on dimension of porous measures we introduce a notion of mean porosity for measures and obtain dimensional bounds for mean porous and porous measures as a corollary we generalize to all porous borel measures the estimate obtained recently by j p eckmann e jarvenpaa and m jarvenpaa for porous measures satisfying the doubling condition we also discuss various generalizations of this notion and possible applications

harmonic measure versus hausdorff measures on repellers for holomorphic maps this paper is a continuation of a joint paper of the author with f przytycki and m urbanski we study a harmonic measure on a boundary of so called repelling boundary domain an important example is a basin of a sink for a rational map using the results of the above mentioned paper we prove that either the boundary of the domain is an analytically embedded circle or interval or else the harmonic measure is singular with respect to the hausdors measure corresponding to the function xc t t exp c log t logloglog 1 for some c 0 0 introduction and statement of results let q be a simply connected domain in the riemann sphere card q 2 let r d q be the riemann map from the unit disc d onto q since r has nontangential limits almost everywhere on s1 ad d the image r l of the length measure i on s1 can be considered this measure denoted usually by co and called a harmonic measure on aq has been studied a long time see mk2 and puz for the corresponding references with the most interesting results obtained recently in two remarkable papers mkl and mk2 for an increasing function 0 di di 0 0 o we define an outer measure for a c c a a lim inf diambj where the infimum is taken over all coverings of a with balls of a diameter smaller than a the diameter is computed with respect to the spherical metrics on in particular for 0 t ta we obtain the ol dimensional hausdorff measure ata will be denoted by aa let u be a probability measure on borel subsets of hausdorff dimension of u is defined as

thermodynamics dimension and the weil petersson metric in this paper we study branched coverings of metrized simplicial trees f t t which arise from polynomial maps f c c with disconnected julia sets we show that the collection of all such trees up to scale forms a contractible space ptd compactifying the moduli space of polynomials of degree d that f records the asymptotic behavior of the multipliers of f and that any meromorphic family of polynomials over can be completed by a unique tree at its central fiber in the cubic case we give a combinatorial enumeration of the trees that arise and show that pt3 is itself a tree resume dans ce travail nous etudions des revetements ramifies d arbres metriques simpliciaux f t t qui sont obtenus a partir d applications polynomiales f c c possedant un ensemble de julia non connexe nous montrons que la collection de tous ces arbres a un facteur d echelle pres forme un espace contractile ptd qui compactifier l espace des modules des polynomes de degre d nous montrons aussi que f enrigistre le comportement asymptotique des multiplicateurs de f et que tout famille meromorphe de polynomes definis sur peut etre completee par un unique arbre comme sa fibre centrale dans le cas cubique nous donnons une enumeration combinatoire des arbres ainsi obtenus et montrons que pt3 est lui meme un arbre

uniform rectifiability and harmonic measure ii poisson kernels in l p imply uniform rectifiability the first author was supported by nsf grants dms 0801079 and dms 1101244 the second author was supported by micinn grant mtm2010 16518 and by csic pie 200850i015 the third author was partially supported by grants dms 0901524 career dms 1056965 us nsf sloan research foundation and mtm2010 16232 mtm2009 14694 c02 01 spain

perturbations of elliptic operators in 1 sided chord arc domains part i small and large perturbation for symmetric operators the first and third authors acknowledge financial support from the spanish ministry of economy and competitiveness through the severo ochoa programme for centres of excellence in r d sev 2015 0554 they also acknowledge that the research leading to these results has received funding from the european research council under the european union s seventh framework programme fp7 2007 2013 erc agreement no 615112 hapdegmt the second author was supported by nsf grant dms 1664047

harmonic measure and quantitative connectivity geometric characterization of the l p solvability of the dirichlet problem it is well known that quantitative scale invariant absolute continuity more precisely the weak a infty property of harmonic measure with respect to surface measure on the boundary of an open set omega subset mathbb r n 1 with ahlfors david regular boundary is equivalent to the solvability of the dirichlet problem in omega with data in l p partial omega for some p infty in this paper we give a geometric characterization of the weak a infty property of harmonic measure and hence of solvability of the l p dirichlet problem for some finite p this characterization is obtained under background hypotheses an interior corkscrew condition along with ahlfors david regularity of the boundary that are natural and in a certain sense optimal we provide counter examples in the absence of either of them or even one of the two upper or lower ahlfors david bounds moreover the examples show that the upper and lower ahlfors david bounds are each quantitatively sharp

