Related papers: Dual pairs for matrix Lie groups
For all non-equivalent matrix systems of Painlev\'e-4 type found by authors in arXiv:2107.11680, isomonodromic Lax pairs are presented. Limiting transitions from these systems to matrix Painlev\'e-2 equations are found.
Based on a PML for the advective wave equation, we propose two PML models for the linearized Euler equations. The derivation of the first model can be applied to other physical models. The second model was implemented. Numerical results are…
We define the group of almost periodic diffeomorphisms on $\mathbb{R}^n$ and on an arbitrary Lie group. We then study the properties of its Riemannian and Lie group exponential maps and provide applications to fluid equations. In…
We promote Lazard's Poincar\'e duality for p-adic Lie groups to spectrum coefficients. The key aspect is the determination of the dualizing object in terms of "linear" data, namely the adjoint representation.
We introduce a mixed discontinuous/continuous finite element pair for ocean modelling, with continuous quadratic pressure/layer depth and discontinuous velocity. We investigate the finite element pair applied to the linear shallow-water…
We show that double Lie algebroids, together with a chosen linear splitting, are equivalent to pairs of 2-term representations up to homotopy satisfying compatibility conditions which extend the notion of matched pair of Lie algebroids. We…
The primary goal of this paper is to explicitly write down all semisimple dual pairs in the exceptional Lie algebras. (A dual pair in a reductive Lie algebra $\mathfrak{g}$ is a pair of subalgebras such that each member equals the other's…
We define and study the Weil pairing on the moduli of twisted curves. If $X$ is a twisted curve, then we can combinatorially describe a certain subgroup and a quotient group of $\text{Pic}(X)[2]$ that are Weil dual. Moreover, the pairing…
A sequence inverse relationship can be defined by a pair of infinite inverse matrices. If the pair of matrices are the same, they define a dual relationship. Here presented is a unified approach to construct dual relationships via…
Let $X(\Gamma)$ be the space of all finite Borel measure $\mu$ in $\mathbb R^2$ which is supported on the curve $\Gamma$ and absolutely continuous with respect to the arc length of $\Gamma$. For $\Lambda\subset\mathbb R^2,$ the pair…
Arnold showed that the Euler equations of an ideal fluid describe geodesics in the Lie algebra of incompressible vector fields. We will show that helicity induces a splitting of the Lie algebra into two isotropic subspaces, forming a Manin…
A Lax pair for the 2D Euler equation is found.
Lagrangian formulation for perfect fluid equations which hold invariant under the $\ell$-conformal Galilei group with half-integer $\ell$ is proposed. It is based on a Clebsch-type parametrization and reproduces Lagrangian description of…
Let $G$ be a linear Lie group acting properly and isometrically on a $G$-spin$^c$ manifold $M$ with compact quotient. We show that Poincar\'e duality holds between $G$-equivariant $K$-theory of $M$, defined using finite-dimensional…
This paper surveys results found by the authors in the previous papers (see for example, A. Duyunova, V. Lychagin, S. Tychkov, Differential invariants for spherical layer flows of a viscid fluid, Journal of Geometry and Physics, 130,…
Using methods from algebraic topology and group cohomology, I pursue Grothendieck's question on equality of geometric and cohomological Brauer groups in the context of complex-analytic spaces. The main result is that equality holds under…
We use a connection between relativistic hydrodynamics and scalar field theory to generate exact analytic solutions describing non-stationary inhomogeneous flows of the perfect fluid with one-parametric equation of state (EOS) $p =…
We derived one-fluid equations based on a relativistic two-fluid approximation of e$^\pm$ pair plasma and electron-ion plasma to reveal the specific relativistic nature of their behavior. Assuming simple condition on the relativistic…
We study the interpolation group whose elements are suitable pairs of formal power series. This group has a faithful representation into infinite lower triangular matrices and carries thus a natural structure as a Lie group. The matrix…
We obtain a complete solution to the problem of classifying all two-dimensional ideal fluid flows with harmonic Lagrangian labelling maps; thus, we explicitly provide all solutions, with the specified structural property, to the…