theoretical analysis of non gaussian heterogeneity effects on subsurface flow and transport miur italian ministry of education university and research water jpi waterworks project we need water needs availability quality and sustainability european union 640979 university of arizona vanderbilt university under the consortium for risk evaluation with stakeholder participation cresp iii u s department of energy

waves and bumps in neuronal networks with axo dendritic synaptic interactions we consider a firing rate model of a neuronal network continuum that incorporates axo dendritic synaptic processing and the finite conduction velocities of action potentials the model equation is an integral one defined on a spatially extended domain apart from a spatial integral mixing the network connectivity function with space dependent delays arising from non instantaneous axonal communication the integral model also includes a temporal integration over some appropriately identified distributed delay kernel these distributed delay kernels are biologically motivated and represent the response of biological synapses to spiking inputs they are interpreted as green s functions of some linear dierential operator exploiting this green s function description we discuss formal reductions of this non local system to equivalent partial dierential equation pde models we distinguish between those spatial connectivity functions that give rise to local pde models and those that give rise to pde models with delayed non local terms for cases in which local pdes are derived we investigate traveling wave solutions in a comoving frame by numerically computing global heteroclinic connections for sigmoidal firing rate functions we also calculate exact solutions parameterized by axonal conduction velocity for the heaviside firing rate function the sigmoidal firing rate function in the limit of infinte gain the inclusion of synaptic adaptation is shown to alter traveling wave fronts to traveling pulses which we study analytically and numerically in terms of a global homoclinic orbit finally we consider the impact of dendritic interactions on waves and on static spatially localized solutions exact analysis for

two bump solutions of amari type models of neuronal pattern formation abstract we study a partial integro differential equation defined on a spatially extended domain that arises in the modeling of pattern formation in neuronal networks for a one dimensional domain we develop criteria for the existence and stability of equal width 2 bump solutions under the assumption that the firing rate function is the heaviside function we apply these criteria to an example for which the connectivity is of lateral inhibition type i e the coupling function has one positive zero and find that families of 2 bump solutions exist but none of the solutions are stable extensive numerical searches suggest that this is true for all coupling functions of this form however for a large class of coupling functions which have three positive zeros we find the coexistence of both stable and unstable 2 bump solutions we also extend our investigation to two spatial dimensions and give numerical evidence for the coexistence of 1 bump and 2 bump solutions our results imply that lateral inhibition type coupling is not sufficient to produce stable patterns that are more complex than single isolated patches of high activity

relationship between juxtaposed overlapping and fractal representations of multimodal spatial variability 1 it has been shown by neuman 1990 di federico and neuman 1997 and di federico et al 1999 that the power variogram of a statistically isotropic or anisotropic fractal field can be constructed as a weighted integral from zero to infinity of exponential or gaussian variograms of overlapping homogeneous random fields modes having mutually uncorrelated increments and variance proportional to a power 2h of the integral spatial correlation scale where h is the hurst coefficient low and high frequency cutoffs are related to length scales of the sampling window domain and data support sample volume respectively intermediate cutoffs account for lacunarity due to gaps in the multiscale hierarchy created by a hiatus of modes associated with discrete ranges of scales alternative mathematical representations of multimodal spatial variability were formulated by various authors in which space is filled by a discrete number of juxtaposed mutually exclusive materials or categories each having its own architecture and attributes the spatial distribution of categories is characterized by indicator random variables and their attributes by random fields this paper focuses on expressions developed by lu and zhang 2002 in which the indicator variables and their attributes are mutually uncorrelated while each is autocorrelated and cross correlated within and between categories upon rewriting their expressions for statistically homogeneous and anisotropic media it is demonstrated mathematically that in the limit as the categories stretch to occupy each point in space overlap in a way that preserves their local architecture their attributes become mutually uncorrelated categories are said to overlap if they are found in fixed proportions within a representative sampling volume centered about any mathematical point throughout space the idea is analogous to the well known dual continuum concept in which two categories most commonly fractures and porous blocks are considered to overlap in reality the categories do not overlap but are considered to preserve their relative architectural arrangement throughout space the variogram of an attribute sampled jointly over overlapping statistically homogeneous and anisotropic categories is the sum of the variograms of statistically homogeneous and anisotropic components of this attribute sampled over individual categories weighted by their volumetric proportions it is further demonstrated that when the overlapping categories represent an infinite or truncated hierarchy of modes whose attributes have exponential or gaussian variograms such that the product of category probability density function pdf and attribute variance is proportional to a power 2h 1 of the attribute integral scale they represent a fractal field characterized by a power or truncated power variogram as in the work of di federico and neuman any category pdf that satisfies this condition and any attribute pdf having a variance that satisfies the same condition are admissible neither of them need to be gaussian to form fractional brownian motion or levy like to approximate fractional levy motion for the attribute to form a random fractal intermediate cutoffs render the latter model discrete as is that of lu and zhang hence the two multimodal models are mutually consistent and complementary

combinatorial characterization of sub hyperbolic rational maps in 1980 s thurston established a combinatorial characterization for post critically finite rational maps among post critically finite branched coverings of the two sphere to itself a completed proof was written by douady and hubbard in their paper a douady j h hubbard a proof of thurston s topological characterization of rational functions acta math 171 1993 263 297 this criterion was then extended by cui jiang and sullivan to sub hyperbolic rational maps among sub hyperbolic semi rational branched coverings of the two sphere to itself the goal of this paper is to present a new but simpler proof for the combinatorial characterization of sub hyperbolic rational maps by adapting some arguments in the proof in douady and hubbard s paper

