the algebraic theory of quasicrystals with five fold symmetries an algebraic binary operation is introduced into quasicrystals admitting five fold symmetry in terms of this many quasicrystals displaying full pentagonal or icosahedral symmetry are seen to be finitely generated examples are given in dimensions 1 2 3 and 4 the operation of left quasicrystal addition is affine linear the monoid generated by these operators is discussed and a presentation for it given in the generic case

independence properties in subalgebras of ultraproduct ii1 factors let mn be a sequence of finite factors with dim mn and denote by m mn their ultraproduct over a free ultrafilter we prove that if q m is either an ultraproduct q qn of subalgebras qn mn with qn mnqn mn n or the centralizer q b m of a separable amenable subalgebra b m then for any separable subspace x m q m there exists a diffuse abelian von neumann subalgebra in q which is free independent to x relative to q m some related independence properties for subalgebras in ultraproduct ii1 factors are also discussed

groups associated to ii1 factors we extend recent work of the first named author constructing a natural hom semigroup associated to any pair of ii1 factors this semigroup always satisfies cancelation hence embeds into its grothendieck group when the target is an ultraproduct of a mcduff factor e g r this grothendieck group turns out to carry a natural vector space structure in fact it is a banach space with natural actions of outer automorphism groups

on the affine self similarities of the three dimensional penrose pattern we prove that the vertex set of the usual three dimensional penrose tiling admits an infinite number of independent scaling factors and an infinite number of inflation centres more exactly we prove that there exist an infinite number of real numbers and an infinite number of points such that

nonnuclear subalgebras of af algebras we show that any non type i separable unital af algebra b can be modeled from inside by a nonnuclear c algebra and from outside by a nonexact c algebra more precisely there exist unital separable quasidiagonal c algebras a b c of real rank zero stable rank one such that a is nonnuclear c is nonexact and both a and c are asymptotically homotopy equivalent to b in particular a b and c have the same ordered k theory groups hence isomorphic ideal lattices and a and b have affinely homeomorphic trace spaces

amenability tubularity and embeddings into mathcal r omega suppose m is a tracial von neumann algebra embeddable into mathcal r omega the ultraproduct of the hyperfinite ii1 factor and x is an n tuple of selfadjoint generators for m denote by x m k the microstate space of x of order m k we say that x is tubular if for any e 0 there exist m in mathbb n and 0 such that if x 1 ldots x n y 1 ldots y n in gamma x m k gamma then there exists a k k unitary u satisfying ux iu y i 2 epsilon for each 1 i n we show that the following conditions are equivalent m is amenable i e injective x is tubular any two embeddings of m into mathcal r omega are conjugate by a unitary u in mathcal r omega

a new application of random matrices ext c _ mathrm red f_2 is not a group in the process of developing the theory of free probability and free entropy voiculescu introduced in 1991 a random matrix model for a free semicircular system since then random matrices have played a key role in von neumann algebra theory cf v8 v9 the main result of this paper is the following extension of voiculescui s random matrix result let x n 1 x n r be a system of r stochastically independent n i n gaussian self adjoint random matrices as in voiculescui s random matrix paper v4 and let x1 xr be a semi circular system in a c probability space then for every polynomial p in r noncommuting variables lim niaeia p x n 1 o x n r o p x1 xr for almost all o in the underlying probability space we use the result to show that the ext invariant for the reduced c algebra of the free group on 2 generators is not a group but only a semi group this problem has been open since anderson in 1978 found the first example of a c algebra a for which ext a is not a group

convex sets associated to c algebras abstract for a separable unital c algebra a and a separable mcduff ii 1 factor m we show that the space h om w a m of weak approximate unitary equivalence classes of unital homomorphisms a m may be considered as a closed bounded convex subset of a separable banach space a variation on n brown s convex structure h om n r u many separable unital c algebras including all separable unital nuclear c algebras have the property that for any mcduff ii 1 factor m h om w a m is affinely homeomorphic to the trace space of a in general h om w a m and the trace space of a do not share the same data several examples are provided we characterize extreme points of h om w a m in many cases and we give two different conditions one necessary and the other sufficient for extremality in general the universality of c f is reflected in the fact that for any unital separable a h om w a m may be embedded as a face in h om w c f m we also extend brown s construction to apply more generally to h om a m u

goodness of fit testing in mathbb r m based on the weighted empirical distribution of certain nearest neighbor statistics let x1 x be a random sample in r from an unknown density f x recently bickel and breiman have introduced a goodness of fit test for this situation based on the empirical distribution function of the variables wi expt ng x1 v ri i 1 n where g x is the hypothesized density and v r represents the volume of the nearest neighbor sphere centered at xi under the null hypothesis h f x g x the empirical process is asymptotically independent of g in this paper a weighted version of the empirical process is shown to produce tests which are still essentially distribution free under h but in addition have asymptotic power against sequences of alternatives contiguous to g the optimal weight function is obtained as a function of the particular sequence of alternatives chosen and consistency behavior against fixed alternatives is determined monte carlo results illustrate the power performance of the tests for various densities and weight functions

