Semi-supervised dialogue summarization (SSDS) leverages model-generated summaries to reduce reliance on human-labeled data and improve the performance of summarization models. While addressing label noise, previous works on semi-supervised learning primarily focus on natural language understanding tasks, assuming each sample has a unique label. However, these methods are not directly applicable to SSDS, as it is a generative task, and each dialogue can be summarized in different ways. In this work, we propose a novel scoring approach, SiCF, which encapsulates three primary dimensions of summarization model quality: Semantic invariance (indicative of model confidence), Coverage (factual recall), and Faithfulness (factual precision). Using the SiCF score, we select unlabeled dialogues with high-quality generated summaries to train summarization models. Comprehensive experiments on three public datasets demonstrate the effectiveness of SiCF scores in uncertainty estimation and semi-supervised learning for dialogue summarization tasks. Our code is available at \url{https://github.com/amazon-science/summarization-sicf-score}.

Formulas involving fundamental mathematical constants had a great impact on various fields of science and mathematics, for example aiding in proofs of irrationality of constants. However, the discovery of such formulas has historically remained scarce, often perceived as an act of mathematical genius by great mathematicians such as Ramanujan, Euler, and Gauss. Recent efforts to automate the discovery of formulas for mathematical constants, such as the Ramanujan Machine project, relied on exhaustive search. Despite several successful discoveries, exhaustive search remains limited by the space of options that can be covered and by the need for vast amounts of computational resources. Here we propose a fundamentally different method to search for conjectures on mathematical constants: through analysis of integer sequences. We introduce the Enumerated Signed-continued-fraction Massey Approve (ESMA) algorithm, which builds on the Berlekamp-Massey algorithm to identify patterns in integer sequences that represent mathematical constants. The ESMA algorithm found various known formulas for $e, e^2, tan(1)$, and ratios of values of Bessel functions. The algorithm further discovered a large number of new conjectures for these constants, some providing simpler representations and some providing faster numerical convergence than the corresponding simple continued fractions. Along with the algorithm, we present mathematical tools for manipulating continued fractions. These connections enable us to characterize what space of constants can be found by ESMA and quantify its algorithmic advantage in certain scenarios. Altogether, this work continues in the development of augmenting mathematical intuition by computer algorithms, to help reveal mathematical structures and accelerate mathematical research.