We give simple formulas for the canonical metric, gradient, Lie derivative, Riemannian connection, parallel translation, geodesics and distance on the Grassmann manifold of p-planes in Rn. In these formulas, p-planes are represented as the column space of n £ p matrices. The Newton method on abstract Riemannian manifolds proposed by S. T. Smith is made explicit on the Grassmann manifold. Two ap...

In this note we offer a short summary of the content of the lecture on polar Grassmann codes given at the Indian Intitute of Technology, Mumbai, during the International Conference on Algebraic Geometry and Coding Theory in December 2013. More precisely, we consider the codes arising from the projective system determined by the image ε k (∆k) of the Grassmann embedding ε gr k of an orthogonal G...

Let p : E → B be a Serre fibration with E compact, B a connected finite CW -complex, and fiber either the real Grassmann manifold O(n)/O(2) × O(n − 2) or the complex Grassmann manifold U(n)/U(2) × U(n− 2), where n ≥ 4. We prove that if n is odd, then the fiber is totally non-homologous to zero in E with respect to Z2.

In this paper we investigate linear error correcting codes and projective caps related to the Grassmann embedding ε k of an orthogonal Grassmannian ∆k. In particular, we determine some of the parameters of the codes arising from the projective system determined by ε k (∆k). We also study special sets of points of ∆k which are met by any line of ∆k in at most 2 points and we show that their imag...

In this report we present computational methods for the best multilinear rank approximation problem. We consider algorithms build on quasi-Newton methods operating on product of Grassmann manifolds. Specifically we test and compare methods based on BFGS and L-BFGS updates in local and global coordinates with the Newton-Grassmann and alternating least squares methods. The performance of the quas...

Abstract Cross-domain classification refers to completing the corresponding task in a target domain which lacks label information, by exploring useful knowledge related source but with different data distribution. Domain adaptation can deal such cross-domain classification, reducing divergence of domains and transferring relevant from target. To mine discriminant information samples geometric s...

Tensor, Clifford and Grassmann algebras belong to a wide class of non-commutative algebras that have a Poincaré-Birkhoff-Witt (PBW) “monomial” basis. The necessary and sufficient condition for an algebra to have the PBW basis has been established by T. Mora and then V. Levandovskyy as the so called “non-degeneracy condition”. This has led V. Levandovskyy to a re-discovery of the so called G-alg...

The aim of this paper is to give a hint for thinking to graduate or undergraduate students in Mathematical Physics who are interested in both Geometry and Quantum Computation. First I make a brief review of some properties on Grassmann manifolds and next I show a path between Grassmann manifolds and Quantum Computation which is related to the efficiency of quantum computing. ∗E-mail address : f...

The Landau-Lifshitz equation is an example of soliton equations with a zerocurvature representation defined on an elliptic curve. This equation can be embedded into an integrable hierarchy of evolution equations called the Landau-Lifshitz hierarchy. This paper elucidates its status in Sato, Segal and Wilson’s universal description of soliton equations in the language of an infinite dimensional ...