piola - kirchhoff stress tensor ) , is the internal energy , is the gradient or divergence operator expressed in the material coordinates , and denotes the deformation gradient ,    the sph approximation for a scalar function in terms of the lagrangian coordinates can be written as the sum extends over all particles within the range of a scalar weight function , which is centered at position and depends on the distance between coordinates and , . is the volume associated with a particle in the reference configuration .the weight function is typically chosen to be radially symmetric and have compact support , i.e. , it includes only neighbors within a certain radial distance .this domain of influence is denoted .the sph approximation of a derivative of is obtained by operating directly with the gradient operator on the kernel functions , where the gradient of the kernel function is defined as follows :   the conditions for the zeroth- and first - order completeness of the sph approximation are stated as follows : in the simple form as stated here , neither of the completeness conditions are fulfilled by the sph approximation .ad - hoc _ improvement consists in adding eqn .( [ eqn : first_order_completeness ] ) to eqn .[ eqn : sph_derivative_approximation ] , such that a _ symmetrized _ approximation for the derivative of a function is obtained , the symmetrization does not result in first - order completeness , however , it yields zeroth - order completeness for the derivatives of a function .   in order to fulfill first - order completeness, the sph approximation has to reproduce the constant gradient of a linear field .a number of correction techniques exploit this condition as the basis for correcting the gradient of the sph weight function , where is the diagonal unit matrix .based on this expression , a corrected kernel gradient can be defined : which uses the correction matrix , defined as :   by construction , the corrected kernel gradient now satisfies eqn .( [ eqn : first_order_completeness_vector ] ) ,    for calculating the internal forces of a solid body subjected to a deformation , expressions are required for ( i ) the deformation gradient , ( ii ) a constitutive equation which provides a stress tensor as function of the deformation gradient , and ( iii ) an expression for transforming the stresses into forces acting on the nodes which serve as the discrete representation of the body .the deformation gradient is obtained by calculating the derivative of the displacement field , i.e. , by using the symmetrized sph derivative approximation , eqn .( [ eqn : sph_derivative_approximation_symmetrized ] ) , for eqn .( [ eqn : defgrad ] ) : note that in the above equation , the corrected kernel gradient has been introduced via the matrix . the sph approximation of the stress divergence , eqn .( [ eqn : divergence ] ) , is not so clear .depending on how it is performed , several different approximations can be obtained .the most frequently used form , which is variationally consistent with respect to an energy minimization principle , is the following : for a radially symmetric kernel which depends only on distance , the antisymmetry property holds .therefore , the above force expression will conserve linear momentum exactly , as .we use the antisymmetry property of the kernel gradient to rewrite the force approximation as follows : replacing the uncorrected kernel gradient with the corrected gradient ( c.f .( [ eqn : correct_sph_gradient ] ) , the following expression is obtained : this corrected force evaluation conserves linear momentum due to its antisymmetry with respect to interchange of the particle indices and , i.e. , .the here constructed antisymmetric force expression is usually not seen in the literature .in contrast , it seems to be customary to directly insert the corrected kernel gradient into eqn .( [ eqn : sph_force_uncorrected ] ) , which destroys the local conservation of linear momentum .we note that the above construction of the sph expression that conserves linear momentum is arbitrary , and in similar spirit to the _ ad - hoc _ symmetrization procedure encountered in eqn .( [ eqn : sph_derivative_approximation_symmetrized ] ) .this section provides a concise derivation of the peridynamic approximation of the deformation gradient and the forces acting on particles .in contrast to the preceding section , where this quantities have been derived for sph , we employ a different nomenclature here , which is consistent with the most relevant peridynamic literature . in peridynamics , each particle defines the origin of an influence domain , termed neighborhood , of radius . within this neighborhood ,vectors from the origin to any point in exist , see the following figure .is defined by a radial cutoff around a particle .bonds exist between and all particles within .these bonds are denoted here by , of which only one is shown above .note that this representation already employs discrete particle locations whereas the peridynamic theory assumes a continuum of points.,scaledwidth=75.0% ]   central to the state - based peridynamic theory is the concept of _ states_. states are functions that act on on vectors .this is written in the following way , where the state has acted on the vector to produce a different vector .angular brackets indicate which vector the state acts upon .the mapping results produce tensors of different order , depending on the nature of the state . _ vector states _ of order one map a vector to a different vector ._ scalar states _ of order zero produce a scalar for every vector they act on . for the purpose of this paper we only need to be concerned with the following states :_ the reference position vector state _ returns the original bond vector ._ the deformation state _ returns the deformed image of the original bond vector .let the original vector go from to , i.e. , .upon deformation , the coordinates of these two points change to and . the deformation state vector returns the deformed image of the original vector , i.e. ,   the _ influence function _ is a scalar state that returns a number that depends only on the magnitude of .  vector and scalar states can be combined using the usual mathematical operations addition , subtraction , multiplication , etc .the conventions are detailed in . here, we only need the definition of the _ tensor product _ between two vector states and , this tensor product is required for the following peridynamic concepts .firstly , the _ shape tensor _ is defined :  is symmetric and real and thus positive - definite , implying that it can be inverted .the shape tensor is then used in the concept of _ tensor reduction _ , which is an operation that produces a second order tensor from two vector states :    the deformation gradient tensor may be approximated as a tensor reduction of deformation vector state and reference position vector state according to eqn .( 134 ) in : the necessary peridynamic calculus for evaluating this quantity has been presented above .we now make the transition from the continuum representation to a discrete nodal , or particle representation , converting integrals to sums .a piecewise constant discretization of the integrals defined above yields the discrete shape tensor as here , the sum includes all particles within the neighborhood of and is the volume of particle .the discrete expression for the approximate deformation gradient is obtained as :    the fundamental equation of peridynamics is an integral equation of force densities ( force per volume ) which ensures local conservation of impulse , see eqn .