The classification of thick representations of the symmetric group
We give the classification of thick representations and dense representations of the symmetric group over a field of characteristic zero.
math.RT↗
arXiv subjects
Kazunori Nakamoto
Publications and source records attributed to Kazunori Nakamoto.
Enhancing Pancreatic Cancer Staging with Large Language Models: The Role of Retrieval-Augmented Generation
Purpose: Retrieval-augmented generation (RAG) is a technology to enhance the functionality and reliability of large language models (LLMs) by retrieving relevant information from reliable external knowledge (REK). RAG has gained interest in radiology, and we previously reported the utility of NotebookLM, an LLM with RAG (RAG-LLM), for lung cancer staging. However, since the comparator LLM differed from NotebookLM's internal model, it remained unclear whether its advantage stemmed from RAG or inherent model differences. To better isolate RAG's impact and assess its utility across different cancers, we compared NotebookLM with its internal LLM, Gemini 2.0 Flash, in a pancreatic cancer staging experiment. Materials and Methods: A summary of Japan's pancreatic cancer staging guidelines was used as REK. We compared three groups - REK+/RAG+ (NotebookLM with REK), REK+/RAG- (Gemini 2.0 Flash with REK), and REK-/RAG- (Gemini 2.0 Flash without REK) - in staging 100 fictional pancreatic cancer cases based on CT findings. Staging criteria included TNM classification, local invasion factors, and resectability classification. In REK+/RAG+, retrieval accuracy was quantified based on the sufficiency of retrieved REK excerpts. Results: REK+/RAG+ achieved a staging accuracy of 70%, outperforming REK+/RAG- (38%) and REK-/RAG- (35%). For TNM classification, REK+/RAG+ attained 80% accuracy, exceeding REK+/RAG- (55%) and REK-/RAG- (50%). Additionally, REK+/RAG+ explicitly presented retrieved REK excerpts, achieving a retrieval accuracy of 92%. Conclusion: NotebookLM, a RAG-LLM, outperformed its internal LLM, Gemini 2.0 Flash, in a pancreatic cancer staging experiment, suggesting that RAG may improve LLM's staging accuracy. Furthermore, its ability to retrieve and present REK excerpts provides transparency for physicians, highlighting its applicability for clinical diagnosis and classification.
cs.CL↗
Extension of the one-sample Kolmogorov-Smirnov test
We propose here a new goodness-of-fit test, named the one-sample OVL-q test (q = 1, 2, . . .), which can be considered an extension of the one-sample Kolmogorov-Smirnov test (equivalent to the one-sample OVL-1 test). We have analyzed the asymptotic properties of the one-sample OVL-2 test statistic and enabled the calculation of asymptotic p-values for the test statistic. We further conducted numerical experiments and demonstrated that the one-sample OVL-2 test can sometimes exceed the detection power of conventional goodness-of-fit tests.
math.ST↗
Hochschild cohomology of the quadratic monomial algebra ${\rm N}_m$
Let ${\rm N}_m(R) = \{ (a_{ij}) \in {\rm M}_m(R) \mid a_{11} = a_{22} = \cdots = a_{mm} \mbox{ and } a_{ij} = 0 \mbox{ for any } i > j \}$ for a commutative ring $R$. Then ${\rm N}_m(R)$ is a quadratic monomial algebra over $R$. We calculate ${\rm HH}^{\ast}({\rm N}_m(R), {\rm M}_m(R)/{\rm N}_m(R))$ as $R$-modules. We also determine the $R$-algebra structure of the Hochschild cohomology ring ${\rm HH}^{\ast}({\rm N}_m(R), {\rm N}_m(R))$. For $m \ge 3$, ${\rm HH}^{\ast}({\rm N}_m(R), {\rm N}_m(R))$ is an infinitely generated algebra over $R$ and has no Batalin-Vilkovisky algebra structure giving the Gerstenhaber bracket.
math.RA↗
A novel statistical approach for two-sample testing based on the overlap coefficient
Here we propose a new nonparametric framework for two-sample testing, named as the OVL-$q$ ($q = 1, 2, \ldots$). This can be regarded as a natural extension of the Smirnov test, which is equivalent to the OVL-1. We specifically focus on the OVL-2, implement its fast algorithm, and show its superiority over other statistical tests in some experiments.
