multiple imputation and posterior simulation for multivariate missing data in longitudinal studies summary this paper outlines a multiple imputation method for handling missing data in designed longitudinal studies a random coefficients model is developed to accommodate incomplete multivariate continuous longitudinal data multivariate repeated measures are jointly modeled specifically an i i d normal model is assumed for time independent variables and a hierarchical random coefficients model is assumed for time dependent variables in a regression model conditional on the time independent variables and time with heterogeneous error variances across variables and time points gibbs sampling is used to draw model parameters and for imputations of missing observations an application to data from a study of startle reactions illustrates the model a simulation study compares the multiple imputation procedure to the weighting approach of robins rotnitzky and zhao 1995 journal of the american statistical association90 106 121 that can be used to address similar data structures

on a class of three dimensional integrable lagrangians we characterize non degenerate lagrangians of the form such that the corresponding euler lagrange equations are integrable by the method of hydrodynamic reductions the integrability conditions constitute an over determined system of fourth order pdes for the lagrangian density f which is in involution and possesses interesting differential geometric properties the moduli space of integrable lagrangians factorized by the action of a natural equivalence group is three dimensional familiar examples include the dispersionless kadomtsev petviashvili dkp and the boyer finley lagrangians f u x 3 3 u y 2 u x u t and respectively a complete description of integrable cubic and quartic lagrangians is obtained up to the equivalence transformations the list of integrable cubic lagrangians reduces to three examples there exists a unique integrable quartic lagrangian we conjecture that these examples exhaust the list of integrable polynomial lagrangians which are essentially three dimensional it was verified that there exist no polynomial integrable lagrangians of degree five we prove that the euler lagrange equations are integrable by the method of hydrodynamic reductions if and only if they possess a scalar pseudopotential playing the role of a dispersionless lax pair

explicit combinatorial interpretation of kerov character polynomials as numbers of permutation factorizations abstract we find an explicit combinatorial interpretation of the coefficients of kerov character polynomials which express the value of normalized irreducible characters of the symmetric groups s n in terms of free cumulants r 2 r 3 of the corresponding young diagram our interpretation is based on counting certain factorizations of a given permutation

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

growth processes related to the dispersionless lax equations this paper is a short review of the connection between certain types of growth processes and the integrable systems theory written from the viewpoint of the latter starting from the dispersionless lax equations for the 2d toda hierarchy we interpret them as evolution equations for conformal maps in the plane this provides a unified approach to evolution of smooth domains such as laplacian growth and growth of slits we show that the lowner differential equation for a parametric family of conformal maps of slit domains arises as a consistency condition for reductions of the dispersionless toda hierarchy it is also demonstrated how the both types of growth processes can be simulated by the large n limit of the dyson gas picture for the model of normal random matrices

coisotropic deformations of associative algebras and dispersionless integrable hierarchies the paper is an inquiry of the algebraic foundations of the theory of dispersionless integrable hierarchies like the dispersionless kp and modified kp hierarchies and the universal whitham hierarchy of genus zero it stands out for the idea of interpreting these hierarchies as equations of coisotropic deformations for the structure constants of certain associative algebras it discusses the link between the structure constants and hirota s tau function and shows that the dispersionless hirota bilinear equations are within this approach a way of writing the associativity conditions for the structure constants in terms of the tau function it also suggests a simple interpretation of the algebro geometric construction of the universal whitham equations of genus zero due to krichever

a note about models for selectivity bias on decrit une representation tabulaire simplifiee qui fait ressortir une hypothese structurale implicite importante dans des modeles de selection bases sur la censure

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

constructing multiply connected quadrature domains multiply connected bounded quadrature domains with finite connectivity are re constructed from their quadrature data using conformal mappings that are ratios of products of schottky klein prime functions this method provides the natural generalization of the conformal maps to simply and doubly connected quadrature domains constructed by the first author in a num ber of physical applications the efficacy of the method is demonstrated by the explicit construction of a range of examples as well as by comparison with alternative constructive methods recently in troduced by crowdy r soc lond proc ser a math phys eng sci 457 2001 pp 2337 2359 and richardson european j appl math 12 2001 pp 571 599