domain theory for concurrency a simple domain theory for concurrency is presented based on a categorical model of linear logic and associated comonads it highlights the role of linearity in concurrent computation two choices of comonad yield two expressive metalanguages for higher order processes both arising from canonical constructions in the model their denotational semantics are fully abstract with respect to contextual equivalence one language derives from an exponential of linear logic it supports a straightforward operational semantics with simple proofs of soundness and adequacy the other choice of comonad yields a model of affine linear logic and a process language with a tensor operation to be understood as a parallel composition of independent processes the domain theory can be generalised to presheaf models providing a more refined treatment of nondeterministic branching the article concludes with a discussion of a broader programme of research towards a fully fledged domain theory for concurrency

riemann map and holomorphic dynamics f f r a f r a and f u c c l a f r a i e fra repels to the side of a n o assume that u is always sufficiently small so that f has no singularities in u c a let r d z a be a riemann map a conformal homeomorphism of the unit disc onto a then there exists a holomorphic extension g of r l o f o r to a neighbourhood of s 1 and gls happens to be an expanding map i e there exists n 0 such that for every z s l l g z l although the proofs are straightforward we include them for sceptics in the last section sect 7 together with a remark about examples of a and f satisfying the above assumptions denote by gthe non tangential limit of r and by k s 1 the domain where it exists denote by ijl g the space of all borel probability g invariant ergodic positive entropy measures on s 1 it is known see p that for every egji g k exists almost everywhere so the f invariant measure r on fra can be considered the functions log g log lf are z respectively r 0 integrable see 1 5 denote these integrals by u g g f in this paper we consider derivatives with respect to the riemannian metric on 2 define the hausdorff dimension of any probability measure v

analytical representations of the spread harmonic measure density we study the spread harmonic measure that characterizes the spatial distribution of reaction events on a partially reactive surface for euclidean domains in which brownian motion can be split into independent lateral and transverse displacements we derive analytical formulas for the spread harmonic measure density and analyze its asymptotic behavior this analysis is applicable to slab domains general cylindrical domains and a half space we investigate the spreading effect due to multiple reflections on the surface and the underlying role of finite reactivity we discuss further extensions and applications of analytical results to describe laplacian transfer phenomena such as permeation through semipermeable membranes secondary current distribution on partially blocking electrodes and surface relaxation in nuclear magnetic resonance

dahlberg s theorem in higher co dimension abstract in 1977 the celebrated theorem of b dahlberg established that the harmonic measure is absolutely continuous with respect to the hausdorff measure on a lipschitz graph of dimension n 1 in r n and later this result has been extended to more general non tangentially accessible domains and beyond in the present paper we prove the first analogue of dahlberg s theorem in higher co dimension on a lipschitz graph of dimension d in r n d n 1 with a small lipschitz constant we construct a linear degenerate elliptic operator l such that the corresponding harmonic measure l is absolutely continuous with respect to the hausdorff measure on more generally we provide sufficient conditions on the matrix of coefficients of l which guarantee the mutual absolute continuity of l and the hausdorff measure

σ finiteness of elliptic measures for quasilinear elliptic pde in space abstract in this paper we study the hausdorff dimension of a elliptic measure f in space associated to a positive weak solution to a certain quasilinear elliptic pde in an open subset and vanishing on a portion of the boundary of that open set we show that this measure is concentrated on a set of finite n 1 dimensional hausdorff measure for p n and the same result holds for p n with an assumption on the boundary we also construct an example of a domain in space for which the corresponding measure has hausdorff dimension n 1 for p n for some which depends on various constants including p the first result generalizes the authors previous work in 3 when the pde is the p laplacian and the second result generalizes the well known theorem of wolff in 24 when p 2 and n 2

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

rectifiability and elliptic measures on 1 sided nta domains with ahlfors david regular boundaries let omega subset mathbb r n 1 n geq 2 be 1 sided nta domain aka uniform domain i e a domain which satisfies interior corkscrew and harnack chain conditions and assume that partial omega is n dimensional ahlfors david regular we characterize the rectifiability of partial omega in terms of the absolute continuity of surface measure with respect to harmonic measure we also show that these are equivalent to the fact that partial omega can be covered mathcal h n a e by a countable union of portions of boundaries of bounded chord arc subdomains of omega and to the fact that partial omega possesses exterior corkscrew points in a qualitative way mathcal h n a e our methods apply to harmonic measure and also to elliptic measures associated with real symmetric second order divergence form elliptic operators with locally lipschitz coefficients whose derivatives satisfy a natural qualitative carleson condition

