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In this work, we develop a modelling framework for granular flows based on the shallow water moment equations on inclined planes. Under the assumption of a polynomial expansion of the velocity field, the model extends the classical shallow…
This is the first in a series of two papers that develops a theory of relatively Anosov representations using the original "contracting flow on a bundle" definition of Anosov representations introduced by Labourie and Guichard-Wienhard. In…
The present thesis represents developments in two main directions related to the simple Lie algebras. The first one is devoted to the representation theory of the simple Lie algebras. Specifically, we present recent results, which include…
An alternative representation of an exact statistical field theory for simple fluids, based on the method of collective variables, is presented. The results obtained are examined from the point of another version of theory that was…
Equations for a perfect fluid can be obtained by means of the variational principle both in the Lagrangian description and in the Eulerian one. It is known that we need additional fields somehow to describe a rotational isentropic flow in…
We use Chern-Weil theory for Hermitian holomorphic vector bundles with canonical connections for explicit computation of the Chern forms of trivial bundles with special non-diagonal Hermitian metrics. We prove that every del-dellbar exact…
A variational principle is derived for two-dimensional incompressible rotational fluid flow with a free surface in a moving vessel when both the vessel and fluid motion are to be determined. The fluid is represented by a stream function and…
We summarize the progress made during the last few years on the study of Vassiliev invariants from the point of view of perturbative Chern-Simons gauge theory. We argue that this approach is the most promising one to obtain a combinatorial…
General features of the formalism describing hydrodynamic evolution of transversally thermalized matter possibly produced at the very early stages of ultra-relativistic heavy-ion collisions are presented. Thermodynamical consistency of the…
Relativistic Riemannian superfluid hydrodynamics used in general relativity to investigate superfluids in pulsars is extended to non-Riemannian background spacetime endowed with Cartan torsion. From the Gross-Pitaeviskii (GP) it is shown…
This paper presents an exterior-algebra generalization of electromagnetic fields and source currents as multivectors of grades $r$ and $r-1$ respectively in a flat space-time with $n$ space and $k$ time dimensions. Formulas for the Maxwell…
We prove a generalization of the classical Poincar\'e-Lelong formula. Given a holomorphic section $f$, with zero set $Z$, of a Hermitian vector bundle $E\to X$, let $S$ be the line bundle over $X\setminus Z$ spanned by $f$ and let $Q=E/S$.…
We use entropy theory as a new tool to study sectional hyperbolic flows in any dimension. We show that for $C^1$ flows, every sectional hyperbolic set $\Lambda$ is entropy expansive, and the topological entropy varies continuously with the…
It is known that $C^r$ Morse-Smale vector fields form an open dense subset in the space of vector fields on orientable closed surfaces and are structurally stable for any $r \in \mathbb{Z}_{>0}$. In particular, $C^r$ Morse vector fields…
We introduce transverse Chern-Ricci flow for transversely Hermitian foliations, which is analogous to the Chern-Ricci flow. We show that when $\mathcal{F}$ is homologically orientable and the basic first Bott-Chern class is zero, starting…
We introduce the notion of a generalized flow on a graph with coefficients in a R-representation and show that the module of flows is isomorphic to the first derived functor of the colimit. We generalize Kirchhoff's laws and build an exact…
This paper gives a framework to study a continuum limit of a gradient flow on a graph where the number of vertices increases in an appropriate way. As examples we prove the convergence of a discrete total variation flow and a discrete…
We give a new and self-contained proof of the existence and unicity of the flow for an arbitrary (not necessarily homogeneous) smooth vector field on a real supermanifold, and extend these results to the case of holomorphic vector fields on…
A rigorous formulation of Vessiot's vector field approach to the analysis of general systems of partial differential equations is provided. It is shown that this approach is equivalent to the formal theory of differential equations and that…
A generalized continuum framework for the theory of solute transport in fluids is proposed and systematically developed. This framework rests on the introduction of a generic force balance for the solute, a balance distinct from the…