dynamics of closed interfaces in two dimensional laplacian growth we study the process of two dimensional laplacian growth in the limit of zero surface tension for cases with a closed interface around a growing bubble exterior problem with circular geometry using the timedependent conformal map technique we obtain a class of fingerlike solutions that are characterized by a finite number of poles we find the conditions under which these solutions remain smooth for all times these solutions allow the description of the system in terms of a finite number of degrees of freedom at least in the limit of zero surface tension we believe that whenever they remain smooth they can also be used as a nonlinear basis even when surface tension is included s1063 651x 98 03102 x

solvable vector nonlinear riemann problems exact implicit solutions of dispersionless pdes and wave breaking we have recently solved the inverse spectral problem for integrable partial differential equations pdes in arbitrary dimensions arising as commutation of multidimensional vector fields depending on a spectral parameter the associated inverse problem in particular can be formulated as a nonlinear riemann hilbert nrh problem on a given contour of the complex plane the most distinguished examples of integrable pdes of this type like the dispersionless kadomtsev petviashivili dkp the heavenly and the two dimensional dispersionless toda equations are real pdes associated with hamiltonian vector fields the corresponding nrh data satisfy suitable reality and symplectic constraints in this paper generalizing the examples of solvable nrh problems illustrated in manakov and santini 2009 j phys a math theor 42 095203 2008 j phys a math theor 41 055204 2009 j phys a math theor 42 404013 we present a general procedure to construct solvable nrh problems for integrable real pdes associated with hamiltonian vector fields allowing one to construct exact implicit solutions of such pdes parametrized by an arbitrary number of real functions of a single variable then we illustrate this theory on few distinguished examples for the dkp and heavenly equations for the dkp case we characterize a class of similarity solutions of solutions constant on their parabolic wave front and breaking simultaneously on it of localized solutions whose breaking point travels with constant speed along the wave front and of localized solutions breaking in a point of the x y plane for the heavenly equation we characterize two classes of symmetry reductions

the loewner equation maps and shapes an approach called schramm loewner evolution sle provides a new method for dealing with a wide variety of scale invariant problems in two dimensions this approach is based upon an older method called loewner evolution le which connects analytic and geometrical constructions in the complex plane in this paper the bases of le and sle are described and some simple applications are discussed in relatively non technical form a bibliography of the subject is presented

on a competitive model of laplacian growth we introduce a competitive model of laplacian growth in both stochastic and deterministic versions this defines two different aggregation laws with probabilities and 1 the parameter varying from 0 to 1 is used to weight a ratio between the inner and outer harmonic measures that leads to a competition between the eden like process and the dla solved with site sticking conditions we perform numerical and qualitative analysis of the competitive growth

taylor s power law and fluctuation scaling explained by a central limit like convergence a power function relationship observed between the variance and the mean of many types of biological and physical systems has generated much debate as to its origins this taylor s law or fluctuation scaling has been recently hypothesized to result from the second law of thermodynamics and the behavior of the density of states this hypothesis is predicated on physical quantities like free energy and an external field the correspondence of these quantities with biological systems though remains unproven questions can be posed as to the applicability of this hypothesis to the diversity of observed phenomena as well as the range of spatial and temporal scales observed with taylor s law we note that the cumulant generating functions derived from this thermodynamic model correspond to those derived over a quarter century earlier for a class of probabilistic models known as the tweedie exponential dispersion models these latter models are characterized by variance to mean power functions their phenomenological basis rests with a central limit theorem like property that causes many statistical systems to converge mathematically toward a tweedie form we review evaluations of the tweedie poisson gamma model for taylor s law and provide three further cases to test the clustering of single nucleotide polymorphisms snps within the horse chromosome 1 the clustering of genes within human chromosome 8 and the mertens function this latter case is a number theoretic function for which a thermodynamic model cannot explain taylor s law but where tweedie convergence remains applicable the tweedie models are applicable to diverse biological physical and mathematical phenomena that express power variance functions over a wide range of measurement scales they provide a probabilistic description for taylor s law that allows mechanistic insight into complex systems without the assumption of a thermodynamic mechanism

density of eigenvalues of random normal matrices the relation between random normal matrices and conformal mappings discovered by wiegmann and zabrodin is made rigorous by restricting normal matrices to have spectrum in a bounded set it is shown that for a suitable class of potentials the asymptotic density of eigenvalues is uniform with support in the interior domain of a simple smooth curve

