astala s conjecture on distortion of hausdorff measures under quasiconformal maps in the plane let e subset mathbb c be a compact set g mathbb c to mathbb c be a k quasiconformal map and let 0 t 2 let mathcal h t denote t dimensional hausdorff measure then mathcal h t e 0 quad rightarrow quad mathcal h t left ge right 0 quad t frac 2kt 2 left k 1 right t

equivalence of q bosons using the exponential phase operator various forms of the q boson are explained and their hidden symmetry revealed by transformations using the exponential phase operator both the one component and the multicomponent q bosons are discussed as a byproduct we obtain a new boson algebra having a shifted vacuum structure and define a global operator u 1 gauge transformation

improved bounds for the overall properties of a nonlinear composite dielectric the construction of generalized hashin shtrikman bounds for nonlinear composite problems relies on the introduction of a comparison material in recent work a nonlinear comparison medium has been used however this requires detailed knowledge of the properties of the trial fields that are employed the fields used have the property of bounded mean oscillation and this enables the size of the penalty exacted by using a nonlinear comparison material to be controlled some recent results concerning riesz transforms and the beurling operator are used here to reduce the effect of the penalty and hence to generate improved bounds in addition an exact solution is established for a particular class of composites

on astala s theorem for martingales and fourier multipliers abstract we exhibit a large class of symbols m on r d d 2 for which the corresponding fourier multipliers t m satisfy the following inequality if d e are measurable subsets of r d with e d and d then d e t m e x d x e e ln d 2 e if e d 2 d e 1 2 d e ln e d e if e d 2 here denotes the lebesgue measure on i r d when d 2 these multipliers include the real and imaginary parts of the beurling ahlfors operator b and hence the inequality is also valid for b with the right hand side multiplied by 2 the inequality is sharp for the real and imaginary parts of b this work is motivated by k astala s celebrated results on the gehring reich conjecture concerning the distortion of area by quasiconformal maps the proof rests on probabilistic methods and exploits a family of appropriate novel sharp inequalities for differentially subordinate martingales these martingale bounds are of interest on their own right

correlated binary regression using a quadratic exponential model summary a model of quadratic exponential form is parameterized in terms of marginal means and pairwise correlations for the regression analysis of correlated binary data pseudo maximum likelihood methods using a special case termed the multiplicative model are proposed but are noted to be computationally unattractive if the blocks of correlated responses are at all large on the other hand score estimating functions for mean and correlation parameters are shown to be of a particularly simple form under the quadratic exponential family special cases of such estimating functions having attractive computational properties are identified and illustrated

quasiconformal distortion of riesz capacities and hausdorff measures in the plane in this paper we prove the sharp distortion estimates for the quasi conformal mappings in the plane both in terms of the riesz capacities from non linear potential theory and in terms of the hausdorff measures

pucci s conjecture and the alexandrov inequality for elliptic pdes in the plane the inequality of alexandrov bakel man and pucci is a basic tool in the theory of linear elliptic partial differential equations pdes which are not in divergence form as well as in the more general theory of nonlinear elliptic pdes here in two dimensions we prove the sharp form of the maximum principle as conjectured by pucci in 1966 give sharp forms of removable singularity results and prove a number of results for the degenerate elliptic setting these results make use of the substantial recent advances in the planar theory of quasiconformal mappings astala was supported in part by the academy of finland projects 34082 and 41933 iwaniec by the national science foundation grants dms 0301582 and dms 0244297 martin by the n z marsden fund and the n z royal society james cook fellowship ams 1991 classification primary 30c60 35j15 35j70

burkholder integrals morrey s problem and quasiconformal mappings inspired by morrey s problem on rank one convex functionals and the burkholder integrals of his martingale theory we find that the burkholder functionals bp p 2 are quasiconcave when tested on deformations of identity f id c with bp df x 0 pointwise or equivalently deformations such that df 6 p p 2jf in particular this holds in explicit neighbourhoods of the identity map among the many immediate consequences this gives the strongest possible l estimates for the gradient of a principal solution to the beltrami equation fz z fz for any p in the critical interval 2 6 p 6 1 1 f examples of local maxima lacking symmetry manifest the intricate nature of the problem

