dense morphisms in commutative banach algebras using a new notion of stability we compute exactly the stable rank of the polydisc algebra extend oka s extension theorem to tuples of functions without common zeros and give an estimation for a question raised by swan concerning the stable rank of a dense subalgebra of a given banach algebra 0 introduction the stable rank of a ring a 2 is an algebraic invariant of a which turns out to be closely related to the topology of the spectrum of a when a is a complex commutative banach algebra see 4 7 8 in this paper we study certain approximation and interpolation problems which are related to that notion our main result theorem 2 7 whose statement is too technical to be described here has several applications first we find exactly the stable rank of the polydisc algebra an of c sr an n 2 1 moreover for every generated algebra a sr a n 2 1 another application is a c dimensional version of oka s extension theorem finally we give a partial answer to a question of swan 23 remark p 206 if there exists a morphism f a b with dense image such that a invertible a invertible then sr a 4 sr b sr a 2

a remark concerning percolation thresholds in polydisperse systems of finite diameter rods a lattice based analysis of the percolation threshold for randomly distributed cylindrical particles is generalized to consider arbitrary joint distributions over the radii and lengths of the rods effects due to the finite hard core diameter of the particles are accounted for an analogy to site percolation on a modified bethe lattice is exploited to yield a result for the percolation threshold that is equivalent to one that has been obtained from integral equation methods in the limit of large aspect ratios for the rods

the stable rank of topological algebras and a problem of r g swan abstract various notions of stable ranks are studied for topological algebras some partial answers to r g swan s problem have two banach or good frechet algebras as in the density theorem in k theory the same stable rank are obtained for example a frechet dense subalgebra a of a c algebra b closed under c functional calculus of self adjoint elements has the same bass stable rank as b

continuum percolation threshold for interpenetrating squares and cubes monte carlo simulations are performed to determine the critical percolation threshold for interpenetrating square objects in two dimensions and cubic objects in three dimensions simulations are performed for two cases i objects whose edges are aligned parallel to one another and ii randomly oriented objects for squares whose edges are aligned the critical area fraction at the percolation threshold ensuremath varphi c 0 6666 ifmmode pm else textpm fi 0 0004 while for randomly oriented squares ensuremath varphi c 0 6254 ifmmode pm else textpm fi 0 0002 6 smaller for cubes whose edges are aligned the critical volume fraction at the percolation threshold ensuremath varphi c 0 2773 ifmmode pm else textpm fi 0 0002 while for randomly oriented cubes ensuremath varphi c 0 2168 ifmmode pm else textpm fi 0 0002 22 smaller

the fractal dimension and other percolation exponents in four and five dimensions for percolation the fractal dimension df which is identical to the magnetic field scaling power yh has never been calculated for hypercubic lattices of dimension d 5 percolation is studied for systems of high dimensionality using the method of large cell monte carlo position space renormalisation group the estimate df yh 3 69 or 0 02 d 5 and 3 12 or 0 02 d 4 is obtained the thermal scaling power yt 1 nu where nu is the correlation length exponent is calculated yt 1 nu 0 51 or 0 05 d 5 and nu 0 64 or 0 02 d 4 the results are compared with the epsilon expansions of df and nu

percolation thresholds and excluded area for penetrable rectangles in two dimensions the percolation threshold for penetrable rectangles in two dimensions is examined by way of an analogy to a lattice model that employs the averaged pairwise excluded area between particles as a key ingredient percolation thresholds are calculated as functions of i the aspect ratio of the rectangles for the case of a random isotropic orientational distribution and ii the orientational order parameter for situations in which the particles exhibit varying degrees of alignment the threshold volume fraction is found to rise with increases in the degree of particle alignment and our results for isotropically distributed systems are in close agreement with findings from recent monte carlo mc simulations

relatively spectral morphisms and applications to k theory abstract spectral morphisms between banach algebras are useful for comparing their k theory and their noncommutative dimensions as expressed by various notions of stable ranks in practice one often encounters situations where the spectral information is only known over a dense subalgebra we investigate such relatively spectral morphisms we prove a relative version of the density theorem regarding isomorphism in k theory we also solve swan s problem for the connected stable rank in fact for an entire hierarchy of higher connected stable ranks that we introduce