( 28 ) . in : \left < \bm{x } ' - \bm{x } \right > - \underline{\bm{t}}[\bm{x } ' ]\left < \bm{x } - \bm{x ' } \right > \right \ } \mathrm{d}v_{x'}\ ] ] here ,  ] ensures balance of linear momentum and is the volume associated with the coordinate .the vector state is related the classical stress tensor , which can be computed using the deformation gradient and a classical material model .specifically , this relation is stated as follows , see eqn .( 142 ) . in : \left < \bm{\xi } \right > = \underline{\omega } \left < \xi \right > \bm{p}_{\bm{x } } \bm{k}^{-1}_{\bm{x } } \bm{\xi}\ ] ] the stress tensor above is the nominal stress tensor which applies here as the stress is expressed in the reference configuration .the dependence on signifies that both stress and shape tensor have to be evaluated at coordinate . converting to a discrete particle expression ,one obtains : discrete peridynamic approximation of both the deformation gradient and the particle forces arising from a classical stress tensor are equal to the total - lagrangian sph expressions with linear kernel gradient correction . in order to show this equivalence , we introduce some changes in nomenclature : note that the postulated equivalence in the last line above states that the derivative of the sph weight function has to equal the peridynamic weight function . however , no implications arise from this requirement , as suitable weight functions can be chosen which fulfills this requirement .   with these changes ,the peridynamic shape tensor becomes equal to the correction matrix required for first - order consistent sph :    in a similar manner , the peridynamic expression of the deformation gradient can be shown to be equal to the sph approximation : thus , the peridynamic concept of reduction leads to the approximation of a tensor field , which is correct to first order .using the same rules for changing the peridynamic notation into sph notation , one obtains from the peridynamic force expression : equality of total - lagrangian sph and peridynamic for classical material models with nodal integration raises some advantageous implementation issues .the first point concerns the weight function and its derivative , which is usually attributed great influence in sph .in contrast , peridynamic implementations typically use a constant unit weight function , i.e. , . using the peridynamics to sph translation , eqn .( [ eqn : pd_sph_weight_function ] ) , this corresponds to a constant gradient for the sph scheme .such a constant implies that the associated weight function is apparently not a proper sph weight function , i.e. , it is certainly not normalized : .this apparent contradiction is solved by the first - order correction scheme used here : the _ ansatz _ used for deriving the corrected gradient , eqn .( [ eqn : correct_sph_gradient ] ) does not depend on any normalization factor as the multiplication the matrix inverse of cancels any constant factor in the weight function .in fact , it can be shown that this route to a corrected kernel gradient represents a moving least squares formulation with a linear basis .it can be concluded that the weight function ( or indeed its derivative ) can be chosen quite arbitrarily within this scheme . for the purpose of illustrating an implementation of the first - order corrected scheme , the uncorrected kernel gradient is simply defined as .algorithm 1 outlines an efficient implementation which moves the multiplication with the shape matrix outside of the inner loop ( loop over neighbors of particle ) , in order to save unnecessary calculations .  +  +  +   +   this algorithm exhibits two main features : a first loop over all pairs of interacting particles is required to define the shape matrix and deformation gradient .naive implementations might split up this loop into two independent loops and first calculate only the shape matrix .however , as the kernel gradient correction depends only on the shape matrix of particle , c.f .( [ eqn : sph_defgrad ] ) , these operations can be combined and the correct deformation gradient obtained by post - multiplying with the shape matrix . with the corrected deformation gradient at hand , a stress tensor is computed using the constitutive law . here , a finite - strain theory is employed , which provides a linear relationship between strain and stress , based on the lam parameters and .the stress measure obtained this way is the second piola - kirchhoff stress , which is applicable to the reference configuration .this stress measure is pushed forward to the current configuration ( first piola - kirchhoff stress ) , as forces are applied to the current particle positions .the calculation of the forces requires a second loop over all pairs of interacting particles . to avoid multiplication of the kernel gradients with the shape matrix inside the second loop ,the shape matrices are multiplied with the stress tensors before entering this loop .once the forces are computed , the particle positions can updated according to any chosen time - integration scheme .the aim of this work is to show the equivalence of first - order corrected total - lagrangian sph and peridynamics applied to classical material models in combination with nodal integration . while the equivalence has been mathematically proven in sec .( [ sec : prove_equivalence ] ) , it is also of interest to demonstrate the effects of the instability which emerges if nodal integrations used . to this end , the implementation according to algorithm 1 , in combination with a unit weight function and a leap - frog time integrator is used . for comparison purposes , reference solutions which do not suffer from rank - deficiency due to nodal integration are shown which are obtained using a meshfree code that employs a stabilization technique based on higher - order derivatives , similar to .the patch test can be regarded as the most basic test for a solid mechanics simulation code .a strip of some material that is discretized using several elements ( or sph / peridynamic particles ) is subjected to a uniform strain field and the stress is computed .regardless of the discretization , the stress should be uniform everywhere .sph / peridynamics with first - order corrected kernel gradients is well - known to pass this test . here ,however , we are interested in the subsequent time - integration following such an initial perturbation . to this end, a random initial particle configuration is obtained by discretizing a square patch of edge length 1 m and uniform area mass density into 444 quadliteral area elements using a stochastic algorithm based on triangular voronoi tesselation and recombination of the triangles into quadliterals .particle volumes and masses are obtained from the quadliteral area and the prescribed mass density .the kernel radius is adjusted individually for each particle such that approximately 12 neighbors are within the kernel range .while the patch test can be easily simulated with an ordered initial configuration of particles , a non - uniform particle configuration poses extreme stability challenges .figure [ fig : quad_mesh ] shows the initial particle configuration .the material model is chosen as linear elastic with and .the patch is uniformly stretched by 10% along both axes .after this initial perturbation , the system is integrated in time using a standard leap - frog algorithm with a cfl - stable timestep , resulting in a periodic contraction and expansion mode .figure [ fig : quad_patch_comparison ] compares plain first - order corrected total - lagrangian sph ( equivalent to peridynamics with nodal integration ) with a meshfree solution that is stabilized against the effects of rank - deficiency . in the unstabilized system ,particle motion is not coherent and particle disorder is observed already after the first contraction .after eight contractions , particle disorder is very pronounced and the motion of the patch is entirely dominated by numerical artifacts .in contrast , the stabilized simulation exhibits a completely coherent particle motion which can can be continued for hundreds of oscillations .this example considers a large deflection of a simply supported 2d beam with dimensions , see fig .