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We give a counterexample to the Arithmetic Puncturing Conjecture and Geometric Puncturing Conjecture of Hassett-Tschinkel using symmetric powers of uniruled surfaces, and propose a corrected conjecture inspired by Campana's conjectures on special varieties. We verify Campana's conjecture on potential density for symmetric powers of products of curves. As a by-product, we obtain an example of a surface without a potentially dense set of rational points, but for which some symmetric power does have a dense set of rational points, and even satisfies Corvaja-Zannier's version of the Hilbert property., Comment: 28 pages. Comments welcome

Physical models often contain unknown functions and relations. In order to gain more insights into the nature of physical processes, these unknown functions have to be identified or reconstructed. Mathematically, we can formulate this research question within the framework of inverse problems. In this work, we consider optimization techniques to solve the inverse problem using Tikhonov regularization and data from laboratory experiments. We propose an iterative strategy that eliminates the need for laboratory experiments. Our method is applied to identify the coupling and damping coefficients in a system of oscillators, ensuring an efficient and experiment-free approach. We present our results and compare them with those obtained from an alternative, purely experimental approach. By employing our proposed strategy, we demonstrate a significant reduction in the number of laboratory experiments required.

Deformable object manipulation remains a key challenge in developing autonomous robotic systems that can be successfully deployed in real-world scenarios. In this work, we explore the challenges of deformable object manipulation through the task of sculpting clay into 3D shapes. We propose the first coarse-to-fine autonomous sculpting system in which the sculpting agent first selects how many and where to place discrete chunks of clay into the workspace to create a coarse shape, and then iteratively refines the shape with sequences of deformation actions. We leverage large language models for sub-goal generation, and train a point cloud region-based action model to predict robot actions from the desired point cloud sub-goals. Additionally, our method is the first autonomous sculpting system that is a real-world text-to-3D shaping pipeline without any explicit 3D goals or sub-goals provided to the system. We demonstrate our method is able to successfully create a set of simple shapes solely from text-based prompting. Furthermore, we explore rigorously how to best quantify success for the text-to-3D sculpting task, and compare existing text-image and text-point cloud similarity metrics to human evaluations for this task. For experimental videos, human evaluation details, and full prompts, please see our project website: https://sites.google.com/andrew.cmu.edu/hierarchicalsculpting

Topological correctness is critical for segmentation of tubular structures. Existing topological segmentation loss functions are primarily based on the persistent homology of the image. They match the persistent features from the segmentation with the persistent features from the ground truth and minimize the difference between them. However, these methods suffer from an ambiguous matching problem since the matching only relies on the information in the topological space. In this work, we propose an effective and efficient Spatial-Aware Topological Loss Function that further leverages the information in the original spatial domain of the image to assist the matching of persistent features. Extensive experiments on images of various types of tubular structures show that the proposed method has superior performance in improving the topological accuracy of the segmentation compared with state-of-the-art methods.

Global internal symmetries act unitarily on local observables or states of a quantum system. In this note, we aim to generalise this statement to extended observables by considering unitary actions of finite global 2-group symmetries $\mathcal{G}$ on line operators. We propose that the latter transform in unitary 2-representations of $\mathcal{G}$, which we classify up to unitary equivalence. Our results recover the known classification of ordinary 2-representations of finite 2-groups, but provide additional data interpreted as a type of reflection anomaly for $\mathcal{G}$., Comment: 36 pages

Starting from an Enriques surface over $\mathbb{Q}(t)$ considered by Lafon, we give the first examples of smooth projective weakly special threefolds which fiber over the projective line in Enriques surfaces (resp. K3 surfaces) with nowhere reduced, but non-divisible, fibres and general type orbifold base. We verify that these families of Enriques surfaces (resp. K3 surfaces) are non-isotrivial and compute their fundamental groups by studying the behaviour of local points along certain \'etale covers. The existence of the above threefolds implies that the Weakly Special Conjecture formulated in 2000 contradicts the Orbifold Mordell Conjecture, and hence the abc conjecture. Using these examples, we can also easily disprove several complex-analytic analogues of the Weakly Special Conjecture. Finally, the existence of such threefolds shows that Enriques surfaces and K3 surfaces can have non-divisible but nowhere reduced degenerations, thereby answering a question raised in 2005., Comment: 29 pages; comments welcome; v2: minor changes to abstract and introduction

