on tracial approximation let c be a class of unital c algebras the class tac of c algebras which can be tracially approximated in the egorov like sense first considered by lin by the c algebras in c is studied lin considered the case that c consists of finite dimensional c algebras or the tensor products of such with c 0 1 in particular the question is considered whether for any simple separable a tac there is a c algebra b which is a simple inductive limit of certain basic homogeneous c algebras together with c algebras in c such that the elliott invariant of a is isomorphic to the elliott invariant of b an interesting case of this question is answered in the final part of the paper the question is also considered which properties of c algebras are inherited by tracial approximation results of this kind are obtained which are used in the proof of the main theorem of the paper and also in the proof of the classification theorem of the second author given in z niu a classification of tracially approximately splitting tree algebra in preparation and z niu a classification of certain tracially approximately subhomogeneous c algebras phd thesis university of toronto 2005 which also uses the main result of the present paper

geometry of fermat adeles if l a s n c n a n s is a family of geometric l functions depending on a parameter a then the function p a c p a where p runs through the set of prime integers is not a rational function and hence is not a function belonging to algebraic geometry the aim of the paper is to show that if one enlarges algebraic geometry by adjoining a fermat quotient operation then the functions c p a become functions in the enlarged geometry at least for l functions of curves and abelian varieties

universal coefficient theorems for the stable ext groups let a be a unital separable nuclear c algebra and let b be a stable c algebra using k theory and kk theory we establish universal coefficient theorems for the stable ext groups of unital extensions of a by b when a and b have certain properties which generalize a result of l brown and m dadarlat for the strong ext groups the class of extensions being studied are also enlarged

abelian mathrm c ast subalgebras of mathrm c ast algebras of real rank zero and inductive limit mathrm c ast algebras

mathbb z actions on ah algebras and mathbb z 2 actions on af algebras we consider mathbb z actions single automorphisms on a unital simple ah algebra with real rank zero and slow dimension growth and show that the uniform outerness implies the rohlin property under some technical assumptions moreover two mathbb z actions with the rohlin property on such a c algebra are shown to be cocycle conjugate if they are asymptotically unitarily equivalent we also prove that locally approximately inner and uniformly outer mathbb z 2 actions on a unital simple af algebra with a unique trace have the rohlin property and classify them up to cocycle conjugacy employing the orderext group as classification invariants

a new finite difference scheme for the rosenau burgers equation abstract in this paper a linear implicit finite difference scheme is given for the initial boundary problem of rosenau burgers equation which is convergent and unconditionally stable the unique solvability of numerical solutions has been shown a priori estimate and second order convergence of the finite difference approximate solution are discussed using energy method numerical results demonstrate that the scheme is efficient and accurate

an approximate universal coefficient theorem an approximate universal coefficient theorem auct for certain c algebras is established we present a proof that kirchberg phillips s classification theorem for separable nuclear purely infinite simple c algebras is valid for c algebras satisfying the auct instead of the uct it is proved that two versions of auct are in fact the same we also show that c algebras that are locally approximated by c algebras satisfying the auct satisfy the auct as an application we prove that certain simple c algebras which are locally type i are in fact isomorphic to simple ah algebras as another application we show that a sequence of residually finite dimensional c algebras which are asymptotically nuclear and which asymptotically satisfies the auct can be embedded into the same simple af algebra

c algebras generated by stable relations abstract presentations of certain c algebras such as c0 0 1 mn by generators and relations are proven to be stable by the weakest definition of stability a set p1 pn of star polynomials is stable if operators a1 an which almost satisfy these relations meaning pi a1 an can be perturbed within c a1 an to satisfy these relations exactly homotopy and perturbation results for these c algebras are corollaries in particular we prove some new results about the dimension drop c algebras a n c 0 1 m n 0 1 c i we show that an is semiprojective answering a question of effros and kaminker also we show that whenever an b is a homomorphism approximately contained in a sub c algebra b0 b there is a nearby homomorphism of an into b0 this answers a question raised by elliott we prove technical results regarding mn c0 0 1 which should be useful elsewhere first we show this is projective meaning a star homomorphism mn c0 0 1 b j always lifts to a star homomorphism mn c0 0 1 b secondly we show that for a finitely generated m n a m n c 0 0 1 c 0 0 1 a if the amalgamation is done properly