dimensions maximal growth sites and optimization in the dielectric breakdown model we study the growth of fractal clusters in the dielectric breakdown model dbm by means of iterated conformal mappings in particular we investigate the fractal dimension and the maximal growth site measured by the hoelder exponent ensuremath alpha min as a function of the growth exponent ensuremath eta of the dbm model we do not find evidence for a phase transition from fractal to nonfractal growth for a finite ensuremath eta value simultaneously we observe that the limit of nonfractal growth d ensuremath rightarrow 1 is consistent with ensuremath alpha min ensuremath rightarrow 1 2 finally using an optimization principle we give a recipe on how to estimate the effective value of ensuremath eta from temporal growth data of fractal aggregates

multi particle diffusion limited aggregation we consider a stochastic aggregation model on mathbb z d start with particles distributed according to the product bernoulli measure with parameter mu in addition start with an aggregate at the origin non aggregated particles move as continuous time simple random walks obeying the exclusion rule whereas aggregated particles do not move the aggregate grows by attaching particles to its surface whenever a particle attempts to jump onto it this evolution is called multi particle diffusion limited aggregation our main result states that if on d 1 the initial density of particles is large enough then with positive probability the aggregate has linearly growing arms that is there exists a constant c 0 so that at time t the aggregate contains a point of distance at least ct from the origin for all t the key conceptual element of our analysis is the introduction and study of a new growth process consider a first passage percolation process called type 1 starting from the origin whenever type 1 is about to occupy a new vertex with positive probability instead of doing it it gives rise to another first passage percolation process called type 2 which starts to spread from that vertex each vertex gets occupied only by the process that arrives to it first this process may have three phases extinction type 1 gets eventually surrounded by type 2 coexistence infinite clusters of both types emerge and strong survival type 1 produces an infinite cluster which entraps all type 2 clusters understanding the various phases of this process is of mathematical interest on its own right we establish the existence of a strong survival phase and use this to show our main result

renormalization theory of stochastic growth an analytical renormalization group treatment is presented of a model which for one value of parameters is equivalent to diffusion limited aggregation the fractal dimension of dla is computed to be 2 1 2 1 5 1 7 higher multifractal exponents are also calculated and found in agreement with numerical results it may be possible to use this technique to describe the dielectric breakdown model as well which is given by different parameter values

estimation of the negative binomial parameter κ by maximum quasi likelihood we investigate estimation of the parameter k of the negative binomial distribution for small samples using a method of moments estimate mme and a maximum quasi likelihood estimate mqle previous work is reviewed the importance of indirect estimation of k through its reciprocal a and of allowance for negative estimates of k or a are discussed samples of size 50 are simulated 10 000 times for each of several parameter combinations to examine the properties of the estimates samples of sizes 10 20 30 and 50 are simulated 1 000 times to investigate the effect of sample size both estimators perform reasonably well except when the mean is small and the sample size does not exceed 20 three examples are given one of a designed experiment for which the mqle is especially suited confidence limits are derived for the mqle further work along these lines is required for adequate assessment of the usual maximum likelihood estimate

on the solutions of the second heavenly and pavlov equations we have recently solved the inverse scattering problem for one parameter families of vector fields and used this result to construct the formal solution of the cauchy problem for a class of integrable nonlinear partial differential equations connected with the commutation of multidimensional vector fields such as the heavenly equation of plebanski the dispersionless kadomtsev petviashvili dkp equation and the two dimensional dispersionless toda 2ddt equation as well as with the commutation of one dimensional vector fields such as the pavlov equation we also showed that the associated riemann hilbert inverse problems are powerful tools to establish if the solutions of the cauchy problem break at finite time to construct their long time behaviour and characterize classes of implicit solutions in this paper using the above theory we concentrate on the heavenly and pavlov equations i establishing that their localized solutions evolve without breaking unlike the cases of dkp and 2ddt ii constructing the long time behaviour of the solutions of their cauchy problems iii characterizing a distinguished class of implicit solutions of the heavenly equation

viscous shocks in hele shaw flow and stokes phenomena of the painleve i transcendent in hele shaw flows at vanishing surface tension the boundary of a viscous fluid develops cusp like singularities in recent papers lee et al 2009 2008 we have showed that singularities trigger viscous shocks propagating through the viscous fluid here we show that the weak solution of the hele shaw problem describing viscous shocks is equivalent to a semiclassical approximation of a special real solution of the painleve i equation we argue that the painleve i equation provides an integrable deformation of the hele shaw problem which describes flow passing through singularities in this interpretation shocks appear as stokes level lines of the painlevelinear problem

