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Given a positive integer $n$ and a partition $(n_1,\ldots,n_r)$ of $n$, one can consider the $n$-dimensional multiprojective space $\mathbb{P}^{n_1}\times \cdots \times \mathbb{P}^{n_r}$ corresponding to that partition. In this paper, we classify these multiprojective spaces. To be precise, we prove that given any two distinct partitions of any positive integer $n$, corresponding multiprojective spaces are not isomorphic using a decomposition of tensor products of irreducible representations of simple Lie algebras., Comment: All comments are welcome

Large language models (LLMs) with retrieval augmented-generation (RAG) have been the optimal choice for scalable generative AI solutions in the recent past. Although RAG implemented with AI agents (agentic-RAG) has been recently popularized, its suffers from unstable cost and unreliable performances for Enterprise-level data-practices. Most existing use-cases that incorporate RAG with LLMs have been either generic or extremely domain specific, thereby questioning the scalability and generalizability of RAG-LLM approaches. In this work, we propose a unique LLM-based system where multiple LLMs can be invoked to enable data authentication, user-query routing, data-retrieval and custom prompting for question-answering capabilities from Enterprise-data tables. The source tables here are highly fluctuating and large in size and the proposed framework enables structured responses in under 10 seconds per query. Additionally, we propose a five metric scoring module that detects and reports hallucinations in the LLM responses. Our proposed system and scoring metrics achieve >90% confidence scores across hundreds of user queries in the sustainability, financial health and social media domains. Extensions to the proposed extreme RAG architectures can enable heterogeneous source querying using LLMs., Comment: 5 pages, 4 tables, IEEE Big Data, 2024

In this paper, we describe the entire structure of the vector space $Sym_2^0$ of all symmetric matrices of size $2$ having trace zero. This is motivated by the geometrical interpretation of any arbitrary element of $Sym_2^0$. We further study the orbits and stable sets of these elements. As an application of the obtained structure of $Sym_2^0$, we obtain the symmetric matrices of size $2$, trace of whose product with any trace zero symmetric matrix is zero. Finally some well known trigonometric formulas are interpreted geometrically incorporating the anatomy of $Sym_2^0$., Comment: All comments are welcome

In this paper, we introduce the notion of the diagonal property and the weak point property for an ind-variety. We prove that the ind-varieties of higher rank divisors of integral slopes on a smooth projective curve have the weak point property. Moreover, we show that the ind-variety of $(1,n)$-divisors has the diagonal property. Furthermore, we obtain that the Hilbert schemes associated to the good partitions of a constant polynomial satisfy the diagonal property. In the process of obtaining this, we provide the exact number of such Hilbert schemes up to isomorphism by proving that the multi symmetric products associated to two distinct partitions of a positive integer $n$ are not isomorphic., Comment: All comments are welcome. arXiv admin note: substantial text overlap with arXiv:2212.04238

Multiple Deep Neural Networks (DNNs) integrated into single Deep Learning (DL) inference pipelines e.g. Multi-Task Learning (MTL) or Ensemble Learning (EL), etc., albeit very accurate, pose challenges for edge deployment. In these systems, models vary in their quantization tolerance and resource demands, requiring meticulous tuning for accuracy-latency balance. This paper introduces an automated heterogeneous quantization approach for DL inference pipelines with multiple DNNs.

Large Language Models (LLMs) have been applied to build several automation and personalized question-answering prototypes so far. However, scaling such prototypes to robust products with minimized hallucinations or fake responses still remains an open challenge, especially in niche data-table heavy domains such as financial decision making. In this work, we present a novel Langchain-based framework that transforms data tables into hierarchical textual data chunks to enable a wide variety of actionable question answering. First, the user-queries are classified by intention followed by automated retrieval of the most relevant data chunks to generate customized LLM prompts per query. Next, the custom prompts and their responses undergo multi-metric scoring to assess for hallucinations and response confidence. The proposed system is optimized with user-query intention classification, advanced prompting, data scaling capabilities and it achieves over 90% confidence scores for a variety of user-queries responses ranging from {What, Where, Why, How, predict, trend, anomalies, exceptions} that are crucial for financial decision making applications. The proposed data to answers framework can be extended to other analytical domains such as sales and payroll to ensure optimal hallucination control guardrails., Comment: 11 pages, 5 figures, 4 tables

