On geometrical structures, associated with linear differential operators of the 1st order

On the Geometrical Gyro-Kinetic Theory

Considering a Hamiltonian Dynamical System describing the motion of charged particle in a Tokamak or a Stellarator, we build a change of coordinates to reduce its dimension. This change of coordinates is in fact an intricate succession of mappings that are built using Hyperbolic Partial Differential Equations, Differential Geometry, Hamiltonian Dynamical System Theory and Symplectic Geometry, Lie Transforms and a new tool which is here introduced : Partial Lie Sums.

Note on a Differential-Geometrical Construction of Optimal Directions in Linearly-Constrained Systems

This note presents an analytic construction of the optimal unit-norm direction hat(x) = x/|x| that maximizes or minimizes the objective linear expression, B . hat(x), subject to a system of linear constraints of the form [A] . x = 0, where x is an unknown n-dimensional real vector to be determined up to an overall normalization constant, 0 is an m-dimensional null vector, and the n-dimensional real vector B and the m\times n-dimensional real matrix [A] (with 0 =< m < n) are given. The analytic solution to this problem can be expressed in terms of a combination of double wedge and Hodge-star pr…

关于作为微分几何对象的类流形多折叠及其在复几何中的应用

On manifold-like polyfolds as differential geometrical objects with applications in complex geometry

本文提出类流形多折叠作为微分几何对象，通过引入张量和证明各种几何结构的存在，推动复几何领域的发展。

AV-differential geometry: Euler-Lagrange equations

A general, consistent and complete framework for geometrical formulation of mechanical systems is proposed, based on certain structures on affine bundles (affgebroids) that generalize Lie algebras and Lie algebroids. This scheme covers and unifies various geometrical approaches to mechanics in the Lagrangian and Hamiltonian pictures, including time-dependent lagrangians and hamiltonians. In our approach, lagrangians and hamiltonians are, in general, sections of certain $\R$-principal bundles, and the solutions of analogs of Euler-Lagrange equations are curves in certain affine bundles. The cor…

Logic of differential calculus and the zoo of geometric strujctures

Since the discovery of differential calculus by Newton and Leibniz and the subsequent continuous growth of its applications to physics, mechanics, geometry, etc, it was observed that partial derivatives in the study of various natural problems are (self-)organized in certain structures usually called geometric. Tensors, connections, jets, etc, are commonly known examples of them. This list of classical geometrical structures is sporadically and continuously widening. For instance, Lie algebroids and BV-bracket are popular recent additions into it. Our goal is to show that the "zoo" of all geom…

Elie Cartan's geometrical vision or how to avoid expression swell

The aim of the paper is to demonstrate the superiority of Cartan's method over direct methods based on differential elimination for handling otherwise intractable equivalence problems. In this sens, using our implementation of Cartan's method, we establish two new equivalence results. Weestablish when a system of second order ODE's is equivalent to flat system (second derivations are zero), and when a system of holomorphic PDE's with two independent variables and one dependent variables is flat. We consider the problem of finding transformation that brings a given equation to the target one. W…

Modified action and differential operators on the 3-D sub-Riemannian sphere

Our main aim is to present a geometrically meaningful formula for the fundamental solutions to a second order sub-elliptic differential equation and to the heat equation associated with a sub-elliptic operator in the sub-Riemannian geometry on the unit sphere $\mathbb S^3$. Our method is based on the Hamiltonian approach, where the corresponding Hamitonian system is solved with mixed boundary conditions. A closed form of the modified action is given. It is a sub-Riemannian invariant and plays the role of a distance on $\mathbb S^3$.

On the internal approach to differential equations 3. Infinitesimal symmetries

The geometrical theory of partial differential equations in the absolute sense, without any additional structures, is developed. In particular the symmetries need not preserve the hierarchy of independent and dependent variables. The order of derivatives can be changed and the article is devoted to the higher--order infinitesimal symmetries which provide a simplifying "linear aproximation" of general groups of higher--order symmetries. The classical Lie's approach is appropriately adapted.

Finite reflection groups and the Dunkl-Laplace differential-difference operators in conformal geometry

For a finite reflection subgroup $G\leq O(n+1,1,\mR)$ of the conformal group of the sphere with standard conformal structure $(S^n,[g_0])$, we geometrically derive differential-difference Dunkl version of the series of conformally invariant differential operators with symbols given by powers of Laplace operator. The construction can be regarded as a deformation of the Fefferman-Graham ambient metric construction of GJMS operators.

Classification of Material G-structures

A geometrical interpretation of the $G$-structures associated to elastic material bodies is given. In addition, characterizations of their integrability are obtained. Since the lack of integrability is a geometrical measure of the lack of homogeneity, the corresponding inhomogeneity conditions are obtained

Differential geometric aspects of parametric estimation theory for states on finite-dimensional C*-algebras

A geometrical formulation of estimation theory for finite-dimensional $C^{\star}$-algebras is presented. This formulation allows to deal with the classical and quantum case in a single, unifying mathematical framework. The derivation of the Cramer-Rao and Helstrom bounds for parametric statistical models with discrete and finite outcome spaces is presented.