[ fig : beam_sketch ] . the material model of the beam is taken as linear elastic with youngs modulus and . a large boundary displacement of applied , and plain total - lagrangian sph results ( equivalent to peridynamics with nodal integration ) are qualitatively compared against a stabilized simulation code .snapshots of the simulations corresponding to the same boundary displacements , are shown in fig .[ fig : beam_comparison ] . in the first snapshot at , no obvious difference between both simulations can be observed . for the second snapshot at , the unstabilized simulation shows a pattern where pairs of particles in the interior of the beam move together and do not follow the rotation of the beam s neutral line . clearly , the unstabilized system tries to minimize its elastic energy by assuming a configuration which is not in agreement with a linear displacement field .this leads to an obviously wrong particle configuration for the last snapshot at .in contrast , the stabilized system shows particle displacements which appear very reasonable and exhibit no instability .   and.,scaledwidth=75.0% ]  -component of the second piola - kirchhoff stress tensor , with red signifying tension and blue compression.,scaledwidth=99.0% ]  we conclude this test case by noting that total - lagrangian sph , or equivalently peridynamics for classical material models based on the deformation gradient and with nodal integration , can become unstable for certain loading conditions even with perfectly uniform reference particle configurations .hence , stabilization techniques , e.g. based on higher - order derivatives are required to stabilize the solution .this work shows that peridynamics when used with classical material models based on the deformation gradient tensor and nodal integration as discretization technique is equivalent to total lagrangian sph with corrected gradients .this result allows to characterize this flavour of peridynamics using the large body of studies published for sph .sph is a collocation method with nodal integration that suffers from two major problems : ( i ) the tensile instability , i.e. , a numerical instability which results in particle clumping under conditions of negative pressure , and ( ii ) susceptibility to zero - energy modes which are caused by rank - deficiency due to nodal integration . of these problems ,the tensile instability is resolved by the total - lagrangian formulation , and the rank - deficiency can be eliminated by using additional integration points .peridynamics with classical material models as presented here is a total - lagrangian method and therefore shows no tensile instability .however , if due to the use of nodal integration , it is still a rank - deficient method .  the rank - deficiency does usually not manifest itself until the reference configuration is updated .such an update can be required as the total - lagrangian nature imposes restrictions on the magnitude of deformations that can be handled with the approximate deformation gradient .alternatively , features of the material model , e.g. plasticity or failure , may require updates of the reference configuration .these updates result in zero - energy modes which need to be treated with dissipative mechanisms .  with the equivalence demonstrated in this work , peridynamics applied to classical material models and in combination with nodal integration does not result in a new numerical method , but is instead a new derivation of an existing meshless method that is not free of problems .what are the implications of these findings for the other peridynamic theories ?the bond - based theory , which can be interpreted as an upscaling of the established meshless method molecular dynamics , does not require the calculation of a deformation gradient and thus does not suffer from the associated problems due to rank - deficiency .the state - based peridynamic theory has also been formulated for some material models which are not based on the classical deformation gradient .if these unconventional material models , e.g. , the linear peridynamic solid , are discretized using nodal integration for computing dilatation and shear strain , rank deficiency problems are also likely to emerge .recently , work relating the reproducing kernel particle method ( rkpm ) to state - based peridynamics was published by bessa _et al _ . in their work , the authors conclude that state - based peridynamics is equivalent to rkpm if nodal integration , i.e. , collocation , is used .these authors positively emphasize that the peridynamic state - based approach is much faster compared to rkpm using more elaborate ( cell based ) integration techniques .these results are also understandable in light of this publication , because , it is known that rkpm with nodal integration is equivalent to moving - least - squares sph ( mlsph ) , which is known to be equivalent to sph with corrected derivatives . for a comprehensive review on this equivalence , which also addresses the question as to how many really different meshless methodsare known , see . however , bessa _ et al _ disregard the unfortuitous implication of the equivalence of peridynamics , sph , and rkpm with nodal integration : we have learned from a number of sph publications , that nodal integration causes problems , as it gives rise to zero - energy modes .therefore the use of nodal integration for peridynamics is a bad thing to do .it is worthwhile to emphasize that the mathematical foundation of peridynamics is clear and straightforward .correct equations of motion emerge from this theory which conserve linear and angular momentum , and approximate linear fields accurately .all of these desirable features can only introduced into the sph approximation by _ ad - hoc _ procedures .e.g. explicit symmetrization .thus , if anything , the peridynamic theory provides us with a better route to deriving meshless discretizations than the sph method .if the simplest form of meshless discretization , namely nodal integration , is used , the advantages of peridynamics over sph vanish and the discrete expressions of both methods become equal . stabilization schemes which address the rank - deficiency problem such as the scheme due to vidal _ then need to be employed in order to keep the simulation stable .future work should therefore address enhanced integration schemes for the peridynamic theory .t. l. warren , s. a. silling , a. askari , o. weckner , m. a. epton , j. xu , a non - ordinary state - based peridynamic method to model solid material deformation and fracture , international journal of solids and structures 46 ( 5 ) ( 2009 ) 11861195 .j. bonet , s. kulasegaram , alternative total lagrangian formulations for corrected smooth particle hydrodynamics ( csph ) methods in large strain dynamic problems , revue europnne des lments 11 ( 7 - 8 ) ( 2002 ) 893912 .m. mller , r. keiser , a. nealen , m. pauly , m. gross , m. alexa , http://dx.doi.org/10.1145/1028523.1028542[point based animation of elastic , plastic and melting objects ] , in : proceedings of the 2004 acm siggraph / eurographics symposium on computer animation , sca 04 , eurographics association , aire - la - ville , switzerland , switzerland , 2004 , pp .http://dx.doi.org/10.1145/1028523.1028542 [ ] .

