An extension of heat hierarchy

We propose a formally completely integrable extension of heat hierarchy based on the space of symmetries isomorphic to the Weyl algebra $\mathcal{A}_1$. The extended heat hierarchy will be the basic model for the analysis of the extension of KP hierarchy, and other integrable equations.Comment: This note is incorporated into arXiv:1409.7024, arXiv:1408.324

Legendrian Gronwall conjecture

The Gronwall conjecture states that a planar 3-web of foliations which admits more than one distinct linearizations is locally equivalent to an algebraic web. We propose an analogue of the Gronwall conjecture for the 3-web of foliations by Legendrian curves in a contact three manifold. The Legendrian Gronwall conjecture states that a Legendrian 3-web admits at most one distinct local linearization, with the only exception when it is locally equivalent to the dual linear Legendrian 3-web of the Legendrian twisted cubic in \,\PP^3. We give a partial answer to the conjecture in the affirmative for the class of Legendrian 3-webs of maximum rank. We also show that a linear Legendrian 3-web which is sufficiently flat at a reference point is rigid under local linear Legendrian deformation.Comment: 15 page

Projectively deformable Legendrian surfaces

Consider an immersed Legendrian surface in the five dimensional complex projective space equipped with the standard homogeneous contact structure. We introduce a class of fourth order projective Legendrian deformation called \emph{$\,\Psi$-deformation}, and give a differential geometric characterization of surfaces admitting maximum three parameter family of such deformations. Two explicit examples of maximally $\, \Psi$-deformable surfaces are constructed; the first one is given by a Legendrian map from \, \PP^2 blown up at three distinct collinear points, which is an embedding away from the -2-curve and degenerates to a point along the -2-curve. The second one is a Legendrian embedding of the degree 6 del Pezzo surface, \, \PP^2 blown up at three non-collinear points. In both cases, the Legendrian map is given by a system of cubics through the three points, which is a subsystem of the anti-canonical system.Comment: 33 page

Selected Essays on the Conflict of Laws. By Brainerd Currie.

Most subspace identication algorithms are not applicable to closed-loop identication because they require future input to be uncorrelated with pastinnovation. In this paper, we propose a new subspace identication method that remove this requirement by using a parsimonious model formulation with innovation estimation. A simulation example is included to show the effectiveness of the proposed method