We present FastLexRank\footnote{https://github.com/LiMaoUM/FastLexRank}, an efficient and scalable implementation of the LexRank algorithm for text ranking. Designed to address the computational and memory complexities of the original LexRank method, FastLexRank significantly reduces time and memory requirements from $\mathcal{O}(n^2)$ to $\mathcal{O}(n)$ without compromising the quality or accuracy of the results. By employing an optimized approach to calculating the stationary distribution of sentence graphs, FastLexRank maintains an identical results with the original LexRank scores while enhancing computational efficiency. This paper details the algorithmic improvements that enable the processing of large datasets, such as social media corpora, in real-time. Empirical results demonstrate its effectiveness, and we propose its use in identifying central tweets, which can be further analyzed using advanced NLP techniques. FastLexRank offers a scalable solution for text centrality calculation, addressing the growing need for efficient processing of digital content.

As robotic systems become increasingly integrated into complex real-world environments, there is a growing need for approaches that enable robots to understand and act upon natural language instructions without relying on extensive pre-programmed knowledge of their surroundings. This paper presents PLATO, an innovative system that addresses this challenge by leveraging specialized large language model agents to process natural language inputs, understand the environment, predict tool affordances, and generate executable actions for robotic systems. Unlike traditional systems that depend on hard-coded environmental information, PLATO employs a modular architecture of specialized agents to operate without any initial knowledge of the environment. These agents identify objects and their locations within the scene, generate a comprehensive high-level plan, translate this plan into a series of low-level actions, and verify the completion of each step. The system is particularly tested on challenging tool-use tasks, which involve handling diverse objects and require long-horizon planning. PLATO's design allows it to adapt to dynamic and unstructured settings, significantly enhancing its flexibility and robustness. By evaluating the system across various complex scenarios, we demonstrate its capability to tackle a diverse range of tasks and offer a novel solution to integrate LLMs with robotic platforms, advancing the state-of-the-art in autonomous robotic task execution. For videos and prompt details, please see our project website: https://sites.google.com/andrew.cmu.edu/plato, Comment: 7 pages, 4 figures, submitted to ICRA 2025

Semantic segmentation is an essential step for many vision applications in order to understand a scene and the objects within. Recent progress in hyperspectral imaging technology enables the application in driving scenarios and the hope is that the devices perceptive abilities provide an advantage over RGB-cameras. Even though some datasets exist, there is no standard benchmark available to systematically measure progress on this task and evaluate the benefit of hyperspectral data. In this paper, we work towards closing this gap by providing the HyperSpectral Semantic Segmentation benchmark (HS3-Bench). It combines annotated hyperspectral images from three driving scenario datasets and provides standardized metrics, implementations, and evaluation protocols. We use the benchmark to derive two strong baseline models that surpass the previous state-of-the-art performances with and without pre-training on the individual datasets. Further, our results indicate that the existing learning-based methods benefit more from leveraging additional RGB training data than from leveraging the additional hyperspectral channels. This poses important questions for future research on hyperspectral imaging for semantic segmentation in driving scenarios. Code to run the benchmark and the strong baseline approaches are available under https://github.com/nickstheisen/hyperseg., Comment: Accepted for publication at 2024 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS 2024)

