strongly self absorbing c algebras say that a separable unital c algebra v c is strongly self absorbing if there exists an isomorphism v such that lx ai e approximately unitarily equivalent homomorphisms we study this class of algebras which includes the cuntz algebras 2 ooo the uhf algebras of infinite type the jiang su algebra z and tensor products of 00 with uhf algebras of infinite type given a strongly self absorbing c algebra v we characterise when a separable c algebra absorbs v tensorially i e is p stable and prove closure properties for the class of separable testable c algebras finally we compute the possible if groups and prove a number of classification results which suggest that the examples listed above are the only strongly self absorbing c algebras

divisibility properties for c algebras we consider three notions of divisibility in the cuntz semigroup of a c algebra and show how they reflect properties of the c algebra we develop methods to construct simple and non simple c algebras with specific divisibility behaviour as a byproduct of our investigations we show that there exists a sequence an of simple unital infinite dimensional c algebras such that the product n 1an has a character

liftings of diagrams of semilattices by diagrams of dimension groups we investigate categorical and amalgamation properties of the functor idc assigning to every partially ordered abelian group g its semilattice of compact ideals idc g our main result is the following theorem 1 every diagram of finite boolean semilattices indexed by a finite dismantlable partially ordered set can be lifted with respect to the idc functor by a diagram of pseudo simplicial vector spaces pseudo simplicial vector spaces are a special kind of finite dimensional partially ordered vector spaces over the rationals with interpolation the methods introduced make it also possible to prove the following ring theoretical result theorem 2 for any countable distributive join semilattices s and t and any field k any v 0 homomorphism f s to t can be lifted with respect to the idc functor on rings by a homomorphism f a to b of k algebras for countably dimensional locally matricial algebras a and b over k we also state a lattice theoretical analogue of theorem 2 with respect to the conc functor and we provide counterexamples to various related statements in particular we prove that the result of theorem 1 cannot be achieved with simplicial vector spaces alone

clustering spatiotemporal data an augmented fuzzy c means in spatiotemporal data commonly encountered in geographical systems biomedical signals and the like each datum is composed of features comprising a spatial component and a temporal part clustering of data of this nature poses challenges especially in terms of a suitable treatment of the spatial and temporal components of the data in this study proceeding with the objective function based clustering such as e g fuzzy c means we revisit and augment the algorithm to make it applicable to spatiotemporal data an augmented distance function is discussed and the resulting clustering algorithm is provided two optimization criteria i e a reconstruction error and a prediction error are introduced and used as a vehicle to optimize the performance of the clustering method experimental results obtained for synthetic and real world data are reported

projection decomposition in multiplier algebras in this paper we present new structural information about the multiplier algebra mathcal m mathcal a of a unital purely infinite simple c algebra mathcal a by characterizing the positive elements a in mathcal m mathcal a that are strict sums of projections belonging to mathcal a if a not in mathcal a and a itself is not a projection then the necessary and sufficient condition for a to be a strict sum of projections belonging to mathcal a is that a 1 and that the essential norm a ess geq 1 based on a generalization of the perera rordam weak divisibility of separable simple c algebras of real rank zero to all unital simple c algebras of real rank zero we show that every positive element of mathcal a with norm 1 can be approximated by finite sums of projections based on block tri diagonal approximations we decompose any positive element a in mathcal m mathcal a with a 1 and a ess geq 1 into a strictly converging sum of positive elements in mathcal a with norm 1

time series k means existing clustering algorithms are weak in extracting smooth subspaces for clustering time series data in this paper we propose a new k means type smooth subspace clustering algorithm named time series k means tskmeans for clustering time series data the proposed tskmeans algorithm can effectively exploit inherent subspace information of a time series data set to enhance clustering performance more specifically the smooth subspaces are represented by weighted time stamps which indicate the relative discriminative power of these time stamps for clustering objects the main contributions of our work include the design of a new objective function to guide the clustering of time series data and the development of novel updating rules for iterative cluster searching with respect to smooth subspaces based on a synthetic data set and five real life data sets our experimental results confirm that the proposed tskmeans algorithm outperforms other state of the art time series clustering algorithms in terms of common performance metrics such as accuracy fscore randindex and normal mutual information

