random generation of stochastic area integrals the authors describe a method of random generation of the integrals a 1 2 t t h int t t h int t s dw 1 r dw 2 s int t t h int t s dw 2 r dw 1 s together with the increments w 1 t h w 1 t and w 2 t h w 2 t of a two dimensional brownian path w 1 t w 2 t the method chosen is based on marsaglia s rectangle wedge tail method generalised to higher dimensions the motivation is the need for a numerical scheme for simulation of strong solutions of general multidimensional stochastic differential equations with an order of convergence o h where h is the stepsize previously no method has obtained an order of convergence better than o sqrt h in the general case

graphs groupoids and cuntz krieger algebras we associate to each locally finite directed graphgtwo locally compact groupoidsgandg the unit space ofgis the space of one sided infinite paths ing andg is the reduction ofgto the space of paths emanating from a distinguished vertex we show that under certain conditions theirc algebras are morita equivalent the groupoidc algebrac g is the cuntz krieger algebra of an infinite 0 1 matrix defined byg and that the algebrasc g contain thec algebras used by doplicher and roberts in their duality theory for compact groups we then analyse the ideal structure of these groupoidc algebras using the general theory of renault and calculate theirk theory

more on minimal invariant sets for nonexpansive mappings minimal invariant sets for nonexpansive mappings share some singular geometrical properties here we present some seemingly unknown ones

pseudogroups and their etale groupoids abstract a pseudogroup is a complete infinitely distributive inverse monoid such inverse monoids bear the same relationship to classical pseudogroups of transformations as frames do to topological spaces the goal of this paper is to develop the theory of pseudogroups motivated by applications to group theory c algebras and aperiodic tilings our starting point is an adjunction between a category of pseudogroups and a category of etale groupoids from which we are able to set up a duality between spatial pseudogroups and sober etale groupoids as a corollary to this duality we deduce a non commutative version of stone duality involving what we call boolean inverse semigroups and boolean etale groupoids as well as a generalization of this duality to distributive inverse semigroups non commutative stone duality has important applications in the theory of c algebras it is the basis for the construction of cuntz and cuntz krieger algebras and in the case of the cuntz algebras it can also be used to construct the thompson groups we then define coverages on inverse semigroups and the resulting presentations of pseudogroups as applications we show that paterson s universal groupoid is an example of a booleanization and reconcile exel s recent work on the theory of tight maps with the work of the second author

higher dimensional generalizations of the thompson groups abstract motivated by the goal of generalizing the cuntz krieger algebras and making heavy use of the pioneering work of robertson and steger kumjian and pask generalized the notion of a directed graph to what they termed a higher rank graph the c algebras constructed from such higher rank graphs have proved to be highly interesting higher rank graphs are in fact a class of cancellative categories and so the one vertex higher rank graphs are therefore a class of cancellative monoids in this paper we call them k monoids finite direct products of free monoids belong to this family but there are many examples not of this form it is well known that the classical thompson groups arise from the free monoids and it is easy to show that brin s higher dimensional thompson groups arise from finite direct products of free monoids in this paper we generalize these constructions and show how to construct a family of new groups with simple commutator subgroups from arbitrary aperiodic k monoids thereby subsuming all the above examples although we give simple direct constructions of our groups a key feature of our paper is that we also show that they arise as topological full groups of the usual etale groupoids associated with k monoids in this way our groups are naturally associated with c algebras the question of the mutual isomorphisms amongst this family of groups appears to be a delicate one and is not handled here

a groupoid approach to discrete inverse semigroup algebras abstract let k be a commutative ring with unit and s an inverse semigroup we show that the semigroup algebra ks can be described as a convolution algebra of functions on the universal etale groupoid associated to s by paterson this result is a simultaneous generalization of the author s earlier work on finite inverse semigroups and paterson s theorem for the universal c algebra it provides a convenient topological framework for understanding the structure of ks including the center and when it has a unit in this theory the role of gelfand duality is replaced by stone duality using this approach we construct the finite dimensional irreducible representations of an inverse semigroup over an arbitrary field as induced representations from associated groups generalizing the case of an inverse semigroup with finitely many idempotents more generally we describe the irreducible representations of an inverse semigroup s that can be induced from associated groups as precisely those satisfying a certain finiteness condition this finiteness condition is satisfied for instance by all representations of an inverse semigroup whose image contains a primitive idempotent

approximation of multiple stochastic integrals and its application to stochastic differential equations abstract as multiple stochastic integrals are not very easy to simulate we would like to treat them as solutions of systems of stochastic differential equations and solve them successively and recursively approximated by the stochastic taylor expansion as a chen series in terms of a philip hall basis or lyndon basis we can save sufficient values of multiple stochastic integrals with independent sample paths in a look up table for future use the table can be used to implement high order schemes to solve stochastic differential equations numerically a numerical example will be shown to illustrate the efficiency

