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Abstract: This is the final one in the series of papers where we introduce and study the -algebras associated with topological graphs. In this paper, we get a sufficient condition on topological graphs so that the associated -algebras are simple and purely infinite. Using this result, we give one method to construct all Kirchberg algebras as -algebras associated with topological graphs. [Copyright &y& Elsevier]

Abstract: Let be a class of unital C*-algebras. The class of C*-algebras which can be tracially approximated (in the Egorov-like sense first considered by Lin) by the C*-algebras in is studied (Lin considered the case that consists of finite-dimensional C*-algebras or the tensor products of such with ). In particular, the question is considered whether, for any simple separable , there is a C*-algebra B which is a simple inductive limit of certain basic homogeneous C*-algebras together with C*-algebras in , 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.) [Copyright &y& Elsevier]

Abstract: In this paper we study uniqueness properties of solutions of the so-called k-generalized Korteweg–de Vries equations. Our goal is to obtain sufficient conditions on the behavior of the difference of two solutions of (1.1) at two different times and which guarantee that . [Copyright &y& Elsevier]

Abstract: Let B be a Douglas algebra and let be the algebra on the disk generated by the harmonic extensions of the functions in B. In this paper we show that is generated by and the complex conjugates of the harmonic extensions of the interpolating Blaschke products invertible in B. Every element S in the Toeplitz algebra generated by Toeplitz operators (on the Bergman space) with symbols in has a canonical decomposition for some R in the commutator ideal ; and S is in iff the Berezin transform vanishes identically on the union of the maximal ideal space of the Douglas algebra B and the set of trivial Gleason parts. [Copyright &y& Elsevier]

Abstract: In this paper we study several qualitative properties of the Degasperis–Procesi equation. We first established the precise blow-up rate and then determine the blow-up set of blow-up strong solutions to this equation for a large class of initial data. We finally prove the existence and uniqueness of global weak solutions to the equation provided the initial data satisfies appropriate conditions. [Copyright &y& Elsevier]

Abstract: The Lévy–Khintchine formula or, more generally, Courrège''s theorem characterizes the infinitesimal generator of a Lévy process or a Feller process on . For more general Markov processes, the formula that comes closest to such a characterization is the Beurling–Deny formula for symmetric Dirichlet forms. In this paper, we extend these celebrated structure results to include a general right process on a metrizable Lusin space, which is supposed to be associated with a semi-Dirichlet form. We start with decomposing a regular semi-Dirichlet form into the diffusion, jumping and killing parts. Then, we develop a local compactification and an integral representation for quasi-regular semi-Dirichlet forms. Finally, we extend the formulae of Lévy–Khintchine and Beurling–Deny in semi-Dirichlet forms setting through introducing a quasi-compatible metric. [Copyright &y& Elsevier]

Abstract: In this paper we consider the blow up phenomenon of critical nonlinear Schrödinger equations in dimension 1D and 2D. We define the minimal mass as the norm necessary to ignite a wave collapse and we stress its role in the blow up mechanism. Asymptotic compactness properties and -concentration are proved. The proof relies on linear and nonlinear profile decompositions. [Copyright &y& Elsevier]

In this paper we define a non-commutative residue for the algebra of classical anisotropic pseudo-differential operators on Rn modulo regularizing operators. Furthermore, by means of the Weyl formula, we show that this trace coincides with the Dixmier trace on operators of order −|M|. [Copyright &y& Elsevier]