Related papers: Assumption Digging in Euclidean Geometry
Machine learning often aims to produce latent embeddings of inputs which lie in a larger, abstract mathematical space. For example, in the field of 3D modeling, subsets of Euclidean space can be embedded as vectors using implicit neural…
Calculus and geometry are ubiquitous in the theoretical modelling of scientific phenomena, but have historically been very challenging to apply directly to real data as statistics. Diffusion geometry is a new theory that reformulates…
Exploration has been a crucial part of reinforcement learning, yet several important questions concerning exploration efficiency are still not answered satisfactorily by existing analytical frameworks. These questions include exploration…
Manifold learning is a popular and quickly-growing subfield of machine learning based on the assumption that one's observed data lie on a low-dimensional manifold embedded in a higher-dimensional space. This thesis presents a mathematical…
The flipped classroom technique has recently been a focus of attention for many math instructors and pedagogical researchers. Although research on the subject has greatly increased in recent years, it is still debated whether the flipped…
Debugging is a vital but challenging skill for beginner programmers to learn. It is also a difficult skill to teach. For secondary school teachers, who may lack time or programming experience, honing students' understanding of debugging can…
The resources compiled in this document provide an approach to embed and teach Ethics in Mathematics at the undergraduate level. We provide mathematical exercises and homework problems that teach students ethical awareness and transferable…
In this paper I present a kind of proof for classical Euclidean geometric problems which relies on both synthetic and analytic geometry. Using the elementary tools of polynomial algebra and multivariate calculus we manage to reduce the…
Various topological techniques and tools have been applied to neural networks in terms of network complexity, explainability, and performance. One fundamental assumption of this line of research is the existence of a global (Euclidean)…
This is an introduction to geometric algebra, an alternative to traditional vector algebra that expands on it in two ways: 1. In addition to scalars and vectors, it defines new objects representing subspaces of any dimension. 2. It defines…
Non-Euclidean foundation models increasingly place representations in curved spaces such as hyperbolic geometry. We show that this geometry creates a boundary-driven asymmetry that backdoor triggers can exploit. Near the boundary, small…
In this note we study numerically the combinatorics of curves and geodesics on the torus with one boundary component. A potential computational difficulty is avoided by counting inside specific orbits of the mapping class group up to a…
This is a survey of our research on geometric structures of projective embeddings and includes some topics of our talks in several symposia during 1990-99. We clarify our main problem, which is to construct a kind of geometric composition…
The aim of this paper is to develop a new axiomatization of planar geometry by reinterpreting the original axioms of Euclid. The basic concept is still that of a line segment but its equivalent notion of betweenness is viewed as a…
When education researchers describe newly developed curricular materials, they typically concentrate on the research base behind their design, and the efficacy of the final products, but do not highlight the initial stages of creating the…
We review and extend existing frameworks on modeling to develop a new framework that describes model-based reasoning in upper-division physics labs. Constructing and using models are core scientific practices that have gained significant…
We describe student difficulties in applying the superposition principle in combination with Gauss's law. We addressed these difficulties by developing a tutorial that uses guided inquiry. Students who used this tutorial following…
The leading idea of the paper is to treat the theorem of Wigner with methods inspired by geometry. The exercise mentionned in the title has two functions: On the one hand it can serve as a pedagogical text in order to make the reader…
A pedagogical but concise overview of Riemannian geometry is provided, in the context of usage in physics. The emphasis is on defining and visualizing concepts and relationships between them, as well as listing common confusions,…
Cosmology and GR remain largely inaccessible to high-school teaching due to the advanced prerequisites to master these topics. Integrating them into upper secondary teaching is a significant challenge that remains unresolved. This…