We study the following Neumann boundary problem related to the stationary solutions of the Keller-Segel system, a basic model of chemotaxis phenomena: \[ -\Delta_g u +\beta u =\lambda\left(\frac{Ve^u}{\int_{\Sigma} Ve^u d v_g}-\frac{1}{|\Sigma|_g}\right) \text { in } \mathring\Sigma \] with $\partial_{ \nu_g} u=0, \text { on } \partial \Sigma$, where $(\Sigma, g)$ is a compact Riemann surface with the interior $\mathring\Sigma$ and the smooth boundary $\partial \Sigma$. Here, $\lambda, \beta\geq 0$ are non-negative parameters, and $V$ is a smooth non-negative function with a finite zero set. For any given integers $m\geq k\geq 0$, we establish a sufficient condition on $V$ for the existence of a sequence of blow-up solutions as $\lambda$ approaches the critical values $4\pi(m+k)$. Moreover, the study expands to the corresponding singular problem.

This work studies the partial blow-up phenomena for the $SU(3)$ Toda system on compact Riemann surfaces with smooth boundary. We consider the following coupled Liouville system with Neumann boundary conditions: $$ -\Delta_g u_1 = 2\rho_1\left( \frac{V_1 e^{u_1}}{\int_{\Sigma} V_1 e^{u_1} \, dv_g} - \frac 1 {|\Sigma|_g}\right) - \rho_2\left( \frac{V_2 e^{u_2}}{\int_{\Sigma} V_2 e^{u_2} \, dv_g} - \frac{1}{|\Sigma|_g}\right) \text{in} \,\mathring\Sigma$$ and $$ -\Delta_g u_2 = 2\rho_2\left( \frac{V_2 e^{u_2}}{\int_{\Sigma} V_2 e^{u_2} \, dv_g} - \frac{1}{|\Sigma|_g}\right) - \rho_1\left( \frac{V_1 e^{u_1}}{\int_{\Sigma} V_1 e^{u_1} \, dv_g} - \frac{1}{|\Sigma|_g}\right) \text{in} \,\mathring\Sigma$$ with boundary conditions $ \partial_{\nu_g} u_1 = \partial_{\nu_g} u_2 = 0 \text{ on} \, \partial \Sigma,$ where $(\Sigma, g)$ is a compact Riemann surface with the interior $\mathring\Sigma$ and smooth boundary $\partial\Sigma$, $\rho_i$ is a non-negative parameter and $V_i$ is a smooth positive function for $i=1,2$. We construct a family of blow-up solutions via the Lyapunov-Schmidt reduction and variational methods, wherein one component remains uniformly bounded from above, while the other exhibits partial blow-ups at a prescribed number of points, both in the interior and on the boundary. This construction is based on the existence of a non-degeneracy solution of a so-called shadow system. Moreover, we establish the existence of partial blow-up solutions in three cases: (i) for any $\rho_2>0$ sufficiently small; (ii) for generic $V_1, V_2$ and any $\rho_2\in (0,2\pi)$; (iii) for generic $V_1, V_2$, the Euler characteristic $\chi(\Sigma)<1$ and any $\rho_2\in (2\pi,+\infty)\setminus 2\pi \mathbb{N}_+$.

On a compact Riemann surface $(\Sigma, g)$ with a smooth boundary $\partial \Sigma$, we consider the following mean field equations with Neumann boundary conditions: $$ -\Delta_g u = \lambda \left(\frac{Ve^u}{\int_{\Sigma} Ve^u \, dv_g} - \frac{1}{|\Sigma|_g}\right) \text{ in } \Sigma \text{ with } \partial_{\nu_g} u = 0 \text{ on } \partial \Sigma, $$ We find conditions on the potential function $V: \Sigma \to \mathbb{R}^+$ such that solutions exist for the parameter $\lambda$ when it is in a small right (or left) neighborhood of a critical value $4\pi(m+k)$ for $k \leq m \in \mathbb{N}_+$ and blow up as $\lambda$ approaches the critical parameter. The blow-up occurs exactly at $k$ points in the interior of $\Sigma$ and $(m-k)$ points on the boundary $\partial \Sigma$.

