stability in the cuntz semigroup of a commutative c algebra let a be ac algebra the cuntz semigroup w a is an analogue for positive elements of the semigroup v a of murray von neumann equivalence classes of projections in matrices over a we prove stability theorems for the cuntz semigroup of a commutative c algebra which are analogues of classical stability theorems for topological vector bundles over compact hausdorff spaces let sdg denote the class of simple unital and infinite dimensional ah algebras with slow dimension growth and let a be an element of sdg we apply our stability theorems to obtain the following i a has strict comparison of positive elements ii w a is recovered functorially from the elliott invariant of a iii the lower semicontinuous dimension functions on a are weak dense in the dimension functions on a iv the dimension functions on a form a choquet simplex statement ii confirms a conjecture of perera and the author while statements iii and iv confirm for sdg conjectures of blackadar and handelman from the early 1980s

multifractal anisotropic scaling of the hydraulic conductivity 1 we analyzed the scaling properties of the hydraulic conductivity k at three sites in northern america made borden and cape cod we found that k at all sites exhibits multifractality fractal and multiscaling in both the vertical and horizontal directions though the multiscaling was within a range smaller than that of the maximum distance between measurements in the vertical direction the k data for made borden and cape cod were multiscaling from 0 15 to 1 35 m 0 05 to 0 5 m and 0 15 to 0 9 m respectively they were multiscaling in the horizontal direction from 9 to 45 m 1 to 10 m and 1 to 17 m respectively the multiscaling was also anisotropic evidence of scaling was poorest for the horizontal direction of the made site and it spanned half an order of magnitude such results compel one to treat multifractality in the horizontal direction of made as supported by heuristic arguments rather than by pure statistical evaluation we fitted a multifractal model to the data and estimated its parameters we found the underlying statistics of all data to be non gaussian and the model capable of reproducing the probability distribution of k data especially the negative skewness of log k we also generated two dimensional isotropic multifractal fields illustrating the role of the parameters of the selected multifractal model

riemannian symmetric superspaces and their origin in random matrix theory gaussian random matrix ensembles defined over the tangent spaces of the large families of cartan s symmetric spaces are considered such ensembles play a central role in mesoscopic physics as they describe the universal ergodic limit of disordered and chaotic single particle systems the generating function for the spectral correlations of each ensemble is reduced to an integral over a riemannian symmetric superspace in the limit of large matrix dimension such a space is defined as a pair g h m r where g h is a complex analytic graded manifold homogeneous with respect to the action of a complex lie supergroup g and m r is a maximal riemannian submanifold of the support of g h

anomalous diffusion exponents in continuous two dimensional multifractal media we study diffusion in heterogeneous multifractal continuous media that are characterized by the second order dimension of the multifractal spectrum d 2 while the fractal dimension of order 0 d 0 is equal to the embedding euclidean dimension 2 we find that the mean anomalous and fracton dimensions d mathrm w and d mathrm s are equal to those of homogeneous media showing that on average the key parameter is the fractal dimension of order 0 d 0 equal to the euclidean dimension and not to the correlation dimension d 2 beyond their average the anomalous diffusion and fracton exponents d mathrm w and d mathrm s are highly variable and consistently range in the interval 1 4 d mathrm w can be consistently either larger or lower than 2 indicating possible subdiffusive and superdiffusive regimes on a realization basis we show that the exponent variability is related to the local conductivity at the medium inlet through the conductivity scaling