modular units and cuspidal divisor classes on x0 n2m with n 24 and m squarefree abstract for a positive integer n let c n be the subgroup of j 0 n generated by the equivalence classes of cuspidal divisors of degree 0 and c n q c n j 0 n q be its q rational subgroup let also c q n be the subgroup of c n q generated by q rational cuspidal divisors we prove that when n n 2 m for some integer n dividing 24 and some squarefree integer m the two groups c n q and c q n are equal to achieve this we show that all modular units on x 0 n on such n are products of functions of the form m k h m h 2 n and k z and determine the necessary and sufficient conditions for products of such functions to be modular units on x 0 n

cuspidal class number of the tower of modular curves x1 npn we consider a cuspidal class number which is the order of a subgroup of the full cuspidal divisor class group of x1 npn with p nmid n and n 1 by studying the second generalized bernoulli numbers we obtain results similar to ones ferrero and washington in ann math 2 109 2 377 395 1979 washington in invent math 49 87 97 1978 about the relative class numbers of cyclotomic mathbb z p extension of an abelian number field

integrality of a ratio of petersson norms and level lowering congruences we prove integrality of the ratiof f g g outside an explicit finite set of primes where g is an arithmetically normalized holomorphic newform on a shimura curve f is a normalized hecke eigenform on gl 2 with the same hecke eigenvalues as g and denotes the petersson inner product the primes dividing this ratio are shown to be closely related to certain level lowering con gruences satisfied by f and to the central values of a family of rankin selberg l functions finally we give two applications the first to proving the integral ity of a certain triple product l value and the second to the computation of the faltings height of jacobians of shimura curves

a distributive semilattice not isomorphic to the maximal semilattice quotient of the positive cone of any dimension group abstract we construct a distributive 0 semilattice which is not isomorphic to the maximal semilattice quotient of the positive cone of any dimension group the size of the semilattice is 2

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

the cuspidal class number formula for the modular curves x1 pm abstract the purpose of the paper is to determine the cuspidal class number of the modular curve x 1 2 2 n 1 it is a continuation of the previous work of the determination of the cuspidal class number for the modular curves x 1 p m for the case p 2

non simple purely infinite c algebras the hausdorff case abstract local and global definitions of pure infiniteness for a c algebra a are compared and equivalence between them is obtained if the primitive ideal space of a is hausdorff and of finite dimension if a has real rank zero or if a is approximately divisible sufficient criteria are given for local pure infiniteness of tensor products they yield that exact simple tensorially non prime c algebras are purely infinite if they have no semi finite lower semi continuous trace one obtains that a is isomorphic to a o if a is 1 purely infinite separable stable nuclear and prim a is a hausdorff space not necessarily of finite dimension

infinite non simple c algebras absorbing the cuntz algebra o abstract the first named author has given a classification of all separable nuclear c algebras a that absorb the cuntz algebra o we say that a absorbs o if a is isomorphic to a o motivated by this classification we investigate here if one can give an intrinsic characterization of c algebras that absorb o this investigation leads us to three different notions of pure infiniteness of a c algebra all given in terms of local algebraic conditions on the c algebra the strongest of the three properties strongly purely infinite is shown to be equivalent to absorbing o for separable nuclear c algebras that either are stable or have an approximate unit consisting of projections in a previous paper 2000 amer j math 122 637 666 we studied an intermediate and perhaps more natural condition purely infinite that extends a well known property for simple c algebras the weakest condition of the three weakly purely infinite is shown to be equivalent to the absence of quasitraces in an ultrapower of the c algebra the three conditions may be equivalent for all c algebras and we prove this to be the case for c algebras that are either simple of real rank zero or approximately divisible

