arXiv subjects

Chandan Maity

Publications and source records attributed to Chandan Maity.

15 recordsLinked to original sources

arXiv2024

HITgram: A Platform for Experimenting with n-gram Language Models

Large language models (LLMs) are powerful but resource intensive, limiting accessibility. HITgram addresses this gap by offering a lightweight platform for n-gram model experimentation, ideal for resource-constrained environments. It supports unigrams to 4-grams and incorporates features like context sensitive weighting, Laplace smoothing, and dynamic corpus management to e-hance prediction accuracy, even for unseen word sequences. Experiments demonstrate HITgram's efficiency, achieving 50,000 tokens/second and generating 2-grams from a 320MB corpus in 62 seconds. HITgram scales efficiently, constructing 4-grams from a 1GB file in under 298 seconds on an 8 GB RAM system. Planned enhancements include multilingual support, advanced smoothing, parallel processing, and model saving, further broadening its utility.

cs.CL↗

arXiv2024

Strongly real adjoint orbits of complex symplectic Lie group

We consider the adjoint action of the symplectic Lie group $\mathrm{Sp}(2n,\mathbb{C})$ on its Lie algebra $\mathfrak{sp}(2n,\mathbb{C})$. An element $X \in \mathfrak{sp}(2n,\mathbb{C})$ is called $\mathrm{Ad}_{\mathrm{Sp}(2n,\mathbb{C})}$-real if $ -X = \mathrm{Ad}(g)X$ for some $g \in \mathrm{Sp}(2n,\mathbb{C})$. Moreover, if $ -X = \mathrm{Ad}(h)X $ for some involution $h \in \mathrm{Sp}(2n,\mathbb{C})$, then $X \in \mathfrak{sp}(2n,\mathbb{C})$ is called strongly $\mathrm{Ad}_{\mathrm{Sp}(2n,\mathbb{C})}$-real. In this paper, we prove that for every element $X \in \mathfrak{sp}(2n,\mathbb{C})$, there exists a skew-involution $g \in \mathrm{Sp}(2n,\mathbb{C})$ such that $-X =\mathrm{Ad}(g)X$. Furthermore, we classify the strongly $\mathrm{Ad}_{\mathrm{Sp}(2n,\mathbb{C})}$-real elements in $\mathfrak{sp}(2n,\mathbb{C})$. We also classify skew-Hamiltonian matrices that are similar to their negatives via a symplectic involution.

math.GR↗

arXiv2024

Product of two involutions in quaternionic special linear group

An element of a group is called reversible if it is conjugate to its own inverse. Reversible elements are closely related to strongly reversible elements, which can be expressed as a product of two involutions. In this paper, we classify the reversible and strongly reversible elements in the quaternionic special linear group $\mathrm{SL}(n,\mathbb{H})$ and quaternionic projective linear group $ \mathrm{PSL}(n,\mathbb{H})$. We prove that an element of $ \mathrm{SL}(n,\mathbb{H})$ (resp. $ \mathrm{PSL}(n,\mathbb{H})$) is reversible if and only if it is a product of two skew-involutions (resp. involutions).

math.GR↗

arXiv2024

A note on adjoint reality in simple complex Lie algebras

Let $G$ be a Lie group with Lie algebra $\mathfrak g$. In the paper "Reality of unipotent elements in simple Lie groups, Bull. Sci. Math., 185, 2023, 103261" by K. Gongopadhyay and C. Maity, an infinitesimal version of the notion of classical reality, namely adjoint reality, has been introduced. An element $X \in \mathfrak g$ is adjoint real if $-X$ belongs to the adjoint orbit of $X$ in $\mathfrak g$. In this paper, we investigate the adjoint real and the strongly adjoint real semisimple elements in complex simple classical Lie algebras. We also prove that every element in a complex symplectic Lie algebra is adjoint real.

math.GR↗

arXiv2023

Strongly reversible classes in $\mathrm{SL}(n,\mathbb{C})$

An element of a group is called $\textit{strongly reversible}$ or $\textit{strongly real}$ if it can be expressed as a product of two involutions. We provide necessary and sufficient conditions for an element of $\mathrm{SL}(n,\mathbb{C})$ to be a product of two involutions. In particular, we classify the strongly reversible conjugacy classes in $\mathrm{SL}(n,\mathbb{C})$.