geometric approach to stationary shapes in rotating hele shaw flows abstract we consider the geometry of a special family of curves associated with the motion of a two fluid interface in a rotating hele shaw cell this family of stationary exact solutions with surface tension consists of interface shapes which balance exactly the competing capillary and centrifugal forces the result is the formation of patterned structures presenting fingers that eventually assume a teardrop like shape and tend to be detached from the main body of the rotating fluid occurrence of a pinch off event by using the vortex sheet formalism we approach the problem analytically and show that the curvature of these particular curves can be very simply expressed as a function of the radial distance to the rotation axis motivated by this fact and through a simple geometric interpretation we show that the exact solutions for the rotating hele shaw problem satisfy a first order ordinary differential equation which is readily solved so that the shape solutions are determined up to quadratures the ability to probe a number of key morphological features of such solutions analytically is demonstrated and a gallery of plots is provided to highlight these findings the possibility of accessing a criterion to predict the occurrence of pinch off is also discussed finally we prove that the general solutions are dense and use this fact to guarantee the very existence of the neat highly symmetric physical patterns

laplacian growth as one dimensional turbulence abstract a new model of laplacian stochastic growth is formulated using conformal mappings the model describes two growth regimes stable and turbulent separated by a sharp phase transition the first few fourier components of the mapping define the web an envelope of the cluster the web is used to study the transition and the dynamics of large scale features of the cluster characterized by evolution from macro to micro scales also we derive scaling laws for the cluster size

lowner equations hirota equations and reductions of the universal whitham hierarchy this paper reconsiders finite variable reductions of the universal whitham hierarchy of genus zero in the perspective of dispersionless hirota equations in the case of one variable reduction dispersionless hirota equations turn out to be a powerful tool for understanding the mechanism of reduction all relevant equations describing the reduction lowner type equations and diagonal hydrodynamic equations can be thereby derived and justified in a unified manner the case of multi variable reductions is not so straightforward nevertheless the reduction procedure can be formulated in a general form and justified with the aid of dispersionless hirota equations as an application previous results of guil manas and martinez alonso are reconfirmed in this formulation

finite size effect of harmonic measure estimation in a dla model variable size of probe particles abstract a finite size effect in the probing of the harmonic measure in simulation of diffusion limited aggregation dla growth is investigated we introduce a variable size of probe particles to estimate harmonic measure and extract the fractal dimension of dla clusters taking two limits of vanishingly small probe particle size and of infinitely large size of a dla cluster we generate 1000 dla clusters consisting of 50 million particles each using an off lattice killing free algorithm developed in the early work the introduced method leads to unprecedented accuracy in the estimation of the fractal dimension we discuss the variation of the probability distribution function with the size of probing particles

the inverse spectral transform for the dunajski hierarchy and some of its reductions i cauchy problem and longtime behavior of solutions in this paper we apply the formal inverse spectral transform for integrable dispersionless partial differential equations pdes arising from the commutation condition of pairs of one parameter families of vector fields recently developed by s v manakov and one of the authors to one distinguished class of equations the so called dunajski hierarchy we concentrate for concreteness i on the system of pdes characterizing a general anti self dual conformal structure in neutral signature ii on its first commuting flow and iii on some of their basic and novel reductions we formally solve their cauchy problem and we use it to construct the longtime behavior of solutions showing in particular that unlike the case of soliton pdes different dispersionless pdes belonging to the same hierarchy of commuting flows evolve in time in very different ways exhibiting either a smooth dynamics or a gradient catastrophe at finite time

the resultant on compact riemann surfaces we introduce a notion of the resultant of two meromorphic functions on a compact riemann surface and demonstrate its usefulness in several respects for example we exhibit several integral formulas for the resultant relate it to potential theory and give explicit formulas for the algebraic dependence between two meromorphic functions on a compact riemann surface as a particular application the exponential transform of a quadrature domain in the complex plane is expressed in terms of the resultant of two meromorphic functions on the schottky double of the domain

convergent calculation of the asymptotic dimension of diffusion limited aggregates scaling and renormalization of small clusters diffusion limited aggregation dla is a model of fractal growth that had attained a paradigmatic status due to its simplicity and its underlying role for a variety of pattern forming processes we present a convergent calculation of the fractal dimension d of dla based on a renormalization scheme for the first laurent coefficient of the conformal map from the unit circle to the expanding boundary of the fractal cluster the theory is applicable from very small 2 3 particles to asymptotically large n infinity clusters the computed dimension is d 1 713 0 003

conformal maps and integrable hierarchies we show that conformal maps of simply connected domains with an analytic boundary to a unit disk have an intimate relation to the dispersionless 2d toda integrable hierarchy the maps are determined by a particular solution to the hierarchy singled out by the conditions known as string equations the same hierarchy locally solves the 2d inverse potential problem i e reconstruction of the domain out of a set of its harmonic moments this is the same solution which is known to describe 2d gravity coupled to c matter we also introduce a concept of the function for analytic curves

