A nonsingular real algebraic plane projective curve is called a dividing curve or a curve of type I if its real point set divides its complex point set. Rokhlin’s formula, which holds for such curves, is an important step in order to classify nonsingular real algebraic plane projective curves. It gives prohibitions on the complex orientations of a curve of type I and also on its real scheme. Th...

and Hans Schneider· Department of Mathematics University of Wisconsin Madison, Wisconsin 53706 An elementary proof is given that a bounded multiplicative group of complex (real) n X n nonsingular matrices is similar to a unitary (orthogonal) group. Given a norm on a complex n-space, it follows that there exists a nonsingular n X n matrix L (the lAewner-John matrix for the norm) such that LHL -1...

A description is given of those ray-patterns, which will be called inverse closed ray-nonsingular, of complex matrices that contain invertible matrices only and are closed under inversion. Here, two n × n matrices are said to belong to the same ray-pattern if in each position either the entries of both matrices are zeros, or both entries are nonzero and their quotient is positive. Possible Jord...

Let X be a rational nonsingular compact connected real algebraic surface. Denote by Aut(X) the group of real algebraic automorphisms of X. We show that the group Aut(X) acts n-transitively on X, for all natural integers n. As an application we give a new and simpler proof of the fact that two rational nonsingular compact connected real algebraic surfaces are isomorphic if and only if they are h...

The computational function of a matchgate is represented by its character matrix. In this article, we show that all nonsingular character matrices are closed under matrix inverse operation, so that for every k, the nonsingular character matrices of k-bit matchgates form a group, extending the recent work of Cai and Choudhary [1] of the same result for the case of k = 2, and that the single and ...

The paper mainly discusses the lower bounds for the rank of matrices and sufficient conditions for nonsingular matrices. We first present a new estimation for [Formula: see text] ([Formula: see text] is an eigenvalue of a matrix) by using the partitioned matrices. By using this estimation and inequality theory, the new and more accurate estimations for the lower bounds for the rank are deduced....

Matrix We take a detour to introduce abstract matrices. We want to acquire a good understanding of such matrices, since they not only represent matroids, but 74 Chapter 3. From Graphs to Matroids also display a lot of structural information about matroids that other ways do not. An abstract matrix B is a {0, 1} matrix with row and column indices plus a function called abstract determinant and d...