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

on a class of geometry driven free boundary problems an abstract class of two dimensional geometry driven free boundary problems is considered in which the evolution of the cauchy transform of the time evolving domain satisfies a partial differential equation of conservation type this idea generalizes an association between free boundary problems and the inverse gravitational problem it is shown that this class of free boundary problems admits exact solutions expressible in terms of a finite set of time evolving parameters in addition even when exact solutions are not available a theorem is established which shows that solutions corresponding to certain initial conditions possess nontrivial conserved quantities two distinct problems of physical interest are shown to fall within this abstract class the evolution of a fluid blob in a rotating hele shaw cell and the evolution of highly viscous fluid under the effects of surface tension various aspects of these two problems are discussed

hydrodynamic reductions of the dispersionless harry dym hierarchy we investigate the reductions of the dispersionless harry dym hierarchy to systems of finitely many partial differential equations these equations must satisfy the compatibility condition and they are diagonalizable and semi hamiltonian by imposing a further constraint the compatibility is reduced to a system of algebraic equations whose solutions are described

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

characters of symmetric groups sharp bounds and applications we provide new estimates on character values of symmetric groups which hold for all characters and which are in some sense best possible it follows from our general bound that if a permutation s n has at most n o 1 cycles of length m then 1 1 m o 1 for all irreducible characters of s n this is a far reaching generalization of a result of fomin and lulov

a dynamical mother body in a hele shaw problem it is well known that in the framework of the internal hele shaw problem the fluid cannot be fully removed through a single point sink because of a cusp formation occurring prior to the moving boundary reaching the sink unless the initial domain is a circle with a sink located in its center we give a definition of a dynamical mother body and use it for developing an algorithm of complete removal of a fluid droplet having an algebraic boundary to illustrate our theory we consider examples where fluid can be completely removed through the sinks distributed along arcs of curves and or points

upper bound on the characters of the symmetric groups let c be a conjugacy class in the symmetric group s n and be a partition of n let f be the degree of the irreducible representation s c the character of s at c and r c the normalized character c f we prove that there exist constants b 0 and 1 q 0 such that for n 4 for every conjugacy class c in s n and every irreducible representation s of s n r c max q 1 n 1 n b supp c where supp c is the number of non fixed digits under the action of a permutation in c 1 is the size of the largest part in and 1 is the number of parts in the proof is obtained by enumeration of rim hook tableaux the hook formula and probabilistic arguments combinatorial algebraic and statistical applications follow this result in particular we estimate the rate of mixing of random walks on the alternating groups with respect to conjugacy classes

a classification of integrable quasiclassical deformations of algebraic curves a previously introduced scheme for describing integrable deformations of algebraic curves is completed lenard relations are used to characterize and classify these deformations in terms of hydrodynamic type systems a general solution of the compatibility conditions for consistent deformations is given and expressions for the solutions of the corresponding lenard relations are provided

positivity results for stanley s character polynomials in stanley r p stanley irreducible symmetric group characters of rectangular shape sem lothar combin 50 b50d 2003 11 pp the author introduces expressions for the normalized characters of the symmetric group and states some positivity conjectures for these expressions here we give an affirmative partial answer to stanley s positivity conjectures about the expressions using results on kerov polynomials in particular we use new positivity results in goulden and rattan i p goulden a rattan an explicit form for kerov s character polynomials trans amer math soc in press math co 0505317 november 2005 we shall see that the generating series c t introduced in i p goulden a rattan an explicit form for kerov s character polynomials trans amer math soc in press math co 0505317 november 2005 is critical to our discussion

detection of influential observation in linear regression a new measure based on confidence ellipsoids is developed for judging the contribution of each data point to the determination of the least squares estimate of the parameter vector in full rank linear regression models it is shown that the measure combines information from the studentized residuals and the variances of the residuals and predicted values two examples are presented