fluctuation results for hastings levitov planar growth we study the fluctuations of the outer domain of hastings levitov clusters in the small particle limit these are shown to be given by a continuous gaussian process mathcal f taking values in the space of holomorphic functions on z 1 of which we provide an explicit construction the boundary values mathcal w of mathcal f are shown to perform an ornstein uhlenbeck process on the space of distributions on the unit circle mathbb t which can be described as the solution to the stochastic fractional heat equation begin aligned frac partial partial t mathcal w t vartheta delta 1 2 mathcal w t vartheta sqrt 2 xi t vartheta end aligned where delta denotes the laplace operator acting on the spatial component and xi t vartheta is a space time white noise as a consequence we find that when the cluster is left to grow indefinitely the boundary process mathcal w converges to a log correlated fractional gaussian field which can be realised as delta 1 4 w for w complex white noise on mathbb t

weak convergence of the localized disturbance flow to the coalescing brownian flow we define a new state space for the coalescing brownian flow also known as the brownian web on the circle the elements of this space are families of order preserving maps of the circle depending continuously on two time parameters and having a certain weak flow property the space is equipped with a complete separable metric a larger state space allowing jumps in time is also introduced and equipped with a skorokhod type metric also complete and separable we prove that the coalescing brownian flow is the weak limit in this larger space of a family of flows which evolve by jumps each jump arising from a small localized disturbance of the circle a local version of this result is also obtained in which the weak limit law is that of the coalescing brownian flow on the line our set up is well adapted to time reversal and our weak limit result provides a new proof of time reversibility of the coalescing brownian flow we also identify a martingale associated with the coalescing brownian flow on the circle and use this to make a direct calculation of the laplace transform of the time to complete coalescence

laplacian growth with separately controlled noise and anisotropy conformal mapping models are used to study the competition of noise and anisotropy in laplacian growth for this purpose a family of models is introduced with the noise level and directional anisotropy controlled independently fractalization is observed in both anisotropic growth and growth with varying noise the fractal dimension is determined from the cluster size scaling with cluster area for isotropic growth d 1 7 at both high and low noise for anisotropic growth with reduced noise the dimension can be as low as d 1 5 and apparently is not universal also we study the fluctuations of particle areas and observe in agreement with previous studies that exceptionally large particles may appear during growth leading to pathologically irregular clusters this difficulty is circumvented by using an acceptance window for particle areas

selected topics on quadrature domains this is a selection of facts old and recent about quadrature domains the text written in the form of a survey is addressed to non experts and covers a variety of phenomena related to quadratur

vortex patch equilibria of the euler equation and random normal matrices a new analogy is drawn between vortex patch or v state equilibria of the euler equations for ideal fluids and the planar limit of random normal matrix models physically the former is a quite different fluid dynamical problem to hele shaw flow or laplacian growth to which an analogy with matrix models has become well known in recent years the connection of random matrices with vortex dynamics is made via the so called modified schwarz potential this theoretical link while interesting in itself has immediate ramifications for random matrix theory by virtue of a transfer of mathematical technology already well developed for vortex dynamics hence for multi support matrix models in the planar limit we describe a constructive approach to the inverse problem of finding the shapes of the eigenvalue supports for a given potential using automorphic conformal mappings within a schottky model of the underlying algebraic curves and use of the schottky klein prime function two support eigenvalue distributions in a quartic potential are given as an illustrative example

dispersionless dkp hierarchy and the elliptic lowner equation we show that the dispersionless dkp hierarchy the dispersionless limit of the pfaff lattice admits a suggestive reformulation through elliptic functions we also consider one variable reductions of the dispersionless dkp hierarchy and show that they are described by an elliptic version of the lowner equation with a particular choice of the driving function the latter appears to be closely related to the painleve vi equation with a special choice of parameters

stochastic model for dielectric breakdown we discuss a model for the development of discharge patterns in dielectric breakdown based on the laplace equation associated with a probability field the model gives rise to random fractals with well defined hausdorff dimensions the relations of this model with the diffusion limited aggregation are discussed in detail the possibility of application to other stochastic phenomena like fracture propagation is proposed

two matrix model with semiclassical potentials and extended whitham hierarchy we consider the two matrix model with potentials whose derivatives are arbitrary rational functions of fixed pole structure and the support of the spectra of the matrices are union of intervals hard edges we derive an explicit formula for the planar limit of the free energy and we derive a calculus which allows us to compute derivatives of arbitrarily high order by extending classical rauch s variational formulae the four point correlation functions are explicitly worked out the formalism extends naturally to the computation of residue formulae for the tau function of the so called universal whitham hierarchy studied mainly by i krichever our setting extends the moduli space in that there are certain extra data