Abstract: this paper discusses the similarity of meshless discretizations of peridynamics and smooth - particle - hydrodynamics ( sph ) , if peridynamics is applied to classical material models based on the deformation gradient . we show that the discretized equations of both methods coincide if nodal integration is used . this equivalence implies that peridynamics reduces to an old meshless method and all instability problems of collocation - type particle methods apply . these instabilities arise as a consequence of the nodal integration scheme , which causes rank - deficiency and leads to spurious zero - energy modes . as a result of the demonstrated equivalence to sph , enhanced implementations of peridynamics should employ more accurate integration schemes .   meshless methods , peridynamics , smooth - particle hydrodynamics


people have been trying to make life structured and organized throughout recorded time ( and probably before ) .but , nature is not orderly and the social world is not orderly .a good example are the capital markets .models have been created to explain them . these models are , of necessity , simplifications of reality . by making a few simplifying assumption about the way investors behave, an entire analytic framework has been created to help us understand the markets .the models have not worked well .studies of economic forecasts show that economists have made serious forecasting errors at every major turning points since the early 1970s , when the studies began . included in the groupstudied was townsend - greenspan , run by fed chairman alan greenspan .forecasters tend to be out as a group at these turning points .what went wrong ?econometric analysis assumes that , without outside influences everything balances out .supply equals demand .if exogenous factors perturb the system , taking it away from equilibrium , the system reacts inmediately reverting to equilibrium in a linear fashion .but a free - market economy is an evolving structure with emotional forces , such as greed and fear , which cause the economy to develop far from equilibrium " conditions sometimes .an efficient market " is one in which assets are fairly priced and neither buyers nor sellers have advantage .however , new financial instruments with low interest eventually die , even if they are fairly priced . any trader will confirm that a healthy market is one with volatility , but not necessarily fair price .we may say that a healthy economy and a healthy market _ do not _ tend to equilibrium but are , instead , far from equilibrium and equilibrium theories are likely to produce dubious results .another problem is that with the econometric view of the world , the markets and the economy have no memory , or only limited memory of the past . as an example, let us say that interest rates depend solely on the current rate of inflation and the money supply . a simple model would be :   if the coefficients and are fixed , then depends on current levels of and . it does not matter whether and are rising or falling . what is missing , is the feedback effect produced by the fact that , in human decision making , the expectations of the future are influenced by recent experiences .feedback systems are characterized by long - term correlations and trends .+ these characteristics - far from equilibrium conditions and feedback mechanisms - are symptomatic of nonlinear dynamic systems .nonlinear differential - or difference - equations are complex and have multiple , messy solutions .life is messy , there are many possibilities .let s illustrate with a simple , nonlinear model related with a social system .suppose a new tv program with audience ( normalized ) .the audience rises at a rate . considering only this effect, the audience would increase as :   but there are spectators that may see the tv show and do not like it .we may suppose that this effect cause a reduction of audience at .the evolution of the audience rating would then be :   although this is a simple model , it explains that at low levels of audience growing ( ) , the audience goes to zero ( and the tv managers get rid of the show ) and , at higher levels of audience growing , the audience will converge to a steady value .let us see what happens for .suppose a growth rate of , and . by iterating equation ( 3 ) , a steady audience of is reached .you can try with your personal computer , and a spreadsheet , copying equation ( 3 ) down for 200 cells approximately . repeating this experiment for different initial values the same final audience of is obtained . if the growth rate is increased to , two possible final audiences appear , and the system oscillates between them . as the growth rate continues to rise , each possible audience bifurcates and 4 , 8 , and 16 final audiences appear .finally , at , the system displays an infinite number of possible final values , fluctuating between them in a chaotic way .the changes from one audience to the next one seem absolutely random , as though blown about by environmental noise .yet in the middle of this complexity , stable cycles return .even though the parameter is rising , meaning that the nonlinearity is driving the system harder , a window will suddenly appear with a regular period : an odd period , like or .then the period - doubling bifurcations begin all over at a faster rate , rapidly passing through cycles of 3 , 6 , 12 , ... or 7 , 14 , 28 , ... , and then breaking off again to chaos .the final audience value behavior with increasing growth rate is shown in fig . 1 .  equation ( 3 ) is the famous `` logistic equation '' , which was formulated in 1845 by pierre verhulst to model the growth of populations limited by finite resources .its strange behavior discussed before was discovered by robert may in 1976 and it was later used by mitchell feigenbaum for his ground breaking work on the universality of the period - doubling route to chaos .this model is , obviously , too simplified , because , for instance , it assumes that the audience decreasing is directly related to the growth rate .however , it shows us the most important characteristics of nonlinear dynamic systems .the first one is the _ extremely sensitive dependence on initial conditions _ which is the signature of chaos : a slight change in in the chaotic region will result in a completely different audience after steps . around 1960 ,edward lorenz discovered ( by accident ) this characteristic in the models used for numerical weather forecasting ; and it was he who coined the term `` butterfly effect '' .he worked with twelve equations to simulate the atmosphere , by solving them with an mit computer . in order to examine some of the results in more detail, he used a small computer that he had in his office to introduce intermediate conditions which the big computer had printed out as new initial conditions to start a new computation .but the solution that came out was completely different from that obtained with the big computer .lorenz found that the reason was that the numbers that he used as new initial conditions were not the same as the original ones , they had been rounded off from six decimal places to three and the difference had amplified until being as big as the signal itself .you can repeat easily the lorenz experiment with the logistic equation ( 3 ) , taking , using a computer or even a pocket calculator by making two series of iterations with the same starting point .after iterations , in one of the series the output is truncated to three decimal places and taken as input for the following iteration . soon afterwards( around steps ) the outputs will be completely different .another nice examples of this kind of experiments with different calculators and with different implementations of the same quadratic law on the same calculator may be found in .the other important characteristic that may be appreciated in fig .1 is that it is a _fractal_. in the windows of stability , inside each figure is a smaller figure , identical to the larger figure . enlarging the smaller figure , another window of stabilitymay be