af embeddings into c algebras of real rank zero abstract it is proved that every separable c algebra of real rank zero contains an af sub c algebra such that the inclusion mapping induces an isomorphism of the ideal lattices of the two c algebras and such that every projection in a matrix algebra over the large c algebra is equivalent to a projection in a matrix algebra over the af sub c algebra this result is proved at the level of monoids using that the monoid of murray von neumann equivalence classes of projections in a c algebra of real rank zero has the refinement property as an application of our result we show that given a unital c algebra a of real rank zero and a natural number n then there is a unital homomorphism mn1 mnr a for some natural numbers r n1 nr with nj n for all j if and only if a has no representation of dimension less than n

on topologically finite dimensional simple c algebras we show that if a simple c algebra a is topologically finite dimensional in a suitable sense then not only k0 a has certain good properties but a is even accessible to elliott s classification program more precisely we prove the following results

summarizing a set of time series by averaging from steiner sequence to compact multiple alignment summarizing a set of sequences is an old topic that has been revived in the last decade due to the increasing availability of sequential datasets the definition of a consensus object is on the center of data analysis issues since it crystallizes the underlying organization of the data dynamic time warping dtw is currently the most relevant similarity measure between sequences for a large panel of applications since it makes it possible to capture temporal distortions in this context averaging a set of sequences is not a trivial task since the average sequence has to be consistent with this similarity measure the steiner theory and several works in computational biology have pointed out the connection between multiple alignments and average sequences taking inspiration from these works we introduce the notion of compact multiple alignment which allows us to link these theories to the problem of summarizing under time warping having defined the link between the multiple alignment and the average sequence the second part of this article focuses on the scan of the space of compact multiple alignments in order to provide an average sequence of a set of sequences we propose to use a genetic algorithm based on a specific representation of the genotype inspired by genes this representation of the genotype makes it possible to consistently paint the fitness landscape experiments carried out on standard datasets show that the proposed approach outperforms existing methods

nonsmooth analysis and subgradient methods for averaging in dynamic time warping spaces abstract time series averaging in dynamic time warping dtw spaces has been successfully applied to improve pattern recognition systems this article proposes and analyzes subgradient methods for the problem of finding a sample mean in dtw spaces the class of subgradient methods generalizes existing sample mean algorithms such as dtw barycenter averaging dba we show that dba is a majorize minimize algorithm that converges to necessary conditions of optimality after finitely many iterations empirical results show that for increasing sample sizes the proposed stochastic subgradient ssg algorithm is more stable and finds better solutions in shorter time than the dba algorithm on average therefore ssg is useful in online settings and for non small sample sizes the theoretical and empirical results open new paths for devising sample mean algorithms nonsmooth optimization methods and modified variants of pairwise averaging methods

a solution to dilworth s congruence lattice problem we construct a distributive algebraic lattice d that is not isomorphic to the congruence lattice of any lattice this solves a long standing open problem traditionally attributed to r p dilworth from the forties the lattice d has compact top element and aleph omega 1 compact elements our results extend to all algebras possessing a polynomially definable structure of a join semilattice with a largest element

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

modular units and cuspidal divisor class groups of x 1 n abstract in this article we consider the group f 1 n of modular units on x 1 n that have divisors supported on the cusps lying over of x 0 n called the cusps for each positive integer n we will give an explicit basis for the group f 1 n this enables us to compute the group structure of the rational torsion subgroup c 1 n of the jacobian j 1 n of x 1 n generated by the differences of the cusps in addition based on our numerical computation we make a conjecture on the structure of the p primary part of c 1 p n for a regular prime p

purely infinite corona algebras of simple c algebras in this paper we study the problem of when the corona algebra of a non unital c algebra is purely infinite a complete answer is obtained for stabilisations of simple and unital algebras that have enough comparison of positive elements our result relates the pure infiniteness condition from its strongest to weakest forms to the geometry of the tracial simplex of the algebra and to the behaviour of corona projections despite the fact that there is no real rank zero condition

