Fundamental mathematical constants appear in nearly every field of science, from physics to biology. Formulas that connect different constants often bring great insight by hinting at connections between previously disparate fields. Discoveries of such relations, however, have remained scarce events, relying on sporadic strokes of creativity by human mathematicians. Recent developments of algorithms for automated conjecture generation have accelerated the discovery of formulas for specific constants. Yet, the discovery of connections between constants has not been addressed. In this paper, we present the first library dedicated to mathematical constants and their interrelations. This library can serve as a central repository of knowledge for scientists from different areas, and as a collaborative platform for development of new algorithms. The library is based on a new representation that we propose for organizing the formulas of mathematical constants: a hypergraph, with each node representing a constant and each edge representing a formula. Using this representation, we propose and demonstrate a systematic approach for automatically enriching this library using PSLQ, an integer relation algorithm based on QR decomposition and lattice construction. During its development and testing, our strategy led to the discovery of 75 previously unknown connections between constants, including a new formula for the `first continued fraction' constant $C_1$, novel formulas for natural logarithms, and new formulas connecting $\pi$ and $e$. The latter formulas generalize a century-old relation between $\pi$ and $e$ by Ramanujan, which until now was considered a singular formula and is now found to be part of a broader mathematical structure. The code supporting this library is a public, open-source API that can serve researchers in experimental mathematics and other fields of science.

Machine learning (ML) models serve critical functions, such as classifying loan applicants as good or bad risks. Each model is trained under the assumption that the data used in training, and the data used in field come from the same underlying unknown distribution. Often this assumption is broken in practice. It is desirable to identify when this occurs in order to minimize the impact on model performance. We suggest a new approach to detect change in the data distribution by identifying polynomial relations between the data features. We measure the strength of each identified relation using its R-square value. A strong polynomial relation captures a significant trait of the data which should remain stable if the data distribution does not change. We thus use a set of learned strong polynomial relations to identify drift. For a set of polynomial relations that are stronger than a given desired threshold, we calculate the amount of drift observed for that relation. The amount of drift is estimated by calculating the Bayes Factor for the polynomial relation likelihood of the baseline data versus field data. We empirically validate the approach by simulating a range of changes in three publicly-available data sets, and demonstrate the ability to identify drift using the Bayes Factor of the polynomial relation likelihood change.

In this short note we observe that the sample complexity of PAC machine learning of various concepts, including learning the maximum (EMX), can be exactly determined when the support of the probability measures considered as models satisfies an a-priori bound. This result contrasts with the recently discovered undecidability of EMX within ZFC for finitely supported probabilities (with no a priori bound). Unfortunately, the decision procedure is at present, at least doubly exponential in the number of points times the uniform bound on the support size.