found , where another smaller version of the main figure appears . at smaller and smaller scaleshappens the same .this is called the _ self similar _ property , which is characteristic ofthe father of fractals , benoit mandelbrot , begun thinking in them studying the distribution of large and small incomes in economy in the s .he noticed that an economist s article of faith seemed to be wrong .it was the conviction that small , transient changes had nothing in common with large , long term changes . instead of separating tiny changes from grand ones , his picture bound them together .he found patterns across every scale .each particular change was random and unpredictable .but the sequence of changes was independent of scale : curves of daily changes and monthly changes matched perfectly .mandelbrot worked in ibm , and after his study in economy , he came upon the problem of noise in telephone lines used to transmit information between computers .the transmission noise was well known to come in clusters .periods of errorless communication would be followed by periods of errors .mandelbrot provided a way of describing the distribution of errors that predicted exactly the pattern of errors observed . his description worked by making deeper and deeper separations between periods with errors and periods without errors .but within periods of errors ( no matter how short ) some periods completely clean will be found .mandelbrot was duplicating the cantor set , created by the century mathematician georg cantor . to make a cantor set ( see fig .2 ) , you start with the line segment from to . then you remove the middle third .that leaves two segments , and you remove the middle third from each .that leaves four segments , and you remove the middle third from each - and so on to infinity .the cantor set is the strange `` dust '' of points that remains .they are arranged in clusters , infinitely many yet infinitely sparse and with total length .  in the _ fractal way _ of looking nature roughness and asymmetryare not just accidents on the classic and smooth shapes of euclidian geometry .mandelbrot has said that `` mountains are not cones and clouds are not spheres '' .fractals have been named the _ geometry of nature _ because they can be found everywhere in nature : mammalian lungs , trees , and coastlines , to name just a few .an english scientist , lewis richardson around 1920 checked encyclopedias in spain , portugal , belgium and the netherlands and discovered discrepancies of twenty percent in the lengths of their common frontiers . in the authors measured the coast of britain on a geographical map with different compass settings by counting the number of steps along the coast with each setting .the smaller the setting they used , the longer the length of the coastline they obtained . if this experiment is done to measure the perimeter of a circle , ( or any other euclidean shape ), the length obtained converges with smaller compass settings . in his famous book mandelbrot states that the length of a coastline can never be actually measured , because it depends on the length of the ruler we use .since length is not a valid way to compare coastlines , mandelbrot proposes fractal dimension to measure the degree of roughness or irregularity of coastlines and other rough objects .the idea is that the degree of irregularity remains constant over different scales .we have previously discussed that fractals are self - similar , but a segment , or a square can be divided into small copies ( see fig .3 ) . these structures ,although self - similar are not fractals .there is a relation between the reduction or scaling factor and the number of scaled down pieces into which the structure is divided . where for the line and for the square , which agree exactly with the known ( topological ) dimensions of the segment and the square .it may be easily seen that in the cantor set , scales as and .then for the cantor set . is called the _ self similarity dimension_. it is a special form of mandelbrot s fractal dimension .  the most popular version of mandelbrot s fractal dimension is the _ box - counting dimension _ , which is a concept related to the self - similarity dimension and it is used in cases such as the coastlines where there is no exact self - similarity .the recipe of box - counting dimension calculation is to put the structure onto a regular mesh with mesh size , and simply count the number of grid boxes which contain some of the structure .this gives a number which , of course will depend on the size of the mesh .then change to progressively smaller sizes counting the corresponding .the scaling relation linking and the box - counting dimension is next make a log - log diagram and try to fit a straight line to the plotted points and measure its slope .the box - counting dimension and the self - similarity dimension give the same numbers in many cases , as , for example , the cantor set .fractal sets show self similarity with respect to space .fractal time series have statistical self similarity with respect to time . in the social and economic fields , time seriesare very common . in a simple way to demonstrate self - similarity in a time series of stock returns is devised by asking the reader to guess which graph corresponds to daily , weeekly and monthly returns between three different graphs with no scale on the axes .an important statistics used to characterize time series is the _hurst exponent _ .hurst was a hydrologist who worked on the nile river dam project in the first decades of this century .at that time , it was common to assume that the uncontrollable influx of water from rainfall followed a random walk , in which each step is random and independent from previous ones .the random walk is based on the fundamental concept of brownian motion .brownian motion refers to the erratic displacements of small solid particles suspended in a liquid .the botanist robert brown , about 1828 , realized that the motion of the particles is due to light collisions with the molecules of the liquid .hurst measured how the reservoir level fluctuated around its average level over time .the range of this fluctuation depends on the length of time used for measurement .if the series were produced by a random walk , the range would increase with the square root of time as .hurst found that the random walk assumption was wrong for the fluctuations of the reservoir level as well as for most natural phenomena , like temperatures , rainfall and sunspots .the fluctuations for all this phenomena may be characterized as a `` biased random walk''-a trend with noise- with range increasing as , with .mandelbrot called this kind of generalized random walk _fractional brownian motion_. in high - school statistical courses we have been taught that nature follows the gaussian distribution which corresponds to random walk and .hurst s findings show that it is wrong .the proper range for is from , corresponding to very rough random fractal curves , to 1 corresponding to rather smooth looking fractals .in fact , there is a relation between and the fractal dimension of the graph of a random fractal : thus , when the exponent vary from to , yields dimensions decreasing from to , which correspond to more or less wiggly lines drawn in two dimensions .fractional brownian motion can be divided into three distinct categories : , and .the case is the ordinary random walk or brownian motion with independent increments which correspond to normal distribution .  for is a negative correlation between the increments .this type of system is called _antipersistent_. if the system has been up in some period , it is more likely to be down in the next period .conversely , if it was down before , it is more likely to be up next .the antipersistence strength depends on how far is from .  