math.GR↗

arXiv2023

On the third and fourth Betti numbers of a homogeneous space of a Lie group

In the paper "The second cohomology of nilpotent orbits in classical Lie algebras, Kyoto J. Math. 60 (2020), no. 2, 717-799" by I. Biswas, P. Chatterjee, and C. Maity, explicit descriptions of the second and first real de Rham cohomology groups of a general homogeneous space of a Lie group are given, extending an earlier result in "On the exactness of Kostant-Kirillov form and the second cohomology of nilpotent orbits, Internat. J. Math. 23 (2012), no. 8, 1250086" by I. Biswas and P. Chatterjee. From the computational viewpoint, they turned out to be new and very useful, and in fact played a crucial role in determining the second cohomology of nilpotent orbits as done in the above two papers. In this paper, we give computable and explicit descriptions of the third and fourth real de Rham cohomologies of a general homogeneous space, in terms of the associated Lie-theoretic data, along the lines mentioned above. We also draw numerous corollaries of our main results in important special settings. Moreover, as a consequence, we obtain a new and interesting invariant by showing that for a large class of homogeneous spaces, the difference between the third and fourth Betti numbers coincides with the difference between the numbers of simple factors of the ambient group and the associated closed subgroup.

math.GR↗

arXiv2023

Reversibility and Real Adjoint Orbits of Linear Maps

We extend classical results on the classification of reversible elements of the group $\mathrm{GL}(n, \mathbb{C})$ (and $\mathrm{GL}(n, \mathbb{R})$) to $\mathrm{GL}(n, \mathbb{H})$ using an infinitesimal version of the classical reversibility, namely adjoint reality in the Lie algebra set-up. We also provide a new proof of such a classification for the general linear groups over $\mathbb{R}$ and $\mathbb{C}$. Further, we classify the real adjoint orbits in the Lie algebra $\mathfrak{gl}(n, \mathbb{D})$ for $ \mathbb{D}=\mathbb{R}, \mathbb{C}$ or $\mathbb{H} $.

math.GR↗

arXiv2022

Reversibility of Affine Transformations

An element $g$ in a group $G$ is called reversible if $g$ is conjugate to $g^{-1}$ in $ G $. An element $g$ in $G$ is strongly reversible if $ g $ is conjugate to $g^{-1}$ by an involution in $G$. The group of affine transformations of $\mathbb{D}^n$ may be identified with the semi-direct product $\mathrm{GL}(n, \mathbb{D}) \ltimes \mathbb{D}^n $, where $\mathbb{D}:=\mathbb{R}, \mathbb{C}$ or $ \mathbb{H} $. This paper classifies reversible and strongly reversible elements in the affine group $\mathrm{GL}(n, \mathbb{D}) \ltimes \mathbb{D}^n $.

math.GR↗

arXiv2022

Real adjoint orbits of the unipotent subgroup

Let $G$ be a linear Lie group that acts on it's Lie algebra $\mathfrak{g}$ by the adjoint action: $\mathrm{Ad}(g)X=gXg^{-1}$. An element $X\in \mathfrak {g}$ is called $\mathrm{Ad}_G$-real if $-X = \mathrm{Ad}(g)X $ for some $g\in G$. An $\mathrm{Ad}_G$-real element $X$ is called strongly $\mathrm{Ad}_G $-real if $-X = \mathrm{Ad}(τ) X $ for some involution $τ\in G$. Let $K=\mathbb{R}$, $\mathbb{C}$ or $\mathbb{H}$. Let $\mathrm{U}_n(K)$ be the group of unipotent upper-triangular matrices over $K$. Let $\mathfrak{u}_n (K)$ be the Lie algebra of $\mathrm{U}_n(K)$ that consists of $n \times n$ upper triangular matrices with $0$ in all the diagonal entries. In this paper, we consider the $\mathrm{Ad}$-reality of the Lie algebra $ \mathfrak{u}_n(K) $ that comes from the adjoint action of the Lie group $\mathrm{U}_n(K)$ on $ \mathfrak{u}_n(K)$. We prove that there is no non-trivial $\mathrm{Ad}_{\mathrm{ U}_n(K)}$-real element in $\mathfrak{u}_n (K)$. We also consider the adjoint action of the extended group $\mathrm{U}_n^\pm(K)$ that consists of all upper triangular matrices over $K$ having diagonal elements as $1$ or $-1$, and construct a large class of $\mathrm{Ad} _{\mathrm{ U}_n^\pm( K)} $-real elements. As applications of these results, we recover related results concerning classical reality in these groups.

