Mean dimension for AH-algebras with diagonal maps is introduced. It is shown that if a simple unital AH-algebra with diagonal maps has mean dimension zero, then it has strict comparison on positive elements. In particular, the strict order on projections is determined by traces. Moreover, a lower bound of the mean dimension is given in term of Toms’ comparison radius. Using classification resul...

We investigate dispersive estimates for the Schrödinger operator H = −∆+V with V is a real-valued decaying potential when there are zero energy resonances and eigenvalues in four spatial dimensions. If there is a zero energy obstruction, we establish the low-energy expansion eχ(H)Pac(H) = O(1/(log t))A0 +O(1/t)A1 +O((t log t) )A2 +O(t (log t))A3. Here A0, A1 : L (R) → L∞(Rn), while A2, A3 are o...

In this article we outline the methods that are used to prove undecidability of Hilbert’s Tenth Problem for function fields of characteristic zero. Following Denef we show how rank one elliptic curves can be used to prove undecidability for rational function fields over formally real fields. We also sketch the undecidability proofs for function fields of varieties over the complex numbers of di...

A classification is given of certain separable nuclear C*algebras not necessarily of real rank zero, namely, the class of separable simple C*-algebras which are inductive limits of continuoustrace C*-algebras whose building blocks have spectrum homeomorphic to the closed interval [0, 1], or to a disjoint union of copies of this space. Also, the range of the invariant is calculated. 1991 Mathema...

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 beha...

Let (A; ') be the reduced free product of innnitely many C {algebras (A ; ') with respect to faithful states. Assume that the A are not too small, in a speciic sense. If ' is a trace then the positive cone of K 0 (A) is determined entirely by K 0 ('). If, furthermore, the image of K 0 (') is dense in R, then A has real rank zero. On the other hand, if ' is not a trace then A is simple and purel...

Linear models identification from data with missing values is posed as a weighted low-rank approximation problem with weights related to the missing values equal to zero. Alternating projections and variable projections methods for solving the resulting problem are outlined and implemented in a literate programming style, using Matlab/Octave’s scripting language. The methods are evaluated on sy...

Abstract. The positive semidefinite minimum rank of a simple graph G is defined to be the smallest possible rank over all 1 positive semidefinite real symmetric matrices whose ijth entry (for i 6= j) is nonzero whenever {i, j} is an edge in G and is zero 2 otherwise. The computation of this parameter directly is difficult. However, there are a number of known bounding parameters 3 and technique...

we derive the formulas of the maximal andminimal ranks of four real matrices $x_{1},x_{2},x_{3}$ and $x_{4}$in common solution $x=x_{1}+x_{2}i+x_{3}j+x_{4}k$ to quaternionmatrix equations $a_{1}x=c_{1},xb_{2}=c_{2},a_{3}xb_{3}=c_{3}$. asapplications, we establish necessary and sufficient conditions forthe existence of the common real and complex solutions to the matrixequations. we give the exp...

Rank/Select dictionaries are data structures for an ordered set S f0; 1; : : : ; n 1g to compute rank(x; S) (the number of elements in S which are no greater than x), and select(i; S) (the i-th smallest element in S), which are the fundamental components of succinct data structures of strings, trees, graphs, etc. In those data structures, however, only asymptotic behavior has been considered an...