strong asymptotics of the orthogonal polynomials with respect to a measure supported on the plane we consider the orthogonal polynomials pn z with respect to the measure z a 2nce n z 2da z over the whole complex plane we obtain the strong asymptotic of the orthogonal polynomials in the complex plane and the location of their zeros in a scaling limit where n grows to infinity with n the asymptotics are described in terms of three probability measures associated with the problem the first measure is the limit of the counting measure of zeros of the polynomials which is captured by the g function much in the spirit of ordinary orthogonal polynomials on the real line the second measure is the equilibrium measure that minimizes a certain logarithmic potential energy supported on a region k of the complex plane the third measure is the harmonic measure of kc with a pole at this appears as the limit of the probability measure given up to the normalization constant by the squared modulus of the nth orthogonal polynomial times the orthogonality measure i e pn z 2 z a 2nce n z 2da z the compact region k that is the support of the second measure undergoes a topological transition under the variation of the parameter t n n in a double scaling limit near the critical point given by tc a a 2c we observe the hastings mcleod solution to painleve ii in the asymptotics of the orthogonal polynomials 2014 wiley periodicals inc

gaussian fluctuations of characters of symmetric groups and of young diagrams we study asymptotics of reducible representations of the symmetric groups s q for large q we decompose such a representation as a sum of irreducible components or alternatively young diagrams and we ask what is the character of a randomly chosen component or what is the shape of a randomly chosen young diagram our main result is that for a large class of representations the fluctuations of characters and fluctuations of the shape of the young diagrams are asymptotically gaussian in this way we generalize kerov s central limit theorem the considered class consists of representations for which the characters almost factorize and this class includes for example the left regular representation plancherel measure irreducible representations and tensor representations this class is also closed under induction restriction outer product and tensor product of representations our main tool in the proof is the method of genus expansion well known from the random matrix theory

semiclassical expansions in the toda hierarchy and the hermitian matrix model an iterative algorithm for determining a class of solutions of the dispersionful 2 toda hierarchy characterized by string equations is developed this class includes the solution which underlies the large n limit of the hermitian matrix model in the one cut case it is also shown how the double scaling limit can be naturally formulated in this scheme

multipolar vortices and algebraic curves this paper demonstrates that analytical solutions to the steady two dimensional euler equations possessing all the qualitative properties of multipolar vortices observed experimentally and numerically can be constructed using the theory of algebraic curves and quadrature domains the solutions consist of a finite set of line vortices superposed on finite area patches of uniform vorticity by way of example new solutions are presented in which the support of the vorticity is a quintuply connected vortex patch with five vorticity maxima modelling a symmetric pentapolar vortex consisting of a central core vortex four satellite vortices and four enclosed zones of irrotational fluid between the core and the satellite vortices the method of construction is amenable to the derivation of exact solutions of the two dimensional euler equations having vorticity distributions of even greater geometrical complexity

the double scaling limit method in the toda hierarchy critical points of semiclassical expansions of solutions to the dispersionful toda hierarchy are considered and a double scaling limit method of regularization is formulated the analogues of the critical points characterized by the strong conditions in the hermitian matrix model are analysed and the property of doubling of equations is proved a wide family of sets of critical points is introduced and the corresponding double scaling limit expansions are discussed

