A q-analogue and a symmetric function analogue of a result by Carlitz, Scoville and Vaughan

We derive an equation that is analogous to a well-known symmetric function identity: $\sum_{i=0}^n(-1)^ie_ih_{n-i}=0$. Here the elementary symmetric function $e_i$ is the Frobenius characteristic of the representation of $\mathcal{S}_i$ on the top homology of the subset lattice $B_i$, whereas our identity involves the representation of $\mathcal{S}_n\times \mathcal{S}_n$ on the Segre product of $B_n$ with itself. We then obtain a q-analogue of a polynomial identity given by Carlitz, Scoville and Vaughan through examining the Segre product of the subspace lattice $B_n(q)$ with itself. We recognize the connection between the Euler characteristic of the Segre product of $B_n(q)$ with itself and the representation on the Segre product of $B_n$ with itself by recovering our polynomial identity from specializing the identity on the representation of $\mathcal{S}_i\times \mathcal{S}_i$

Experimental Validation of Uncertainty Quantification Methods for Robot Perception

In real-world settings, from woking in manufacturing plants to self driving on highway, robots empowered by Machine Learning (ML) models are tasked with complex, dynamic tasks that demand high levels of precision and adaptability. The reliability of these systems hinges on the perception capabilities of ML model, making uncertainty quantification methods vital. Conformal prediction is a user-friendly paradigm for creating statistically rigorous uncertainty sets/intervals for the predictions of such models. It ensures that robots can effectively assess and respond to varying conditions with safe and trustworthy actions, reducing the risk of errors and enhancing overall system performance. The purpose of this research project is to experimentally validate the effectiveness conformal prediction in object detection of a control algorithm on a ground robot platform