geometrical aspects of stability theory for hill s equations where a and b are real parameters apparently the first stability diagram was drawn in the classical paper by b van der pol m j o sti utt 15 since then many papers and textbooks have appeared on the subject we mention j j stoker 13 j meixner f w sch fke 12 d m levy 8 j b keller 11 m i weinstein j b keller 16 17 j poschel e trubowltz 9 and v i arnold 3 5 6 this literature contains extensive estimates on the order of tangency of resonance tongues and the behavior of stability boundaries at infinity a full understanding of the stability diagrams is still lacking for instance the appearance of instability pockets in some cases see figure 1 cf 15 calls for a geometric explanation it should be noted that this phenomenon does not occur in the classical mathieu case cf 12 for a preliminary study near resonances using singularity theory we refer to afsharnejad 1 related bifurcational aspects of nonlinear perturbations of mathieu s equation near resonances were studied by broer vegter 7 the global geometry of the symplectic group was used for a stability proof of a hill equation in levi 10 we introduce the main object of study of this paper

dynamics of cubic siegel polynomials we study the one dimensional parameter space of cubic polynomials in the complex plane which have a fixed siegel disk of rotation number where is a given irrational number of brjuno type the main result of this work is that when is of bounded type the boundary of the siegel disk is a quasicircle which contains one or both critical points of the cubic polynomial we also show that these boundaries vary continuously as one moves in the parameter space this is most nontrivial near the set of cubics with both critical points on the boundary of their siegel disk we prove that this locus is a jordan curve in the parameter space most of the techniques and results can be generalized to polynomials of higher degrees

stability and bifurcations in neural fields with finite propagation speed and general connectivity a stability analysis is presented for neural field equations in the presence of finite propagation speed along axons and for a general class of connectivity kernels and synaptic proper ties sufficient conditions are given for the stability of equilibrium solutions it is shown that the propagation delays play a significant role in nonstationary bifurcations of equilibria whereas the stationary bifurcations depend only on the connectivity kernel in the case of nonstationary bifurca tions bounds are determined on the frequencies of the resulting oscillatory solutions a perturbative scheme is used to calculate the types of bifurcations leading to spatial patterns oscillations and traveling waves for high propagation speeds a simple method is derived that allows the determi nation of the bifurcation type by visual inspection of the fourier transforms of the kernel and its first moment results are numerically illustrated on a class of neurologically plausible systems with combinations of gaussian excitatory and inhibitory connections

rational rotation numbers for maps of the circle we consider families of maps of the circle of degree 1 which are homeomorphisms but not diffeomorphisms that is maps like x to x t frac c 2 pi sin 2 pi x bmod 1 withc 1 we prove that the set of parameter values corresponding to irrational rotation numbers has lebesgue measure 0 in other words the intervals on which frequency locking occurs fill up the set of full measure

the combinatorial rigidity conjecture is false for cubic polynomials we show that there exist two cubic polynomials with connected julia sets which are combinatorially equivalent but not topologically conjugate on their julia sets this disproves a conjecture by mcmullen from 1995

flow in multiscale log conductivity fields with truncated power variograms in a previous paper we offered an interpretation for the observation that the log hydraulic conductivity of geologic media often appears to be statistically homogeneous but with variance and integral scale which grow with the size of the region window being sampled we did so by demonstrating that such behavior is typical of any random field with a truncated power semi variogram and that this field can be viewed as a truncated hierarchy of mutually uncorrelated homogeneous fields with either exponential or gaussian spatial autocovariance structures the low and high frequency cutoff scales l and u are related to the length scales of the sampling window and data support respectively we showed how this allows the use of truncated power variograms to bridge information about a multiscale random field across windows of different sizes either at a given locale or between different locales in this paper we investigate mean uniform steady state groundwater flow in unbounded domains where the log hydraulic conductivity forms a truncated multiscale hierarchy of gaussian fields each associated with an exponential autocovariance we start by deriving an expression for effective hydraulic conductivity as a function of the hurst coefficient h and the cutoff scales in one two and three dimensional domains which is qualitatively consistent with experimental data we then develop leading order analytical expressions for two and three dimensional autocovariance and cross covariance functions of hydraulic head velocity and log hydraulic conductivity versus h l and u examine their behavior and compare them with those corresponding to an exponential log hydraulic conductivity autocovariance our results suggest that it should be possible to bridge information about hydraulic heads and groundwater velocities across windows of disparate scales in particular when l u the variance of head is infinite in two dimensions and grows in proportion to l2 2h in three dimensions while the variance and longitudinal integral scale of velocity grow in proportion to l2h and l respectively in both cases