quasiregular mappings in even dimensions o introduction 1 notation 2 some exterior algebra 3 differential forms in lnm i2 a n 4 differential systems for quasiregular mappings 5 liouville theorem in even dimensions 6 hodge theory in lp r n 7 the beltrami equation in even dimensions 8 the beurling ahlfors operator 9 regularity theorems for quasiregular mappings 10 the caccioppoli type estimate 11 removability theorems for quasiregular mappings 12 some examples appendix the 4 dimensional case

integral estimates of conformal derivatives and spectral properties of the neumann laplacian abstract in this paper we study integral estimates of derivatives of conformal mappings d of the unit disc d c onto bounded domains that satisfy the ahlfors condition these integral estimates lead to estimates of constants in sobolev poincare inequalities and by the rayleigh quotient we obtain spectral estimates of the neumann laplace operator in non lipschitz domains quasidiscs in terms of the quasi conformal geometry of the domains specifically the lower estimates of the first non trivial eigenvalues of the neumann laplace operator in some fractal type domains snowflakes were obtained

sobolev regular flows of non lipschitz vector fields abstract we show that vector fields with exponentially integrable derivatives admit a well defined flow of homeomorphisms x t w 1 p t l o c for some p t 1 at least for small times when the field is certain riesz potential of a bounded function the result becomes global in time due to techniques from geometric function theory the local result also applies to the flows arising from yudovich solutions to the planar euler system with bounded vorticity

distributional solutions of the beltrami equation abstract we study the distributional solutions to the generalized beltrami equation under sobolev assumptions on the beltrami coefficients in this setting we prove that these distributional solutions are true quasiregular maps and they are smoother than expected that is they have second order derivatives in l l o c 1 e for some e 0

the geometry of the weil petersson metric in complex dynamics in this work we study an analogue of the weil petersson metric on the space of blaschke products of degree 2 proposed by mcmullen we show that the weilpetersson metric is incomplete and study its metric completion our work parallels known results for the teichmuller space of a punctured torus

the quantum group suq 2 and a q analogue of the boson operators a new realisation of the quantum group suq 2 is constructed by means of a q analogue to the jordan schwinger mapping determining thereby both the complete representation structure and q analogues to the wigner and racah operators to achieve this realisation a new elementary object is defined a q analogue to the harmonic oscillator the uncertainty relation for position and momentum in a q harmonic oscillator is quite unusual

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

l p bounds for the beurling ahlfors transform let b denote the beurling ahlfors transform defined on l p c 1 p 00 the celebrated conjecture of t iwaniec states that its l p norm b p p 1 1 where p max p p p 1 in this paper the new upper estimate b p 1 575 p 1 1 p is found

the spectral estimates for the neumann laplace operator in space domains abstract in this paper we prove discreteness of the spectrum of the neumann laplacian the free membrane problem in a large class of non convex space domains the lower estimates of the first non trivial neumann eigenvalue are obtained in terms of geometric characteristics of sobolev mappings the suggested approach is based on sobolev poincare inequalities that are obtained with the help of a geometric theory of composition operators on sobolev spaces these composition operators are induced by generalizations of conformal mappings that are called as mappings of bounded 2 distortion weak 2 quasiconformal mappings

quasiconformal planes with bi lipschitz pieces and extensions of almost affine maps abstract a quasiplane f v is the image of an n dimensional euclidean subspace v of r n 1 n n 1 under a quasiconformal map f r n r n we give sufficient conditions in terms of the weak quasisymmetry constant of the underlying map for a quasiplane to be a bi lipschitz n manifold and for a quasiplane to have big pieces of bi lipschitz images of r n one main novelty of these results is that we analyze quasiplanes in arbitrary codimension n n to establish the big pieces criterion we prove new extension theorems for almost affine maps which are of independent interest this work is related to investigations by tukia and vaisala on extensions of quasisymmetric maps with small distortion

