unitary orbits in 1 dimensional nccw complexes abstract let a be a 1 dimensional nccw complex in this paper it is shown that the usual distance d u defined on the approximate unitary equivalence classes unitary orbits of self adjoints in a is equal to a distance d w defined on morphisms from the cuntz semigroup of c 0 0 1 to cu a in addition another distance d p on approximately unitary equivalence classes related to spectral information of positive elements in the classes is proved to be equal to d u and d w on simple unital inductive limits of 1 dimensional nccw complexes

automatic average case analysis of algorithms abstract many probabilistic properties of elementary discrete combinatorial structures of interest for the average case analysis of algorithms prove to be decidable this paper presents a general framework in which such decision procedures can be developed it is based on a combination of generating function techniques for counting and complex analysis techniques for asymptotic estimations we expose here the theory of exact analysis in terms of generating functions for four different domains the iterative recursive and unlabelled labelled data type domains we then present some major components of the associated asymptotic theory and exhibit a class of naturally arising functions that can be automatically analyzed a fair fragment of this theory is also incorporated into a system called lambda upsilon omega in this way using computer algebra one can produce automatically non trivial average case analyses of algorithms operating over a variety of decomposable combinatorial structures at a fundamental level this paper is part of a global attempt at understanding why so many elementary combinatorial problems tend to have elementary asymptotic solutions in several cases it proves possible to relate entire classes of elementary combinatorial problems whose structure is well defined with classes of elementary special functions and classes of asymptotic forms relative to counting probabilities or average case complexity

classification of c homomorphisms from c0 0 1 to a c algebra abstract a class of c algebras is described for which the c homomorphisms from c 0 0 1 to the algebra may be classified by means of the cuntz semigroup functor examples are given of algebras simple and non simple for which this classification fails it is shown that a suitable suspension of the cuntz semigroup functor deals successfully with some of these counterexamples

the cuntz semigroup the elliott conjecture and dimension functions on c algebras we prove that the cuntz semigroup is recovered functorially from the elliott invariant for a large class of c algebras in particular our re sults apply to the largest class of simple c algebras for which k theoretic classification can be hoped for this work has three significant consequences first it provides new conceptual insight into elliott s classification program proving that the usual form of the elliott conjecture is equivalent among z stable algebras to a conjecture which is in general substantially weaker and for which there are no known counterexamples second and third it resolves for the class of algebras above two conjectures of blackadar and handelman concerning the basic structure of dimension functions on c algebras we also prove in passing that the cuntz pedersen semigroup is recovered functorially from the elliott invariant for a large class of simple unital c algebras

the dixmier property and tracial states for c algebras a t was partially supported by an nserc postdoctoral fellowship and through the epsrc grant ep n00874x 1 acknowledgements we are grateful to luis santiago for helpful discussions at an early stage of this investigation we would also like to thank the referee for providing helpful comments which have led to a number of improvements

nuclear dimension and mathcal z stability simple separable unital monotracial and nuclear mathrm c algebras are shown to have finite nuclear dimension whenever they absorb the jiang su algebra mathcal z tensorially this completes the proof of the toms winter conjecture in the unique trace case

analytic models and ambiguity of context free languages we establish that several classical context free languages are inherently ambiguous by proving that their counting generating functions when considered as analytic functions exhibit some characteristic form of transcendental behaviour to that purpose we survey some general results on elementary analytic properties and enumerative uses of algebraic functions in relation to formal languages in particular the paper contains a general density theorem for unambiguous context free languages

unitary orbits of self adjoint operators in simple z stable c algebras abstract we prove that in a simple unital exact z stable c algebra the distance between the unitary orbits of self adjoint elements with connected spectrum is completely determined by spectral data this fails without the assumption of z stability