statistical physics of fracture surfaces morphology experiments on fracture surface morphologies offer increasing amounts of data that can be analyzed using methods of statistical physics one finds scaling exponents associated with correlation and structure functions indicating a rich phenomenology of anomalous scaling we argue that traditional models of fracture fail to reproduce this rich phenomenology and new ideas and concepts are called for we present some recent models that introduce the effects of deviations from homogeneous linear elasticity theory on the morphology of fracture surfaces successfully reproducing the multiscaling phenomenology at least in 1 1 dimensions for surfaces in 2 1 dimensions we introduce novel methods of analysis based on projecting the data on the irreducible representations of the so 2 symmetry group it appears that this approach organizes effectively the rich scaling properties we end up proposing new experiments in which the rotational symmetry is not broken such that the scaling properties should be particularly simple

on the uniform boundedness conjecture for drinfeld modules in analogy with the famous theorems of mazur and merel on the torsion subgroups of elliptic curves one can formulate similar conjectures for the torsion points of drinfeld modules we prove some partial results for rank 2 drinfeld q t modules for example the uniform boundedness of the primary torsion

the dispersionless 2d toda equation dressing cauchy problem longtime behaviour implicit solutions and wave breaking we have recently solved the inverse spectral problem for one parameter families of vector fields and used this result to construct the formal solution of the cauchy problem for a class of integrable nonlinear partial differential equations in multidimensions including the second heavenly equation of plebanski and the dispersionless kadomtsev petviashvili dkp equation arising as commutation of vector fields in this paper we make use of the above theory i to construct the nonlinear riemann hilbert dressing for the so called two dimensional dispersionless toda equation elucidating the spectral mechanism responsible for wave breaking ii we present the formal solution of the cauchy problem for the wave form of it exp tt xx yy iii we obtain the longtime behaviour of the solutions of such a cauchy problem showing that it is essentially described by the longtime breaking formulae of the dkp solutions confirming the expected universal character of the dkp equation as prototype model in the description of the gradient catastrophe of two dimensional waves iv we finally characterize a class of spectral data allowing one to linearize the rh problem corresponding to a class of implicit solutions of the pde

non laplacian growth algebraic domains and finite reflection groups dynamics of planar domains with moving boundaries driven by the gradient of a scalar field that satisfies an elliptic pde is studied we consider the question for which kind of pdes are the domains algebraic provided the field has singularities at a fixed point inside the domain the construction reveals a direct connection with the theory of the calogero moser systems related to finite reflection groups and their integrable deformations

conformal mappings and dispersionless toda hierarchy ii general string equations in this article we classify the solutions of the dispersionless toda hierarchy into degenerate and non degenerate cases we show that every non degenerate solution is determined by a function mathcal h z 1 z 2 of two variables we interpret these non degenerate solutions as defining evolutions on the space mathfrak d of pairs of conformal mappings g f where g is a univalent function on the exterior of the unit disc f is a univalent function on the unit disc normalized such that g f 0 0 and f 0 g 1 for each solution we show how to define the natural time variables t n n in mathbb z as complex coordinates on the space mathfrak d we also find explicit formulas for the tau function of the dispersionless toda hierarchy in terms of mathcal h z 1 z 2 imposing some conditions on the function mathcal h z 1 z 2 we show that the dispersionless toda flows can be naturally restricted to the subspace of mathfrak d defined by f w 1 overline g 1 bar w this recovers the result of zabrodin theor math phy 12 1511 1525 2001

the τ function of the universal whitham hierarchy matrix models and topological field theories the universal witham hierarchy is considered from the point of view of topological field theories the function for this hierarchy is defined it is proved that the algebraic orbits of whitham hierarchy can be identified with various topological matter models coupled with topological gravity

fingered growth in channel geometry a loewner equation approach a simple model of laplacian growth is considered in which the growth takes place only at the tips of long thin fingers following carleson and makarov l carleson and n makarov j anal math 87 103 2002 the evolution of the fingers is studied with use of the deterministic loewner equation the method is then extended to study the growth in a linear channel with reflecting sidewalls one and two finger solutions are found and analyzed it turns out that the presence of the walls has a significant influence on the shapes of the fingers and the dynamics of the screening process in which longer fingers suppress the growth of the shorter ones