multifractal scaling of the intrinsic permeability existing fractal studies dealing with subsurface heterogeneity treat the logarithm of the permeability k as the variable of concern we treat k as a multifractal and investigate its scaling and fractality using measured horizontal k data from two locations in the united states the first data set was from a shoreline sandstone near coalinga california and the second was from an eolian sandstone goggin 1988 by applying spectral analyses and computing the scaling of moments of various orders using the double trace moment method lavallee 1991 lavallee et al 1992 we found that k is multiscaling i e scaling and multifractal we also found that the so called universal multifractal um schertzer and lovejoy 1987 model essentially a log levy multifractal was able to reproduce the multiscaling behavior reasonably well the um model has three parameters a s and h representing the multifractality index the codimension of the mean field and the distance to stationary multifractal respectively we found a 5 1 7 s 5 0 23 h 5 0 22 and a 5 1 6 s 5 0 11 h 5 0 075 for the shoreline and eolian data sets respectively the fact that a values were less than 2 indicates that the underlying statistics are non gaussian we generated stationary and nonstationary multifractals and illustrated the role of the um parameters on simulated fields studies that treated log k as the variable of concern have pointed out the necessity for large data records especially when the underlying distribution is levy stable our investigation revealed that even larger data records are required when treating k as a multifractal because log k is less intermittent or irregular than k

simple c algebras with perforation abstract simple c algebras are constructed for which the ordered k 0 group is not weakly unperforated the algebras are approximately homogeneous and do not have slow dimension growth

on a simple unital projectionless c algebra we construct a unital separable c algebra z as an analog of the hyperfinite type ii 1 factor besides being nuclear simple projectionless and infinite dimensional z has a unique tracial state and is kk equivalent to inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i the algebra of complex numbers it is shown that unital endomorphisms on z are approximately inner and that z is isomorphic to the infinite tensor product of its replicas it is also shown that a z a for certain interesting classes of unital simple nuclear c algebras a of real rank zero

the infinite number of generalized dimensions of fractals and strange attractors abstract we show that fractals in general and strange attractors in particular are characterized by an infinite number of generalized dimensions dq q 0 to this aim we develop a rescaling transformation group which yields analytic expressions for all the quantities dq we prove that lim q 0 dq fractal dimension d limq 1dq information dimension and dq 2 correlation exponent v dq with other integer q s correspond to exponents associated with ternary quaternary and higher correlation functions we prove that generally dq dq for any q q for homogeneous fractals dq dq a particularly interesting dimension is dq for two examples feigenbaum attractor generalized baker s transformation we calculate the generalized dimensions and find that d is a non trivial number all the other generalized dimensions are bounded between the fractal dimension and d

c algebras of real rank zero abstract the concept of real rank of a c algebra is introduced as a non commutative analogue of dimension it is shown that real rank zero is equivalent to the previously defined conditions fs and hp and that it is invariant under strong morita equivalence in particular under stable isomorphism real rank zero is also invariant under inductive limits and split extensions and the class may well be regarded as the conceptual completion of the af algebras in some cases notably when the algebra is matroid it is shown that the multiplier algebra also has real rank zero although that is not true in general by a result of g j murphy this implies a weyl von neumann type result for self adjoint multiplier elements in these cases

dyson maleev representation of nonlinear sigma models for nonlinear sigma models in the unitary symmetry class the nonlinear target space can be parameterized with cubic polynomials this choice of coordinates has been known previously as the dyson maleev parameterization for spin systems and we show that it can be applied to a wide range of sigma models the practical use of this parameterization includes simplification of diagrammatic calculations in perturbative methods and of algebraic manipulations in non perturbative approaches we illustrate the use and specific issues of the dyson maleev parameterization with three examples the keldysh sigma model for time dependent random hamiltonians the supersymmetric sigma model for random matrices and the supersymmetric transfer matrix technique for quasi one dimensional disordered wires we demonstrate that nonlinear sigma models of unitary like symmetry classes c and b d also admit the dyson maleev parameterization