classification of unital extensions of at algebras abstract we prove that the six term exact sequence in k theory together with the elliott invariants of the ideal and quotient is a complete invariant for certain unital extensions of a t algebras up to stable isomorphism we also give some necessary and sufficient conditions for such extensions being a t algebras or quasidiagonal extensions and prove the above invariant with the equivalence class of unit is a complete invariant for unital quasidiagonal extensions of a t algebras up to isomorphism

computing isogeny covariant differential modular forms we present the computation modulo p 2 and explicit formulas for the unique isogeny covariant differential modular form of order one and weight x p 1 p called f jet an isogeny covariant differential modular form of order two and weight x p2 p 1 1 denoted by f jet h jet and an isogeny covariant differential modular form h jet of order two and weight x 1 p2 0 1

regularity properties in the classification program for separable amenable c algebras we report on recent progress in the program to classify separable amenable c algebras our emphasis is on the newly apparent role of regularity properties such as finite decomposition rank strict comparison of positive elements and z stability and on the importance of the cuntz semigroup we include a brief history of the program s successes since 1989 a more detailed look at the villadsen type algebras which have so dramatically changed the landscape and a collection of announcements on the structure and properties of the cuntz semigroup

arithmetic partial differential equations ii abstract we continue the study of arithmetic partial differential equations initiated in 7 by classifying arithmetic convection equations on modular curves and by describing their space of solutions certain of these solutions involve the fourier expansions of the eisenstein modular forms of weight 4 and 6 while others involve the serre tate expansions mori 1995 13 buium 2003 4 of the same modular forms in this sense our arithmetic convection equations can be seen as unifying the two types of expansions the theory can be generalized to one of arithmetic heat equations on modular curves but we prove that they do not carry arithmetic wave equations finally we prove an instability result for families of arithmetic heat equations converging to an arithmetic convection equation

distance between unitary orbits of normal elements in simple c algebras of real rank zero let x y x y be two normal elements in a unital simple c c algebra a we introduce a function d c x y d c x y and show that in a unital simple af algebra there is a constant 1 c 0 1 c 0 such that c d c x y dist u x u y d c x y c d c x y dist u x u y d c x y where u x u x and u y u y are the closures of the unitary orbits of x and of y respectively we also generalize this to unital simple c c algebras with real rank zero stable rank one and weakly unperforated k 0 k 0 group more complicated estimates are given in the presence of non trivial k 1 k 1 information

non commutative spheres iii irrational rotations leta be the irrational rotation algebra i e thec algebra generated by two unitariesu v satisfyingvu e 2 i uv with irrational and consider the fixed point subalgebrab under the flip automorphismu u 1 v v 1 we prove thatb is an af algebra

solutions to arithmetic differential equations in algebraically closed fields abstract arithmetic differential equations or geometry exploits analogies between derivations and p derivations arising from lifts of frobenius to study problems in arithmetic geometry along the way two main classes such functions describable as series arose prominently namely characters of abelian schemes and isogeny covariant modular forms however the theory of these functions is not as straightforward in ramified settings overconvergence was introduced in 13 to account for these issues which essentially imposes growth conditions extensions of these series to a fixed level of ramification necessary as such extensions have non trivial fractional coefficients in this article we introduce a rescaling process which identifies a class of functions we call totally overconvergent which extend all the way to the algebraic closure of ring of integers of the maximally unramified extension of q p applications built on these functions allow one to remove boundedness assumptions on ramification the bulk of the article is devoted to establishing that most functions arising in practice namely those in the applications described in 5 7 8 are totally overconvergent which essentially extends results in 13 to unbounded ramification

isocrystals associated to arithmetic jet spaces of abelian schemes abstract using buium s theory of arithmetic differential characters we construct a filtered f isocrystal h a k associated to an abelian scheme a over a p adically complete discrete valuation ring with perfect residue field as a filtered vector space h a k admits a natural map to the usual de rham cohomology of a but the frobenius operator comes from arithmetic differential theory and is not the same as the usual crystalline one when a is an elliptic curve we show that h a k has a natural integral model h a which implies an integral refinement of a result of buium s on arithmetic differential characters the weak admissibility of h a k depends on the invertibility of an arithmetic differential modular parameter thus the fontaine functor associates to suitably generic a a local galois representation of an apparently new kind