statistical graph space analysis the sample mean is one of the most fundamental concepts in statistics properties of the sample mean that are well defined in euclidean spaces become unclear in graph spaces this paper proposes conditions under which the following properties are valid existence uniqueness and consistency of means the midpoint property necessary conditions of optimality and convergence results of mean algorithms the theoretical results address common misconceptions about the graph mean in graph edit distance spaces serve as a first step towards a statistical analysis of graph spaces and result in a theoretically well founded mean algorithm that outperformed six other mean algorithms with respect to solution quality on different graph datasets representing images and molecules highlightsa first step towards a theory of statistical graph space analysis is proposed mmm algorithm is proposed that outperformed six other mean algorithms necessary conditions of optimality are proved convergence of mmm algorithm is shown basic statistical and geometrical properties are shown

simple riesz groups of rank one having wild intervals abstract we prove that every partially ordered simple group of rank one which is not riesz embeds into a simple riesz group of rank one if and only if it is not isomorphic to the additive group of the rationals using this result we construct examples of simple riesz groups of rank one g containing unbounded intervals d n n 1 and d that satisfy a for each n 1 t d n g for every t q n but q n d n g where q n is a sequence of relatively prime integers b for every n 1 n d g we sketch some potential applications of these results in the context of k theory

dimension growth for c algebras we introduce the growth rank of a c algebra a valued invariant whose minimal instance is equivalent to the condition that an algebra absorbs the jiang su algebra tensorially and prove that its range is exhausted by simple nuclear c algebras as consequences we obtain a well developed theory of dimension growth for approximately homogeneous ah c algebras establish the existence of simple nuclear and non stable c algebras which are not tensorially prime and show the assumption of stability to be particularly natural when seeking classification results for nuclear c algebras via k theory the properties of the growth rank lead us to propose a universal property which can be considered inside any class of unital and nuclear c algebras we prove that satisfies this universal property inside a large class of locally subhomogeneous algebras representing the first uniqueness theorem for which does not depend on the classification theory of nuclear c algebras

a global averaging method for dynamic time warping with applications to clustering mining sequential data is an old topic that has been revived in the last decade due to the increasing availability of sequential datasets most works in this field are centred on the definition and use of a distance or at least a similarity measure between sequences of elements a measure called dynamic time warping dtw seems to be currently the most relevant for a large panel of applications this article is about the use of dtw in data mining algorithms and focuses on the computation of an average of a set of sequences averaging is an essential tool for the analysis of data for example the k means clustering algorithm repeatedly computes such an average and needs to provide a description of the clusters it forms averaging is here a crucial step which must be sound in order to make algorithms work accurately when dealing with sequences especially when sequences are compared with dtw averaging is not a trivial task starting with existing techniques developed around dtw the article suggests an analysis framework to classify averaging techniques it then proceeds to study the two major questions lifted by the framework first we develop a global technique for averaging a set of sequences this technique is original in that it avoids using iterative pairwise averaging it is thus insensitive to ordering effects second we describe a new strategy to reduce the length of the resulting average sequence this has a favourable impact on performance but also on the relevance of the results both aspects are evaluated on standard datasets and the evaluation shows that they compare favourably with existing methods the article ends by describing the use of averaging in clustering the last section also introduces a new application domain namely the analysis of satellite image time series where data mining techniques provide an original approach