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

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

an index of local sensitivity to nonignorable drop out in longitudinal modelling in longitudinal studies with potentially nonignorable drop out one can assess the likely effect of the nonignorability in a sensitivity analysis troxel et al proposed a general index of sensitivity to nonignorability or isni to measure sensitivity of key inferences in a neighbourhood of the ignorable missing at random mar model they derived detailed formulas for isni in the special case of the generalized linear model with a potentially missing univariate outcome in this paper we extend the method to longitudinal modelling we use a multivariate normal model for the outcomes and a regression model for the drop out process allowing missingness probabilities to depend on an unobserved response the computation is straightforward and merely involves estimating a mixed effects model and a selection model for the drop out together with some simple arithmetic calculations we illustrate the method with three examples

string equations in whitham hierarchies τ functions and virasoro constraints a scheme for solving whitham hierarchies satisfying a special class of string equations is presented the function of the corresponding solutions is obtained and the differential expressions of the underlying virasoro constraints are characterized illustrative examples of exact solutions of whitham hierarchies are derived and applications to conformal maps dynamics are indicated

representations of symmetric groups and free probability abstract we consider representations of symmetric groupssqfor largeq we give the asymptotic behaviour of the characters when the corresponding young diagrams rescaled by a factorq 1 2 converge to some prescribed shape this behaviour can be expressed in terms of the free cumulants for a probability measure associated with the limit shape of the diagram we also show that the basic operations of representation theory like taking tensor products restriction or induction have a limiting behavior which can be described using the free probability theory of d voiculescu

integrable quasiclassical deformations of algebraic curves a general scheme for determining and studying integrable deformations of algebraic curves based on the use of lenard relations is presented the method is illustrated with the analysis of the hyperelliptic case an associated multi hamiltonian hierarchy of systems of hydrodynamic type is characterized

models for nonresponse in sample surveys abstract the literature on the analysis of incomplete data using models is reviewed in the context of nonresponse in sample surveys the modeling approach provides a large body of methods for handling unit and item nonresponse some of which cannot be derived from the randomization theory of inference for surveys key concepts from the literature on incomplete data such as factorizations of the likelihood for special data patterns the em algorithm for general data patterns and ignorability of the response mechanism are discussed within the survey context model based procedures are related to common methods for handling nonresponse in surveys such as weighting or imputation of means within subclasses of the population

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

the cutoff phenomenon in finite markov chains abstract natural mixing processes modeled by markov chains often show a sharp cutoff in their convergence to long time behavior this paper presents problems where the cutoff can be proved card shuffling the ehrenfests urn it shows that chains with polynomial growth drunkard s walk do not show cutoffs the best general understanding of such cutoffs high multiplicity of second eigenvalues due to symmetry is explored examples are given where the symmetry is broken but the cutoff phenomenon persists

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

algebraic varieties in the birkhoff strata of the grassmannian gr 2 harrison cohomology and integrable systems the local properties of the families of algebraic subsets wg in the birkhoff strata 2g of gr 2 containing the hyperelliptic curves of genus g are studied it is shown that the tangent spaces tg for wg are isomorphic to the linear spaces of 2 coboundaries particular subsets in wg are described by the integrable dispersionless coupled kdv systems of hydrodynamical type defining a special class of 2 cocycles and 2 coboundaries in tg it is demonstrated that the blows ups of such 2 cocycles and 2 coboundaries and gradient catastrophes for associated integrable systems are interrelated

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

hodograph solutions of the dispersionless coupled kdv hierarchies critical points and the euler poisson darboux equation it is shown that the hodograph solutions of the dispersionless coupled kdv dckdv hierarchies describe critical and degenerate critical points of a scalar function which obeys the euler poisson darboux equation singular sectors of each dckdv hierarchy are found to be described by solutions of higher genus dckdv hierarchies concrete solutions exhibiting shock type singularities are presented

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

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

zonal polynomials via stanleyʼs coordinates and free cumulants abstract we study zonal characters which are defined as suitably normalized coefficients in the expansion of zonal polynomials in terms of power sum symmetric functions we show that the zonal characters just like the characters of the symmetric groups admit a nice combinatorial description in terms of stanley s multirectangular coordinates of young diagrams we also study the analogue of kerov polynomials namely we express the zonal characters as polynomials in free cumulants and we give an explicit combinatorial interpretation of their coefficients in this way we prove two recent conjectures of lassalle for jack polynomials in the special case of zonal polynomials