critical exponents for the percolation problem and the yang lee edge singularity gives details of a calculation of critical exponents for a class of field theory models that have an interaction cubic in the fields the results have already been reported give the exponents to third order in epsilon where epsilon 6 d and d is the dimensionality of space the class of models includes the percolation problem and the yang lee edge singularity so the authors give explicit results for the exponents to order epsilon 3 in these cases by resummation methods based on the symptotic behaviour of the epsilon expansion they obtain numerical estimates for these exponents for a number of interesting values of d

computation of the critical exponents of percolation computes the critical indices of percolation using a method in which the integrals are computed at fixed dimension at the order of two loops following this method the authors obtain the series giving the critical exponents expanded with respect to the coupling constant while in the epsilon expansion method the expansion of the series is performed with respect to the coupling constant and to the number of dimensions d thus the authors think that they have obtained better control of the position of the fixed point at a dimension not too near to size they are also able to improve the numerical values of the exponents at d 3

amplitude ratios and β estimates from general dimension percolation moments low concentration series are generated for moments of the percolation cluster size distribution gamma j sj 1 s is the number of sites on a cluster for j 2 8 and general dimensionality d these diverge at pc as gamma j approximately aj pc p gamma j with gamma j gamma j gamma j 2 delta where delta gamma beta is the gap exponent the series yield new accurate values for delta and beta delta 2 23 or 0 05 2 10 or 0 04 2 03 or 0 05 and beta 0 44 or 0 15 0 66 or 0 09 0 83 or 0 08 at d 3 4 5 in addition ratios of the form ajak aman with j k m n are shown to be universal new values for some of these ratios are evaluated from the series from the epsilon expansion epsilon 6 d and exactly in d 1 and on the bethe lattice the results are in excellent agreement with each other for all dimensions results for different lattices at d 2 3 agree very well these amplitude ratios are much better behaved than other ratios considered in the past and should thus be more useful in characterising percolating systems

banach algebras and rational homotopy theory let a be a unital commutative banach algebra with maximal ideal space max a we determine the rational h type of gln a the group of invertible n n matrices with coefficients in a in terms of the rational cohomology of max a we also address an old problem of j l taylor let lcn a denote the space of last columns of gln a we construct a natural isomorphism max a q 2n 1 s lcn a q for n 1 2 s 1 which shows that the rational cohomology groups of max a are determined by a topological invariant associated to a as part of our analysis we determine the rational h type of certain gauge groups f x g for g a lie group or more generally a rational h space

dependency of percolation critical exponents on the exponent of power law size distribution abstract the standard percolation theory uses objects of the same size moreover it has long been observed that the percolation properties of the systems with a finite distribution of sizes are controlled by an effective size and consequently the universality of the percolation theory is still valid in this study the effect of power law size distribution on the critical exponents of the percolation theory of the two dimensional models is investigated two different object shapes i e stick shaped and square are considered these two shapes are the representative of the fractures in fracture reservoirs and the sandbodies in clastic reservoirs the finite size scaling arguments are used for the connectivity to determine the dependency of the critical exponents on the power law exponent in particular the deviations of percolation exponents from their universal values as well as the connectivity behavior of such systems are investigated numerically as a result this extends the applicability of the conventional percolation approach to study the connectivity of systems with a very broad size distribution

continuum percolation thresholds in two dimensions a wide variety of methods have been used to compute percolation thresholds in lattice percolation the most powerful of these methods consists of microcanonical simulations using the union find algorithm to efficiently determine the connected clusters and in two dimensions using exact values from conformal field theory for the probability at the phase transition that various kinds of wrapping clusters exist on the torus we apply this approach to percolation in continuum models finding overlaps between objects with real valued positions and orientations in particular we find precise values of the percolation transition for disks squares rotated squares and rotated sticks in two dimensions and confirm that these transitions behave as conformal field theory predicts the running time and memory use of our algorithm are essentially linear as a function of the number of objects at criticality

