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Abstract The progenitor system of Type Ia supernovae (SNe Ia) is expected to be a close binary system consisting of a carbon/oxygen white dwarf (WD) and a nondegenerate star or another WD. Here, we present results from high-cadence monitoring observations of SN 2021hpr in a spiral galaxy, NGC 3147, and constraints on the progenitor system based on its early multicolor light-curve data. First, we classify SN 2021hpr as a normal SN Ia from its long-term photometric and spectroscopic data. More interestingly, we found a significant “early excess” in the light curve over a simple power-law ∼ t 2 evolution. The early light curve evolves from blue to red to blue during the first week. To explain this, we fitted the early part of the BVRI -band light curves with a two-component model consisting of ejecta–companion interaction and a simple power-law model. The early excess and its color can be explained by shock-cooling emission due to a companion star having a radius of 8.84 ± 0.58 R ⊙ . We also examined Hubble Space Telescope preexplosion images, finding no detection of a progenitor candidate, consistent with the above result. However, we could not detect signs of a significant amount of stripped mass from a nondegenerate companion star (≲0.003 M ⊙ for H α emission). The early excess light in the multiband light curve supports a nondegenerate companion in the progenitor system of SN 2021hpr. At the same time, the nondetection of emission lines opens the door for other methods to explain this event. 

A fundamental challenge in automated reasoning about programming assignments at scale is clustering student submissions based on their underlying algorithms. State-of-the-art clustering techniques are sensitive to control structure variations, cannot cluster buggy solutions with similar correct solutions, and either require expensive pair-wise program analyses or training efforts. We propose a novel technique that can cluster small imperative programs based on their algorithmic essence: (A) how the input space is partitioned into equivalence classes and (B) how the problem is uniquely addressed within individual equivalence classes. We capture these algorithmic aspects as two quantitative semantic program features that are merged into a program's vector representation. Programs are then clustered using their vector representations. The computation of our first semantic feature leverages model counting to identify the number of inputs belonging to an input equivalence class. The computation of our second semantic feature abstracts the program's data flow by tracking the number of occurrences of a unique pair of consecutive values of a variable during its lifetime. The comprehensive evaluation of our tool SemCluster on benchmarks drawn from solutions to small programming assignments shows that SemCluster (1) generates far fewer clusters than other clustering techniques, (2) precisely identifies distinct solution strategies, and (3) boosts the performance of clustering-based program repair, all within a reasonable amount of time. 

Abstract Following the method of Seifert surfaces in knot theory, we define arithmetic linking numbers and height pairings of ideals using arithmetic duality theorems, and compute them in terms of $$n$$-th power residue symbols. This formalism leads to a precise arithmetic analogue of a “path-integral formula” for linking numbers.