quasicrystals and icosians a family of quasicrystals of dimensions 1 2 3 4 governed by the root lattice e8 is constructed the use of the icosian ring found in the quaternions with coefficients in q square root 5 allows simultaneous interpretation of the construction both in physical space and as a result of the standard cut and projection method in double dimension icosians are seen to provide a natural co ordination scheme for these quasicrystals nested sequences of quasicrystals form systems whose symmetries are all derivable from inflational and reflective symmetries directly related to the arithmetic of the icosians the use of coxeter diagrams clarifies the amazing relationship of e8 and quasicrystal symmetries and leads to the fundamental chain e8 contains implies d6 contains implies a4 contains implies a1 a1 that underlies five fold symmetry in quasicrystals decomposition of quasicrystals into concentric shells and a counting formula for the cardinalities of these shells is discussed

operator algebraic superridigity for sl n ℤ n 3 for n 3 let sln we prove the following superridigity result for in the context of operator algebras let l be the von neumann algebra generated by the left regular representation of let m be a finite factor and let u m be its unitary group let u m be a group homomorphism such that m then either

g model sets and their self similarities the model sets also called cut and project sets first defined by yves meyer in harmonic analysis play a central role in quasicrystal modelling each of them is defined by using a cut and project scheme containing two projectors and a lattice we present a method which can be used to study the self similarities of a model set based on the matrices of these projectors in a basis of the lattice this method also allows one to study the self similarities of the diffraction spectrum of a model set because generally the bragg peaks with intensity above a given threshold also form a model set the diffraction pattern corresponding to a quasicrystal is invariant under a finite group g and the local structure of the quasicrystal can be described by using a finite union of orbits of g called a g cluster the neighbours of each atom belong to some orbits of g and the quasicrystal can be regarded as a union of interpenetrating partially occupied translations of the corresponding g cluster we present a method to obtain a model set called the g model set by starting from a g cluster the experimental diffraction patterns allow one to determine the symmetry group g and high resolution electron microscopy images enable one to choose a g cluster describing the local structure the existing computer programs for the cut and project method allow one to pass directly from the local structure of the quasicrystal to a mathematical model to compute the theoretical diffraction spectrum and to compare it with the experimental data

on the self similarities of a model set we present a method to determine certain self similarities of model sets the collection of spaces and mappings r n 1 r n r k n 2 r k n

inflation centres of the cut and project quasicrystals a rather general form of the conventional cut and project scheme is used to define quasicrystals as point sets in real n dimensional euclidean space the inflation or equivalently the self similarity properties of such quasicrystals are studied here assuming only the convexity of the acceptance window our result is a description of inflation centres of all types in a quasicrystal and a proof that our description is complete there are no other inflation centres for any chosen quasicrystal point internal inflation centre u its inflation properties are given as a set of scaling factors it turns out that the scaling factors form a one dimensional quasicrystal with a u dependent acceptance window scaling window the intersection of the scaling windows associated with all points of a quasicrystal is the one dimensional quasicrystal of universal internal scaling symmetries its acceptance window is the interval external inflation centres of a cut and project quasicrystal are those which are not among quasicrystal points their complete description is given analogically to the description of the internal ones imposing some additional requirements on the scaling factors between any two adjacent quasicrystal points one finds a countable infinity of external inflation centres the scaling factors belonging to any such centre u form an infinite u dependent subset of points of the quasicrystal with acceptance window containing

a multidimensional goodness of fit test based on interpoint distances abstract distributional assumptions can be examined with multidimensional goodness of fit tests we propose a conceptually simple test with an appealing logic and accessible asymptotic properties which is generalizable to a variety of problems and appears to work well against diverse alternatives to test whether a k dimensional random sample x1 x n follows the distribution g consider a triangle formed by two randomly selected data points x i and x j and a variable y g our statistic estimates the likelihood that the side formed by the line from x i to x j is the smallest the middle or the largest side of the triangle

close point spatial tests and their application to random number generators we study statistical tests of uniformity based on thel p distances between them nearest pairs of points forn points generated uniformly over thek dimensional unit hypercube or unit torus the number of distinct pairs at distance no more thant fort 0 is a stochastic process whose initial part after an appropriate transformation and asn 8 is asymptotically a poisson process with unit rate convergence to this asymptotic is slow in the hypercube as soon ask exceeds 2 or 3 due to edge effects but is reasonably fast in the torus we look at the quality of approximation of the exact distributions of the tests statistics by their asymptotic distributions discuss computational issues and apply the tests to random number generators linear congruential generators fail decisively certain variants of the tests as soon asn approaches the square root of the period length