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 brownian excursion area a numerical analysis abstract using an explicit expression for the laplace transform of the brownian excursion area generating function a numerical analysis gives moments density and distribution function an asymptotic formula is given for small argument

nuclear dimension mathcal z stability and algebraic simplicity for stably projectionless c algebras the main result here is that a simple separable c algebra is mathcal z stable where mathcal z denotes the jiang su algebra if i it has finite nuclear dimension or ii it is approximately subhomogeneous with slow dimension growth this generalizes the main results of toms invent math 183 2 225 244 2011 and winter invent math 187 2 259 342 2012 to the nonunital setting as a consequence finite nuclear dimension implies mathcal z stability even in the case of a separable c algebra with finitely many ideals algebraic simplicity is established as a fruitful weakening of being simple and unital and the proof of the main result makes heavy use of this concept

closed unitary and similarity orbits of normal operators in purely infinite c algebras abstract we will investigate the norm closure of the unitary and similarity orbits of normal operators in unital simple purely infinite c algebras a simple proof to the classification of when two normal operators are approximately unitarily equivalent in said algebras with trivial k 1 group will be given using previously known techniques using these techniques some upper and lower bounds for the distance between unitary orbits of normal operators will be obtained in addition a complete characterization of when one normal operator is in the closed similarity orbit of another normal operator will be given for unital simple purely infinite c algebras and type iii factors with separable predual

uniform random generation of strings in a context free language let s be the set of all strings of length n generated by a given context free grammar a uniform random generator is one which produces strings from s with equal probability in generating these strings care must be taken in choosing the disjuncts that form the right hand side of a grammar rule so that the produced string will have the specified length uniform random generators have applications in studying the complexity of parsers in estimating the average efficiency of theorem provers for the propositional calculus in establishing a measure of ambiguity of a grammar etc two methods are presented for generating uniform random strings in an unambiguous context free language the first method will generate a random string of length n in linear time but must use a precomputed table of size o n r 1 where r is the number of nonterminals in the grammar used to specify the language the second method precomputes part of the table and calculates the other entries as they are called for it requi

une theorie combinatoire des series formelles abstract this paper presents a combinatorial theory of formal power series the combinatorial interpretation of formal power series is based on the concept of species of structures a categorical approach is used to formulate it a new proof of cayley s formula for the number of labelled trees is given as well as a new combinatorial proof due to g labelle of lagrange s inversion formula polya s enumeration theory of isomorphism classes of structures is entirely renewed recursive methods for computing cycle index polynomials are described a combinatorial version of the implicit function theorem is stated and proved the paper ends with general considerations on the use of coalgebras in combinatorics

spectral variation of normal matrices abstract it has been a durable conjecture that the distance appropriately defined between the spectra of two normal matrices is bounded by the operator norm of their difference we report on some numerical studies that show that this conjecture is in general false in fact even in the 3 x 3 case the spectral distance may exceed the norm distance by between 1 and 2 percent we also describe the background of the conjecture and offer an analysis of extremal counterexamples

perturbation of spectral subspaces and solution of linear operator equations abstract let a and b be normal operators on a hilbert space let k a and k b be subsets of the complex plane at distance at least from each other let e be the spectral projector for a belonging to k a and let f be the spectral projector for b belonging to k b our main results are estimates of the form ef c e a b f in some special situations the constant c is as low as 1 as an application we prove for an absolute constant d that if the space is finite dimensional and if a and b are normal with a b d then the spectrum of b can be obtained from that of a multiplicities counted by moving each eigenvalue by at most our main results have equivalent formulations as statements about the operator equation aq qb s let a and b be normal operators on perhaps different hilbert spaces assume a k a and b k b where k a k b and are as before then we give estimates of the forms q c aq qb

resurrecting the asymptotics of linear recurrences abstract once on the forefront of mathematical research in america the asymptotics of the solutions of linear recurrence equations is now almost forgotten especially by the people who need it most namely combinatorists and computer scientists here we present this theory in a concise form and give a number of examples that should enable the practicing combinatorist and computer scientist to include this important technique in her or his asymptotics tool kit

quasidiagonality of nuclear c algebras we prove that faithful traces on separable and nuclear c algebras in the uct class are quasidiagonal this has a number of consequences firstly by results of many hands the classification of unital separable simple and nuclear c algebras of finite nuclear dimension which satisfy the uct is now complete secondly our result links the finite to the general version of the toms winter conjecture in the expected way and hence clarifies the relation between decomposition rank and nuclear dimension finally we confirm the rosenberg conjecture discrete amenable groups have quasidiagonal c algebras