dynamic instabilities in scalar neural field equations with space dependent delays in this paper we consider a class of scalar integral equations with a form of space dependent delay these non local models arise naturally when modelling neural tissue with active axons and passive dendrites such systems are known to support a dynamic oscillatory turing instability of the homogeneous steady state in this paper we develop a weakly nonlinear analysis of the travelling and standing waves that form beyond the point of instability the appropriate amplitude equations are found to be the coupled mean field ginzburg landau equations describing a turing hopf bifurcation with modulation group velocity of o 1 importantly we are able to obtain the coefficients of terms in the amplitude equations in terms of integral transforms of the spatio temporal kernels defining the neural field equation of interest indeed our results cover not only models with axonal or dendritic delays but those which are described by a more general distribution of delayed spatio temporal interactions we illustrate the predictive power of this form of analysis with comparison against direct numerical simulations paying particular attention to the competition between standing and travelling waves and the onset of benjamin feir instabilities

on the spectra of certain integro differential delay problems with applications in neurodynamics abstract we investigate the spectrum of certain integro differential delay equations iddes which arise naturally within spatially distributed nonlocal pattern formation problems our approach is based on the reformulation of the relevant dispersion relations with the use of the lambert function as a particular application of this approach we consider the case of the amari delay neural field equation which describes the local activity of a population of neurons taking into consideration the finite propagation speed of the electric signal we show that if the kernel appearing in this equation is symmetric around some point a 0 or consists of a sum of such terms then the relevant dispersion relation yields spectra with an infinite number of branches as opposed to finite sets of eigenvalues considered in previous works also in earlier works the focus has been on the most rightward part of the spectrum and the possibility of an instability driven pattern formation here we numerically survey the structure of the entire spectra and argue that a detailed knowledge of this structure is important within neurodynamical applications indeed the amari idde acts as a filter with the ability to recognise and respond whenever it is excited in such a way so as to resonate with one of its rightward modes thereby amplifying such inputs and dampening others finally we discuss how these results can be generalised to the case of systems of iddes

moduli space of cubic newton maps abstract in this article we study the topology and bifurcations of the moduli space m 3 of cubic newton maps it s a subspace of the moduli space of cubic rational maps carrying the riemann orbifold structure c 2 3 we prove two results the boundary of the unique unbounded hyperbolic component is a jordan arc and the boundaries of all other hyperbolic components are jordan curves the head s angle map is surjective and monotone the fibers of this map are characterized completely the first result is a moduli space analogue of the first author s dynamical regularity theorem 37 the second result confirms a conjecture of tan lei

multiple bumps in a neuronal model of working memory we study a partial integro differential equation defined on a spatially extended domain that arises from the modeling of working or short term memory in a neuronal network the equation is capable of supporting spatially localized regions of high activity which can be switched on and off by transient external stimuli we analyze the effects of coupling between units in the network showing that if the connection strengths decay monotonically with distance then no more thanon e regionof high activity canpersist whereas if they decay inanoscillatory fashion then multiple regions can persist

spatially structured activity in synaptically coupled neuronal networks ii lateral inhibition and standing pulses we consider the existence and stability of standing pulse solutions to a system of integro differential equations used to describe the activity of synaptically coupled networks of excitatory and inhibitory neurons in a single spatial domain assuming an arrangement of synaptic connections described by lateral inhibition previous formal arguments have demonstrated the existence of both stable and unstable standing pulses s amari biol cybern 27 1977 pp 77 87 these results have formed the basis for several recent hypotheses regarding the generation of sustained activity patterns in prefrontal cortex and other brain regions implicit in the lateral inhibition arrangement however is the assumption that the dynamics of inhibition are instantaneous here we present two arguments demonstrating the loss of stability of standing pulse solutions through a hopf bifurcation when more realistic inhibitory dynamics are considered the first argument parallels amari s formal presentation while the seco

quasiconformal homeomorphisms and dynamics i solution of the fatou julia problem on wandering domains on demontre un resultat dynamique sur l iteration d une application analytique complexe sur la sphere de riemann

blueprint for perturbative solution of flow and transport in strongly heterogeneous composite media using fractal and variational multiscale decomposition 1 this paper offers a blueprint for extending perturbative ensemble moment and potentially other e g gaussian or perturbative karhunen loeve solutions of stochastic flow and transport problems in composite randomly heterogeneous porous media to cases in which medium heterogeneity is arbitrarily large the proposed approach is based on a novel idea of combined fractal and variational multiscale decomposition it considers log hydraulic conductivity to be a composite random field with piecewise statistically homogeneous increments characterized by a truncated power variogram i e a truncated fractal this consideration is supported either exactly or approximately by a growing amount of observational data it allows decomposing the field into two or more mutually uncorrelated components having relatively small variances associated with a hierarchy of spatial correlation scales combining this fractal decomposition with a variational multiscale decomposition formalism motivated by the multiscale nature of turbulence allows decomposing flow and transport problems into two or more coupled sets of variational problems each of which is amenable to perturbative solution regardless of how heterogeneous the original field was the problems are defined on a hierarchy of computational grids having discretization intervals proportional to the correlation scales the proposed approach does not require introducing effective parameters on any scale instead it allows resolving flow and transport ensemble moments fully on multiple scales of spatial resolution the approach nevertheless allows subgrid closure another idea inspired by the literature on turbulence through subgrid refinement or the use of effective subgrid parameters it additionally offers the critically important capability of conditioning flow and transport analyses on multiscale measurements details are presented in the context of steady state single phase flow in a composite medium

siegel disks with smooth boundaries we show the existence of angles r z such that the quadratic polynomial p z e2i z z2 has a siegel disk with c smooth boundary this result was first announced by r perez marco in 1993