exponential mixtures and quadratic exponential families summary conditions are derived under which quadratic and polynomial exponential models can be generated as mixtures of linear exponential models the conditions are highly restrictive for continuous sample spaces but less restrictive in the discrete case some properties of binary quadratic exponential models are explored with a view towards finding models that have properties suitable for epidemiological applications

self similarity of siegel disks and hausdorff dimension of julia sets let f z ez z where is an irrational number of bounded type according to siegel f is linearizable on a disk containing the origin in this paper we show the hausdorff dimension of the julia set j f is strictly less than two and if is a quadratic irrational such as the golden mean then the siegel disk for f is self similar about the critical point in the latter case we also show the rescaled first return maps converge exponentially fast to a system of commuting branched coverings of the complex plane

testing covariance structure in multivariate models application to family disease data abstract recent interest in modeling multivariate responses for members of groups has emphasized the need for testing goodness of fit here we describe a way to test the covariance structure of a multivariate distribution parameterized by a vector the idea is to extend this distribution the null distribution to a more general distribution that depends on an additional scalar and a specific quadratic function of the response vector chosen to capture features of an alternative covariance structure when 0 the more general distribution reduces to the null one standard likelihood theory yields a score test for 0 that is a test of fit of the null distribution the score statistic is the standardized difference between observed and expected values of the quadratic function where the expectation is taken with respect to the null distribution with replaced by its maximum likelihood estimate applying the methods to case control data on familial cancers of the ovary and breast we illu

a martingale study of the beurling ahlfors transform inrn abstract we apply the theory of martingale transforms to study the beurling ahlfors transform s in dimensions n 2 this operator reveals a rich structure through its representation as a martingale and we obtain new results concerning the operator norm of s acting on the class of differential forms having l p coefficients in particular we show that its norm is independent of the dimension when restricted to k forms and we present new essen type norm inequalities related to this martingale structure finally we suggest a purely analytic method to further investigate these norms which up to now has been lacking

deformed algebras from the inverse schwinger method we consider a problem which may be viewed as inverse to the schwinger realization of lie algebra and suggest a procedure for deforming the algebra so obtained we illustrate the method through a few simple examples extending schwinger s su 1 1 construction as results various q deformed algebras are re produced as well as their undeformed counterparts some extensions of the method are briefly pointed out

quasihyperbolic boundary conditions and poincare domains we prove that a domain in bbb r n whose quasihyperbolic metric satisfies a logarithmic growth condition with coefficient beta le 1 is a q p poincare domain for all p and q satisfying p in 1 infty cap n n beta n and q in p beta p where p np n p denotes the sobolev conjugate exponent an elementary example shows that the given ranges for p and q are sharp the proof makes use of estimates for a variational capacity when p 2 we give an application to the solvability of the neumann problem on domains with irregular boundaries we also discuss the relationship between this growth condition on the quasihyperbolic metric and the s john condition

dimension gap under conformal mappings abstract we give an estimate for the hausdorff gauge dimension of the boundary of a simply connected planar domain under p integrability of the hyperbolic metric p 1 this estimate does not degenerate when p tends to one for p 1 the boundary can even have positive area the same phenomenon is extended to general planar domains in terms of the quasihyperbolic metric we also give an example which shows that our estimates are essentially sharp

quasiconformal homeomorphisms and dynamics ii structural stability implies hyperbolicity for kleinian groups sous groupe de psl 2 c familles holomorphes d homeomorphismes de c lemme champs de ligne invariants familles holomorphes de groupes abstraitement isomorphes car finiment genere la stabilite structurelle implique geometriquement fini ou rigide theoreme a avec paraboles stabilite structurelle au sens de la dynamique differentiable

harmonic measure on locally flat domains we will review work with tatiana toro yielding a characterizati on of those domains for which the harmonic measure has a density whose logarithm has vanishing mean oscillation