algebraic languages and polyominoes enumeration abstract in this paper the use of algebraic languages theory in solving an open problem in combinatorics is shown by constructing a bijection between convex polyominoes and words of an algebraic language and by solving the corresponding algebraic system we prove that the number of convex polyominoes with perimeter 2n 8 is 2n 11 4 n 4 2n 1 2n n

nuclear dimension and mathcal z stability of pure c algebras in this article i study a number of topological and algebraic dimension type properties of simple c algebras and their interplay in particular a simple c algebra is defined to be tracially m bar m pure if it has strong tracial m comparison and is tracially bar m almost divisible these notions are related to each other and to nuclear dimension

recognizable formal power series on trees trees are a very basic object in computer science they intervene in nearly a ly domain and they are studied for their own or used to represent conveniently a given situation there are at least three directions where investigations on trees themselves are motivated and this for different reasons first the notion of tree is the basis of algebraic semantics nivat 19 rosen 2 2 etc in this context the study of special languages of trees i e forests their classification and their behaviour under various types of transformations are of great importance arnold fl dauchet 8 j l ilin 1161 mongy 18 by essence work in this area is an extension of the algebraic theory of languages trees and ianguages are in fact directly related via the derivation trees of an algebraic grammar thatcher 25 a second topic heavily related to trees concerns dzra structures trees mainly binary trees and its variants constitute one of the most widely known data structures see e g hlnuth 15 the analysis of the worst cale expected or average running time t ehzvgrr nfi certain algorithms requires sometimes long and delicate computations flajolet lo kemp 14 flajolet and steyaert 12 finally trees occupy a distinguished place in the enumeration of graphs and maps both because of the simplicity of their structure and for the relationship between their el lcodings and aigebraic languages the nature of the enumerating series and especially the question whether they are algebraic or not is one of the central problems in this domain cori 7 chottin 4 we propose here a theory of formal power series on trees and present some of their basic properties together with various examples of applications which as we hope will show the interest of its development within the framework we just sketched a formal power series on trees is a function which gssociates a number to esch tree thus we could also have called them tree functions ii in analogy with the term word function used by several authors paz and salomaa zo cobham 5 as an equivalent denomination for formal power series on words the main goal of a formal power series is to count or to represent the result of some computation on

classification of inductive limits of 1 dimensional nccw complexes abstract a classification result is obtained for the c algebras that are stably isomorphic to inductive limits of 1 dimensional noncommutative cw complexes with trivial k 1 group the classifying functor cu is defined in terms of the cuntz semigroup of the unitization of the algebra for the simple c algebras covered by the classification cu reduces to the ordered k 0 group the cone of traces and the pairing between them as an application of the classification it is shown that the crossed products by a quasi free action o 2 r are all isomorphic for a dense set of positive irrational numbers

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

the cone of lower semicontinuous traces on a c algebra the cone of lower semicontinuous traces is studied with a view to its use as an invariant its properties include compactness hausdorffness and continuity with respect to inductive limits a suitable notion of dual cone is given the cone of lower semicontinuous 2 quasitraces on a non exact c algebra is considered as well these results are applied to the study of the cuntz semigroup it is shown that if a c algebra absorbs the jiang su algebra then the subsemigroup of its cuntz semigroup consisting of the purely non compact elements is isomorphic to the dual cone of the cone of lower semicontinuous 2 quasitraces this yields a computation of the cuntz semigroup for the following two classes of c algebras c algebras that absorb the jiang su algebra and have no non zero simple subquotients and simple c algebras that absorb the jiang su algebra 1 introduction the most standard invariants in the classification of nu clear c algebras are their k groups and their traces the traces at least for simple c algebras are assumed to be bounded in the unital case and lower semicontinuous and densely finite in the non unital case if one has in mind the classification of non simple c algebras it is clear that these two kinds of traces will not suffice and a broader class should be considered in this paper we study the properties of the cone of all lower semicontinuous traces on a c algebra with the purpose of applying our results to questions in the classification of non simple c algebras one sign that this is natural is that this cone in particular determines the lattice of closed two sided ideals an important invariant in its own right see 3 4 we shall also consider lower semicontinuous 2 quasitraces since these appear naturally as functionals on the cuntz semigroup of the algebra if the