proving congruence of bisimulation in functional programming languages we give a method for proving congruence of bisimulation like equivalences in functional programming languages the method applies to languages that can be presented as a set of expressions together with an evaluation relation we use this method to show that some generalizations of abramsky s applicative bisimulation are congruences whenever evaluation can be specified by a certain natural form of structured operational semantics one of the generalizations handles nondeterminism and diverging computations

capacity and principal eigenvalues the method of enlargement of obstacles revisited we describe a coarse graining method which provides lower bounds on the principal dirichlet eigenvalue of the laplacian in regions receiving small obstacles and sharpens the previous method of enlargement of obstacles based on a quantitative wiener criterion one replaces the actual obstacles by obstacles of a much larger size controls on the shift of principal eigenvalues and capacity estimates on the locus where the wiener criterion breaks down are derived the results are written in a self contained fashion

on the hausdorff dimension of harmonic measure in higher dimension assume a i n e a domain in f n where e is a compact set denote o a a x the harmonic measure for a of a evaluated at x r d according to 0ksendal s theorem o o e o a x is singular with respect to d dimensional lebesgue measure for d 2 and general domains this result seemed to be so far the only known localization property recently for d 2 it has been shown by p jones and t wolff j w that s o9 has dimension at most 1 this result completes previous work due to n g makarov and l carleson see i m and c2 let g be the green s function of a r with pole at some point of c r o9 in both the carleson and jones wolff arguments the integral g g nn log nn as boundary

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

prime algebraicity a prime algebraic lattice can be characterised as isomorphic to the downwards closed subsets ordered by inclusion of its complete primes it is easily seen that the downwards closed subsets of a partial order form a completely distributive algebraic lattice when ordered by inclusion the converse also holds any completely distributive algebraic lattice is isomorphic to such a set of downwards closed subsets of a partial order the partial order can be recovered from the lattice as the order of the lattice restricted to its complete primes consequently prime algebraic lattices are precisely the completely distributive algebraic lattices the result extends to scott domains several consequences are explored briefly the representation of berry s di domains by event structures a simplified form of information systems for completely distributive scott domains and a simple domain theory for concurrency

conformal distortion of boundary sets conformal maps f of the disk into itself have the property that dim f 1 f dim f for any set f on the unit circle

null sets of harmonic measure on nta domains lipschitz approximation revisited we show the david jerison construction of big pieces of lipschitz graphs inside a corkscrew domain does not require surface measure be upper ahlfors regular thus we can study absolute continuity of harmonic measure and surface measure on nta domains of locally finite perimeter using lipschitz approximations a partial analogue of the f and m riesz theorem for simply connected planar domains is obtained for nta domains in space as one consequence every wolff snowflake has infinite surface measure

p harmonic measure in simply connected domains revisited let be a bounded simply connected domain in the complex plane c let n be a neighborhood of let p be xed 1 p 1 and let be a positive weak solution to the p laplace equation in n assume that u has zero boundary values on

boundary structure and size in terms of interior and exterior harmonic measures in higher dimensions in this work we introduce the use of powerful tools from geometric measure theory gmt to study problems related to the size and structure of sets of mutual absolute continuity for the harmonic measure omega of a domain omega omega subset mathbb r n and the harmonic measure omega of omega omega int omega c in dimension n ge 3

complex projective structures lyapunov exponent degree and harmonic measure we study several new invariants associated to a holomorphic projective struc ture on a riemann surface of finite analytic type the lyapunov exponent of its holonomy which is of probabilistic dynamical nature and was introduced in our previous work the degree which measures the asymptotic covering rate of the developing map and a family of harmonic measures on the riemann sphere previously introduced by hussenot we show that the degree and the lyapunov exponent are related by a simple formula and give estimates for the hausdorff dimension of the harmonic measures in terms of the lyapunov exponent in accordance with the famous sullivan dictionary this leads to a descrip tion of the space of such projective structures that is reminiscent of that of the space of polynomials in holomorphic dynamics

improved kinetic renormalisation group approach to diffusion limited aggregation an improved kinetic renormalisation group approach to diffusion limited aggregation is presented this approach is based on the growth process itself and accounts for the dispersity in the growth probabilities it yields the multifractal spectrum d q with better values at smaller q on the 2d square lattice the authors find df 1 694 for the fractal dimensions of the cluster and that of its interface and d 1 1 01 for the information dimension the former agrees with the simulation results df approximately 1 70 and the latter compares very well with the exact value d 1 1

