on the hull of two dimensional percolation clusters a recently proven equivalence between certain random walks with memory and the hull of percolation clusters in two dimensions is used to estimate the fractal dimension of the latter by means of monte carlo simulations the authors calculate the value dh 1 750 or 0 002 in agreement with a recent conjecture that dh 7 4 exactly

the complex lorentzian leech lattice and the bimonster ii let d be the incidence graph of the projective plane over f3 the artin group of the graph d maps onto the bimonster and a complex hyperbolic reflection group acting on 13 dimensional complex hyperbolic space y the generators of the artin group are mapped to elements of order 2 resp 3 in the bimonster resp let y y be the complement of the union of the fixed points of reflections in daniel allcock has conjectured that the orbifold fundamental group of y surjects onto bimonster in this article we study the reflection group we show that th e artin group of d maps to the orbifold fundamental group of y thus answering a question in allcock s article a monstrous proposal and taking one step towards the proof of allcock s conjecture the finite group aut d acts on y we make a detailed study of the complex hyperbolic line fixed by the subgroup l3 3 aut d we define meromorphic automorphic forms on y invariant under with poles along mirrors we show that the restriction of these forms to the complex hyperbolic line fixed by l3 3 gives meromorphic modular forms of level 13

structure and perimeters of percolation clusters site percolation clusters are simulated at the percolation threshold on the square lattice an algorithm for walks around each cluster is used to obtain information on its fractal geometry the fractal dimensionality of the external perimeter is found to depend on the size of adsorbent particles used to measure it if the vacant perimeter sites have only nearest neighbour connectivity then the perimeter has dimension de 1 37 or 0 03 instead of that of the hull dh 1 75 the authors also measured the mass of internal dangling sites surrounded by the backbone s blobs singly and doubly connected sites the number of blobs and the average linear distance between entry into the exit from a blob

on the y555 complex reflection group abstract we give a computer free proof of a theorem of basak describing the group generated by 16 complex reflections of order 3 satisfying the braid and commutation relations of the y 555 diagram the group is the full isometry group of a certain lattice of signature 13 1 over the eisenstein integers z 1 3 along the way we enumerate the cusps of this lattice and classify the root and niemeier lattices over z 1 3

monte carlo series analysis of irreversible self avoiding walks i the indefinitely growing self avoiding walk igsaw high precision monte carlo data are used to estimate the exponents which govern the asymptotic behaviour of the recently introduced indefinitely growing self avoiding walk in two dimensions for this walk the exponent gamma is by definition equal to one applying the same methods which are used to extract the exponents from exact series enumeration the authors give an estimate for the exponent nu of 0 567 or 0 003 the leading corrections to this asymptotic behaviour are also calculated

on scaling relations in growth models for percolating clusters and diffusion fronts employing analogies between the growth of the incipient percolating cluster and the growth of a diffusion front the authors develop a scaling theory for this latter model their scaling assumptions support the sapoval rosso gouyet conjecture dh 1 1 nu which relates in two dimensions the fractal dimension dh of the hull of the incipient percolating cluster to the correlation length exponent nu in percolation and yield relations between the static and dynamic exponents of the growth of a diffusion front

the chemical distance distribution in percolation clusters the authors study the conditional probability density p r mod l for the geometrical distance r corresponding to a given chemical distance l for percolation clusters in two dimensions they argue that i p r mod l alxg exp axdelta where x r lnu g 2 5 or 0 3 delta 9 8 or 0 5 and nu 0 88 or 0 02 and ii there is a relation delta 1 nu 1 which is in good agreement with numerical data these results are derived by considering a special class of self avoiding walks consisting of chains which are chemical paths shortest paths on critical percolation clusters

locally toroidal polytopes of rank 6 and sporadic groups abstract we augment the list of finite universal locally toroidal regular polytopes of type 3 3 4 3 3 due to p mcmullen and e schulte adding as well as removing entries this disproves a related long standing conjecture our new universal polytope is related to a well known y shaped presentation for the sporadic simple group fi 22 and admits s 4 o 8 2 s 3 as the automorphism group we also discuss further extensions of its quotients in the context of y shaped presentations as well we note that two known examples of finite universal polytopes of type 3 3 4 3 3 are related to y shaped presentations of orthogonal groups over f 2 mixing construction is used in a number of places to describe covers and 2 covers

the fractal dimension of percolation cluster hulls large simulations of 2d site percolation on a square lattice provide several estimates of the fractal dimension dh of the hull or outer boundary of percolation clusters close to criticality statistical estimates based on an ensemble of more than 106 clusters with varying sizes yield dh 1 74 or 0 02 geometrical measurements of the hull two point correlation for individual clusters with more than 105 sites give dh 1 76 or 0 01 the observed scaling behaviour is the same both slightly above and below pc