averages over percolation clusters on a cayley tree averages over percolation clusters gamma of the form sigma gamma p gamma x gamma n gamma k identical to chi k p are considered where n gamma is the number of sites in the cluster x gamma is some property of the cluster and p gamma is the probability per site of forming such a cluster at bond concentration p

spreading of percolation in three and four dimensions the authors present monte carlo simulations of the spreading of bond percolation on three and four dimensional simple hyper cubic lattices the algorithm is the same as that applied previously in two dimensions with the spreading proceeding from a hyperplane they find the spreading dimension to be d 1 82 or 0 02 three dimensional and d 1 88 or 0 03 four dimensional also they obtain values of the percolation probability and of the static exponents with errors at least comparable to the best values from the literature

thin spectra and stable range conditions abstract the topological stable rank of a banach algebra a is the least integer n tsr a such that un a is dense in an where un a a1 an an i 1naai a we give a spectral characterization of the condition tsr a n valid for arbitrary banach algebras for a commutative banach algebra a the condition tsr a n is related to the presence of some kind of discs in the spectrum x a

percolation phase diagrams for multi phase models built on the overlapping sphere model abstract the overlapping sphere os percolation model gives a two phase microstructure matrix plus inclusions that is useful for testing composite material ideas and other applications as well as serving as a paradigm of overlapping object percolation and phase transitions real materials often have more than two phases so it is of interest to extend the applicability of the os model a flexible variant of the os model can be constructed by randomly assigning the spheres different phase labels according to a uniform probability distribution as they are inserted one by one into the matrix the resulting three or more phase models can have different amounts of percolating and non percolating phases depending on the volume fraction of each phase and the total os volume fraction a three dimensional digital image approach is used to approximately map out the percolation phase diagram of such models explicitly up to four phases one matrix plus three spherical inclusion phases and implicitly for n 4 phases for the three phase model it was found that a single os sub phase has a percolation threshold that ranges from about a volume fraction of 0 16 when the matrix volume fraction is about 0 01 to about 0 30 at a matrix volume fraction of about 0 7 the approximate analytical dependence of this sub phase percolation threshold on the defining model parameters serves to guide the building of the percolation phase diagram for the n phase model and is used to determine the maximum value of n n 6 at which all n phases can be simultaneously percolated

low concentration series in general dimension we discuss recent work on the development and analysis of low concentration series for many models the recent breakthrough in the extremely efficient no free end method of series generation facilitates the derivation of 15th order series for multiple moments in general dimension the 15th order series have been obtained for lattice animals percolation and the edwards anderson ising spin glass in the latter cases multiple moments have been found from complete graph tables through to 13th order general dimension 13th order series have been derived for the resistive susceptibility the moments of the logarithms of the distribution of currents in resistor networks and the average transmission coefficient in the quantum percolation problem 11th order series have been found for several other systems including the crossover from animals to percolation the full resistance distribution nonlinear resistive susceptibility and current distribution in dilute resistor networks diffusion on percolation clusters the dilute ising model dilute antiferromagnet in a field and random field ising model and self avoiding walks on percolation clusters series for the dilute spin 1 2 quantum heisenberg ferromagnet are in the process of development analysis of these series gives estimates for critical thresholds amplitude ratios and critical exponents for all dimensions where comparisons are possible our series results are in good agreement with bothe expansion results near the upper critical dimension and with exact results when available in low dimensions and are competitive with other numerical approaches in intermediate realistic dimensions

percolation thresholds of two dimensional continuum systems of rectangles the present paper introduces an efficient monte carlo algorithm for continuum percolation composed of randomly oriented rectangles by conducting extensive simulations we report high precision percolation thresholds for a variety of homogeneous systems with different rectangle aspect ratios this paper verifies and extends the excluded area theory it is confirmed that percolation thresholds are dominated by the average excluded areas for both homogeneous and heterogeneous rectangle systems except for some special heterogeneous systems where the rectangle lengths differ too much from one another in terms of the excluded areas generalized formulas are proposed to effectively predict precise percolation thresholds for all these rectangle systems this paper is therefore helpful for both practical applications and theoretical studies concerning relevant systems