homology and topological full groups of etale groupoids on totally disconnected spaces for almost finite groupoids we study how their homology groups reflect dynamical properties of their topological full groups it is shown that two clopen subsets of the unit space has the same class in h0 if and only if there exists an element in the topological full group which maps one to the other it is also shown that a natural homomorphism called the index map from the topological full group to h1 is surjective and any element of the kernel can be written as a product of four elements of finite order in particular the index map induces a homomorphism from h1 to k1 of the groupoid c algebra explicit computations of homology groups of af groupoids and etale groupoids arising from subshifts of finite type are also given

graph inverse semigroups their characterization and completion abstract graph inverse semigroups generalize the polycyclic inverse monoids and play an important role in the theory of c algebras in this paper we characterize such semigroups and show how they may be completed under suitable conditions to form what we call the cuntz krieger semigroup of the graph this semigroup is proved to be the ample semigroup of a topological groupoid associated with the graph and the semigroup analogue of the leavitt path algebra of the graph

topological equivalence of tilings we introduce a notion of equivalence on tilings which is formulated in terms of their local structure we compare it with the known concept of locally deriving one tiling from another and show that two tilings of finite type are topologically equivalent whenever their associated groupoids are isomorphic

the fixed point property in banach spaces with the nus property abstract in this paper we show that the weak nearly uniform smooth banach spaces have the fixed point property for nonexpansive mappings

quantales and fell bundles abstract we study fell bundles on groupoids from the viewpoint of quantale theory given any saturated upper semicontinuous fell bundle e g on an etale groupoid g with g 0 locally compact hausdorff equipped with a suitable completion c algebra a of its convolution algebra we obtain a map of involutive quantales p max a g where max a consists of the closed linear subspaces of a and g is the topology of g we study various properties of p which mimick to various degrees those of open maps of topological spaces these are closely related to properties of g and a such as g being hausdorff principal or topological principal or being a line bundle under suitable conditions which include g being hausdorff but without requiring saturation of the fell bundle a is an algebra of sections of the bundle if and only if it is the reduced c algebra c r g e we also prove that max a is stably gelfand this implies the existence of a pseudogroup i b and of an etale groupoid b associated canonically to any sub c algebra b a we study a correspondence between fell bundles and sub c algebras based on these constructions and compare it to the construction of weyl groupoids from cartan subalgebras

etale groupoids and their quantales abstract we establish close and previously unknown relations between quantales and groupoids in particular to each etale groupoid either localic or topological there is associated a unital involutive quantale we obtain a bijective correspondence between localic etale groupoids and their quantales which are given a rather simple characterization and here are called inverse quantal frames we show that the category of inverse quantal frames is equivalent to the category of complete and infinitely distributive inverse monoids and as a consequence we obtain a non functorial correspondence between these and localic etale groupoids that generalizes more classical results concerning inverse semigroups and topological etale groupoids this generalization is entirely algebraic and it is valid in an arbitrary topos as a consequence of these results we see that a localic groupoid is etale if and only if its sublocale of units is open and its multiplication map is semiopen and an analogue of this holds for topological groupoids in practice we are provided with new tools for constructing localic and topological etale groupoids as well as inverse semigroups for instance via presentations of quantales by generators and relations the characterization of inverse quantal frames is to a large extent based on a new quantale operation here called a support whose properties are thoroughly investigated and which may be of independent interest

the local structure of tilings and their integer group of coinvariants the local structure of a tiling is described in terms of a multiplicative structure on its pattern classes the groupoid associated to the tiling is derived from this structure and its integer group of coinvariants is defined this group furnishes part of the k0 group of the groupoid c algebra for tilings which reduce to decorations of the group itself as well as the image of its state is computed for substitution tilings in case the substitution is locally invertible and v primitive this yields in particular the set of possible gap labels predicted by k theory for schrodinger operators describing the particle motion in such a tiling

a perspective on non commutative frame theory abstract this paper extends the fundamental results of frame theory to a non commutative setting where the role of locales is taken over by etale localic categories this involves ideas from quantale theory and from semigroup theory specifically ehresmann semigroups restriction semigroups and inverse semigroups we prove several main results to start with we establish a duality between the category of complete restriction monoids and the category of etale localic categories the relationship between monoids and categories is mediated by a class of quantales called restriction quantal frames this result builds on the work of pedro resende on the connection between pseudogroups and etale localic groupoids but in the process we both generalize and simplify for example we do not require involutions and in addition we render his result functorial a wider class of quantales called multiplicative ehresmann quantal frames is put into a correspondence with those localic categories where the multiplication structure map is semiopen and all the other structure maps are open we also project down to topological spaces and as a result extend the classical adjunction between locales and topological spaces to an adjunction between etale localic categories and etale topological categories in fact varying morphisms we obtain several adjunctions just as in the commutative case we restrict these adjunctions to spatial sober and coherent spectral equivalences the classical equivalence between coherent frames and distributive lattices is extended to an equivalence between coherent complete restriction monoids and distributive restriction semigroups consequently we deduce several dualities between distributive restriction semigroups and spectral etale topological categories we also specialize these dualities for the setting where the topological categories are cancellative or are groupoids our approach thus links unifies and extends the approaches taken in the work by lawson and lenz and by resende