for is a positive correlation between the increments .this is a _ persistent _ series . if the system has been up ( down ) in the last period , it will likely continue positive ( negative ) in the next period .trends are characteristics of persistent series .the strength of persistence increases as approaches .persistent time series are plentiful in nature and in social systems . as an examplethe hurst exponent of the nile river is , a long range pesistence that requires unusually high barriers , such as the aswn high dam to contain damage in the floods .it can be said that a set s defining relation is an _ indicator function _ associated to a point which can only take two values : _ true _ or if the point belongs to the set ; and _ false _ or if the point does not belong to the set . however , most facts about nature demand more general mathematical objects to embody the idea of _ shades of grey_. those objects are called _ measures_.  a simple example of multifractal may be obtained by considering a map of a continent . a possible measure is _ the number of people_. to each subset of the map , the measure lays a quantity , which is the number of people on .if we divide the map into two equal size parts and , and respectfully will be different .the division can be done several times giving .some countries have more people than others parts of a country contain more people than others and so on . of people _ is a measure irregular at many scales . when the irregularity is ( at least statistically ) the same at all scales , the measure is _ self similar _ or _multifractal_.  for euclidean support of a self - similar measure , the box counting dimension only confirms that there is nothing fractal about this support .thus gives not enough quantitative description about the self similar measure supported by this set .what we are seeking is a measure given by a weight which can be thought as the average density of probability in each box , defined as in a euclidean space of dimension ( or in a space of embedding dimension ) .for fractals , instead of density , one speaks in terms of the _ coarse hlder exponent _ :   for a multifractal will be restricted to an interval while for a fractal there will be an unique . to obtain the frequency distribution , one must count the number of boxes of size that have a coarse hlder exponent .now suppose that a box of side has been selected at random among boxes whose total number is proportional to .the probability of hitting is . in the case of interest to us , this distribution no longer tends to a limit as , .thus , instead of we use . as , becomes the _ singularity exponent _ and tends to the _ singularity spectrum _ .this implies that for each the number of boxes increases for decreasing as . is an upsidedown bell shaped curve , which values could be interpreted as a fractal dimension of the subsets of boxes of size . when there are infinite subsets , each characterized by its own and a fractal dimension .the aim of this formalism is to determinate the f( ) singularity spectrum of a measure .a partition function can be defined from this spectrum ( it is the same model as the thermodynamic one ) .  both functions , and , describe the same aspects of a multifractal , and they are related to each other .in fact , the relationships are    where is given as a function of by the solution of the equation    these two equations represent a _ legendre transform _ from the variables and to the variables and .the spectrum of is obtained from the spectrum   the capacity or box dimension of the support of the distribution is given by . corresponds to the scaling behavior of the information and is called _information dimension_. plays an important role in the analysis of nonlinear dynamic systems , especially in describing the loss of information as a chaotic system evolves in time .  for , and the _ q - point correlation integrals _ are related . is called _ correlation dimension _ because it is associated with the `` correlation function '' of the fractal set , that is , the probability of finding , within a distance of a given member of the set , another member .as we will show in the following section the wavelet transform is especially suited to analyze a time series as a multifractal .the work of jean morlet , a geophysicist with the oil company elf - aquitaine , who developed wavelets as a tool for oil prospecting , is usually taken as the starting point of the history of wavelets .the standard way to look for underground oil is to send vibrations under ground and to analyze their echos to obtain the deepness and thickness of the layers and what materials they are made of .the problem is that there are hundreds of layers and all the signals interfere with each other .fourier analysis was used to get information from the interference of the echos . as more powerfulwere the computers available , more fourier windows were placed here and there .but the finer local definition needed to have access to information on different thicknesses layers could nt be achieved .  in windowed fourier analysis ,a small window is `` blind '' to low frequencies , which correspond to signals too large for the window . on the other hand ,large windows lose information about a brief change . instead of keeping the size of the window fixed and change the frequencies of oscillations that filled the window , morlet did the reverse : he kept constant the number of oscillations in the window and stretched or compressed the window like an accordion .this makes it possible to analyze a signal at different scales .the _ wavelet transform ( wt ) _ is sometimes called a `` mathematical microscope '' : big wavelets give an approximate image of the signal , while smaller wavelets zoom in on details .the wavelet transform of a signal consists in decomposing it into frequency and time coefficients , asociated to the wavelets .the analyzing wavelet , by means of translations and dilations , generates the so called family of wavelets .the wavelet transform turns the signal into a function (a , b) ] .this means that around     where is an order polynomial , and    provided the first moments are zero .  if we have , the first moments are vanishing .the wavelet modulus function (a , t)|$ ] will have a local maximum around the points where the signal is singular .these local maximum points make a geometric place called modulus maxima line .(a , b_l(a))| \sima^\alpha(b_l(a))\:\:\:\:\ : for \:\ : a \to 0,\ ] ]  where is the position at the scale of the maximum belonging to the the line .  the wavelet transform modulus maxima method consists in the analysis of the scaling behavior of some partition functions that can be defined as :  (a , b_l(a))|^q,\ ] ]  and will scale like  .this partition function works like the previously defined partition function for singular measures . for will prevail the most pronounced modulus maxima and , on the other hand , for will survive the lower ones .the most pronounced modulus take place when very deep singularities are detected , while the others correspond to smoother singularities .we can get ( eq . 3 ) andthe lagrange transform can be applied , obtaining and spectra , like in the previous section .the shape of is a hump that has a maximum value for the associated with the general behavior of the series .so , this particular singularity exponent can be thought like the hurst exponent for the series .before applying the wtmm method to the analysis of social signals in time , it is important to see how it works on `` simple '' functions , like self - similar measures lying on `` generalized '' cantor set .  the _ devil staircase _ is a self - similar measure constructed recursively with the cantor set , but giving different weights or probabilities to the different segments at each step ( see ) .the devil staircase is the distribution function associated with the final probabilities .let be a generalized devil staircase constructed recursively as follows : each interval at each step of the construction is divided into four subintervals of the same length on which we distribute respectively the weights , and .4 displays .two wavelet transforms of different scaling factor are shown in fig . and ) singularity spectra are displayed in fig .6 a ) and b ) .this generalized devil staircase is thus an everywhere singular signal that displays multifractal properties .as we mentioned above , persistent processes are common in nature . applying the wtmmwe will be able to verify that the electrical demand is a persistent time series and to compare the quality of the process in two different places .as an example of application of wavelets to a fractal time series , we chose the electrical demand of two completely different places : australia , a whole continent , and mar del plata a touristic city of argentina .