math.ST↗
Decision tree-based estimation of the overlap of two probability distributions
A new nonparametric approach, based on a decision tree algorithm, is proposed to calculate the overlap between two probability distributions. The devised framework is described analytically and numerically. The convergence of the estimated overlap to the true value is proved along with some experimental results.
math.ST↗
The classification of thick representations of simple Lie groups
We characterize finite-dimensional thick representations over ${\Bbb C}$ of connected complex semi-simple Lie groups by irreducible representations which are weight multiplicity-free and whose weight posets are totally ordered sets. Moreover, using this characterization, we give the classification of thick representations over ${\Bbb C}$ of connected complex simple Lie groups.
math.RT↗
Applications of Hochschild cohomology to the moduli of subalgebras of the full matrix ring
Let ${\rm Mold}_{n, d}$ be the moduli of rank $d$ subalgebras of ${\rm M}_n$ over ${\Bbb Z}$. For $x \in {\rm Mold}_{n, d}$, let ${\mathcal A}(x) \subseteq {\rm M}_n(k(x))$ be the subalgebra of ${\rm M}_n$ corresponding to $x$, where $k(x)$ is the residue field of $x$. In this article, we apply Hochschild cohomology to ${\rm Mold}_{n, d}$. The dimension of the tangent space $T_{{\rm Mold}_{n, d}/{\Bbb Z}, x}$ of ${\rm Mold}_{n, d}$ over ${\Bbb Z}$ at $x$ can be calculated by the Hochschild cohomology $H^{1}({\mathcal A}(x), {\rm M}_n(k(x))/{\mathcal A}(x))$. We show that $H^{2}({\mathcal A}(x), {\rm M}_n(k(x))/{\mathcal A}(x)) = 0$ is a sufficient condition for the canonical morphism ${\rm Mold}_{n, d} \to {\Bbb Z}$ being smooth at $x$. We also calculate $H^{i}(A, {\rm M}_n(k)/A)$ for several $R$-subalgebras $A$ of ${\rm M}_n(R)$ over a commutative ring $R$. In particular, we summarize the results on $H^{i}(A, {\rm M}_n(k)/A)$ for all $k$-subalgebras $A$ of ${\rm M}_n(k)$ over an algebraically closed field $k$ in the case $n=2, 3$.
math.RA↗
Special classes of irreducible representations I
We introduce special classes of irreducible representations of groups: thick representations and dense representations. Denseness implies thickness, and thickness implies irreducibility. We show that absolute thickness and absolute denseness are open conditions for representations. Thereby, we can construct the moduli schemes of absolutely thick representations and absolutely dense representations. We also describe several results and several examples on thick representations for developing theory of thick representations.
math.AG↗
The moduli of representations of degree 2
There are 6 types of 2-dimensional representations in general. For any groups and any monoids, we can construct the moduli of 2-dimensional representations for each type: the moduli of absolutely irreducible representations, representations with Borel mold, representations with semi-simple mold, representations with unipotent mold, representations with unipotent mold over ${\Bbb F}_2$, and representations with scalar mold. We can also construct them for any associative algebras.
math.AG↗
Virtual Hodge polynomials of the moduli spaces of representations of degree 2 for free monoids
In this paper we study the topology of the moduli spaces of representations of degree $2$ for free monoids. We calculate the virtual Hodge polynomials of the character varieties for several types of $2$-dimensional representations. Furthermore, we count the number of isomorphism classes for each type of $2$-dimensional representations over any finite field ${\Bbb F}_q$, and show that the number coincides with the virtual Hodge polynomial evaluated at $q$.
math.AG↗
Remarks on kernel Bayes' rule
Kernel Bayes' rule has been proposed as a nonparametric kernel-based method to realize Bayesian inference in reproducing kernel Hilbert spaces. However, we demonstrate both theoretically and experimentally that the prediction result by kernel Bayes' rule is in some cases unnatural. We consider that this phenomenon is in part due to the fact that the assumptions in kernel Bayes' rule do not hold in general.
stat.ML↗
The moduli of representations with Borel mold
The author constructs the moduli of representations whose images generate the subalgebra of upper triangular matrices (up to inner automorphisms) of the full matrix ring for any groups and any monoids.
math.AG↗