The present paper provides an alternative proof of a theorem of Suslin-Voevodsky. We use more classical tools, namely certain Severi-Brauer schemes and continuous étale cohomology.

Let $G$ be a real, connected, noncompact, semisimple Lie group, let $K_{\mathbb{C}}$ be the complexification of a maximal compact subgroup $K$ of $G$, and let $\mathfrak{g} = \mathfrak{k} + \mathfrak{p}$ be the corresponding Cartan decomposition of the complexified Lie algebra of $G$. Sequences of strongly orthogonal noncompact weights are constructed and classified for each real noncompact simple Lie group of classical type. We show that for each partial subsequence {$\gamma _{1},\ldots ,\gamma _{i}$} there is a corresponding family of nilpotent $K_{\mathbb{C}}$-orbits in $\mathfrak{p}$, ordered by inclusion and such that the representation of $K$ on the ring of regular functions on each orbit is multiplicity-free. The $K$-types of regular functions on the orbits and the regular functions on their closures are both explicitly identified and demonstrated to coincide, with one exception in the Hermitian symmetric case. The classification presented also includes the specification of a base point for each orbit and exhibits a corresponding system of restricted roots with multiplicities. A formula for the leading term of the Hilbert polynomials corresponding to these orbits is given. This formula, together with the restricted root data, allows the determination of the dimensions of these orbits and the algebraic-geometric degree of their closures. In an appendix, the location of these orbits within D. King’s classification of spherical nilpotent orbits in complex symmetric spaces is depicted via signed partitions and Hasse diagrams.

In this paper, by using the sphere characteristic function $T(r,w)$, we define and establish the existence of a new singular direction for the algebroidal function $w$, namely a $T$ direction, for which the characteristic function $T(r,w)$ is used as a comparison function. This extended the previous result due to Guo, Zheng and Ng in [Bull. Austral. Math. Soc. 69(2004), 277-287].

We extend the definition of a semidualizing module to general associative rings. This enables us to define and study Auslander and Bass classes with respect to a semidualizing bimodule $C$. We then study the classes of $C$-flats, $C$-projectives, and $C$-injectives, and use them to provide a characterization of the modules in the Auslander and Bass classes. We extend Foxby equivalence to this new setting. This paper contains a few results which are new even in the commutative, noetherian setting.

We show the strong unique continuation property for certain elliptic systems of second order with two independent variables.

We study Synge type theorems for positively curved Finsler manifolds.

We give an explicit description of rational curves in the product of three copies of complex projective lines, which are transformed into twistor lines in M. Nagata’s example of non-projective complete algebraic variety, viewed as the twistor space of Eguchi-Hanson metric. In particular, we show that there exist two families of such curves and both of them are parameterized by mutually diffeomorphic, connected real 4-dimensional manifolds. We also give a relationship between these two families through a birational transformation naturally associated to the Nagata’s example.

The existence of a unique positive periodic solution for the differential delay equation $x'(t) = -a(t)x(t) + b(t)f (x(t - c(t)))$ is established. Our method is based on Hilbert’s projective metric and the contraction mapping principle.

We construct the universal coverings of spaces of self-holomorphic maps on the complex projective space $\mathbb{C}P_{n}$ by using the resultants, and we study their homotopy types.

We study entire holomorphic curves in the algebraic torus, and show that they can be characterized by the “growth rate” of their derivatives.