In this paper an upper bound for the number of algebraically independent Poincaré-Liapunov constants in a certain basis for planar polynomial differential systems is given. Finally, it is conjectured that an upper bound for the number of functionally independent Poincaré-Liapunov quantities would be m + 3m− 7 where m is the degree of the polynomial differential system. Moreover, the computation...

We investigate the finiteness structure of a complete open Riemannian n-manifold M whose radial curvature at a base point of M is bounded from below by that of a non-compact von Mangoldt surface of revolution with its total curvature greater than π. We show, as our main theorem, that all Busemann functions on M are exhaustions, and that there exists a compact subset of M such that the compact s...

We investigate the combinatorial geometry obtained from algebraic closure over a fixed subfield in an algebraically closed field. The main result classifies the subgeometries which are isomorphic to projective planes. This is applied to give new examples of algebraic characteristic sets of matroids. The main technique used, which is motivated by classical projective geometry, is that a particul...

We propose a new class of mathematical structures called (m, n)-semirings (which generalize the usual semirings) and describe their basic properties. We define partial ordering and generalize the concepts of congruence, homomorphism, and so forth, for (m, n)-semirings. Following earlier work by Rao (2008), we consider systems made up of several components whose failures may cause them to fail a...

We give an algorithmic solution in a simple combinatorial data of Birkhoff′s type problem studied in [14], for the category repft(I, K[t]/(tm)) of filtered I-chains of modules over the K-algebra K[t]/(tm), where m ≥ 2, I is a finite poset with a unique maximal element, and K is an algebraically closed field.

We present counterexamples to Fujita's conjecture in positive characteristic. More precisely, given any algebraically closed field k of characteristic p>0 and integer m, we show there exists a smooth projective surface S over admitting an ample Cartier divisor A such that the adjoint linear system |KS+mA| is not free base points.