finite size scaling in stick percolation this work presents the generalization of the concept of universal finite size scaling functions to continuum percolation a high efficiency algorithm for monte carlo simulations is developed to inv

critical condition of the water retention model we study how much water can be retained without leaking through boundarieswhen each unit square of a two dimensional lattice is randomly assigned a blockof unit bottom area but with different heigh

on the range of validity of the 6 e expansion for percolation in 6 epsilon dimensions the critical properties of percolation can be computed starting from a phi 3 theory having the same symmetry of the one state potts model the authors argue that this fixed point must be unstable with respect to a phi 4 interaction in two dimensions therefore a phi 4 perturbation is relevant this suggestion is confirmed by an explicit two loops computation

a novel method of calculating amplitude ratios by series expansions the author introduces a new method of calculating critical amplitude ratios using series which is both simple and powerful this method which gives estimates for the amplitude ratios that are neither biased by the values of the critical points nor by the critical exponents is applied to several models it is shown that this method produces results where no reliable estimates from series expansion exist in particular one finds 0 025 or 0 001 for at b2 for the 3d ising model and 220 or 10 for c c for the two dimensional percolation model in agreement with and with more accuracy than values obtained by other methods

asymmetry in the percolation thresholds of fully penetrable disks with two different radii we perform simulations of continuum gradient percolation to measure the percolation threshold of systems of fully penetrable disks of two different radii as a function of volumetric proportion v the percolation threshold phic v lambda is approximately symmetric about v 0 5 for a fixed ratio lambda of the radii however the difference from symmetry is strongly statistically significant we also improve by an order of magnitude the measurement of the percolation threshold for disks of equal radius phic 0 676 347 5 6

effect of anisotropy on the scaling of connectivity and conductivity in continuum percolation theory we investigate the effects of anisotropy on the finite size scaling of connectivity and conductivity of continuum percolation in three dimensions we consider a system of size x ifmmode times else texttimes fi y ifmmode times else texttimes fi z in which cubic bodies of size a ifmmode times else texttimes fi b ifmmode times else texttimes fi c are placed randomly we define two aspect ratios to request anisotropy then we expect that the displacement of average connected fraction p averaged over the realizations about the isotropic universal curves will be a function of the two aspect ratios this is accounted by considering an apparent percolation threshold in each direction which leads to 50 of realizations connecting in that direction we find the aspect ratios dependency of the apparent threshold and investigate the finite size scaling transformations for the mean connected fraction and its associated fluctuations moreover we apply a single phase pressure solver to determine the conductivity of various realizations of the system finally we apply the same idea to account for the effect of anisotropy on the conductivity scaling

percolation thresholds of a group of anisotropic three dimensional fracture networks abstract percolation thresholds average number of connections per object of two models of anisotropic three dimensional 3d fracture networks made of mono disperse hexagons have been calculated numerically the first model is when the fracture networks are comprised of two groups of fractures that are distributed in an anisotropic manner about two orthogonal mean directions i e z and x directions we call this model bipolar anisotropic fracture network bfn the second model is when three groups of fractures are distributed about three orthogonal mean directions that is z x and y directions in this model three families of fractures about three orthogonal mean directions are oriented in 3d space we call this model tripolar anisotropic fracture network tfn the finite size scaling method is used to predict the infinite percolation thresholds the effect of anisotropicity on percolation thresholds in x y and z directions is investigated we have revealed that as the anisotropicity of networks increases the percolation thresholds in x y and z directions span the range of 2 3 to 2 0 where 2 3 and 2 0 are extremums of percolation thresholds for isotropic and non isotropic orthogonal fracture networks respectively

the approximate invariance of the average number of connections for the continuum percolation of squares at criticality we perform monte carlo simulations to determine the average excluded area aex of randomly oriented squares randomly oriented widthless sticks and aligned squares in two dimensions we find significant differences between our results for randomly oriented squares and previous analytical results for the same the sources of these differences are explained using our results for aex and monte carlo simulation results for the percolation threshold we estimate the mean number of connections per object bc at the percolation threshold for squares in 2 d we study systems of squares that are allowed random orientations within a specified angular interval our simulations show that the variation in bc is within 1 6 when the angular interval is varied from 0 to 2