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Graph embedding seeks to build a low-dimensional representation of a graph G. This low-dimensional representation is then used for various downstream tasks. One popular approach is Laplacian Eigenmaps (LE), which constructs a graph embedding based on the spectral properties of the Laplacian matrix of G. The intuition behind it, and many other embedding techniques, is that the embedding of a graph must respect node similarity: similar nodes must have embeddings that are close to one another. Here, we dispose of this distance-minimization assumption. Instead, we use the Laplacian matrix to find an embedding with geometric properties instead of spectral ones, by leveraging the so-called simplex geometry of G. We introduce a new approach, Geometric Laplacian Eigenmap Embedding, and demonstrate that it outperforms various other techniques (including LE) in the tasks of graph reconstruction and link prediction.

The damped harmonic oscillator extends the simple harmonic oscillator by incorporating energy dissipation. This paper examines the three characteristic regimes—underdamped, critically damped, and overdamped—and demonstrates how the damping coefficient fundamentally determines system behavior. An interactive simulation enables exploration of parameter space, building intuition for this fundamental physical system.

Graph traversal algorithms systematically visit vertices in a graph. This paper examines the two fundamental approaches: Breadth-First Search (BFS) explores level-by-level, while Depth-First Search (DFS) explores as deep as possible before backtracking. Understanding their distinct characteristics is essential for selecting the appropriate algorithm for path-finding, connectivity analysis, and graph-based problem solving.

The Iris dataset is a classic multivariate dataset used for classification and visualization. This paper provides an interactive exploratory analysis using Python and Plotly, demonstrating species clustering through petal and sepal measurements. Interactive visualizations enable readers to explore the data patterns that make this dataset ideal for demonstrating machine learning classification algorithms.

The spectral theorem establishes that symmetric matrices can be diagonalized by orthogonal transformations. This fundamental result connects linear algebra with geometric intuition and enables applications from optimization to quantum mechanics. We present the theorem with proof and demonstrate its power through concrete examples.