on the stable rank of simple c algebras in 16 m a rieffel introduced the notion of stable rank for c algebras for a a unital c algebra the stable rank denoted by sr a is the least integer n such that the set of n tuples over a which generate a as a left ideal is dense in an if no such n exists we set sr a oo for unital c algebras the stable rank was shown to coincide with the bass stable rank which is defined for rings see 12 the stable rank can also be defined for nonunital c algebras but we shall only consider the unital case stable rank is related to nonstable k theory in 16 theorem 10 12 and 3 corollary 7 14 it was shown that if a is a simple unital c algebra of stable rank one then the natural map u a u a o 4 ki a is an isomorphism see also 17 2 and 18 it is a natural question which values occur as the stable rank of simple c algebras suppose that a is simple and infinite then by 7 proposition 1 5 a contains two isometries with orthogonal ranges and so by 16 proposition 6 5 sr a oo in the case of finite simple c algebras the following is known whenever a is simple and stably finite and b is a uhf algebra the tensor product a0 b has stable rank one see 19 this is also the case for every simple approximately homogeneous ah algebra with slow dimension growth see 4 and 8 so in short sr oo for every infinite simple c algebra and sr 1 for various classes of finite simple c algebras in this paper we answer the question above by showing that every n c n u oo is the stable rank of some simple finite c algebra more precisely it is shown that for every n 2 3 4 oo there exists a simple separable and unital ah algebra a of stable rank n in particular the group of invertible elements in a is not dense and like the examples in 20 a does not have slow dimension growth and is not approximately divisible see 5 i thank marius dadarlat for suggesting that i study ah algebras with connecting homomorphisms as described in the following section and juliana erlijman for drawing my attention to some of the powerful tools of differential topology and i thank george elliott for my stay at the fields institute where this paper was written

z stability and infinite tensor powers of c algebras we prove that under a mild hypothesis the infinite tensor power of a unital separable c algebra absorbs the jiang su algebra z tensorially combining this result with a recent theorem of winter we complete elliott s classification program for strongly self absorbing ash algebras we also give a succinct universal property for z in an ambient category so large that there are no unital separable c algebras without characters that are known to lie outside it this category contains the vast majority of our stock in trade separable amenable c algebras and is closed under passage to quotients and separable superalgebras in particular the category is closed under the formation of unital direct limits unital tensor products and crossed products by countable discrete groups finally we take a significant step toward the confirmation of elliott s classification conjecture for the c algebras of minimal diffeomorphisms

the corona factorization property and refinement monoids the corona factorization property of a c algebra originally defined to study extensions of c algebras has turned out to say something important about intrinsic struc tural properties of the c algebra we show in this paper that a unital c algebra a of real rank zero has the corona factorization property if and only if its monoid v a of murray von neumann equivalence classes of projections in matrix algebras over a has a certain rather weak comparability property that we call the corona factorization property for monoids we show that a projection in such a c algebra is properly infinite if and only if a multiple of it is properly infinite the latter result is obtained from some more general result we establish about conical refinement monoids we show that the set of order units together with the zero element in a conical refinement monoid is again a refinement monoid under the assumption that the monoid satisfies weak divisibility and if u is an element in a refinement monoid such that nu is properly infinite then u can be written as a sum u s t such that ns and nt are properly infinite

exact solution to median surface problem using 3d graph search and application to parameter space exploration in this paper we propose the generalized median surface problem and present an exact solution by applying a 3d graph search algorithm to our knowledge this is a novel class of median computation problem which has not been considered before in the literature in addition to the theoretic interest in median surface computation we also demonstrate its practical value by means of the task of parameter space exploration without ground truth which is an effective means of dealing with the difficult parameter selection problem for image segmentation we present a concrete application for artery boundary detection in sonography ultrasound imaging it will be shown that the median computation can not only avoid the parameter training but also potentially achieve results comparable with trained parameters the median based approach can thus be a good alternate particularly in situations with no ground truth available highlightsthis paper proposes the median surface problem an optimal graph search algorithm is applied to solve the problem efficiently experimental work is conducted to demonstrate its usability it can potentially be used for segmentation parameter exploration

on some arithmetic properties of siegel functions we deal with several arithmetic properties of the siegel functions which are modular units by modifying the ideas in kubert and lang modular units grundlehren der mathematischen wissenschaften vol 244 spinger heidelberg 1981 we establish certain criterion for determining a product of siegel functions to be integral over mathbb z j we also find generators of the function fields mathcal k x 1 n by examining the orders of siegel functions at the cusps and apply them to evaluate the ramanujan s cubic continued fraction systematically furthermore we construct ray class invariants over imaginary quadratic fields in terms of singular values of j and siegel functions

