The aim of the paper is to give an effective formula for the calculation of the probability that a random subset of an affine geometry AG(r−1, q) has rank r. Tables for the probabilities are given for small ranks. The expected time to the first moment at which a random subset of an affine geometry achieves the rank r is derived.

The list of q-Painlevé equations we are going to investigate is given in section 3 below. We remark that these q-Painlevé equations were discovered through various approaches to discrete Painlevé equations, including singularity confinement analysis, compatibility conditions of linear difference equations, affine Weyl group symmetries and τ-functions on the lattices. Also, in Sakai’s framework ...

Affine spheres were introduced by Ţiţeica in [72, 73], and studied later by Blaschke, Calabi, and Cheng-Yau, among others. These are hypersurfaces in affine R which are related to real Monge-Ampère equations, to projective structures on manifolds, and to the geometry of Calabi-Yau manifolds. In this survey article, we will outline the theory of affine spheres their relationships to these topics...

Locally extended affine Lie algebras were introduced by Morita and Yoshii in [J. Algebra 301(1) (2006), 59-81] as a natural generalization of extended affine Lie algebras. After that, various generalizations of these Lie algebras have been investigated by others. It is known that a locally extended affine Lie algebra can be recovered from its centerless core, i.e., the ideal generated by weight...

This paper initially proves mathematically that the solution of the relative orientation of the push broom stereo images leads to a set of ill-posed system of observation equations which means no relative orientation for this kind of imageries is possible. It is then demonstrated that the only way to bypass the inherent rank deficiency in the system of the relative orientation equations, is to ...

Abstract We analyze the tail behavior of solutions to sample average approximations (SAAs) stochastic programs posed in Hilbert spaces. require that integrand be strongly convex with same convexity parameter for each realization. Combined a standard condition from literature on programming, we establish non-asymptotic exponential bounds distance between SAA and program’s solution, without assum...