the weighted majority algorithm we study the construction of prediction algorithms in a situation in which a learner faces a sequence of trials with a prediction to be made in each and the goal of the learner is to make few mistakes we are interested in the case where the learner has reason to believe that one of some pool of known algorithms will perform well but the learner does not know which one a simple and effective method based on weighted voting is introduced for constructing a compound algorithm in such a circumstance we call this method the weighted majority algorithm we show that this algorithm is robust in the presence of errors in the data we discuss various versions of the weighted majority algorithm and prove mistake bounds for them that are closely related to the mistake bounds of the best algorithms of the pool for example given a sequence of trials if there is an algorithm in the pool a that makes at most m mistakes then the weighted majority algorithm will make at most c log a m mistakes on that sequence where c is fixed constant

parallel algorithm and dynamic exponent for diffusion limited aggregation a parallel algorithm for diffusion limited aggregation dla is described and analyzed from the perspective of computational complexity the dynamic exponent z of the algorithm is defined with respect to the probabilistic parallel random access machine pram model of parallel computation according to t sim l z where l is the cluster size t is the running time and the algorithm uses a number of processors polynomial in l it is argued that z d d 2 2 where d is the fractal dimension and d 2 is the second generalized dimension simulations of dla are carried out to measure d 2 and to test scaling assumptions employed in the complexity analysis of the parallel algorithm it is plausible that the parallel algorithm attains the minimum possible value of the dynamic exponent in which case z characterizes the intrinsic history dependence of dla

a implies nta for a class of variable coefficient elliptic operators abstract we consider a certain class of second order variable coefficient divergence form elliptic operators in a uniform domain with ahlfors regular boundary and we show that the a property of the elliptic measure associated to any such operator and its transpose imply that the domain is in fact nta and hence chord arc the converse was already known and follows from work of kenig and pipher 30

julia sets and complex singularities in hierarchical ising models we study the analytical continuation in the complex plane of free energy of the ising model on diamond like hierarchical lattices it is known 12 13 that the singularities of free energy of this model lie on the julia set of some rational endomorphismf related to the action of the migdal kadanoff renorm group we study the asymptotics of free energy when temperature goes along hyperbolic geodesics to the boundary of an attractive basin off we prove that for almost all with respect to the harmonic measure geodesics the complex critical exponent is common and compute it

on the structure of partial balayage abstract we define partial balayage construct it with classical potential theoretic methods and establish the so called structure formula in full generality finally we give a stochastic construction of partial balayage

absolute continuity of harmonic measure for domains with lower regular boundaries abstract we study absolute continuity of harmonic measure with respect to surface measure on domains that have large complements we show that if r d 1 is ahlfors d regular and splits r d 1 into two nta domains then h d on this result is a natural generalization of a result of wu in 49 we also prove that almost every point in is a cone point if is a lipschitz graph combining these results and a result from 8 we characterize sets of absolute continuity with finite h d measure if d 1 for domains with large complements both in terms of the cone point condition and in terms of the rectifiable structure of the boundary even in the plane this extends the results of mcmillan in 38 and pommerenke in 43 which were only known for simply connected planar domains finally we also show our first result holds for elliptic measure associated with real second order divergence form elliptic operators with a mild assumption on the gradient of the matrix

petri nets event structures and domains part i abstract the general aim of this paper is to find a theory of concurrency combining the approaches of petri and scott and others in part i we introduce our formalisms to connect the abstract ideas of events and domains of information we show how casual nets induce certain kinds of domains where the information points are certain sets of events this allows translations between the languages of net theory and domain theory following the idea that events of causal nets are occurrences we generalise causal nets to occurrence nets by adding forwards conflict just as infinite flow charts unfold finite ones so transition nets can be unfolded into occurrence nets next we extend the above connections between nets and domains to these new nets event structures which are intermediate between nets and domains play an important part in all our work finally as an example of how concepts translate from one formalism to the other we show how petri s notion of confusion ties up with kahn and plotkin s concrete domains in part ii we shall continue the job of connecting up notions within net theory and the theory of domains in particular we shall examine the idea of states of computations

online markov decision processes we consider a markov decision process mdp setting in which the reward function is allowed to change after each time step possibly in an adversarial manner yet the dynamics remain fixed similar to the experts setting we address the question of how well an agent can do when compared to the reward achieved under the best stationary policy over time we provide efficient algorithms which have regret bounds with no dependence on the size of state space instead these bounds depend only on a certain horizon time of the process and logarithmically on the number of actions

