Related papers: Why is a soap bubble like a railway?
Given a distribution of pebbles on the vertices of a graph, say that we can pebble a vertex if a pebble is left on it after some sequence of moves, each of which takes two pebbles from some vertex and places one on an adjacent vertex. A…
Bubbles is a fractal-like set related to a circle diffeomorphism; they are a complex analogue to Arnold tongues. In this article, we prove an approximate self-similarity of bubbles.
Twisted hypercubes are generalizations of the Boolean hypercube, obtained by iteratively connecting two instances of a graph by a uniformly random perfect matching. Dudek et al. showed that when the two instances are independent, these…
Graph is a fundamental mathematical structure in characterizing relations between different objects and has been widely used on various learning tasks. Most methods implicitly assume a given graph to be accurate and complete. However, real…
Question Answering for complex questions is often modeled as a graph construction or traversal task, where a solver must build or traverse a graph of facts that answer and explain a given question. This "multi-hop" inference has been shown…
In this article, we briefly describe various tools and approaches that algebraic geometry has to offer to straighten bent objects. Throughout this article we will consider a specific example of a bent or curved piece of paper which in our…
In this note we make use of some properties of vector fields on a manifold to give an alternate proof to [3] for the equivalence between connections and parallel transport on vector bundles over manifolds. Out of the proof will emerge a new…
Using mixtures of soap, water, and long chain polymers, free-floating soap bubbles can be formed with volumes approaching 100 m$^3$. Here we investigate how such thin films are created and maintained over time. We show how the extensional…
We design and conduct a simple experiment to study whether neural networks can perform several steps of approximate reasoning in a fixed dimensional latent space. The set of rewrites (i.e. transformations) that can be successfully performed…
Bubble Sort is simple. Yet, it seems a bit difficult for high school students. This paper presents a pedagogical methodology: Using Design Sketch to visualize the concepts in Bubble Sort, and to evaluate how this approach assists students…
A properly edge-colored graph is a graph with a coloring of its edges such that no vertex is incident to two or more edges of the same color. A subgraph is called rainbow if all its edges have different colors. The problem of finding…
Molecules have seemed like a natural fit to deep learning's tendency to handle a complex structure through representation learning, given enough data. However, this often continuous representation is not natural for understanding chemical…
Covariational reasoning -- reasoning about how changes in one quantity relate to changes in another quantity -- has been examined extensively in mathematics education research. Little research has been done, however, on covariational…
This work proposes a method for effectively computing manipulation paths to rearrange similar objects in a cluttered space. The solution can be used to place similar products in a factory floor in a desirable arrangement or for retrieving a…
Understanding the learning process of artificial neural networks requires clarifying the structure of the parameter space within which learning takes place. A neural network parameter's functional equivalence class is the set of parameters…
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…
We present parity conditions under which a toy rail network is one-way, i.e., whether a direction can be assigned across the network so that all train journeys are completely consistent with it or completely consistent with its opposite. We…
This work uncovers the low-dimensional nature the complex dynamics of actuated separated flows. Namely, motivated by the problem of model-based predictive control of separated flows, we identify the requirements on a model-based observer…
Graphene is the first example of truly two-dimensional crystals - it's just one layer of carbon atoms. It turns out to be a gapless semiconductor with unique electronic properties resulting from the fact that charge carriers in graphene…
The classic double bubble theorem says that the least-perimeter way to enclose and separate two prescribed volumes in $\mathbb{R}^N$ is the standard double bubble. We seek the optimal double bubble in $\mathbb{R}^N$ with density, which we…