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Geometric graphs are a special kind of graph with geometric features, which are vital to model many scientific problems. Unlike generic graphs, geometric graphs often exhibit physical symmetries of translations, rotations, and reflections,…

This is an attempt to present axioms for Euclidean geometry, aiming at the following goals: to work with geometric notions (thus not merely identify points with pairs of numbers, giving a special status to a particular coordinate system);…

When data contains measurement errors, it is necessary to make assumptions relating the observed, erroneous data to the unobserved true phenomena of interest. These assumptions should be justifiable on substantive grounds, but are often…

Probabilistic image segmentation encodes varying prediction confidence and inherent ambiguity in the segmentation problem. While different probabilistic segmentation models are designed to capture different aspects of segmentation…

The paper studies a general scheme for constructing metrics on a product of metric spaces by means of a family of continuous convex functions. This construction includes the conventional $p$-metrics and generates metrics that are…

It is known that statistical model selection as well as identification of dynamical equations from available data are both very challenging tasks. Physical systems behave according to their underlying dynamical equations which, in turn, can…

A reliable language model should be able to signal, prior to generation, when a query falls outside its knowledge. We investigate whether representation geometry can provide such a pre-generation signal by measuring the deviation of hidden…

In [Heimann, Lehrenfeld, Preu{\ss}, SIAM J. Sci. Comp. 45(2), 2023, B139 - B165] new geometrically unfitted space-time Finite Element methods for partial differential equations posed on moving domains of higher-order accuracy in space and…

Mathematical models are used extensively for diverse tasks including analysis, optimization, and decision making. Frequently, those models are principled but imperfect representations of reality. This is either due to incomplete physical…

A classic approach in dynamical systems is to use particular geometric structures to deduce statistical properties, for example the existence of invariant measures with stochastic-like behaviour such as large deviations or decay of…

Probabilistic graphical modeling is a branch of machine learning that uses probability distributions to describe the world, make predictions, and support decision-making under uncertainty. Underlying this modeling framework is an elegant…

Many have wondered how mathematics, which appears to be the result of both human creativity and human discovery, can possibly exhibit the degree of success and seemingly-universal applicability to quantifying the physical world as…

Many mathematical models utilize limit processes. Continuous functions and the calculus, differential equations and topology, all are based on limits and continuity. However, when we perform measurements and computations, we can achieve…

In this note, we present a probabilistic proof of the well-known finite geometric series. The proof follows by taking the moments of the sum and the difference of two independent exponentially distributed random variables.

One of the consequences of persistent technological change is that it force individuals to make decisions under extreme uncertainty. This means that traditional decision-making frameworks cannot be applied. To address this issue we…

An open problem in numerical analysis is to explain why molecular dynamics works. The difficulty is that numerical trajectories are only accurate for very short times, whereas the simulations are performed over long time intervals. It is…

Geometry problem solving, a crucial aspect of mathematical reasoning, is vital across various domains, including education, the assessment of AI's mathematical abilities, and multimodal capability evaluation. The recent surge in deep…

Some divergent trigonometric integrals have appeared in standard tables for many years, listed as converging. We give a simple proof that these integrals diverge and trace their history. The original error was made when a (startlingly)…

Models of physical systems are used to explain and predict experimental results and observations. When students encounter discrepancies between the actual and expected behavior of a system, they revise their models to include the newly…

Understanding the notion of a model is not always easy in logic courses. Hence, tools such as Euler diagrams are frequently applied as informal illustrations of set-theoretical models. We formally investigate Euler diagrams as an…