dimension and stable rank in the k theory of c algebras in topological k theory which can be viewed as the algebraic side of the theory of vector bundles some of the interesting properties which one investigates are for example the conditions under which bundles must possess trivial direct summands or the extent to which the cancellation property for direct whitney sums of bundles holds such properties turn out to be controlled in part by the dimension of the base space and results describing the nature of this control are among what are frequently called stability results 17 during the past few years there has been a sudden flowering of a theory for c algebras see surveys 35 36 37 since c algebras are profitably thought of as non commutative locally compact spaces with the finitely generated projective modules being the appropriate generalization of vector bundles according to swan s theorem 40 it would be natural to look for stability results for c algebras but until now the theory has been focussed on the c groups of c algebras and there has been little discussion of stability properties presumably in part for lack of an appropriate concept of dimension for c algebras one of the main objectives of this paper is to introduce a concept of dimension for c algebras which directly generalizes the classical concept of dimension for compact spaces 16 20 22 and to develop some techniques for calculating it notably for c algebras which are obtained as crossed product algebras for an action of the group of integers in algebraic k theory which is the generalization of topological c theory to rings a substantial number of stability properties have been discovered 2 39 42 48 within that theory the concept which has played a role most analogous to that of dimension is the bass stable rank but despite its many successes the bass stable rank has often been difficult to calculate even for relatively uncomplicated rings another main objective of the present paper is to make accessible for application to c algebras some of the stability results from algebraic c theory this comes about from the fact that the concept of dimension which we introduce dominates the bass stable rank i was led to seek stability results for c algebras by my desire to determine whether the cancellation property for projective modules holds for the irrational rotation c algebras discussed in 30 which as elliott 11 has suggested can appropriately be called 2 dimensional non commutative tori in more concrete terms the question is whether two projections in such an algebra which have the same trace must be unitarily equivalent the present paper provides the general theory which is needed for the affirmative answer which i obtained in a subsequent paper i will derive the

on the independence of k theory and stable rank for simple c algebras jiang and su and independently elliott discovered a simple nuclear infinite dimensional c algebra z having the same elliott invariant as the complex numbers for a nuclear c algebra a with weakly unperforated k group the elliott invariant of a z is isomorphic to that of a thus any simple nuclear c algebra a having a weakly unperforated k group which does not absorb z provides a counterexample to elliott s conjecture that the simple nuclear calgebras will be classified by the elliott invariant in the sequel we exhibit a separable infinite dimensional stably finite instance of such a non z absorbing algebra a and so provide a counterexample to the elliott conjecture for the class of simple nuclear infinite dimensional stably finite separable c algebras

flat dimension growth for c algebras abstract simple and nuclear c algebras which fail to absorb the jiang su algebra tensorially have settled many open questions in the theory of nuclear c algebras but have been little studied in their own right this is due partly to a dearth of invariants sensitive to differences between such algebras we present two new real valued invariants to fill this void the dimension rank ratio for unital ah algebras and the radius of comparison for unital and stably finite algebras we establish their basic properties show that they have natural connections to ordered k theory and prove that the range of the dimension rank ratio is exhausted by simple algebras this last result shows the class of simple nuclear and non z stable c algebras to be uncountable in passing we establish a theory of moderate dimension growth for ah algebras the existence of which was first supposed by blackadar the minimal instances of both invariants are shown to coincide with the condition of being tracially af among simple unital ah algebras of real rank zero and stable rank one whence they may be thought of as generalised measures of dimension growth we argue that the radius of comparison may be thought of as an abstract version of the dimension rank ratio

scaling properties of disordered multifractals abstract in turbulence the simplest phenomenological models of the energy cascade are multiplicative processes constructed on a regular grid in short m p g they have been used mostly for their simplicity allowing many of their properties to be derived analytically and their capacity to reproduce the scale invariance properties of various geophysical fields however these m p g s suffer from the drawback of lacking translation invariance in their spatial statistics spatial homogeneity and therefore they cannot be fully satisfactory models for geophysical fields in this paper we are interested in finding new construction methods for spatially homogeneous random multifractals we investigate the scaling properties of a new family of gridless models of multifractals

scaling of the effective permeability in multifractal porous media abstract the real space renormalization group method is used to calculate analytically the scaling exponents of the effective absolute permeability in multifractal porous media

