Related papers: Flat $(2,3,5)$-Distributions and Chazy's Equations
We transfer the scheme for constructing differential reductions recently developed for the Manakov-Santini hierarchy to the case of the two-component generalization of dispersionless 2DTL hierarchy. We demonstrate that the equation arising…
A Cartan-geometric, jet bundle formulation of curvature-aware variance bounds in parametric statistical estimation is developed. Building on our earlier extrinsic Hilbert space approach to the Cram\'er-Rao and Bhattacharyya-type…
We extend the construction of [19] by introducing spaces of generalized tensor fields on smooth manifolds that possess optimal embedding and consistency properties with spaces of tensor distributions in the sense of L. Schwartz. We thereby…
If $L$ is a semisimple Lie algebra of vector fields on R^N with a split Cartan subalgebra C, then it is proved that the dimension of the generic orbit of C coincides with the dimension of C. As a consequence one obtains a local canonical…
Non-local equations cannot be treated using classical ODE theorems. Nevertheless, several new methods have been introduced in the non-local gluing scheme of our previous article "On higher dimensional singularities for the fractional Yamabe…
There is a one-to-one correspondence between associated families of generic conformally flat (local-)hypersurfaces in 4-dimensional space forms and conformally flat 3-metrics with the Guichard condition. In this paper, we study the space of…
In this paper we construct a CHY representation for all tree-level primitive QCD amplitudes. The quarks may be massless or massive. We define a generalised cyclic factor $\hat{C}(w,z)$ and a generalised permutation invariant function…
We provide classification results for and examples of half conformally flat generalized quasi Einstein manifolds of signature $(2,2)$. This analysis leads to a natural equation in affine geometry called the affine quasi-Einstein equation…
We exploit a natural correspondence between holomorphic $(2,3,5)$-distributions and nondegenerate lines on holomorphic contact manifolds of dimension $5$ to present a new perspective in the study of symmetries of $(2,3,5)$-distributions.…
We consider the ordinary differential equation (ODE) $dx_{t} =b(t,x_{t} ) dt+ dw_{t}$ where $w$ is a continuous driving function and $b$ is a time-dependent vector field which possibly is only a distribution in the space variable. We…
We analyse the automorphisms of solutions to Chazy's equation and the generalised Chazy equation for the parameters $k=2,3,4$ and $9$. These automorphisms are induced by triangular domains with isosceles symmetry. We also prove theorems…
We generalise the Clark-Ocone formula for functions to give analogous representations for differential forms on the classical Wiener space. Such formulae provide explicit expressions for closed and co-closed differential forms and, as a…
In this letter, we introduce a new generalized linearizing transformation (GLT) for second order nonlinear ordinary differential equations (SNODEs). The well known invertible point (IPT) and non-point transformations (NPT) can be derived as…
$\mathcal{HH}$ spaces of type $[\textrm{N}] \otimes [\textrm{N}]$ with twisting congruence of null geodesics defined by the 4-fold undotted and dotted Penrose spinors are investigated. It is assumed that these spaces admit two homothetic…
Bagderina \cite{Bagderina2008} solved the equivalence problem for scalar third-order ordinary differential equations (ODEs), quadratic in the second-order derivative, via point transformations. However, the question is open for the general…
If $\Gamma$ is the nullity space of the curvature tensor of a Riemannian manifold $M^n$, it is well known that if its dimension is constant and if $M^n$ is complete then the distribution $\Gamma$ is completely integrable with flat leaves.…
An interesting result of Veech more than 50 years ago is a parity, or mod $2$, version of the Kronecker--Weyl equidistribution theorem concerning the irrational rotation sequence $\{q\alpha\}$, $q=0,1,2,3,\ldots.$ If $\alpha$ is badly…
For a linear ODE with indeterminate coefficients, we explicitly exhibit a fundamental system of solutions, in terms of the coefficients. We show that the generalized Wronskians of the fundamental system are given by an action of the Schur…
We provide a complete resolution to the question of compactness for the full solution sets of the fourth-order and sixth-order constant $Q$-curvature problems on smooth closed Riemannian manifolds not conformally diffeomorphic to the…
We address the problem of local geometry of third order ODEs modulo contact, point and fibre-preserving transformations of variables. Several new and already known geometries are described in a uniform manner by the Cartan method of…