on powerdomains and modality abstract this note shows a simple connection between powerdomains and modal assertions that can be made about nondeterministic computations we consider three kinds of powerdomains the plotkin powerdomain the smyth powerdomain and one christened the hoare powerdomain by plotkin because it captures the partial correctness of a nondeterministic program the modal operators are for inevitably and for possibly it is shown in a precise sense how the smyth powerdomain is built up from assertions about the inevitable behaviour of a process the hoare powerdomain is built up from assertions about the possible behaviour of a process while the plotkin powerdomain is built up from both kinds of assertions taken together

two representations in multifractal analysis two representations in multifractal analysis the so called q and tau representations are discussed theoretically and computed practically complementary to the standard q representation the so called tau representation is especially suited to resolving the most rarified subsets of the distributed measure moreover these two representations are especially adapted respectively to the well known fixed size and fixed mass box counting algorithms both strategies are first applied to iteratively constructed mathematical measures once tested in this way we use them to analyse the mass distribution and the growth probability distribution of an experimental electrodeposited pattern

the harmonic measure of balls in random trees we study properties of the harmonic measure of balls in typical large discrete trees for a ball of radius nn centered at the root we prove that although the size of the boundary is of order nn most of the harmonic measure is supported on a boundary set of size approximately equal to n n where 0 78 0 78 is a universal constant to derive such results we interpret harmonic measure as the exit distribution of the ball by simple random walk on the tree and we first deal with the case of critical galton watson trees conditioned to have height greater than nn an important ingredient of our approach is the analogous continuous model related to aldous continuum random tree where the dimension of harmonic measure of a level set of the tree is equal to whereas the dimension of the level set itself is equal to 11 the constant is expressed in terms of the asymptotic distribution of the conductance of large critical galton watson trees

carleson sobolev measures for weighted bloch spaces abstract carleson sobolev measures for weighted bloch spaces on the unit ball of c n are described the classical carleson measures for growth spaces on special circular domains are characterized

a fully abstract denotational model for higher order processes a higher order process calculus is defined in which one can describe processes which transmit as messages other processes it may be viewed as a generalisation of the lazy calculus we present a denotational model for the language obtained by generalising the domain equation for abramsky s model of the lazy calculus it is shown to be fully abstract with respect to three different behavioural preorders the first is based on observing the ability of processes to perform an action in all contexts the second on testing and the final one on satisfying certain kinds of modal formulae

algebraic laws for nondeterminism and concurrency since a nondeterministic and concurrent program may in general communicate repeatedly with its environment its meaning cannot be presented naturally as an input output function as is often done in the denotational approach to semantics in this paper an alternative is put forth first a definition is given of what it is for two programs or program parts to be equivalent for all observers then two program parts are said to be observation congruent if they are in all program contexts equivalent the behavior of a program part that is its meaning is defined to be its observation congruence class the paper demonstrates for a sequence of simple languages expressing finite terminating behaviors that in each case observation congruence can be axiomatized algebraically moreover with the addition of recursion and another simple extension the algebraic language described here becomes a calculus for writing and specifying concurrent programs and for proving their properties

heat transport through rough channels abstract we investigate the two dimensional transport of heat through viscous flow between two parallel rough interfaces with a given fractal geometry the flow and heat transport equations are solved through direct numerical simulations and for different conduction convection conditions compared with the behavior of a channel with smooth interfaces the results for the rough channel at low and moderate values of the peclet number indicate that the effect of roughness is almost negligible on the efficiency of the heat transport system this is explained here in terms of makarov s theorem using the notion of active zone in laplacian transport at sufficiently high peclet numbers where convection becomes the dominant mechanism of heat transport the role of the interface roughness is to generally increase both the heat flux across the wall as well as the active length of heat exchange when compared with the smooth channel finally we show that this last behavior is closely related with the presence of recirculation zones in the re entrant regions of the fractal geometry

on the existence of conformal measures a general notion of conformal measure is introduced and some basic properties are studied sufficient conditions for the existence of these measures are obtained using a general construction principle the geometric properties of conformal measures relate equilibrium states and hausdorff measures this is shown for invariant subsets of s under expanding maps