laplacian growth in a channel and hurwitz numbers we study the integrable structure of the 2d laplacian growth problem with zero surface tension in an infinite channel with periodic boundary conditions in a transverse direction similarly to the laplacian growth in radial geometry this problem can be embedded into the 2d toda lattice hierarchy in the zero dispersion limit however the relevant solution to the hierarchy is different we characterize this solution by the string equations and construct the corresponding dispersionless tau function this tau function is shown to coincide with the genus zero part of the generating function for double hurwitz numbers

harmonic moment dynamics in laplacian growth harmonic moments are integrals of integer powers of z x iy over a domain here the domain is an exterior of a bubble of air growing in an oil layer between two horizontal closely spaced plates harmonic moments are a natural basis for such laplacian growth phenomena because unlike other representations these moments linearize the zero surface tension problem s richardson j fluid mech 56 609 1972 so that all moments except the lowest one the area of the bubble are conserved in time in our experiments we directly determine the harmonic moments and show that for nonzero surface tension all moments except the lowest one decay in time rather than exhibiting the divergences of other representations further we derive an expression that relates the derivative of the k th harmonic moment m k to measurable quantities surface tension viscosity the distance between the plates and a line integral over the contour encompassing the growing bubble the laboratory observations are in good accord with the expression we derive for d m k dt which is proportional to the surface tension thus in the zero surface tension limit the moments above k 0 are all conserved in accord with richardson s theory in addition from the measurements of the time evolution of the harmonic moments we obtain a value for the surface tension that is within 20 of the accepted value in conclusion our analysis and laboratory observations demonstrate that an interface dynamics description in terms of harmonic moments is physically realizable and robust

asymptotics of characters of symmetric groups related to stanley character formula we prove an upper bound for characters of the symmetric groups in particular we show that there exists a constant a 0 with a property that for every young diagram with n boxes r rows and c columns tr tr e i amax a r n c n j j n aoj j where j j is the minimal number of factors needed to write 2 sn as a product of transpositions we also give uniform estimates for the error term in the vershik kerov s and biane s character formulas and give a new formula for free cumulants of the transition measure

coisotropic deformations of algebraic varieties and integrable systems coisotropic deformations of algebraic varieties are defined as those for which an ideal of the deformed variety is a poisson ideal it is shown that coisotropic deformations of sets of intersection points of plane quadrics cubics and space algebraic curves are governed in particular by the dkp wdvv dvn d2dtl equations and other integrable hydrodynamical type systems particular attention is paid to the study of two and three dimensional deformations of elliptic curves the problem of an appropriate choice of the poisson structure is discussed

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

the haantjes tensor and double waves for multi dimensional systems of hydrodynamic type a necessary condition for integrability an invariant differential geometric approach to the integrability of 2 1 dimensional systems of hydrodynamic type is developed we prove that the existence of special solutions known as double waves is equivalent to the diagonalizability of an arbitrary matrix of the two parameter familysince the diagonalizability can be effectively verified by calculating the haantjes tensor this provides a simple necessary condition for integrability

geometrical optics in nonlinear media and integrable equations it is shown that the geometrical optics limit of the maxwell equations for certain nonlinear media with slow variation along one axis and particular dependence of the dielectric constant on frequency and fields gives rise to the dispersionless veselov novikov equation for the refractive index it is demonstrated that the dispersionless veselov novikov hierarchy is amenable to the quasiclassical dressing method a connection is noted between the geometrical optics phenomena under consideration and the quasiconformal mappings on the plane

ullemar s formula for the moment map ii we prove the complex analogue of ullemar s formula for the jacobian of the complex moment mapping this formula was previously established in the real case

alternative methods for cps income imputation abstract the u s bureau of the census imputes missing income items in the income supplement of the current population survey cps by a technique commonly known as the cps hot deck this article compares cps hot deck imputations of wages and salary amounts with alternatives based on regression models for the logarithm of wages and salary and for the wage rate comparisons are effected by comparing imputations with an internal revenue service irs wages and salary amount found by an exact match of cps data to irs records although limitations in the matching and in the comparison variable preclude a definitive conclusion we find that a the cps hot deck does not underestimate income aggregates to any serious extent b model based alternatives have slightly smaller mean absolute error than the hot deck when comparable data bases of respondents are used to carry out imputations and c multivariate models for imputing recipiency weeks and hours worked and earnings need to be developed to provide re