the torus parametrization of quasiperiodic li classes the torus parametrization of quasiperiodic local isomorphism classes is introduced and used to determine the number of elements in such a class with special symmetries or inflation properties the method is explained in an illustrative fashion for some widely used tiling classes with golden mean rescaling namely for the fibonacci chain one dimensional the triangle and penrose patterns two dimensional and for kramer s and danzer s icosahedral tilings three dimensional we obtain a rather complete picture of the orbit structure within these classes and also discuss various general results

sums of functions of nearest neighbor distances moment bounds limit theorems and a goodness of fit test abstract the limiting behavior of sums of functions of nearest neighbor distances is studied for an m dimensional sample a central limit theorem and moment bounds for such sums and an invariance principle for the empirical process of nearest neighbor distances are both established as a consequence the asymptotic behavior of a practicable goodness of fit test is obtained based on nearest neighbor distances

multivariate two sample tests based on nearest neighbors abstract a new class of simple tests is proposed for the general multivariate two sample problem based on the possibly weighted proportion of all k nearest neighbor comparisons in which observations and their neighbors belong to the same sample large values of the test statistics give evidence against the hypothesis h of equality of the two underlying distributions asymptotic null distributions are explicitly determined and shown to involve certain nearest neighbor interaction probabilities simple infinite dimensional approximations are supplied the unweighted version yields a distribution free test that is consistent against all alternatives optimally weighted statistics are also obtained and asymptotic efficiencies are calculated each of the tests considered is easily adapted to a permutation procedure that conditions on the pooled sample power performance for finite sample sizes is assessed in simulations

a new proof of the equivalence of injectivity and hyperfiniteness for factors on a separable hilbert space abstract in 1975 a connes proved the fundamental result that injective factors on a separable hilbert space are hyperfinite in this paper a new proof of this result is presented in which the most technical parts of connes proof are avoided particularly the proof does not rely on automorphism group theory the starting point in this approach is wassermann s simple proof of injective semidiscrete together with choi and effros characterization of semidiscrete von neumann algebras as those von neumann algebras n for which the identity map on n has an approximate completely positive factorization through n n matrices

c algebras associated with groups with kazhdan s property t has exhibited block diagonal operators on separable hilbert spaces which are not norm limits of generators of singly generated finite dimensional c algebras it turns out that voiculescu s construction which is based upon a certain class of groups with property t can be put into a more general context involving c algebras which arise very naturally from such groups it is these c algebras which are the subject of this paper as well as giving a natural setting in which to view voiculescu s result these c algebras have almost fortuitously a number of other important properties which are very easily established like the full c algebras of free groups these c algebras are not

a mathematical construction of n dimensional quasicrystals starting from g clusters we present a mathematical construction of n dimensional quasicrystals by starting from its symmetry group an arbitrary finite group and from its local structure described by using a finite union of orbits of g called a g cluster namely we construct the periodization space the g invariant lattice the decomposition into the physical and internal space in a canonical way using a strip and projection method we obtain quasiperiodic patterns corresponding to various generic local structures

approach for a metric space with a convex combination operation and applications abstract in this study we embed a metric space endowed with a convex combination operation which is called a convex combination space into a banach space where the embedding preserves the structures of the metric and convex combination we also establish applications of this embedding for a random element that takes values in this type of space on the one hand we show some useful properties of mathematical expectation such as the representation of expectation through continuous affine mappings and the linearity of expectation on the other hand the notion of conditional expectation is also introduced and discussed using this embedding theorem we establish some basic properties of conditional expectation jensen s inequality the convergences of martingales and ergodic theorem

bayesian representation of stochastic processes under learning de finetti revisited a probability distribution governing the evolution of a stochastic process has infinitely many bayesian representations of the form d among these a natural representation is one whose components s are learnable one can approximate by conditioning on observation of the process and sufficient for prediction s predictions are not aided by conditioning on observation of the process we show the existence and uniqueness of such a representation under a suitable asymptotic mixing condition on the process this representation can be obtained by conditioning on the tail field of the process and any learnable representation that is sufficient for prediction is asymptotically like the tail field representation this result is related to the celebrated de finetti theorem but with exchangeability weakened to an asymptotic mixing condition and with his conclusion of a decomposition into i i d component distributions weakened to components that are learnable and sufficient for prediction

topological dynamical systems associated to ii1 factors abstract if n r is a separable ii1 factor the space h om n r of unitary equivalence classes of unital homomorphisms n r is shown to have a surprisingly rich structure if n is not hyperfinite h om n r is an infinite dimensional complete metrizable topological space with convex like structure and the outer automorphism group out n acts on it by affine homeomorphisms if n r then h om n r is just a point property t is reflected in the extreme points they re discrete in this case for certain free products n r every countable group acts nontrivially on h om n r and we show the extreme points are not discrete for these examples finally we prove that the dynamical systems associated to free group factors are isomorphic