families of distributions for repeated samples of animal counts the criterion of flexibility perry and taylor 1985 journal of animal ecology 54 931 953 that families of distributions with n 2 parameters should fit adequately to all n samples of animal counts over the full range of population densities usually encountered was misinterpreted by kemp 1987 biometrics 43 693 699 this criterion of goodness of fit is examined here for perry and taylor s ades family and one of kemp s alternatives a family of constrained negative binomial c n b distributions using extensive literature data while the ades fitted better than the c n b in 15 of 22 sets the evidence is not overwhelming because in 13 sets the difference was negligible or both families fitted adequately however of the remainder the ades fitted notably better to seven sets these mainly comprised long tailed distributions with large means typical of pest species its use in practical applications with such data is discussed and the importance of its superior reliability stressed kemp s method of fitting is not recommended

on the solutions of the dkp equation the nonlinear riemann hilbert problem longtime behaviour implicit solutions and wave breaking we have recently solved the inverse scattering problem for one parameter families of vector fields and used this result to construct the formal solution of the cauchy problem for a class of integrable nonlinear partial differential equations in multidimensions including the second heavenly equation of plebanski and the dispersionless kadomtsev petviashvili dkp equation we showed in particular that the associated inverse problems can be expressed in terms of nonlinear riemann hilbert problems on the real axis in this paper we make use of the nonlinear riemann hilbert problem of dkp i to construct the longtime behaviour of the solutions of its cauchy problem ii to characterize a class of implicit solutions iii to elucidate the spectral mechanism causing the gradient catastrophe of localized solutions of dkp at finite time as well as in the longtime regime and the corresponding universal behaviours near breaking

martingales of stochastic laplacian growth abstract a family of exponential martingales of a stochastic laplacian growth problem is proposed stochastic laplacian growth describes a regularized interface dynamics in a two fluid system where the viscous fluid is incompressible at a large scale while compressible at a small scale in the vicinity of the interface hence random fluctuations of pressure near the boundary are inevitable by using loewner kufarev equation we study interface dynamics generated by nonlocal random loewner measure which produces the patterns with viscous fingers we use a schottky double construction to introduce a one parametric family of functions of random processes on the double closely connected to the correlation functions of primary operators of the boundary conformal field theory in the coulomb gas framework for a specific value of the parameter these functions are martingales with respect to stochastic loewner flow on the schottky double a connection between the proposed algebraic construction and the physical problem of stochastic interface dynamics in the hele shaw cell relies on the hadamard s variational formula namely the variation of pressure in stochastic laplacian growth in the vicinity of the interface is given by the variance of martingales on the double

integrable structure of the dirichlet boundary problem in two dimensions we study how the solution of the two dimensional dirichlet boundary problem for smooth simply connected domains depends upon variations of the data of the problem we show that the hadamard formula for the variation of the dirichlet green function under deformations of the domain reveals an integrable structure the independent variables corresponding to the infinite set of commuting flows are identified with harmonic moments of the domain the solution to the dirichlet boundary problem is expressed through the tau function of the dispersionless toda hierarchy we also discuss a degenerate case of the dirichlet problem on the plane with a gap in this case the tau function is identical to the partition function of the planar large n limit of the hermitian one matrix model

shocks and finite time singularities in hele shaw flow abstract hele shaw flow at vanishing surface tension is ill defined in finite time the flow develops cusp like singularities we show that this ill defined problem admits a weak dispersive solution when singularities give rise to a graph of shock waves propagating into the viscous fluid the graph of shocks grows and branches velocity and pressure have finite discontinuities across the shock we formulate a few simple physical principles which single out the dispersive solution and interpret shocks as lines of decompressed fluid we also formulate the dispersive weak solution in algebro geometrical terms as an evolution of the krichever boutroux complex curve we study in detail the most generic 2 3 cusp singularity which gives rise to an elementary branching event this solution is self similar and expressed in terms of elliptic functions

universal whitham hierarchy dispersionless hirota equations and multicomponent kp hierarchy abstract the goal of this paper is to identify the universal whitham hierarchy of genus zero with a dispersionless limit of the multicomponent kp hierarchy to this end the multicomponent kp hierarchy is re formulated to depend on several discrete variables called charges these discrete variables play the role of lattice coordinates in underlying toda field equations a multicomponent version of the so called differential fay identity are derived from the hirota equations of the function of this charged multicomponent kp hierarchy these multicomponent differential fay identities have a well defined dispersionless limit the dispersionless hirota equations the dispersionless hirota equations turn out to be equivalent to the hamilton jacobi equations for the s functions of the universal whitham hierarchy the differential fay identities themselves are shown to be a generating functional expression of auxiliary linear equations for scalar valued wave functions of the multicomponent kp hierarchy