non simple purely infinite rings in this paper we introduce the concept of purely infinite rings which in the simple case agrees with the already existing notion of pure infiniteness we establish various permanence proper ties of this notion with respect to passage to matrix rings corners and behaviour under extensions so being purely infinite is preserved under morita equivalence we show that a wealth of examples falls into this class including important analogues of constructions commonly found in operator al gebras in particular for any s unital k algebra having enough nonzero idempotents for example for a von neumann regular algebra its tensor product over k with many non simple leavitt path algebras is purely infinite introduction the notion of pure infiniteness has proved key in the the ory of operator algebras since its conception in the early eighties by j cuntz see 27 this was done for simple algebras and provided a huge list of exam ples one of the milestones of the theory became the classification of separable nuclear unital purely infinite simple algebras by means of k theoretic invariants due to kirchberg 41 and phillips 58

asymmetric learning vector quantization for efficient nearest neighbor classification in dynamic time warping spaces abstract the nearest neighbor method together with the dynamic time warping dtw distance is one of the most popular approaches in time series classification this method suffers from high storage and computation requirements for large training sets as a solution to both drawbacks this article extends learning vector quantization lvq from euclidean spaces to dtw spaces the proposed generic lvq scheme uses asymmetric weighted averaging as update rule we theoretically justify the asymmetric lvq scheme via subgradient techniques and by the margin growth principle in addition we show that the decision boundary of two prototypes from different classes is piecewise quadratic empirical results exhibited superior performance of asymmetric generalized lvq glvq over other state of the art prototype generation methods for nearest neighbor classification

spectral spaces of countable abelian lattice ordered groups a compact topological space x is spectral if it is sober i e every irreducible closed set is the closure of a unique singleton and the compact open subsets of x form a basis of the topology of x closed under finite intersections theorem a topological space x is homeomorphic to the spectrum of some countable abelian l group with unit resp mv algebra iff x is spectral has a countable basis of open sets and for any points x and y in the closure of a singleton z either x is in the closure of y or y is in the closure of x we establish this result by proving that a countable distributive lattice d with zero is isomorphic to the lattice of all principal ideals of an abelian l group we say that d is l representable iff for all a b d there are x y d such that a b a y b x and x y 0 on the other hand we construct a non l representable bounded distributive lattice of cardinality 1 with an l representable countable l elementary sublattice in particular there is no characterization of the class of all l representable distributive lattices in arbitrary cardinality by any class of l sentences

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

semilattices of finitely generated ideals of exchange rings with finite stable rank we find a distributive v 0 1 semilattice s of size aleph 1 that is not isomorphic to the maximal semilattice quotient of any riesz monoid endowed with an order unit of finite stable rank we thus obtain solutions to various open problems in ring theory and in lattice theory in particular there is no exchange ring thus no von neumann regular ring and no c algebra of real rank zero with finite stable rank whose semilattice of finitely generated idempotent generated two sided ideals is isomorphic to s there is no locally finite modular lattice whose semilattice of finitely generated congruences is isomorphic to s these results are established by constructing an infinitary statement denoted here by urpsr that holds in the maximal semilattice quotient of every riesz monoid endowed with an order unit of finite stable rank but not in the semilattice s

the rational torsion subgroups of drinfeld modular jacobians and eisenstein pseudo harmonic cochains let mathfrak n be a square free ideal of mathbb f q t we study the rational torsion subgroup of the jacobian variety j 0 mathfrak n of the drinfeld modular curve x 0 mathfrak n we prove that for any prime number ell not dividing q q 1 the ell primary part of this group coincides with that of the cuspidal divisor class group we further determine the structure of the ell primary part of the cuspidal divisor class group for any prime ell not dividing q 1