The likelihood ratio test of cointegration rank is the most widely used test for cointegration. Many studies have shown by simulation that the small sample distribution is not well approximated by the limiting distribution. We suggest using the bootstrap to generate small sample critical values instead of correcting the test statistics. The idea of bootstrapping the trace test of cointegration ...

The zero forcing number Z(G) is used to study the minimum rank/maximum nullity of the family of symmetric matrices described by a simple, undirected graph G. The positive semidefinite zero forcing number is a variant of the (standard) zero forcing number, which uses the same definition except with a different color-change rule. The positive semidefinite maximum nullity and zero forcing number f...

We introduce the notion of weakly (strongly) infinite real rank for unital C∗-algebras. It is shown that a compact space X is weakly (strongly) infine-dimensional if and only if C(X) has weakly (strongly) infinite real rank. Some other properties of this concept are also investigated. In particular, we show that the group C∗-algebra C∗ (F∞) of the free group on countable number of generators ha...

In this paper I present selected properties of triangular matrices and basic properties of the rank of matrices over a field. I define a submatrix as a matrix formed by selecting certain rows and columns from a bigger matrix. That is in my considerations, as an array, it is cut down to those entries constrained by row and column. Then I introduce the concept of the rank of a m× n matrix A by th...

We establish a congruence formula between p-adic logarithms of Heegner points for two elliptic curves with the same mod p Galois representation. As a first application, we use the congruence formula when p = 2 to explicitly construct many quadratic twists of analytic rank zero (resp. one) for a wide class of elliptic curves E. We show that the number of twists of E up to twisting discriminant X...

The zero forcing number Z(G) is used to study the minimum rank/maximum nullity of the family of symmetric matrices described by a simple, undirected graph G. The positive semidefinite zero forcing number is a variant of the (standard) zero forcing number, which uses the same definition except with a different color-change rule. The positive semidefinite maximum nullity and zero forcing number f...

For a row finite directed graph E, Kumjian, Pask, and Rae-burn proved that there exists a universal C *-algebra C * (E) generated by a Cuntz-Krieger E-family. In this paper we consider two density problems of invertible elements in graph C *-algebras C * (E), and it is proved that C * (E) has stable rank one, that is, the set of all invertible elements is dense in C * (E) (or in its unitization...