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

integrable quasiclassical deformations of cubic curves a general scheme for determining and studying hydrodynamic type systems describing integrable deformations of algebraic curves is applied to cubic curves lagrange resolvents of the theory of cubic equations are used to derive and characterize these deformations

an explicit form for kerov s character polynomials kerov considered the normalized characters of irreducible representations of the symmetric group evaluated on a cycle as a polynomial in free cumulants biane has proved that this polynomial has integer coefficients and made various conjectures recently sniady has proved biane s conjectured explicit form for the first family of nontrivial terms in this polynomial in this paper we give an explicit expression for all terms in kerov s character polynomials our method is through lagrange inversion

on the whitham hierarchy dressing scheme string equations and additional symmetries a new description of the universal whitham hierarchy in terms of a factorization problem in the lie group of canonical transformations is provided this scheme allows us to give a natural description of dressing transformations string equations and additional symmetries for the whitham hierarchy we show how to dress any given solution and prove that any solution of the hierarchy may be undressed and therefore comes from a factorization of a canonical transformation a particularly important function related to the tau function appears as a potential of the hierarchy we introduce a class of string equations which extends and contains previous classes of string equations considered by krichever and by takasaki and takebe the scheme is also applied for a convenient derivation of additional symmetries moreover new functional symmetries of the zakharov extension of the bentley gas equations are given and the action of additional symmetries over the potential in terms of linear pdes is characterized

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

imputation of missing values when the probability of response depends on the variable being imputed abstract a method is developed for imputing missing values when the probability of response depends upon the variable being imputed the missing data problem is viewed as one of parameter estimation in a regression model with stochastic censoring of the dependent variable the prediction approach to imputation is used to solve this estimation problem wages and salaries are imputed to non respondents in the current population survey and the results are compared to the nonrespondents irs wage and salary data the stochastic censoring approach gives improved results relative to a prediction approach that ignores the response mechanism

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

bubble break off in hele shaw flows singularities and integrable structures abstract bubbles of inviscid fluid surrounded by a viscous fluid in a hele shaw cell can merge and break off during the process of break off a thinning neck pinches off to a universal self similar singularity we describe this process and reveal its integrable structure it is a solution of the dispersionless limit of the akns hierarchy the singular break off patterns are universal not sensitive to details of the process and can be seen experimentally we briefly discuss the dispersive regularization of the hele shaw problem and the emergence of the painleve ii equation at the break off

upper bound on the characters of the symmetric groups for balanced young diagrams and a generalized frobenius formula we study asymptotics of an irreducible representation of the symmetric group sn corresponding to a balanced young diagram a young diagram with at most rows and columns for some fixed constant c in the limit as n tends to infinity we show that there exists a constant d which depends only on c with a property that where p denotes the length of a permutation the minimal number of factors necessary to write p as a product of transpositions our main tool is an analogue of the frobenius character formula which holds true not only for cycles but for arbitrary permutations

self similarity in laplacian growth abstract we consider laplacian growth of self similar domains in different geometries self similarity determines the analytic structure of the schwarz function of the moving boundary the knowledge of this analytic structure allows us to derive the integral equation for the conformal map it is shown that solutions to the integral equation obey also a second order differential equation which is the 1d schroedinger equation with the sinh 2 potential the solutions which are expressed through the gauss hypergeometric function characterize the geometry of self similar patterns in a wedge we also find the potential for the coulomb gas representation of the self similar laplacian growth in a wedge and calculate the corresponding free energy

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