scholarly journals NORMAL BGG SOLUTIONS AND POLYNOMIALS

2012 ◽  
Vol 23 (11) ◽  
pp. 1250117 ◽  
Author(s):  
A. ČAP ◽  
A. R. GOVER ◽  
M. HAMMERL
Keyword(s):  
Differential Operators ◽  
Polynomial System ◽  
Homogeneous Model ◽  
Polynomial Systems ◽  
Normal Solution ◽  
Cartan Connection ◽  
Algebraic Sets ◽  
Parabolic Geometry ◽  
Geometric Origin ◽  
Linear Differential

First BGG operators are a large class of overdetermined linear differential operators intrinsically associated to a parabolic geometry on a manifold. The corresponding equations include those controlling infinitesimal automorphisms, higher symmetries and many other widely studied PDE of geometric origin. The machinery of BGG sequences also singles out a subclass of solutions called normal solutions. These correspond to parallel tractor fields and hence to (certain) holonomy reductions of the canonical normal Cartan connection. Using the normal Cartan connection, we define a special class of local frames for any natural vector bundle associated to a parabolic geometry. We then prove that the coefficient functions of any normal solution of a first BGG operator with respect to such a frame are polynomials in the normal coordinates of the parabolic geometry. A bound on the degree of these polynomials in terms of representation theory data is derived. For geometries locally isomorphic to the homogeneous model of the geometry we explicitly compute the local frames mentioned above. Together with the fact that on such structures all solutions are normal, we obtain a complete description of all first BGG solutions in this case. Finally, we prove that in the general case the polynomial system coming from a normal solution is the pull-back of a polynomial system that solves the corresponding problem on the homogeneous model. Thus we can derive a complete list of potential normal solutions on curved geometries. Moreover, questions concerning the zero locus of solutions, as well as related finer geometric and smooth data, are reduced to a study of polynomial systems and real algebraic sets.

PARABOLIC COMPACTIFICATION OF HOMOGENEOUS SPACES

2019 ◽  
pp. 1-38
Author(s):  
Andreas Čap ◽  
A. Rod Gover ◽  
Matthias Hammerl
Keyword(s):  
Symmetric Spaces ◽  
Differential Operators ◽  
Homogeneous Spaces ◽  
Homogeneous Model ◽  
Point Group ◽  
Large Family ◽  
Flag Manifold ◽  
Invariant Differential Operators ◽  
Parabolic Geometry ◽  
Invariant Differential

In this article, we study compactifications of homogeneous spaces coming from equivariant, open embeddings into a generalized flag manifold $G/P$ . The key to this approach is that in each case $G/P$ is the homogeneous model for a parabolic geometry; the theory of such geometries provides a large supply of geometric tools and invariant differential operators that can be used for this study. A classical theorem of Wolf shows that any involutive automorphism of a semisimple Lie group $G$ with fixed point group $H$ gives rise to a large family of such compactifications of homogeneous spaces of $H$ . Most examples of (classical) Riemannian symmetric spaces as well as many non-symmetric examples arise in this way. A specific feature of the approach is that any compactification of that type comes with the notion of ‘curved analog’ to which the tools we develop also apply. The model example of this is a general Poincaré–Einstein manifold forming the curved analog of the conformal compactification of hyperbolic space. In the first part of the article, we derive general tools for the analysis of such compactifications. In the second part, we analyze two families of examples in detail, which in particular contain compactifications of the symmetric spaces $\mathit{SL}(n,\mathbb{R})/\mathit{SO}(p,n-p)$ and $\mathit{SO}(n,\mathbb{C})/\mathit{SO}(n)$ . We describe the decomposition of the compactification into orbits, show how orbit closures can be described as the zero sets of smooth solutions to certain invariant differential operators and prove a local slice theorem around each orbit in these examples.

Higher order Wirtinger inequalities

1980 ◽  
Vol 85 (1-2) ◽  
pp. 87-110 ◽  
Author(s):  
Kurt Kreith ◽  
Charles A. Swanson
Keyword(s):  
Boundary Conditions ◽  
Quadratic Forms ◽  
Differential Operators ◽  
Conjugate Point ◽  
Wirtinger Inequality ◽  
Even Order ◽  
Linear Differential ◽  
Natural Boundary Conditions ◽  
Derivatives Of ◽  
Admissible Functions

SynopsisWirtinger-type inequalities of order n are inequalities between quadratic forms involving derivatives of order k ≦ n of admissible functions in an interval (a, b). Several methods for establishing these inequalities are investigated, leading to improvements of classical results as well as systematic generation of new ones. A Wirtinger inequality for Hamiltonian systems is obtained in which standard regularity hypotheses are weakened and singular intervals are permitted, and this is employed to generalize standard inequalities for linear differential operators of even order. In particular second order inequalities of Beesack's type are developed, in which the admissible functions satisfy only the null boundary conditions at the endpoints of [a, b] and b does not exceed the first systems conjugate point (a) of a. Another approach is presented involving the standard minimization theory of quadratic forms and the theory of “natural boundary conditions”. Finally, inequalities of order n + k are described in terms of (n, n)-disconjugacy of associated 2nth order differential operators.

Discrete differential operators for periodic and two-periodic time functions

2019 ◽  
Vol 38 (1) ◽  
pp. 325-347 ◽  
Author(s):  
Tadeusz Sobczyk ◽  
Michał Radzik ◽  
Natalia Radwan-Pragłowska
Keyword(s):  
Differential Operators ◽  
High Accuracy ◽  
Large Set ◽  
Periodic Functions ◽  
Content Type ◽  
Second Derivatives ◽  
Non Linear ◽  
Discrete Operators ◽  
Linear Differential ◽  
Very High

Purpose To identify the properties of novel discrete differential operators of the first- and the second-order for periodic and two-periodic time functions. Design/methodology/approach The development of relations between the values of first and second derivatives of periodic and two-periodic functions, as well as the values of the functions themselves for a set of time instants. Numerical tests of discrete operators for selected periodic and two-periodic functions. Findings Novel discrete differential operators for periodic and two-periodic time functions determining their first and the second derivatives at very high accuracy basing on relatively low number of points per highest harmonic. Research limitations/implications Reduce the complexity of creation difference equations for ordinary non-linear differential equations used to find periodic or two-periodic solutions, when they exist. Practical implications Application to steady-state analysis of non-linear dynamic systems for solutions predicted as periodic or two-periodic in time. Originality/value Identify novel discrete differential operators for periodic and two-periodic time functions engaging a large set of time instants that determine the first and second derivatives with very high accuracy.

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