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

computational interpretations of linear logic abstract we study girard s linear logic from the point of view of giving a concrete computational interpretation of the logic based on the curry howard isomorphism in the case of intuitionistic linear logic this leads to a refinement of the lambda calculus giving finer control over order of evaluation and storage allocation while maintaining the logical content of programs as proofs and computation as cut elimination in the classical case it leads to a concurrent process paradigm with an operational semantics in the style of berry and boudol s chemical abstract machine this opens up a promising new approach to the parallel implementation of functional programming languages and offers the prospect of typed concurrent programming in which correctness is guaranteed by the typing

random walk perturbations for online combinatorial optimization we study online combinatorial optimization problems that a learner is interested in minimizing its cumulative regret in the presence of switching costs to solve such problems we propose a version of the follow the perturbed leader algorithm in which the cumulative losses are perturbed by independent symmetric random walks in the general setting our forecaster is shown to enjoy near optimal guarantees on both quantities of interest making it the best known efficient algorithm for the studied problem in the special case of prediction with expert advice we show that the forecaster achieves an expected regret of the optimal order o n log n 1 2 where n is the time horizon and n is the number of experts while guaranteeing that the predictions are switched at most o n log n 1 2 times in expectation

hausdorff dimension and σ finiteness of p harmonic measures in space when p n abstract in this paper we study a measure associated with a positive p harmonic function u defined in an open set o r n and vanishing on a portion of o if p n we show is concentrated on a set of finite h n 1 measure while if p n the same conclusion holds provided is uniformly fat in the sense of n capacity our work nearly answers in the affirmative a conjecture in lewis 2015 and also appears to be the natural extension of jones and wolff 1988 wolff 1993 to higher dimensions

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

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

bisimilarity as a theory of functional programming abstract morris style contextual equivalence invariance of termination under any context of ground type is the usual notion of operational equivalence for deterministic functional languages such as pcf contextual equivalence is hard to establish directly instead we define a labelled transition system for call by name pcf and variants and prove that ccs style bisimilarity equals contextual equivalence using co induction we establish equational laws by considering variations of milner s bisimulations up to we obtain a second co inductive characterisation of contextual equivalence in terms of reduction behaviour and production of values hence we use co induction to establish contextual equivalence in a series of stream processing examples we show that milner s context lemma may be extended to our variants of pcf but in fact our form of bisimilarity supports simpler co inductive proofs we sketch how these results extend to variants including eager evaluation and the addition of sums pairs and recursive types

learning permutations with exponential weights we give an algorithm for the on line learning of permutations the algorithm maintains its uncertainty about the target permutation as a doubly stochastic weight matrix and makes predictions using an efficient method for decomposing the weight matrix into a convex combination of permutations the weight matrix is updated by multiplying the current matrix entries by exponential factors and an iterative procedure is needed to restore double stochasticity even though the result of this procedure does not have a closed form a new analysis approach allows us to prove an optimal up to small constant factors bound on the regret of our algorithm this regret bound is significantly better than that of either kalai and vempala s more efficient follow the perturbed leader algorithm or the computationally expensive method of explicitly representing each permutation as an expert

regret in online combinatorial optimization we address online linear optimization problems when the possible actions of the decision maker are represented by binary vectors the regret of the decision maker is the difference between her realized loss and the minimal loss she would have achieved by picking in hindsight the best possible action our goal is to understand the magnitude of the best possible minimax regret we study the problem under three different assumptions for the feedback the decision maker receives full information and the partial information models of the so called semi bandit and bandit problems in the full information case we show that the standard exponentially weighted average forecaster is a provably suboptimal strategy for the semi bandit model by combining the mirror descent algorithm and the inf implicitely normalized forecaster strategy we are able to prove the first optimal bounds finally in the bandit case we discuss existing results in light of a new lower bound and suggest a conjecture on the optimal regret in that case

dirichlet forms quasiregular functions and brownian motion given a quasiregular function on an open setu in n it is shown that there exists a diffusionx t inu such that mapsx t inton dimensional brownian motion the process is constructed from a dirichlet form which can be described explicitly this enables us to apply stochastic methods in the investigation of quasiregular mappings some examples of applications are given including boundary behaviour and value distribution

dissipation in laplacian fields across irregular boundaries the entropy production associated to a laplacian field distributed across irregular boundaries is studied in the context of the active zone approximation an explicit expression is given for the entropy production in terms of geometry whose relation to the variational formulation is discussed it is shown that the entropy production diminishes for successive prefractal generations of the same fractal generator so that the final fractal object is expected to dissipate less than all previous ones the relevance of this result in the abundance of fractal surfaces or interfaces observed in nature is discussed