hausdorff measure of quasicircles abstract s smirnov 2010 10 proved recently that the hausdorff dimension of any k quasicircle is at most 1 k 2 where k k 1 k 1 in this paper we show that if is such a quasicircle then h 1 k 2 b x r c k r 1 k 2 for all x c r 0 where h s stands for the s hausdorff measure on a related note we derive a sharp weak integrability of the derivative of the riemann map of a quasidisk

characterizing the manifest probability distributions of three latent trait models for accuracy and response time in this paper we study the statistical relations between three latent trait models for accuracies and response times the hierarchical model hm of van der linden psychometrika 72 3 287 308 2007 the signed residual time model sm proposed by maris and van der maas psychometrika 77 4 615 633 2012 and the drift diffusion model dm as proposed by tuerlinckx and de boeck psychometrika 70 4 629 650 2005 one important distinction between these models is that the hm and the dm either assume or imply that accuracies and response times are independent given the latent trait variables while the sm does not in this paper we investigate the impact of this conditional independence property or a lack thereof on the manifest probability distribution for accuracies and response times we will find that the manifest distributions of the latent trait models share several important features such as the dependency between accuracy and response time but we also find important differences such as in what function of response time is being modeled our method for characterizing the manifest probability distributions is related to the dutch identity holland in psychometrika 55 6 5 18 1990

on almost fuchsian manifolds almost fuchsian manifold is a class of complete hyperbolic threemanifolds a such three manifold is a quasi fuchsian manifold that contains a closed incompressible minimal surface with principal curvatures everywhere in the range of 1 1 in such a manifold the minimal surface is unique and embedded hence one can parametrize these hyperbolic three manifolds by their minimal surfaces in this paper we obtain estimates on several geometric and analytical quantities of an almost fuchsian manifold m in terms of the data on the minimal surface in particular we obtain an upper bound for the hyperbolic volume of the convex core of m and an upper bound on the hausdorff dimension of the limit set associated to m we also constructed a quasi fuchsian manifold which admits more than one minimal surface and it does not admit a foliation of closed surfaces of constant mean curvature

sign and area in nodal geometry of laplace eigenfunctions the paper deals with asymptotic nodal geometry for the laplace beltrami operator on closed surfaces given an eigenfunction f corresponding to a large eigenvalue we study local asymmetry of the distribution of sign f with respect to the surface area it is measured as follows take any disc centered at the nodal line f 0 and pick at random a point in this disc what is the probability that the function assumes a positive value at the chosen point we show that this quantity may decay logarithmically as the eigenvalue goes to infinity but never faster than that in other words only a mild local asymmetry may appear the proof combines methods due to donnelly fefferman and nadirashvili with a new result on harmonic functions in the unit disc

mappings of finite distortion the degree of regularity this paper investigates the self improving integrability properties of the so called mappings of finite distortion let k x 1 be a measurable function defined on a domain rn n 2 and such that exp k x lloc1 0 we show that there exist two universal constants c1 n c2 n with the following property let f be a mapping in wloc1 1 rn with df x n k x j x f for a e x and such that the jacobian determinant j x f is locally in l1log c1 n l then automatically j x f is locally in l1logc2 n l this result constitutes the appropriate analog for the self improving regularity of quasiregular mappings and clarifies many other interesting properties of mappings of finite distortion namely we obtain novel results on the size of removable singularities for bounded mappings of finite distortion and on the area distortion under this class of mappings

a likelihood based method for analysing longitudinal binary responses summary in this paper we discuss a likelihood based method for analysing correlated binary responses based on a multivariate model it is related to the pseudo maximum likelihood approach suggested recently by zhao prentice 1990 their parameterization results in a simple pairwise model in which the association between responses is modelled in terms of correlations while the present paper uses conditional log odds ratios with this approach higher order associations can be incorporated in a natural way one important advantage of this parameterization is that the maximum likelihood estimates of the marginal mean parameters are robust to misspecification of the time dependence we describe an iterative two stage procedure for obtaining the maximum likelihood estimates two examples are presented to illustrate this methodology