sur la nature non cyclotomique des points d ordre fini des courbes elliptiques appendice par e kowalski et p michel

discrete charges on a two dimensional conductor we investigate the electrostatic equilibria of n discrete charges of size 1 n on a two dimensional conductor domain we study the distribution of the charges on symmetric domains including the ellipse the hypotrochoid and various regular polygons with an emphasis on understanding the distributions of the charges as the shape of the underlying conductor becomes singular we find that there are two regimes of behavior a symmetric regime for smooth conductors and a symmetry broken regime for singular domains for smooth conductors the locations of the charges can be determined to within o left sqrt log n mathord left vphantom n n 2 right kern nulldelimiterspace n 2 right by an integral equation due to pommerenke math ann 179 212 218 1969 we present a derivation of a related but different integral equation which has the same solutions we also solve the equation to obtain asymptotic solutions which show universal behavior in the distribution of the charges in conductors with somewhat smooth cusps conductors with sharp cusps and singularities show qualitatively different behavior where the symmetry of the problem is broken and the distribution of the discrete charges does not respect the symmetry of the underlying domain we investigate the symmetry breaking both theoretically and numerically and find good agreement between our theory and the numerics we also find that the universality in the distribution of the charges near the cusps persists in the symmetry broken regime although this distribution is very different from the one given by the integral equation

genus n algebraic reductions of the benney hierarchy within a schottky model by exploiting a new theoretical connection between reductions of the benney hierarchy and the dirichlet problem for laplace s equation the solution to a spectral problem associated with n parameter algebraic reductions of the benney hierarchy is found explicitly the solutions can be written in terms of the modified green s function associated with reflectionally symmetric n connected planar domains whose holes are all centred on the symmetry axis explicit formulae for the modified green s function in a canonical class of circular domains are constructed using a schottky model of the schottky double of these domains uniformizations of the spectral problem associated with two different types of reductions then follow from these formulae

hydrodynamic reductions of multidimensional dispersionless pdes the test for integrability a d 1 dimensional dispersionless pde is said to be integrable if its n component hydrodynamic reductions are locally parametrized by d 1 n arbitrary functions of one variable the most important examples include the four dimensional heavenly equation descriptive of self dual ricci flat metrics and its six dimensional generalization arising in the context of sdiff 2 self dual yang mills equations given a multidimensional pde which does not pass the integrability test the method of hydrodynamic reductions allows one to effectively reconstruct additional differential constraints which when added to the equation make it an integrable system in fewer dimensions as an example of this phenomenon we discuss the second commuting flow of the dispersionless kp hierarchy considered separately this is a four dimensional pde which does not pass the integrability test however the method of hydrodynamic reductions generates additional differential constraints which reconstruct the full 2 1 dimensional dis

the characterization of two component 2 1 dimensional integrable systems of hydrodynamic type we obtain the necessary and sufficient conditions for a two component 2 1 dimensional system of hydrodynamic type to possess infinitely many hydrodynamic reductions these conditions are in involution implying that the systems in question are locally parametrized by 15 arbitrary constants it is proved that all such systems possess three conservation laws of hydrodynamic type and therefore are symmetrizable in godunov s sense moreover all such systems are proved to possess a scalar pseudopotential which plays the role of the dispersionless lax pair we demonstrate that the class of two component systems possessing a scalar pseudopotential is in fact identical with the class of systems possessing infinitely many hydrodynamic reductions thus establishing the equivalence of the two possible definitions of integrability explicit linearly degenerate examples are constructed

families of discrete distributions satisfying taylor s power law count data in entomological studies can often be described by some form of contagious distribution such as the negative binomial neyman type a or polya aeppli these are all clustered poisson distributions their distributional forms are given in section 2 arising when groups of individuals occur at random i e are poissonian and individuals within a group have their own distribution whilst pahl 1969 found strong evidence that the negative binomial distribution gives good fits to his 35 samples of data on the chrysomelid beetle paropsis atomaria martin and katti 1965 fitted a number of distributions to a variety of insect plant and animal data and observed for instance that mcguire brindley and bancroft s 1957 data on the european cornborer pyrausta nubilalis and beall s 1940 data on the beet webworm loxostege sticticalis are fitted well not only by the negative binomial but also by the neyman type a and other distributions in such circumstances the fits given by different distributions are often very similar and the exact choice is not critical see kemp and kemp 1965 the recent work of tripathi gurland and bhalerao 1986 gives further guidance on the choice of an appropriate distribution when this is not clearly indicated by the ecological context the negative binomial neyman type a and polya aeppli distributions all have two parameters when there are many samples of data in one data set anscombe 1949 and bliss and owen 1958 suggested fitting a model that is parsimonious in parameters by keeping one of them constant from sample to sample their use of a common exponent parameter k when fitting families of negative binomials was criticised by taylor woiwod and perry 1979 who considered that k is an unstable parameter whose relationship with aggregation is doubtful much empirical evidence has accumulated in support of taylor s power law