all bounded type siegel disks of rational maps are quasi disks we prove that every bounded type siegel disk of a rational map must be a quasi disk with at least one critical point on its boundary this verifies douady sullivan s conjecture in the case of bounded type rotation numbers

a characterization of hyperbolic rational maps we give a topological characterization of rational maps with disconnected julia sets our results extend thurston s characterization of postcritically finite rational maps in place of iteration on teichmuller space we use quasiconformal surgery and thurston s original result

hortonian scaling and effect of cutoffs on statistics of self affine river networks 1 the planar structure of river networks exhibits fractal properties in particular available data indicate a tendency for drainage areas a to be distributed according to a power law their boundaries and main channels to form self affine curves the characteristic lengths of drainage areas to be independently self affine in the one dimensional space z of total channel length rendering the network elongated and anisotropic the inverse network density a z to be either constant or self affine in z depending on whether or not the network fills the two dimensional space of a and as found in a recent study areas of given size vegetative cover and mean steady state soil moisture weighted by their flow distance from the basin outlet to be self affine in the one dimensional space of this distance various theoretical and semiempirical relationships have been proposed among exponents defining some of these and other scale dependencies expressions have been proposed for ensemble moments of a and some length measures associated with finite size river basins we present a new model that views any self affine basin property y x as belonging to an infinite hierarchy of mutually uncorrelated statistically homogeneous random functions defined on elementary subbasins each of which is characterized by a unique integral spatial correlation scale we cite a mathematically rigorous hydrologic rationale for our model and use it in conjunction with published scaling relations to obtain the probability density function of to derive analytical expressions for ensemble analogs of horton s scaling laws to deduce from them that streams of any horton strahler order are associated with integral scales 1 where the ratio 1 is a constant independent of to develop analytical expressions relating statistical moments of y x to arbitrary lower and upper cutoff scales that may but need not be taken to represent data support and maximum watershed size to describe ways of estimating the corresponding parameters and to provide a theoretical basis for the heretofore unexplained observation that self affine amplitude fluctuations of basin boundaries and main channels as measured by their variance having a common hurst exponent are larger in the former than in the latter

one hundred years of complex dynamics holomorphic dynamics has been a key area of study for 100 years in this survey we focus on the dynamics of rational functions

transport in multiscale log conductivity fields with truncated power variograms di federico and neuman this issue investigated mean uniform steady state groundwater flow in an unbounded domain with log hydraulic conductivity that forms a truncated multiscale hierarchy of statistically homogeneous and isotropic gaussian fields each associated with an exponential autocovariance here we present leading order expressions for the displacement covariance and dispersion coefficient of an ensemble of solute particles advected through such a flow field in two or three dimensions both quantities are functions of the mean travel distance s the hurst coefficient h and the low and high frequency cutoff integral scales l and u the latter two are related to the length scales of the sampling window region under investigation and sample volume data support respectively if one considers transport to be affected by a finite domain much larger than the mean travel distance so that s l then an early preasymptotic regime develops during which longitudinal and transverse dispersivities grow linearly with s if one considers transport to be affected by a domain which increases in proportion to s then l and s are of similar order and a preasymptotic regime never develops instead transport occurs under a regime that is perpetually close to asymptotic under the control of an evolving scale l s we show that if additionally u l then the corresponding longitudinal dispersivity grows in proportion to l1 2h or equivalently s1 2h both these preasymptotic and asymptotic theoretical growth rates are shown to be consistent with the observed variation of apparent longitudinal fickian dispersivities with scale we conclude by investigating the effect of variable separations between cutoff scales on dispersion

renormalizations and wandering jordan curves of rational maps we realize a dynamical decomposition for a post critically finite rational map which admits a combinatorial decomposition we split the riemann sphere into two completely invariant subsets one is a subset of the julia set consisting of uncountably many jordan curve components most of them are wandering the other consists of components that are pullbacks of finitely many renormalizations together with possibly uncountably many points the quotient action on the decomposed pieces is encoded by a dendrite dynamical system we also introduce a surgery procedure to produce post critically finite rational maps with wandering jordan curves and prescribed renormalizations

on the cycles of components of disconnected julia sets for any integers d ge 3 and n ge 1 we construct a hyperbolic rational map of degree d such that it has n cycles of the connected components of its julia set except single points and jordan curves

a decomposition theorem for herman maps abstract in 1980s thurston established a topological characterization theorem for postcritically finite rational maps in this paper a decomposition theorem for a class of postcritically infinite branched covering termed herman map is developed it s shown that every herman map can be decomposed along a stable multicurve into finitely many siegel maps and thurston maps such that the combinations and rational realizations of these resulting maps essentially dominate the original one this result is motivated by a non expanding version of mcmullen s problem and thurston s theory on characterization of rational maps it enables us to prove a thurston type theorem for rational maps with herman rings

modeling electrocortical activity through improved local approximations of integral neural field equations neural field models of firing rate activity typically take the form of integral equations with space dependent axonal delays under natural assumptions on the synaptic connectivity we show how one can derive an equivalent partial differential equation pde model that properly treats the axonal delay terms of the integral formulation our analysis avoids the so called long wavelength approximation that has previously been used to formulate pde models for neural activity in two spatial dimensions direct numerical simulations of this pde model show instabilities of the homogeneous steady state that are in full agreement with a turing instability analysis of the original integral model we discuss the benefits of such a local model and its usefulness in modeling electrocortical activity in particular we are able to treat patchy connections whereby a homogeneous and isotropic system is modulated in a spatially periodic fashion in this case the emergence of a lattice directed traveling wave predicted by a linear instability analysis is confirmed by the numerical simulation of an appropriate set of coupled pdes