+ data of australia electrical power demand were obtained at the web site : + http://www.tg.nsw.gov.au/sem/statistics .+ mar del plata electrical demand time series was kindly provided by centro operativo de distribucion mar del plata belonging to edea .two time series of 8832 points were taken , as seen in fig .7 a ) and b ) .the fifth derivative of gaussian function was chosen as analyzing wavelet :    twelve wavelet transform data files were obtained applying the wavelet transform to both electrical demand data with , ranging the scaling factor from to in steps of a power of two .then we computed the partition function for each one for , getting , as shown in fig . 8 a ) and b ) . is a nonlinear convex increasing function . for australia , and two slopes which are for and for , while for mar del plata city , and .the corresponding singularity spectra obtained by legendre transforming are displayed in fig .9 a ) and b ) .a multifractal signal is characterized by a single humped shape with a nonunique hlder exponent like each of the graphs shown in fig .as expected from , the support of extends over a finite interval which bounds are and for australia , which is larger than the one for mar del plata ranging from and .the minimum value , , corresponds to the strongest singularity which characterizes the most rarified zone , whereas higher values exhibit weaker singularities until or weakest singularity which corresponds to the densiest zone . and both between and correspond to a persistent process ; although it can be observed a very little less persistence for australia than for mar del plata due to the slighty shift of the curve to the right for the last one , the processes are deeply persistent .the support dimensions are 0.68 and 0.57 for australia and mar del plata respectfully ; which implies that the capacities of the supports are fractional so we are in presence of two chaotic processes .the hlder exponent for the dimension supports , , are 0.74 ( australia ) and 0.73 ( mar del plata ) .these particular corresponds to or which implies that the events with are the most frequent ones . implies we are analyzing a persistent time series which obeys to the joseph effect ( in the bible refers to 7 years of loom , happiness and health and 7 years of hungry and illness ) .this system has long memory effects : what happens now will influence the future , so there is a very deep dependence with the initial conditions .it may be thought like a fractional brownian motion of .a hurst exponent of 0.73 or 0.74 describes a very persistent time series , what is expected in a natural process involved in an inertial system . can be known as hlder exponent or singularity exponent , too .if the distribution is homogeneous there is an unique ( for example fractional brownian motion ) , but if it is not there are several exponents , like in these two cases. the most frequent will characterize the series and will play as hurst exponent . is almost the same for australia and mar del plata ; in fact 0.79 for the first and 0.80 for the second one ; bigger in both cases to ( 0.74 and 0.73 respectfully ) .this implies that the curves are slightly humped to the left , an effect that is more pronounced for the city than for the country and a better precision is obtained for the branch , where the bigger values will prevail ( i.e. high changes in the demand which are more rare ) the asymmetrical shape of the spectrum reveals more pronounced inhomogeneities in the events associates with the branch , asociated with the smaller values of power(i.e slight changes in demand which are more ordinary ) . is other indicator of the behavior . for mar del plata is larger than for australia .the information dimension for australia is which features the scaling behavior of the information while it is for argentina . is a fractional number in both cases .then , in australia and in mar del plata the electricity demand corresponds to chaotic systems with the problems of forecasting associated with them .the correlation dimensions are in the case of mar del plata while for australia .the correlation dimension characterizes a chaotic atractor and , besides , indicates the presence of long - range correlations .the long - range correlations are observed in some biological systems lacking of a characteristic scale of time or length .such behavior may be adaptative because the long - range correlations play the role of the organizing principle for highly complex , non linear processes that generate fluctuations on a wide range of time scales and , in adittion , the lack of characteristic scale helps to prevent excessive mode locking that would restrict the reaction of the organism .as we can see , there is longer - range correlation for australia , implying that for the case of an abrupt change of the demand australia electrical system will have a better answer .we presented a brief introduction to fractals , multifractals and wavelets . since their birth, fractals have shown to be ubiquitous in nature , and , in the last years are finding their way in social and economic systems .as fractals are the geometry of chaos , and chaos is probably present in far from equilibrium processes such as social ones , they will be more frequently found in social systems in the near future .wavelets are a specially useful mathematical tool to sudy fractals . as an example of its application we used the wavelet transform modulus maxima method to compare the electrical power demands of a touristic city , mar del plata , and a whole country , australia .we found that both electrical demands behave , like most ones in nature , as long term memory phenomena . in both cases ,the fractal dimensions obtained correspond to chaotic processes .in particular , the correlation dimensions found by this way tell us that the series observed for australia is longer - range correlated than the one for mar del plata .this lays that australia power generating system is better suited to satisfy oscilations in the demand . in spite of only within the interval , the greater value of mar del plata indicates that the demand varies in a wider ranger , which features the variation in the demography .we think that with these examples the reader can realize that the fractal analysis is especially suited to study the non - linear statistics of social systems .mcnees , _ which forecast should you use ? _ , new england economic review , july / august 1985 ; _ how accurate are macroeconomic forecasts ? _ ,july / august 1988 .linden , _ dreary days in dismal science _ forbes , january 21 , 1991 . r.m .may , _ simple mathematical models with very complicated dynamics _ , nature * 261 * , 459 - 467 ( 1976 ) .feigenbaum , _ quantitative universality for a class of nonlinear transformations _ j. stat .* 19 * , 25 - 52 ( 1978 ) .hinz - otto peitgen , h. jurgens 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Abstract: this is a brief introduction to fractals , multifractals and wavelets in an accessible way , in order that the founding ideas of those strange and intriguing newcomers to science as fractals may be communicated to a wider public . fractals are the geometry of the wildness of nature , where the euclidian geometry fails . the structures of nonlinear dynamics associated with chaos are fractal . fractals may also be used as the geometry of social systems . wavelets are introduced as a tool for fractal analysis . as an example of its application on a social system , we use wavelet fractal analysis to compare electrical power demand of two different places , a touristic city and a whole country .