inverse monoids with a natural semilattice ordering an inverse semigroup is a semigroup s such that for each x e s there exists a unique inverse x s such that both xx x x and x xx x this condition is equivalent to s both being von neumann regular and having all idempotents commute the classic example of such a semigroup is the symmetric inverse semigroup ix of all partial bijections on a set x under the standard composition of partial functions on x more generally for any reasonably endowed mathematical object q the set n of all partial symmetries of q bijections between subobjects of q respecting all relevant structure forms an inverse semigroup the symmetric inverse semigroup of q nearly all such semigroups however are more than mere inverse semigroups clearly such semigroups possess both an identity 1 and a zero 0 so that one has at least inverse monoids with zero but of greater consequence is the fact that the natural partial order on the semigroup given by x y if and only if y yy x xy y usually possesses infima of arbitrary non empty subsets when this occurs every element x has a fixed point idempotent f x 1 ax which is maximal among those idempotents lying beneath x in the natural partial order for example in the symmetric inverse semigroup on a set x infima are given by intersections inf e olj h e h while the fixed point idempotent f a of a partial bijection a is the identity map of its set of fixed points what is more multiplication always distributes over the induced meet operation so that one obtains a type of semiring structure these observations are an indication that the class of inverse monoids which are also meet semilattices with respect to their natural partial order are worthy of further investigation it is the purpose of this paper to initiate just such a study to do so it will be convenient to have a single term to denote this class of algebraic structures the term adopted in this paper is inverse algebra with algebra being understood in the sense of universal algebra the term should not be confused with the semigroup algebra k s of an inverse semigroup s over a field the paper is divided into six sections with the first presenting examples and results which are foundational to the rest of the paper congruences are considered in the next section where congruences of inverse algebras are characterized as those monoid congruences that are generated from their restriction to the subalgebra of idempotents section 3 studies a number of universal constructions including the relatively free generation of inverse algebras from inverse monoids and completions of inverse algebras the structure and consequent classification of monogenic inverse algebras are given

on a problem of common approximate fixed points the nonstandard analysis methods are used to propose some new ideas concerning the problem of common approximate fixed points for a commuting family of nonexpansive mappings

subgroups of the group of homeomorphisms of the cantor space and a duality between a class of inverse monoids and a class of hausdorff etale groupoids abstract under non commutative stone duality there is a correspondence between second countable hausdorff etale groupoids which have a cantor space of identities and what we call tarski inverse monoids that is countable boolean inverse monoids with semilattices of idempotents which are countable and atomless tarski inverse monoids are therefore the algebraic counterparts of the etale groupoids studied by matui and provide a natural setting for many of his calculations under this duality we prove that natural properties of the etale groupoid correspond to natural algebraic properties of the tarski inverse monoid effective groupoids correspond to fundamental tarski inverse monoids and minimal groupoids correspond to 0 simplifying tarski inverse monoids particularly interesting are the principal groupoids which correspond to tarski inverse monoids where every element is a finite join of infinitesimals and idempotents here an infinitesimal is simply a non zero element with square zero the groups of units of fundamental tarski inverse monoids generalize the finite symmetric groups and include amongst their number the thompson groups g n 1 as well as the groups of units of af inverse monoids krieger s ample groups being examples

on fixed point sets of nonexpansive mappings in nonstandard hulls and banach space ultrapowers abstract we present a study on the structure of fixed point sets of nonexpansive mappings in nonstandard hulls banach space ultrapowers initiated by the result of maurey 1981 about metric convexity of fix t our main motivation is a long standing problem concerning the existence of common approximate fixed points for a family of commuting nonexpansive mappings

an adaptive timestepping algorithm for stochastic differential equations we introduce a variable timestepping procedure using local error control for the pathwise strong numerical integration of a system of stochastic differential equations forced by a single wiener process the milstein method is used to advance the numerical solution and the stepsizes are determined via two local error estimates that roughly correspond to leading order deterministic and stochastic local error components one advantage of using two error controls is an increased flexibility that allows for the treatment of both drift and diffusion dominated regimes in a consistent manner numerical results are presented and the generalization of this approach to wider classes of problems and methods is discussed

topological full groups of one sided shifts of finite type we explore the topological full group g of an essentially principal etale groupoid g on a cantor set when g is minimal we show that g and its certain normal subgroup is a complete invariant for the isomorphism class of the etale groupoid g furthermore when g is either almost finite or purely infinite the commutator subgroup d g is shown to be simple the etale groupoid g arising from a onesided irreducible shift of finite type is a typical example of a purely infinite minimal groupoid for such g g is thought of as a generalization of the higman thompson group we prove that g is of type f and so in particular it is finitely presented this gives us a new infinite family of finitely presented infinite simple groups also the abelianization of g is calculated and described in terms of the homology groups of g

neocompact sets and the fixed point property abstract we use a newly introduced concept of neocompactness to study problems from metric fixed point theory in particular we give a sufficient condition for a superreflexive banach space x to have the fixed point property and obtain shorter proofs of some well known results in that theory