arXiv subjects

Sayan Pal

Publications and source records attributed to Sayan Pal.

3 recordsLinked to original sources

arXiv2025

On rational orbits in some prehomogeneous vector spaces

Let $k$ be a field with characteristic different from $2$. In this paper, we describe the $k$-rational orbit spaces in some irreducible prehomogeneous vector spaces $(G,V)$ over $k$, where $G$ is a connected reductive algebraic group defined over $k$ and $V$ is an irreducible rational representation of $G$ with a Zariski dense open orbit. We parametrize all composition algebras over the field $k$ in terms of the orbits in some of these representations. This leads to a parametric description of the reduced Freudenthal algebras of dimensions $6$ and $9$ over $k$ (if $\text{char}(k)\neq 2,3$). We also get a parametrization for the involutions of the second kind defined on a central division $K$-algebra $B$ with center $K$, a quadratic extension of the underlying field $k$.

math.GR↗

arXiv2025

Rational orbits in some prehomogeneous vector spaces associated to $Sp_{6}$ revisited

Let $k$ be a field with $\text{char}(k)\neq 2$. We prove that all maximal flags of composition algebras over $k$, appear as the $k$-rational $Sp_{6}$-orbits in a Zariski-dense $Sp_{6}$-invariant subset $V^{ss}\subset V=\wedge^{3}V_{6}$, where $V_{6}$ is the standard $6$-dimensional irreducible representation of $Sp_{6}$. This gives an arithmetic interpretation for the orbit spaces of the semi-stable sets in the prehomogeneous vector spaces $(Sp_{6}\times GL_{1}^{2},V)$ and $(GSp_{6}\times GL_{1}^{2},V)$. We also get all reduced Freudenthal algebras of dimensions $6$ and $9$, represented by the same orbit spaces.

math.GR↗

arXiv2020

Universal Behaviour of Settling and spreading of particle clusters in quiescent fluids in confined vessels

Here we report experiments on particle cluster settling at high Reynolds number in quiescent fluid contained in a vessel. The particles were observed to settle in a near-circular shape irrespective of the shape of the vessel cross-section and particle shape, size, and types. Effect of different parameters such as mass, type and aspect ratio of the particles, height, and viscosity of liquid was investigated. Formation of the hemispherical bottom cap of the cluster that bounces upon hitting the vessel bottom surface was found to be responsible for the final circular shape of the settled structure. Particle leakage from the cluster was seen in the form of a tail. In the liquid having viscosity beyond 100 cP, cluster breakage was observed that resulted in hindered settling and asymmetric shapes of finally settled particles. The observations are useful to understand the overall area over which settling of such clusters can be observed.

physics.flu-dyn↗