Individual teaching is among the most successful ways to impart knowledge. Yet, this method is not always feasible due to large numbers of students per educator. Quantum computing serves as a prime example facing this issue, due to the hype surrounding it. Alleviating high workloads for teachers, often accompanied with individual teaching, is crucial for continuous high quality education. Therefore, leveraging Large Language Models (LLMs) such as GPT-4 to generate educational content can be valuable. We conducted two complementary studies exploring the feasibility of using GPT-4 to automatically generate tips for students. In the first one students (N=46) solved four multiple-choice quantum computing questions with either the help of expert-created or LLM-generated tips. To correct for possible biases towards LLMs, we introduced two additional conditions, making some participants believe that they were given expert-created tips, when they were given LLM-generated tips and vice versa. Our second study (N=23) aimed to directly compare the LLM-generated and expert-created tips, evaluating their quality, correctness and helpfulness, with both experienced educators and students participating. Participants in our second study found that the LLM-generated tips were significantly more helpful and pointed better towards relevant concepts than the expert-created tips, while being more prone to be giving away the answer. While participants in the first study performed significantly better in answering the quantum computing questions when given tips labeled as LLM-generated, even if they were created by an expert. This phenomenon could be a placebo effect induced by the participants' biases for LLM-generated content. Ultimately, we find that LLM-generated tips are good enough to be used instead of expert tips in the context of quantum computing basics.

In this short survey, we explain Parshin's proof of the geometric Bombieri-Lang conjecture, and show that it can be used to give an alternative proof of Xie-Yuan's recent resolution of the geometric Bombieri-Lang conjecture for projective varieties with empty special locus and admitting a finite morphism to a traceless abelian variety., Comment: 10 pages. Survey note. Jacob Murre SI v2. minor improvements

Paired cluster-randomized experiments (pCRTs) are common across many disciplines because there is often natural clustering of individuals, and paired randomization can help balance baseline covariates to improve experimental precision. Although pCRTs are common, there is surprisingly no obvious way to analyze this randomization design if an individual-level (rather than cluster-level) treatment effect is of interest. Variance estimation is also complicated due to the dependency created through pairing clusters. Therefore, we aim to provide an intuitive and practical comparison between different estimation strategies in pCRTs in order to inform practitioners' choice of strategy. To this end, we present a general framework for design-based estimation in pCRTs for average individual effects. This framework offers a novel and intuitive view on the bias-variance trade-off between estimators and emphasizes the benefits of covariate adjustment for estimation with pCRTs. In addition to providing a general framework for estimation in pCRTs, the point and variance estimators we present support fixed-sample unbiased estimation with similar precision to a common regression model and consistently conservative variance estimation. Through simulation studies, we compare the performance of the point and variance estimators reviewed. Finally, we compare the performance of estimators with simulations using real data from an educational efficacy trial. Our analysis and simulation studies inform the choice of point and variance estimators for analyzing pCRTs in practice.

When humans create sculptures, we are able to reason about how geometrically we need to alter the clay state to reach our target goal. We are not computing point-wise similarity metrics, or reasoning about low-level positioning of our tools, but instead determining the higher-level changes that need to be made. In this work, we propose LLM-Craft, a novel pipeline that leverages large language models (LLMs) to iteratively reason about and generate deformation-based crafting action sequences. We simplify and couple the state and action representations to further encourage shape-based reasoning. To the best of our knowledge, LLM-Craft is the first system successfully leveraging LLMs for complex deformable object interactions. Through our experiments, we demonstrate that with the LLM-Craft framework, LLMs are able to successfully reason about the deformation behavior of elasto-plastic objects. Furthermore, we find that LLM-Craft is able to successfully create a set of simple letter shapes. Finally, we explore extending the framework to reaching more ambiguous semantic goals, such as "thinner" or "bumpy". For videos please see our website: https://sites.google.com/andrew.cmu.edu/llmcraft.