hardy spaces of the conjugate beltrami equation we study hardy spaces of solutions to the conjugate beltrami equation with lipschitz coefficient on dini smooth simply connected planar domains in the range of exponents 1 infty we analyse their boundary behaviour and certain density properties of their traces we derive on the way an analog of the fatou theorem for the dirichlet and neumann problems associated with the equation mbox div sigma nabla u 0 with l p boundary data

discrepancy densities for planar and hyperbolic zero packing we study the problem of geometric zero packing recently introduced by hedenmalm 7 there are two natural densities associated with this problem the discrepancy density h given by h liminfr 1

optimal lower bounds for eigenvalues of linear and nonlinear neumann problems in this paper we prove a sharp lower bound for the first non trivial neumann eigenvalue 1 for the p laplace operator p 1 in a lipschitz bounded domain in n our estimate does not require any convexity assumption on and it involves the best isoperimetric constant relative to in a suitable class of convex planar domains our bound turns out to be better than the one provided by the payne weinberger inequality

non stretch mappings for a sharp estimate of the beurling ahlfors operator abstract in this paper we identify certain classes of non stretch mappings that enjoy a sharp estimate of the beurling ahlfors operator we first make use of a property of subharmonic functions to prove that the banuelos wang conjecture and the iwaniec conjecture are true for a class of mappings that satisfy a quasilinear conjugate beltrami equation by utilizing the principal solutions of beltrami equations we further explicitly construct some classes of non stretch mappings for which the banuelos wang conjecture and the iwaniec conjecture are true

dimension of quasicircles we introduce canonical antisymmetric quasiconformal maps which minimize the quasiconformality constant among maps sending the unit circle to a given quasicircle as an application we prove astala s conjecture that the hausdorff dimension of a k quasicircle is at most 1 k 2

the dimension of the maximal measure for a polynomial map let f c c be a polynomial of degree d 2 the most interesting dynamics of f takes place on its julia set j f an invariant compact perfect set which can be defined for example as the closure of the repelling periodic points brolin constructed the equilibrium distribution m on j f an ergodic measure which is also the weak limit as n x of a probability measure distributed equally over the dn nth inverse images of almost any point we shall prove the following

a q heisenberg algebra and a contact metric structure the actions for two q heisenberg algebras are constructed by extending a contact metric structure of a heisenberg group manifold to the corresponding line bundle the author obtains the q heisenberg algebras under a canonical quantization and regularization also the q parameter is shown to be the regularizing parameter

area distortion of quasiconformal mappings in recent years quasiconformal mappings have been an efficient tool in the study of dynamical systems of the complex plane we show here that in turn methods or ideas from dynamical systems can be used to solve a number of open questions in the theory of planar quasiconformal mappings it has been known since the work of ahlfors a and mori mo that k quasiconformal mappings are locally h51der continuous with exponent 1 k the function

hausdorff dimension and conformal dynamics i strong convergence of kleinian groups this paper investigates the behavior of the hausdorff dimensions of the limit sets n and of a sequence of kleinian groups n where m h is geometrically finite we show if n strongly then a mn h 3 n is geometrically finite for all n 0 b n in the hausdorff topology and c h dim n h dim if h dim 1 on the other hand we give examples showing the dimension can vary discontinuously under strong limits when h dim 1 continuity can be recovered by requiring that accidental parabolics converge radially similar results hold for higher dimensional manifolds applications are given to quasifuchsian groups and their limits