multi cut solutions of laplacian growth a new class of solutions to laplacian growth lg with zero surface tension is presented and shown to contain all other known solutions as special or limiting cases these solutions which are time dependent conformal maps with branch cuts inside the unit circle are governed by a nonlinear integral equation and describe oil fjords with non parallel walls in viscous fingering experiments in hele shaw cells integrals of motion for the multi cut lg solutions in terms of singularities of the schwarz function are found and the dynamics of densities jumps on the cuts are derived the subclass of these solutions with linear cauchy densities on the cuts of the schwarz function is of particular interest because in this case the integral equation for the conformal map becomes linear these solutions can also be of physical importance by representing oil air interfaces which form oil fjords with a constant opening angle in accordance with recent experiments in a hele shaw cell

constrained reductions of two dimensional dispersionless toda hierarchy hamiltonian structure and interface dynamics finite dimensional reductions of the two dimensional dispersionless toda hierarchy constrained by the string equation are studied these include solutions determined by polynomial rational or logarithmic functions which are of interest in relation to the laplacian growth or hele shaw problem governing interface dynamics the consistency of such reductions is proved and the hamiltonian structure of the reduced dynamics is derived the poisson structure of the rationally reduced dispersionless toda hierarchies is also derived

integrable structure of the dirichlet boundary problem in multiply connected domains we study the integrable structure of the dirichlet boundary problem in two dimensions and extend the approach to the case of planar multiply connected domains the solution to the dirichlet boundary problem in the multiply connected case is given through a quasiclassical tau function which generalizes the tau function of the dispersionless toda hierarchy it is shown to obey an infinite hierarchy of hirota like equations which directly follow from properties of the dirichlet green function and from the fay identities the relation to multi support solutions of matrix models is briefly discussed

correction to scaling analysis of diffusion limited aggregation diffusion limited aggregation is consistent with simple scaling however strong subdominant terms are present and these can account for various earlier claims of anomalous scaling we show this in detail for the case of multiscaling

laplacian networks growth local symmetry and shape optimization inspired by river networks and other structures formed by laplacian growth we use the loewner equation to investigate the growth of a network of thin fingers in a diffusion field we first review previous contributions to illustrate how this formalism reduces the network s expansion to three rules which respectively govern the velocity the direction and the nucleation of its growing branches this framework allows us to establish the mathematical equivalence between three formulations of the direction rule namely geodesic growth growth that maintains local symmetry and growth that maximizes flux into tips for a given amount of growth surprisingly we find that this growth rule may result in a network different from the static configuration that optimizes flux into tips

laplacian growth and whitham equations of soliton theory the laplacian growth the hele shaw problem of multiply connected domains in the case of zero surface tension is proven to be equivalent to an integrable system of whitham equations known in soliton theory the whitham equations describe slowly modulated periodic solutions of integrable hierarchies of nonlinear differential equations through this connection the laplacian growth is understood as a flow in the moduli space of riemann surfaces

complex curve of the two matrix model and its tau function we study the hermitian and normal two matrix models in planar approximation for an arbitrary number of eigenvalue supports its planar graph interpretation is given the study reveals a general structure of the underlying analytic complex curve different from the hyperelliptic curve of the one matrix model the matrix model quantities are expressed through the periods of meromorphic generating differential on this curve and the partition function of the multiple support solution as a function of filling numbers and coefficients of the matrix potential is shown to be a quasiclassical tau function the relation to 1 supersymmetric yang mills theories is discussed a general class of solvable multi matrix models with tree like interactions is considered

theory of stochastic laplacian growth we generalize the diffusion limited aggregation by issuing many randomly walking particles which stick to a cluster at the discrete time unit providing its growth using simple combinatorial arguments we determine probabilities of different growth scenarios and prove that the most probable evolution is governed by the deterministic laplacian growth equation a potential theoretical analysis of the growth probabilities reveals connections with the tau function of the integrable dispersionless limit of the two dimensional toda hierarchy normal matrix ensembles and the two dimensional dyson gas confined in a non uniform magnetic field we introduce the time dependent hamiltonian which generates transitions between different classes of equivalence of closed curves and prove the hamiltonian structure of the interface dynamics finally we propose a relation between probabilities of growth scenarios and the semi classical limit of certain correlation functions of light exponential operators in the liouville conformal field theory on a pseudosphere