math.GR↗

arXiv2022

Real adjoint orbits of special linear groups

Let $ G $ be a Lie group with Lie algebra $ \mathfrak{g} $. An element $ X \in \mathfrak{g} $ is called $\mathrm{Ad}_G$-real if $ -X=gXg^{-1} $ for some $ g \in G $. Moreover, if $ -X=gXg^{-1} $ holds for some involution $ g\in G $, then $ X $ is called strongly $\mathrm{Ad}_G$-real. We have classified the $\mathrm{Ad}_G$-real and the strongly $\mathrm{Ad}_G$-real orbits in the special linear Lie algebra $\mathfrak{sl}(n,\mathbb{F}) $ for $ \mathbb{F}=\mathbb{C}$ or $\mathbb{H} $.

math.GR↗

arXiv2022

The second cohomology groups of nilpotent orbits in classical Lie algebras

The second de Rham cohomology groups of nilpotent orbits in non-compact real forms of classical complex simple Lie algebras are explicitly computed. Furthermore, the first de Rham cohomology groups of nilpotent orbits in non-compact classical simple Lie algebras are computed; they are proven to be zero for nilpotent orbits in all the complex simple Lie algebras. A key component in these computations is a description of the second and first cohomology groups of homogeneous spaces of general connected Lie groups which is obtained here. This description, which generalizes a previous theorem of the first two authors, may be of independent interest.

math.GR↗

arXiv2022

Homotopy type of the nilpotent orbits in classical Lie algebras

In the paper "The Second cohomology of nilpotent orbits in classical Lie algebras, Kyoto J. Math. 60 (2020), no. 2, 717-799" by I. Biswas, P. Chatterjee and C. Maity homotopy types of nilpotent orbits are explicitly described in the case of real simple classical Lie algebras for which any maximal compact subgroup in the associated adjoint group is not semisimple. In this paper we extend the above description of homotopy type of nilpotent orbits to the remaining cases of real simple classical Lie algebras for which any maximal compact subgroup in the associated adjoint group is semisimple.

math.GR↗

arXiv2021

Reality of Unipotent elements in Classical Lie Groups

The aim of this paper is to give a classification of real and strongly real unipotent elements in a classical simple Lie group. To do this, we will introduce an infinitesimal version of the notion of classical reality in a Lie group. This notion has been applied to classify real and strongly real unipotent elements in a classical simple Lie group.

math.GR↗

arXiv2018

Phase Mixing of Large Amplitude Relativistic Electron Plasma Oscillation With Inhomogeneous Ion Background

Phase mixing of relativistic large amplitude nonlinear plasma wave in presence of a time independent space periodic ion density profile has been investigated. Inhomogeneous ion along with the relativistic variation of electron mass make the characteristic frequency of the wave to acquire a space dependency and thereby it breaks at arbitrarily small amplitude due to phase mixing. An approximate space time dependent solution is obtained in the weakly relativistic limit by Bogoliuboff and Kryloff method of averaging. We find that the change in the ion density perturbation and also the relativistic electron mass variation have significant effect in modifying the time at which phase mixing occurs.

physics.plasm-ph↗

arXiv2016

On the second cohomology of nilpotent orbits in exceptional Lie algebras

In this paper we consider non-compact non-complex exceptional Lie algebras, and compute the dimensions of the second cohomology groups for most of the nilpotent orbits. For the rest of cases of nilpotent orbits, which are not covered in the above computations, we obtain upper bounds for the dimensions of the second cohomology groups.

math.GR↗