mappings of bmo bounded distortion this paper can be viewed as a sequel to the work 9 where the theory of mappings of bmo bounded distortion is developed largely in even dimensions using singular integral operators and recent developments in the theory of higher integrability of jacobians in hardy orlicz spaces in this paper we continue this theme refining and extending some of our earlier work as well as obtaining results in new directions the planar case was studied earlier by g david 4 in particular he obtained existence theorems modulus of continuity estimates and bounds on area distortion for mappings of bmo distortion in fact in slightly more generality we obtain similar results in all even dimensions one of our main new results here is the extension of the classical theorem of painleve concerning removable singularties for bounded analytic functions to the class of mappings of bmo bounded distortion the setting of the plane is of particular interest and somewhat more can be said here because of the existence theorem or the measurable riemann mapping theorem which is not available in higher dimensions we give a construction to show our results are qualitatively optimal another surprising fact is that there are domains which support no bounded quasiregular mappings but admit

quasiconformal mappings as a tool to study certain two dimensional g closure problems a recent theorem due to astala establishes the best exponent for the area distortion of planar k quasiconformal mappings we use a refinement of astala s theorem due to eremenko and hamilton to prove new bounds on the effective conductivity of two dimensional composites the bounds are valid for composites made of an arbitrary finite number n of possibly anisotropic phases in prescribed volume fractions for n 2 we prove the optimality of the bounds under certain additional assumptions on the g closure parameters

beltrami operators in the plane we determine optimal lp properties for the solutions of the general nonlinear elliptic system in the plane of the form fz h z fz h l c where h is a measurable function satisfying h z w1 h z w2 k w1 w2 and k is a constant k 1 we will also establish the precise invertibility and spectral properties in lp c for the operators i t i t and t where t is the beurling transform these operators are basic in the theory of quasiconformal mappings and in linear and nonlinear elliptic partial differential equations in two dimensions in particular we prove invertibility in lp c whenever 1 p 1 1 we also prove related results with applications to the regularity of weakly quasiconformal mappings

crack initiation in brittle materials in this paper we study the crack initiation in a hyper elastic body governed by a griffith type energy we prove that during a load process through a time dependent boundary datum of the type t tg x and in the absence of strong singularities e g this is the case of homogeneous isotropic materials the crack initiation is brutal that is a big crack appears after a positive time ti 0 conversely in the presence of a point x of strong singularity a crack will depart from x at the initial time of loading and with zero velocity we prove these facts for admissible cracks belonging to the large class of closed one dimensional sets with a finite number of connected components the main tool we employ to address the problem is a local minimality result for the functional epsilon nu gamma int omega f x nabla v rm d x k mathcal h 1 gamma where omega subseteq mathbb r 2 k 0 and f is a suitable caratheodory function we prove that if the uncracked configuration u of relative to a boundary displacement has at most uniformly weak singularities then configurations u with mathcal h 1 gamma small enough are such that epsilon u emptyset epsilon u gamma gamma

bloch functions and asymptotic tail variance let p denote the bergman projection on the unit disk dd p z d w 1 zw 2da w z d where dada is normalized area measure we prove that if z 1 z 1 on dd then the integral i a r 02 exp ar4 p rei 2log 11 r2 d 2 0 r 1 has the bound i a r c a 10 1 a 3 2i a r c a 10 1 a 3 2 for 0 1a 1 no such uniform bound is possible we interpret the theorem in terms the asymptotic tail variance of such a bergman projection p p by the way the asymptotic tail variance induces a seminorm on the bloch space this improves upon earlier work of makarov which covers the range 0 a 264 0 1542 we then apply the theorem to obtain an estimate of the universal integral means spectrum for conformal mappings with a k quasiconformal extension for 0 k 10 k 1 the estimate reads for t ct c and 0 k 10 k 1 b k t 14k2 t 2 1 7k 2 for t 2k 1 7k 2 k t 1 1 7k 2 for t 2k 1 7k 2 which should be compared with the conjecture by prause and smirnov to the effect that for real t with t 2 k t 2 k we should have b k t 14k2t2