Let $X$ denote a noncompact finite volume hyperbolic Riemann surface of genus $g\geq 2$, with only one puncture at $i\infty$ (identifying $X$ with its universal cover $\mathbb{H}$). Let $\overline{X}:=X\cup\lbrace i\infty\rbrace$ denote the Satake compactification of $X$. Let $\Omega_{\overline{X}}$ denote the cotangent bundle on $\overline{X}$. For $k\gg1$, we derive an estimate for $\mu_{\overline{X}}^{\mathrm{Ber},k}$, the Bergman metric associated to the line bundle $\mathcal{L}^{k}:=\Omega_{\overline{X}}\otimes \mathcal{O}_{\overline{X}}\big((k-1)\infty\big)$. For a given $d\geq 1$, the pull-back of the Fubini-Study metric on the Grassmannian, which we denote by $\mu_{\mathrm{Sym}^d(\overline{X})}^{\mathrm{FS},k}$, defines a K\"ahler metric on $\mathrm{Sym}^d(\overline{X})$, the $d$-fold symmetric product of $\overline{X}$. Using our estimates of $\mu_{\overline{X}}^{\mathrm{Ber},k}$, as an application, we derive an estimate for $\mu_{\mathrm{Sym}^d(\overline{X}),\mathrm{vol}}^{\mathrm{FS},k}$, the volume form associated to the (1,1)-form $\mu_{\mathrm{Sym}^d(\overline{X})}^{\mathrm{FS},k}$.

In this paper, we introduce the notion of the diagonal property and the weak point property for an ind-variety. We prove that the ind-varieties of higher rank divisors of integral slopes on a smooth projective curve have the weak point property. Moreover, we show that the ind-variety of $(1,n)$-divisors has the diagonal property. Furthermore, we obtain that the Hilbert schemes associated to the good partitions of a constant polynomial satisfy the diagonal property. On the process of obtaining this, we provide an upper bound on the number of such Hilbert schemes up to isomorphism. Furthermore, we prove that the obtained upper bound is attained in case of genus zero curves and hence conclude that the bound is sharp., Comment: Minor changes in one of the main results, a lemma added. All comments are welcome

Our aim in this paper is to determine the tautological algebra generated by the cohomology classes of the Brill Noether loci in the rational cohomology of the moduli stack $\mathcal{U}_C(n,d)$ of semistable bundles of rank $n$ and degree $d$. When $C$ is a general smooth projective curve of genus $g\geq 2$, $n=2$, $d=2g-2$, the tautological algebra of $ \mathcal{U}_C(2,2g-2)$ (resp. $\mathcal{SU}_C(2,L)$, $deg(L)=2g-2)$) is generated by the divisor classes (resp. the Theta divisor $\Theta$)., Comment: Revised version: statements are proved for moduli stacks, with a discussion on equivariant Chow/Cohomology groups

In this paper, our aim is to find the relations amongst the cohomology classes of Brill-Noether subvarieties of the moduli space of semistable bundles over an elliptic curve. We obtain results similar to the Poincar\'e relations on a Jacobian variety., Comment: All comments are welcome

Automatic question generation (QG) is a challenging problem in natural language understanding. QG systems are typically built assuming access to a large number of training instances where each instance is a question and its corresponding answer. For a new language, such training instances are hard to obtain making the QG problem even more challenging. Using this as our motivation, we study the reuse of an available large QG dataset in a secondary language (e.g. English) to learn a QG model for a primary language (e.g. Hindi) of interest. For the primary language, we assume access to a large amount of monolingual text but only a small QG dataset. We propose a cross-lingual QG model which uses the following training regime: (i) Unsupervised pretraining of language models in both primary and secondary languages and (ii) joint supervised training for QG in both languages. We demonstrate the efficacy of our proposed approach using two different primary languages, Hindi and Chinese. We also create and release a new question answering dataset for Hindi consisting of 6555 sentences., Comment: ACL 2019