Physical models often contain unknown functions and relations. The goal of our work is to answer the question of how one should excite or control a system under consideration in an appropriate way to be able to reconstruct an unknown nonlinear relation. To answer this question, we propose a greedy reconstruction algorithm within an offline-online strategy. We apply this strategy to a two-dimensional semilinear elliptic model. Our identification is based on the application of several space-dependent excitations (also called controls). These specific controls are designed by the algorithm in order to obtain a deeper insight into the underlying physical problem and a more precise reconstruction of the unknown relation. We perform numerical simulations that demonstrate the effectiveness of our approach which is not limited to the current type of equation. Since our algorithm provides not only a way to determine unknown operators by existing data but also protocols for new experiments, it is a holistic concept to tackle the problem of improving physical models.

In this paper we study the Morse property for functions related to limit functions of mean field equations on a smooth, compact surface $\Sigma$ with boundary $\partial\Sigma$. Given a Riemannian metric $g$ on $\Sigma$ we consider functions of the form \[ f_g(x) := \sum_{i=1}^m\sigma_i^2R^g(x_i)+\sum_{i,j=1\\i\ne j}^m\sigma_i\sigma_jG^g(x_i,x_j)+h(x_1,\ldots,x_m), \] where $\sigma_i \neq 0$ for $i=1,\ldots,m$, $G^g$ is the Green function of the Laplace-Beltrami operator on $(\Sigma,g)$ with Neumann boundary conditions, $R^g$ is the corresponding Robin function, and $h \in \mathcal{C}^{2}(\Sigma^m,\mathbb{R})$ is arbitrary. We prove that for any Riemannian metric $g$, there exists a metric $\widetilde g$ which is arbitrarily close to $g$ and in the conformal class of $g$ such that $f_{\widetilde g}$ is a Morse function. Furthermore we show that, if all $\sigma_i>0$, then the set of Riemannian metrics for which $f_g$ is a Morse function is open and dense in the set of all Riemannian metrics., Comment: 19 pages

Recently, Arkani-Hamed et al. proposed the existence of zeros in scattering amplitudes in certain quantum field theories including the cubic adjoint scalar theory Tr($\phi^3$), the $SU(N)$ non-linear sigma model (NLSM) and Yang-Mills (YM) theory. These hidden zeros are special kinematic points where the amplitude vanishes and factorizes into a product of lower-point amplitudes, similar to factorization near poles. In this letter, we show a close connection between the existence of such zeros and color-kinematics duality. In fact, all zeros can be derived from the Bern-Carrasco-Johansson (BCJ) relations. We also show that these zeros extend via the Kawai-Lewellen-Tye (KLT) relations to special Galileon amplitudes and their corrections, evincing that these hidden zeros are also present in permutation-invariant amplitudes., Comment: 1 figure, 2 tables

Manipulating deformable objects remains a challenge within robotics due to the difficulties of state estimation, long-horizon planning, and predicting how the object will deform given an interaction. These challenges are the most pronounced with 3D deformable objects. We propose SculptDiff, a goal-conditioned diffusion-based imitation learning framework that works with point cloud state observations to directly learn clay sculpting policies for a variety of target shapes. To the best of our knowledge this is the first real-world method that successfully learns manipulation policies for 3D deformable objects. For sculpting videos and access to our dataset and hardware CAD models, see the project website: https://sites.google.com/andrew.cmu.edu/imitation-sculpting/home

In this letter, we discuss a generalization of the Adler zero to loop integrands in the planar limit of the $SU(N)$ non-linear sigma model (NLSM). While possible to maintain at one-loop, the Adler zero for integrands is violated starting at the two-loop order and is only recovered after integration. Here we propose a non-zero soft theorem satisfied by loop integrands with any number of loops and legs. This requires a generalization of NLSM integrands to an off-shell framework with certain deformed kinematics. Defining an `algebraic soft limit', we identify a particularly simple non-vanishing soft behavior of integrands, which we call the `algebraic soft theorem'. We find that the proposed soft theorem is satisfied by the `surface' integrand of Arkani-Hamed, Cao, Dong, Figueiredo and He, which is obtained from the shifted ${\rm Tr}\phi^3$ surfacehedron integrand. Finally, we derive an on-shell version of the algebraic soft theorem that takes an interesting form in terms of propagator renormalization factors and lower-loop integrands in a mixed theory of pions and scalars., Comment: 6 pages