weighted sobolev spaces and embedding theorems in the present paper we study embedding operators for weighted sobolev spaces whose weights satisfy the well known muckenhoupt a p condition sufficient conditions for boundedness and compactness of the embedding operators are obtained for smooth domains and domains with boundary singularities the proposed method is based on the concept of generalized quasiconformal homeomorphisms homeomorphisms with bounded mean distortion the choice of the homeomorphism type depends on the choice of the corresponding weighted sobolev space such classes of homeomorphisms induce bounded composition operators for weighted sobolev spaces with the help of these homeomorphism classes the embedding problem for non smooth domains is reduced to the corresponding classical embedding problem for smooth domains examples of domains with anisotropic holder singularities demonstrate the sharpness of our machinery comparatively with known results

bloch functions asymptotic variance and geometric zero packing motivated by a problem in quasiconformal mapping we introduce a problem in complex analysis with its roots in the mathematical physics of the bose einstein condensates in superconductivity the p

harmonic measure and polynomial julia sets there is a natural conjecture that the universal bounds for the dimension spectrum of harmonic measure are the same for simply connected and for non simply connected domains in the plane because of the close relation to conformal mapping theory the simply connected case is much better understood and proving the above statement would give new results concerning the properties of harmonic measure in the general case we establish the conjecture in the category of domains bounded by polynomial julia sets the idea is to consider the coefficients of the dynamical zeta function as subharmonic functions on a slice of teichmuller s space of the polynomial and then to apply the maximum principle 1 dimension spectrum of harmonic measure in this paper we discuss some properties of harmonic measure in the complex plane for a domain and a point a let a denote the harmonic measure of evaluated at a the measure a can be defined for instance as the hitting distribution of a brownian motion started at a if e then a e is the probability that a random brownian path first hits the boundary at a point of e much work has been devoted to describing dimensional properties of when the domain is as general as possible in particular jones and wolff 7 proved that no matter what the domain is harmonic measure is concentrated on a borel set of hausdorff dimension at most one in other words dim 1 for all plane domains 1 1 we are interested in finding similar but stronger universal estimates involving the dimension spectrum of 1 1 universal spectrum for every positive we denote f dim z where z is the lower pointwise dimension of z lim inf 0 log b z log b z is a general notation for the disc with center z and radius the universal dimension spectrum is the function sup f 1 2 where the supremum is taken over harmonic measures of all planar domains the first author is supported by n s f grant dms 9970283 the second author is supported by n s f grant dms 9800714

quasiconformal mappings and sharp estimates for the distance to l in some function spaces abstract we provide several estimates which involve the distance to l in some function spaces the composition operator induced by a quasiconformal mapping and the logarithm of the jacobian of a quasiconformal mapping our results are sharp in the two dimensional case

snowballs are quasiballs we introduce snowballs which are compact sets in r 3 homeomorphic to the unit ball they are 3 dimensional analogs of domains in the plane bounded by snowflake curves for each snowball b a quasiconformal map f colon r 3 to r 3 is constructed that maps b to the unit ball

on the first eigenvalues of free vibrating membranes in conformal regular domains in 1961 g polya published a paper about the eigenvalues of vibrating membranes the free vibrating membrane corresponds to the neumann laplace operator in bounded plane domains in this paper we obtain estimates for the first non trivial eigenvalue of this operator in a large class of domains that we call conformal regular domains this class includes convex domains john domains etc on the basis of our estimates we conjecture that the eigenvalues of the neumann laplace operator depend on the hyperbolic metrics of plane domains we propose a new method for the estimates which is based on weighted poincare sobolev inequalities obtained by the authors recently

relating the approaches to quantised algebras and quantum groups this paper constructs two representations of the quantum groupu q g by exploiting its quotient structure and the quantum double construction here the quantum group is taken as the dual to the quantised algebrau q g a one parameter deformation of the universal enveloping algebra of the lie algebra g as in drinfel d 6 and jimbo 10 from the two representations the hopf structure of the quantised algebrau q g is reexpressed in a matrix format this is the very structure given by faddeev et al 7 in their approach to defining quantum groups and quantised algebras via the quantisation of the function space of the associated lie group to g

continuity of capacity of a holomorphic motion abstract we study the behavior of various set functions under holomorphic motions we show that under such deformations logarithmic capacity varies continuously while analytic capacity may not