the supersymmetric technique for random matrix ensembles with zero eigenvalues the supersymmetric technique is applied to computing the average spectral density near zero energy in the large n limit of the random matrix ensembles with zero eigenvalues b diii odd and the chiral ensembles classes aiii bdi and cii the supersymmetric calculations reproduce the existing results obtained by other methods the effect of zero eigenvalues may be interpreted as reducing the symmetry of the zero energy supersymmetric action by breaking a certain abelian symmetry

representation theory of sl 2 1 abstract in this note we present a complete analysis of finite dimensional representations of the lie superalgebra click to view the mathml source this includes in particular the decomposition of all tensor products into their indecomposable building blocks our derivation makes use of a close relation with the representation theory of click to view the mathml source for which analogous results are described and derived keywords lie superalgebras representation theory

recasting the elliott conjecture let a be a simple unital finite and exact c algebra which absorbs the jiang su algebra mathcal z tensorially we prove that the cuntz semigroup of a admits a complete order embedding into an ordered semigroup which is obtained from the elliott invariant in a functorial manner we conjecture that this embedding is an isomorphism and prove the conjecture in several cases in these same cases mathcal z stable algebras all we prove that the elliott conjecture in its strongest form is equivalent to a conjecture which appears much weaker outside the class of mathcal z stable c algebras this weaker conjecture has no known counterexamples and it is plausible that none exist thus we reconcile the still intact principle of elliott s classification conjecture that mathrm k theoretic invariants will classify separable and nuclear c algebras with the recent appearance of counterexamples to its strongest concrete form

non simple purely infinite c algebras a c algebra a is defined to be purely infiniteif there are no characters on a a nd if for every pair of positive elements a b in a such that b lies in the closed two sided ideal generated by a there exists a sequence rn in a such that rn arn b this definition agrees with the usual definition by j cuntz when a is simple it is shown that the property of being purely infinite is preserved under extensions morita equivalence inductive limits and it passes to quotients and to hereditary sub c algebras it is shown that a is purely infinite for every c algebra a purely infinite c algebras admit no traces and conversely an approximately divisible exact c algebra is purely infinite if it admits no nonzero trace

effective saturated hydraulic conductivity of two dimensional random multifractal fields 0 991 p 0 0001 thisrelationship indicates that hkidecreases with increasing long range spatial correlation among the k values in the field furthermore as hki decreases it becomes increasingly dominated by flow channeling this is because high values of k become more and more clustered as p decreases this approach may prove useful for the prediction of hki from generalized dimensions estimated by multifractal analysis of field measurements of k the results may also be applicable to the design of sampling strategies for multiple small scale slug tests at a given resolution citation koirala s r e perfect r w gentry and j w kim 2008 effective saturated hydraulic conductivity of two dimensional random multifractal fields water resour res 44 w08410 doi 10 1029 2007wr006199

an infinite family of non isomorphic c algebras with identical k theory we exhibit a countably infinite family of simple separable nuclear and mutually non isomorphic c algebras which agree on k theory and traces the algebras do not absorb the jiang su algebra z tensorially answering a question of n c phillips they are also pairwise shape and morita equivalent confirming a conjecture from our earlier work the distinguishing invariant is the radius of comparison a non stable invariant of the cuntz semigroup

flow through porous media with multifractal hydraulic conductivity 1 we make a nonlinear analysis of flow through saturated porous media when the hydraulic conductivity k is an isotropic lognormal field with multifractal scale invariance in this case logk is isotropic gaussian with spectral density where d is the space dimension our main result is that the hydraulic gradient h and specific flow q are also multifractal fields whose renormalization under space contraction involves random rotation of the field and random scaling of its amplitude the scaling properties and marginal distributions of h and q are obtained analytically as functions of the space dimension d and a multifractal parameter of k the codimension ck the fields h and q are anisotropic at large scales but approach isotropy at very small scales using scaling arguments we obtain the effective conductivity of the medium keff as an explicit function of d ck and the scaling range of k