universal scaling of hydraulic conductivities and dispersivities in geologic media an interpretation is offered for the observation that dispersivities increase with scale apparent longitudinal dispersivity data from a variety of hydrogeologic settings are assumed to represent a continuous hierarchy of log hydraulic conductivity fields with mutually uncorrelated increments each field having its own exponential autocovariance associated integral scale and variance that increases as a power of scale such a hierarchy is shown theoretically to form a self similar random field with homogeneous increments regardless of whether or not the underlying assumption is valid one can justify interpreting the apparent dispersivities in a manner consistent with a recent quasi linear theory of non fickian and fickian dispersion in homogeneous media which supports the notion of a self similar hierarchy a posteriori the hierarchy is revealed to possess a semivariogram s cs where c is a constant and a fractal dimension d e 0 75 where e is the topological dimension of interest this can be viewed as a universal scaling rule about which large deviations occur due to local influences including the existence of discrete natural scales at which log hydraulic conductivity is statistically homogeneous as such homogeneity is at best a local phenomenon occurring intermittently over narrow bands of the scale spectrum one must question the utility of associating medium properties with representative elementary volumes and relying on fickian models of dispersion over more than relatively narrow scale intervals porous and fractured media appear to follow the same idealized scaling rule for both flow and transport raising a question about the validity of many distinctions commonly drawn between such media finally the data suggest that conditioning transport models through calibration against hydraulic measurements has the effect of filtering out large scale modes from the hierarchy

on the julia set of a typical quadratic polynomial with a siegel disk let 0 1 be an irrational number with continued fraction expan sion a1 a 2 a 3 and consider the quadratic polynomial p z 7 e 2i z z 2 by performing a trans quasiconformal surgery on an associated blaschke product model we prove that if log an o p n a sn 1 then the julia set of p is locally connected and has lebesgue measure zero in particular it follows that for almost every 0 1 the quadratic p has a siegel disk whose boundary is a jordan curve passing through the critical point of p b y standard renormalization theory these results generalize to the quadratics which have siegel disks of higher periods

real laminations and the topological dynamics of complex polynomials abstract we characterize the laminations associated to complex polynomials with connected julia sets and without irrationally neutral cycles

scaling of random fields by means of truncated power variograms and associated spectra an interpretation is offered for the observation that the log hydraulic conductivity of geologic media often appears to be statistically homogeneous but with variance and integral scale which grow with domain size we first demonstrate that the power semi variogram and associated spectra of random fields having homogeneous isotropic increments can be constructed as weighted integrals from zero to infinity an infinite hierarchy of exponential or gaussian variograms and spectra of mutually uncorrelated homogeneous isotropic fields modes we then analyze the effect of filtering out truncating high and low frequency modes from this infinite hierarchy in the real and spectral domains a low frequency cutoff renders the truncated hierarchy homogeneous with an autocovariance function that varies monotonically with separation distance in a manner not too dissimilar than that of its constituent modes the integral scales of the lowest and highest frequency modes cutoffs are related respectively to the length scales of the sampling window domain and data support sample volume taking each relationship to be one of proportionality renders our expressions for the integral scale and variance of a truncated field dependent on window and support scales in a manner consistent with observations the traditional approach of truncating power spectral densities yields autocovariance functions that oscillate about zero with finite in one and two dimensions or vanishing in one dimension integral scales our hierarchical theory allows bridging across scales at a specific locale by calibrating a truncated variogram model to data observed on a given support in one domain and predicting the autocovariance structure of the corresponding multiscale field in domains that are either smaller or larger one may also venture we suspect with less predictive power to bridge across both domain scales and locales by adopting generalized variogram parameters derived on the basis of juxtaposed hydraulic and tracer data from many sites

a derivation of a macroscopic field theory of the brain from the quasi microscopic neural dynamics abstract the purpose of this paper is twofold first we present a semi quantitative nonlinear field theory of the brain under realistic anatomical connectivity conditions describing the interaction between functional units within the brain this macroscopic field theory is derived from the quasi microscopic conversion properties of neural populations occurring at synapses and somas the quasi microscopic models by wilson cowan 1972 1973 and nunez 1974 can be derived from these functional units are treated as inhomogeneities within a nonlinear one dimensional neural tissue second for the case of the kelso experiment the field equation is treated analytically and numerically and can be reduced to a set of ordinary differential equations which corresponds to a model by jirsa et al 1994 1995 this phenomenological model reproduces the spatio temporal phenomena experimentally observed here the most prominent property of the neural tissue is the parametric excitation the macroscopic field parameters can be expressed by quasi microscopic neural parameters