the fiber - optic distributed measuring systems were being actively developed during the past few years . particularly these systems can be used for acoustic field investigating , for monitoring of stresses in industry , in geophysical researches , etc .the reconstruction of information obtained by measuring system needs solving of tomographic problem .but frequently such solution is complex , needs large computer consumption and can not be performed in real time .therefore previously learned neural network can be utilized for solving of tomographic problem .an optical perceptron , which was implemented using a collection of amplitude holograms recorded on a disk - shaped holographic carrier and processing the output data from a fiber - optic measuring system , is reported in ref .another computer linear neural network was developed by authors of refs .we present the similar computer non - linear feed - forward neural network in this paper .consider two - dimensional fiber - optic distributed measuring system being set of measuring lines .the fiber - optic phase sensors form square lattice also having diagonal lines ( fig .[ square ] ) .the phase shift in this linear sensor is proportional to physical action value integrated over the line while the intensity of the transmitted light .the phase shift in fiber - optic line have to be along one cosine half - period .signal from such measuring system is simulated on input of neural network which is intended to reconstruct the value of physical action at intersection points .the fiber - optic distributed measuring system . ]  for solving of tomographic problem we choose perceptron with the non - linear hidden layer since this network has universal approximation capability .the neurons of the first layer serve as network inputs and feed data from the measuring system to the next layer .the second ( hidden ) layer proceeds following transformation : where are states of neuron inputs being signals from measuring sensors , are states of outputs and are synapses .we used as activation function .the hidden layer of neurons has inputs and outputs .the output layer of the neurons takes the linear transformation : where are states of third layer neuron outputs , are synapses of third layer .we selected gaussian distributions of the some physical quantity with one or two peaks as learning patterns .the following expression was used as objective function : where is superscript indicating number of learning pattern , are output states of the neural network for some learning pattern . for the training pattern formed as squared cosines of the sum of along the corresponding measuring line .we used error back - propagation for the network training , so we had to minimize with respect to , .we combined conjugate gradient and coordinate descent minimization methods for neural network learning .when conjugate gradient method failed we switched to the coordinate descent and after some iterations tried to start conjugate gradient minimization again .also we applied the `` thermal jumps '' technique .the minimization procedure finished after certain iteration count or on given accuracy reaching .on the first step we learned the neural networks to reconstruct the signal having , as well as training patterns , one gaussian peak .examples of initially unknown reconstructed distributions with different variances and are shown in tables [ example - delta ] , [ example - smooth ] .the neural networks was learned independently in all the cases with the similar randomly generated patterns .  on the next step we trained neural networks reconstructing distributions with two gaussian peaks .examples are shown in the tables [ example - delta2 ] , [ example - smooth2 ] . in the case of the smooth distributionthe reconstructed signal is close to the original . for the -like distributions the locations of the peaksare recovered correctly but their magnitude differs from the original .one can see from tables that neural network sufficiently accurately reconstructs unknown pattern and can be used in practice.  presented neural network unlike refs . can work not only with signal from fiber - optic measuring line being within the interval of linear dependence of light intensity on phase shift .the computer model of the neural network solving tomographic problem is proposed .this network can be used in conjunction with the distributed fiber - optic measuring system and partly may be made as electronic or optical hardware .n. kulchin , o. b. vitrik , r. v. romashko , yu . s. petrov , o. v. kirichenko , o. t. kamenev , `` tomographic methods for investigating vector fields with the aid of fibre - optic measuring systems , '' _ quantum electronics _ , 27 ( 5 ) , pp . 455458 , 1997 .n. kulchin , o. t. kamenev , i. v. denisov , o. b. vitrik , o. v. kirichenko , r. v. romashko , `` measuring networks with the tomographic principle of data gathering , '' _ optical memory & neural networks _ , 6 ( 2 ) , pp . 149156 , 1997 .n. kulchin , o. b. vitrik , r. v. romashko , yu .s. petrov , o. v. kirichenko , o. t. kamenev , `` tomography methods for vector field study using space - distributed fiber optic sensors with integral sensitivity , '' _ fiber and integrated optics _ , 17 ( 1 ) ,pp . 7584 , 1998 .a. cichocki , r. undehauen , m. lendl , k. weinzierl , `` neural networks for linear inverse problems with incomplete data especially in applications to signal and image reconstruction , '' _ neurocomputing _ , 8 , pp . 741 , 1995 .n. kulchin , i. v. denisov , and o. t. kamenev , `` optoelectronic neural system for processing the output data from a fiber - optic measuring network , '' _ technical physics letters _ , 25 ( 3 ) , pp . 235236 , 1999 .n. kulchin , i. v. denisov , v. f. obuh , o. b. vitrik , yu . s. petrov , r. v. romashko , o. t. kamenev , `` computer neural networks for processing of optical tomography information , '' _ pacific science review _ , 1 ,pp . 14 , 1999 .

Abstract: a computer model of the feed - forward neural network with the hidden layer is developed to reconstruct physical field investigated by the fiber - optic measuring system . the gaussian distributions of some physical quantity are selected as learning patterns . neural network is learned by error back - propagation using the conjugate gradient and coordinate descent minimization of deviation . learned neural network reconstructs the two - dimensional scalar physical field with distribution having one or two gaussian peaks .   * keywords : * neural networks , optical tomography , fiber - optic measuring systems