A large-volume Time Projection Chamber (TPC) is the main tracking and particle identification (PID) detector of the ALICE experiment at the CERN LHC. PID in the TPC is performed via specific energy-loss measurements (dE/dx), which are derived from the average pulse-height distribution of ionization generated by charged-particle tracks traversing the TPC volume. During Runs 1 and 2, until 2018, the gas amplification stage was based on multiwire proportional chambers (MWPC). Signals from the MWPC show characteristic long negative tails after an initial positive peak due to the long ion drift times in the MWPC amplification region. This so-called ion tail can lead to a significant amplitude loss in subsequently measured signals, especially in the high-multiplicity environment of high-energy Pb-Pb collisions, which results in a degradation of the dE/dx resolution. A detailed study of the signal shapes measured with the ALICE TPC with the Ne-CO2 (90-10) and Ar-CO2 (90-10) gas mixtures is presented, and the results are compared with three-dimensional Garfield simulations. The impact of the ion tail on the PID performance is studied employing the ALICE simulation framework and the feasibility of an offline correction procedure to account for the ion tail is demonstrated., Comment: 15 pages, 16 figures

To study the nonlinear properties of complex natural phenomena, the evolution of the quantity of interest can be often represented by systems of coupled nonlinear stochastic differential equations (SDEs). These SDEs typically contain several parameters which have to be chosen carefully to match the experimental data and to validate the effectiveness of the model. In the present paper the calibration of these parameters is described by nonlinear SDE-constrained optimization problems. In the optimize-before-discretize setting a rigorous analysis is carried out to ensure the existence of optimal solutions and to derive necessary first-order optimality conditions. For the numerical solution a Monte-Carlo method is applied using parallelization strategies to compensate for the high computational time. In the numerical examples an Ornstein-Uhlenbeck and a stochastic Prandtl-Tomlinson bath model are considered.

Large language models (LLMs) that do not give consistent answers across contexts are problematic when used for tasks with expectations of consistency, e.g., question-answering, explanations, etc. Our work presents an evaluation benchmark for self-consistency in cases of under-specification where two or more answers can be correct. We conduct a series of behavioral experiments on the OpenAI model suite using an ambiguous integer sequence completion task. We find that average consistency ranges from 67\% to 82\%, far higher than would be predicted if a model's consistency was random, and increases as model capability improves. Furthermore, we show that models tend to maintain self-consistency across a series of robustness checks, including prompting speaker changes and sequence length changes. These results suggest that self-consistency arises as an emergent capability without specifically training for it. Despite this, we find that models are uncalibrated when judging their own consistency, with models displaying both over- and under-confidence. We also propose a nonparametric test for determining from token output distribution whether a model assigns non-trivial probability to alternative answers. Using this test, we find that despite increases in self-consistency, models usually place significant weight on alternative, inconsistent answers. This distribution of probability mass provides evidence that even highly self-consistent models internally compute multiple possible responses., Comment: BlackboxNLP @ EMNLP 2023

We verify the transcendental part of a conjecture of Campana predicting that the rational points on the weakly-special non-special simply-connected smooth projective threefolds constructed by Bogomolov-Tschinkel are not dense. To prove our result, we establish fundamental properties of moduli spaces of orbifold maps. The crux of the argument relies on a dimension bound for such moduli spaces which relies on a recently established extension of Kobayashi-Ochiai's finiteness theorem to the more general setting of Campana's orbifold pairs., Comment: Minor improvements