numerical solutions of moment equations for flow in heterogeneous composite aquifers 1 we analyze flow in heterogeneous media composed of multiple materials whose hydraulic properties and geometries are uncertain our analysis relies on the composite media theory of winter and tartakovksy 2000 2002 which allows one to derive and solve moment equations even when the medium is highly heterogeneous we use numerical solutions of darcy flows through a representative composite medium to investigate the robustness of perturbation approximations in porous medium with total log conductivity variances as high as 20 we also investigate the relative importance of the two sources of uncertainty in composite media material properties and geometry in our examples the uncertain geometry by itself captures the main features of the mean head estimated by the full composite model even when the within material conductivities are deterministic however neglecting randomness within materials leads to head variance estimates that are qualitatively and quantitatively wrong we compare the composite media approach to approximations that replace statistically inhomogeneous conductivity fields with pseudohomogeneous random fields with deterministic trends we demonstrate that models with a deterministic trend can be expected to give a poor estimate of the statistics of head

on stochastic modeling of flow in multimodal heterogeneous formations 1 most existing stochastic models are developed for unimodal porous media that may be well characterized with only the first two statistical moments however the distribution of hydraulic properties such as hydraulic conductivity may possess a multiplicity of modes thus the first two moments may not be adequate to characterize properties of such porous media in turn the stochastic models developed for unimodal porous media may not be applicable to flow and transport in a multimodal heterogeneous porous medium this study investigates under what circumstances the second order moment based stochastic models are applicable to multimodal heterogeneous porous media we assume that a porous medium is composed of a number of materials categories each of which may have a different mean variance and correlation scale the distribution of materials in the domain is characterized by indicator random variables we first derive analytical expressions for the mean and covariance of the log saturated hydraulic conductivity ln ks of the multimodal porous medium in terms of categorical proportions transition probability among categories and covariances of indicator random variables we express the covariance in terms of the statistics of materials in the porous medium which allows us to accurately evaluate the variance and the correlation length of the composite ln ks field we then solve the second order moment equations for the equivalent unimodal field with an exponential covariance with a single correlation scale computed for the composite field on the other hand we conduct two sets of monte carlo simulations one with multimodal random fields and the other with equivalent unimodal random fields examples for porous media with two materials are given numerical experiments show that a bimodal ln ks field may be well approximated by an equivalent unimodal field when one of the two modes is dominant under which condition the applicability of the second order moment based model is subject to the same limitation of relatively small variance as that for unimodal fields when the bimodal distribution has two more or less equally important modes although it cannot be adequately represented by an equivalent unimodal distribution the second order moment based stochastic model seems to be applicable to systems with larger composite variances than it does for an one mode dominant distribution

groundwater flow in heterogeneous composite aquifers 1 we introduce a stochastic model of flow through highly heterogeneous composite porous media that greatly improves estimates of pressure head statistics composite porous media consist of disjoint blocks of permeable materials each block comprising a single material type within a composite medium hydraulic conductivity can be represented through a pair of random processes 1 a boundary process that determines block arrangement and extent and 2 a stationary process that defines conductivity within a given block we obtain second order statistics for hydraulic conductivity in the composite model and then contrast them with statistics obtained from a standard univariate model that ignores the boundary process and treats a composite medium as if it were statistically homogeneous next we develop perturbation expansions for the first two moments of head and contrast them with expansions based on the homogeneous approximation in most cases the bivariate model leads to much sharper perturbation approximations than does the usual model of flow through an undifferentiated material when both are applied to highly heterogeneous media we make this statement precise we illustrate the composite model with examples of one dimensional flows which are interesting in their own right and which allow us to compare the accuracy of perturbation approximations of head statistics to exact analytical solutions we also show the boundary process of our bivariate model is equivalent to the indicator functions often used to represent composite media in monte carlo simulations

thurston s pullback map on the augmented teichmuller space and applications let f be a postcritically finite branched self cover of a 2 dimensional topological sphere such a map induces an analytic self map f of a finite dimensional teichmuller space we prove that this map extends continuously to the augmented teichmuller space and give an explicit construction for this extension this allows us to characterize the dynamics of thurston s pullback map near invariant strata of the boundary of the augmented teichmuller space the resulting classification of invariant boundary strata is used to prove a conjecture by pilgrim and to infer further properties of thurston s pullback map our approach also yields new proofs of thurston s theorem and pilgrim s canonical obstruction theorem

multivariate sensitivity analysis of saturated flow through simulated highly heterogeneous groundwater aquifers a multivariate analysis of variance anova is used to measure the relative sensitivity of groundwater flow to two factors that indicate different dimensions of aquifer heterogeneity an aquifer is modeled as the union of disjoint volumes or blocks composed of different materials with different hydraulic conductivities the factors are correlation between the hydraulic conductivities of the different materials and the contrast between mean conductivities in the different materials the precise values of aquifer properties are usually uncertain because they are only sparsely sampled yet are highly heterogeneous hence the spatial distribution of blocks and the distribution of materials in blocks are uncertain and are modeled as stochastic processes the anova is performed on a large sample of monte carlo simulations of a simple model flow system composed of two materials distributed within three disjoint blocks our key finding is that simulated flow is much more sensitive to the contrast between mean conductivities of the blocks than it is to the intensity of correlation although both factors are statistically significant the methodology of the experiment anova performed on monte carlo simulations of a multi material flow system constitutes the basis of additional studies of more complicated interactions between factors that define flow and transport in aquifers with uncertain properties