on the relations between monadic semantics we present a simple computational metalanguage with general recursive types and multiple notions of effects through which a variety of concrete denotational semantics can be conveniently factored by suitably interpreting the effects as monads we then propose a methodology for relating two such interpretations of the metalanguage with the aim of showing that the semantics they induce agree for complete programs as a prototypical instance of such a relation we use the framework to show agreement between a direct and a continuation semantics of the simple untyped functional language from reynolds s original paper on the subject

on growth walks with self avoiding constraints the authors study a general class of growth walks in two dimensions with a self avoiding constraint which can be considered as dressed self avoiding walks a walker travels on a square lattice and chooses at each step one of the three forward directions with probability p1 p2 and p3 if the directions are allowed the self avoiding constraint is obtained by dressing the walk with black and white particles after each step such that the walk generates the external perimeter of arbitrary clusters consisting of black or white particles they discuss the critical values of p1 p2 and p3 which allow for infinite walks and study the end to end distance r t and the corresponding distribution function n r t furthermore they introduce a bias field in growth process and discuss how the time evolution of the walk is modified by the bias

generic weakest precondition semantics from monads enriched with order we devise a generic framework where a weakest precondition semantics in the form of indexed posets is derived from a monad whose kleisli category is enriched by posets it is inspired by jacobs recent identification of a categorical structure that is common in various predicate transformers but adds generality in the following aspects 1 different notions of modality such as may vs must are captured by eilenberg moore algebras 2 nested alternating branching like in games and in probabilistic systems with nondeterministic environments is modularly modeled by a monad on the eilenberg moore category of another

percolation and motion in a simple random environment the authors present a particularly simple model of deterministic classical motion in a two dimensional random environment as the parameters of the model are varied a transition occurs from all trajectories being localised to some being extended they construct a mean field theory for this transition and relate the model exactly to percolation models in particular parameter ranges they point out that it is a member of a new class of site percolation analogues of direct bond percolation

probability density of the 2d percolation cluster perimeter the authors study the form of the probability density p r t of the end to end distance r of a segment of length t of the perimeter of the 2d percolation cluster at criticality they find that p r t behaves as p r t varies as t 2dh exp r t1dh delta where delta 1 1 1 dh the exponent dh 7 4 is the fractal dimension of the hull the external perimeter of a percolation cluster this result fits very well with recent numerical data

relating computational effects by ττ lifting we consider the problem of establishing a relationship between two interpretations of base type terms of a l c calculus extended with algebraic operations we show that the given relationship holds if it satisfies a set of natural conditions we apply this result to 1 comparing two monadic semantics related by a strong monad morphism and 2 comparing two monadic semantics of fresh name creation email protected s new name creation monad and the global counter monad we also consider the same problem relating semantics of computational effects in the presence of recursive functions we apply this additional by extending the previous monad morphism comparison result to the recursive case

the leech lattice and complex hyperbolic reflections we construct a natural sequence of finite covolume reflection groups acting on the complex hyperbolic spaces h 13 h 9 and h 5 and show that the 9 dimensional example coincides with the largest of the groups of mostow 11 our reflection groups arise as automorphism groups of certain lorentzian lattices over the eisenstein integers and we obtain our largest example by using the complex leech lattice in a manner inspired by conway 5 we also construct finite covolume reflection groups on the quaternionic hyperbolic spaces h 7 h 5 and h 3 again using the leech lattice and apply results of borcherds 4 to obtain automorphic forms for our groups

generation of percolation cluster perimeters by a random walk a type of self avoiding random walk whish generates the perimeter of two dimensional lattice percolation clusters is given the algorithm has been simulated on a computer yielding the mean perimeter length as a function of occupation probability

on coxeter diagrams of complex reflection groups we study complex coxeter diagrams of some unitary reflection groups using solely the combinatorics of diagrams we give a new proof of the classification of root lattices defined over e z e 2 i 3 there are only four such lattices namely the e lattices whose real forms are a2 d4 e6 and e8 next we address the issue of characterizing the diagrams for unitary reflection groups a question that was raised by broue malle and rouquier to this end we describe an algorithm which given a unitary reflection group g picks out a set of complex reflections if g is the reflection group of a root lattice defined over e the reflections selected by the algorithm form the known diagram for g for the reflection group of the complex coxeter todd lattice k e 12 we find a new diagram that extends to an affine diagram withz 7z symmetry if g is a weyl group the algorithm immediately yields a set of simple roots otherwise experimental evidences indicate that the algorithm selects a minimal generating set of reflections if g is primitive and g has a set of roots whose z span is a discrete subset of the ambient vector space

probability densities of random walks in random systems abstract we review recent results for the probability distribution of random walkers in random systems where diffusion is anomalous and the mean square displacement scales with time as r 2 t t 2 w d w 2 the random systems are characterized by structural disorder and by random transition rates in general the mean distribution function p r t of the random walkers is a stretched gaussian and scales as log p r t p r 0 r r t u where u d w d w 1 on random fractals the fluctuations of the density distribution p r t for fixed distance r and time t have a broad logarithmic distribution the average moments p q scale in a multifractal way as p q where q q l space the fluctuations of p are narrow and p q p q