anisotropy lacunarity and upscaled conductivity and its autocovariance in multiscale random fields with truncated power variograms it has been shown by di federico and neuman 1997 1998a b that observed multiscale behaviors of subsurface fluid flow and transport variables can be explained within the context of a unified stochastic framework which views hydraulic conductivity as a random fractal characterized by a power variogram such a random field is statistically nonhomogeneous but possesses homogeneous spatial increments di federico and neuman 1997 have demonstrated that the power variogram and associated spectra of a statistically isotropic fractal field can be constructed as a weighted integral from zero to infinity an infinite hierarchy of exponential or gaussian variograms and spectra of mutually uncorrelated fields modes that are homogeneous and isotropic we show in this paper that the same holds true when the field and its constituent modes are statistically anisotropic provided the ratios between principal integral spatial correlation scales are the same for all modes we then analyze the effect of filtering out truncating modes of low high and intermediate spatial frequency from this infinite hierarchy in the real and spectral domains a low frequency cutoff renders the truncated hierarchy homogeneous the integral scales of the lowest and highest frequency cutoff modes are related to length scales of the sampling window domain and data support sample volume respectively taking the former to be proportional to the latter renders expressions for the integral scale and variance of the truncated field dependent on window and support scale in a manner previously shown to be consistent with observations in the isotropic case it also allows in principle bridging across scales at a specific locale as well as among locales by adopting either site specific or generalized variogram parameters the introduction of intermediate cutoffs allows us to account in a straightforward manner for lacunarity due to gaps in the multiscale hierarchy created by the absence of modes associated with discrete ranges of scales for example where textural and structural features are associated with distinct ranges of scale such as fractures having discrete ranges of trace length and density which dissect the rock into matrix blocks having corresponding ranges of sizes we explore mathematically and graphically the effects that anisotropy and lacunarity have on the integral scale variance covariance and spectra of a truncated fractal field we then develop an expression for the equivalent hydraulic conductivity of a box shaped porous block embedded within a multiscale log hydraulic conductivity field under mean uniform flow the block is larger than the support scale of the field but is smaller than a surrounding sampling window consequently its equivalent hydraulic conductivity is a random variable whose variance and spatial autocorrelation function conditioned on a known mean value of support scale conductivity across the window are given explicitly by our multiscale theory

spatially structured activity in synaptically coupled neuronal networks i traveling fronts and pulses we consider traveling front and pulse solutions to a system of integro differential equations used to describe the activity of synaptically coupled neuronal networks in a single spatial dimension our first goal is to establish a series of direct links between the abstract nature of the equations and their interpretation in terms of experimental findings in the cortex and other brain regions this is accomplished first by presenting a biophysically motivated derivation of the system and then by establishing a framework for comparison between numerical and experimental measures of activity propagation speed our second goal is to establish the existence of traveling pulse solutions using more rigorous methods two techniques are presented the first a shooting argument reduces the problem from finding a specific solution to an integro differential equation system to finding any solution to an ode system the second a singular perturbation argument provides a construction of traveling pulse solutions un

mating siegel quadratic polynomials 1 1 mating definitions and some history mating quadratic polynomials is a topological construction suggested by douady and hubbard do2 to partially parametrize quadratic rational maps of the riemann sphere by pairs of quadratic polynomials some results on matings of higher degree maps exist but we will not discuss them in this paper while there exist several presumably equivalent ways of describing the construction of mating the following approach is perhaps the most standard consider two monic quadratic polynomials fi and f2 whose filled julia sets k fi are locally connected for each fi let di denote the conformal isomorphism between the basin of infinity c k f and c ed with i 00 00 and v oo 1 these bottcher maps conijugate the polynomials to the squaring map

canonical thurston obstructions abstract we refine douady and hubbard s proof of thurston s topological characterization of rational functions by proving the following theorem let f s 2 s 2 be a branched covering with finite postcritical set p f and hyperbolic orbifold let c denote the set of all homotopy classes of nonperipheral simple closed curves in s 2 p f such that the length of the unique geodesic homotopic to tends to zero under iteration of the thurston map induced by f on teichmuller space then either c is empty and f is equivalent to a rational function or else c is a thurston obstruction

geometrically finite and semi rational branched coverings of the two sphere in 1982 thurston gave a necessary and sufficient condition for a critically finite branched covering of the two sphere to itself to be combinatorially equivalent to a rational map we discuss extending this result to geometrically finite rational maps we give an example to show that thurston s original condition is not sufficient this example is topologically pathological near accumulation points of the postcritical set we give two conditions forbidding such pathology show that they are equivalent and in a sequel to the present paper will show that thurston s condition together with this tameness is both necessary and sufficient to characterize geometrically finite

global topology of hyperbolic components cantor circle case we prove that in the moduli space md of degree d 2 rational maps any hyperbolic component in the disconnectedness locus and of cantor circle type is a finite quotient of r4d 4 n tn where n is determined by dynamics the proof uses some ideas from riemann surface theory abel s theorem dynamical system and algebraic topology

on advective transport in fractal permeability and velocity fields we consider advective transport in a steady state random velocity field with homogeneous increments such a field is self affine with a power law dyadic semivariogram s proportional to d2 where d is distance and is a hurst coefficient it is characterized by a fractal dimension d e 1 where e is topological dimension as the mean and variance of such a field are undefined we condition them on measurement at some point x0 we then introduce a tracer at another point y0 and invoke elementary theoretical considerations to demonstrate that its conditional mean dispersion is local at all times its conditional mean concentration and variance are given explicitly by well established expressions which however have not been previously recognized as being valid in fractal fields once the conditional mean travel distance s of the tracer becomes large compared to the distance between y0 and x0 the corresponding dispersion and dispersivity tensors grow in proportion to s1 2 where 0 1 this supralinear rate of growth is consistent with that exhibited by apparent longitudinal dispersivities obtained by standard methods of interpretation from tracer behavior observed in a variety of geologic media under varied flow and transport regimes filtering out modes from the fractal velocity field with correlation scales larger than some s0 allows an asymptotic transport regime to develop when s s0 the corresponding asymptotic dispersivities grow in proportion to s02 when 0 this linear to sublinear rate of growth is consistent with that exhibited by apparent longitudinal dispersivities obtained from calibrated numerical models in a variety of media a self affine natural log permeability field gives rise to a self affine velocity field while s is sufficiently small to insure that the variance of the log permeabilities which grows as a power of s remains nominally less than one an analysis of published apparent longitudinal dispersivity data in light of the above theoretical results supports my earlier conclusion that when one juxtaposes data from a large number of generally dissimilar geologic media from a variety of locales one observes a tendency toward self affine behavior with a hurst coefficient 0 25 at any given locale such media may or may not exhibit fractal behavior if they do may or may not be close to 0 25

a proof of thurston s topological characterization of rational functions the criterion proved in this paper was stated by thurston in november 1982 thurston lectured on its proof on several occasions notably at the nsf summer conference in duluth 1983 where one of the authors jhh was present using the notes of various attendants at these lectures we have reconstructed a proof that we have made as precise as we could since this required a certain amount of work on our part we thought it might be of some use to present this proof to the reader we thank dennis sullivan for useful conversations and fritz yon haesseler and especially ben wittner for help with the writing and valuable suggestions after the first version was written clifford earle pointed out that better estimates than what we had were to be found in b

analysis of nonlocal neural fields for both general and gamma distributed connectivities abstract this work studies the stability of equilibria in spatially extended neuronal ensembles we first derive the model equation from statistical properties of the neuron population the obtained integro differential equation includes synaptic and space dependent transmission delay for both general and gamma distributed synaptic connectivities the latter connectivity type reveals infinite finite and vanishing self connectivities the work derives conditions for stationary and nonstationary instabilities for both kernel types in addition a nonlinear analysis for general kernels yields the order parameter equation of the turing instability to compare the results to findings for partial differential equations pdes two typical pde types are derived from the examined model equation namely the general reaction diffusion equation and the swift hohenberg equation hence the discussed integro differential equation generalizes these pdes in the case of the gamma distributed kernels the stability conditions are formulated in terms of the mean excitatory and inhibitory interaction ranges as a novel finding we obtain turing instabilities in fields with local inhibition lateral excitation while wave instabilities occur in fields with local excitation and lateral inhibition numerical simulations support the analytical results

components of degree two hyperbolic rational maps we examine the structure of hyperbolic components and some boundary points of these for degree two rational maps rather than using quasi conformal deformation theory we use a technique from a theorem of thurston about the existence of rational maps this involves regarding a rational map as acting on the teichmuller space of punctured sphere

constructing locally connected non computable julia sets a locally connected quadratic siegel julia set has a simple explicit topological model such a set is computable if there exists an algorithm to draw it on a computer screen with an arbitrary resolution we constructively produce parameter values for siegel quadratics for which the julia sets are non computable yet locally connected

a general fractal model of flow and solute transport in randomly heterogeneous porous media 1 one of the challenges of hydrology is modeling flow and solute transport through media that are not uniformly porous this paper proposes a general fractal model of flow and solute transport in randomly heterogeneous porous media we describe the random field using fractional levy motion flm which more effectively represents a field with a high degree of variability understanding flow and solute transport in the flm field expands the applicability of stochastic theory following the work of v di federico and s p neuman 1997 and v di federico et al 1999 we derive truncated isotropic and anisotropic power variograms for the flm fields the results are analogous to but more general than those of fractional brownian motion fbm v di federico and s p neuman 1997 and v di federico et al 1999 for a random field when the levy index in flm is replaced by a constant of 2 fbm is recovered this study employs first order perturbation analyses of flow and solute transport in two dimensional flm fields for the stationary fbm and flm fields macrodispersion coefficients are compared the plume is found to grow slowest in the flm field but will show highest concentration the proposed fractal model is general in the sense that when the levy index of flm equals 2 the results of flow and solute transport for the commonly used fbm model are recovered when the upper cutoff of the truncated power variogram is close to the lower cutoff the results for the stationary exponential or gaussian model can be well approximated since the proposed general fractal model has broader applications than the stationary and fbm models it simulates flow and solute transport in more different scenarios to yield more accurate modeling results

dynamics of mcmullen maps abstract in this article we develop the yoccoz puzzle technique to study a family of rational maps termed mcmullen maps we show that the boundary of the immediate basin of infinity is always a jordan curve if it is connected this gives a positive answer to the question of devaney higher regularity of this boundary is obtained in almost all cases we show that the boundary is a quasi circle if it contains neither a parabolic point nor a recurrent critical point for the whole julia set we show that the mcmullen maps have locally connected julia sets except in some special cases