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We study some properties of graphs (or, rather, graph sequences) defined by demanding that the number of subgraphs of a given type, with vertices in subsets of given sizes, approximatively equals the number expected in a random graph. It…
Mathematics is a mountain, but students need more than descriptions of the view: they need a trail they can actually walk. This paper presents the Math Teaching Atlas, a framework for mathematical exposition built around route units (single…
We developed analytical and numerical methods to study a transport of non-interacting particles in large networks consisting of M d-dimensional containers C_1,...,C_M with radii R_i linked together by tubes of length l_{ij} and radii a_{ij}…
We study random bubble lattices which can be produced by processes such as first order phase transitions, and derive characteristics that are important for understanding the percolation of distinct varieties of bubbles. The results are…
The phenomenon of convection is found in a wide variety of settings on different scales -- from applications in the cooling technology of laptops to heating water on a stove, and from the movement of ocean currents to describing…
Graphs are commonly used in mathematics to represent some relationships between items. However, as simple objects, they sometimes fail to capture all relevant aspects of real-world data. To address this problem, we generalize them and model…
We introduce a colorful version of separating path systems, in which two edges can only be separated from each other by two paths of distinct colors. We calculate the minimum sizes of such systems for various standard classes of graphs and…
Bidirected graphs are a common generalisation of directed graphs where arcs can also be incoming to both their incident nodes, or outgoing from both their incident nodes. Such arcs allow a walk to change direction. Some algorithms can…
We present a system for object recognition based on a semantic graph representation, which the system can learn from image examples. This graph is based on intrinsic properties of objects such as structure and geometry, so it is more robust…
In 1988 Rafla conjectured that every simple drawing of a complete graph $K_n$ contains a plane, i.e., non-crossing, Hamiltonian cycle. The conjecture is far from being resolved. The lower bounds for plane paths and plane matchings have…
Trying to be effective (no matter who exactly and in what field) a person face the problem which inevitably destroys all our attempts to easily get to a desired goal. The problem is the existence of some insuperable barriers for our mind,…
The recently introduced path-star task is a minimal task designed to exemplify limitations to the abilities of language models (Bachmann and Nagarajan, 2024). It involves a path-star graph where multiple arms radiate from a single starting…
Abstract reasoning is a cornerstone of human intelligence, and replicating it with artificial intelligence (AI) presents an ongoing challenge. This study focuses on efficiently solving Raven's progressive matrices (RPM), a visual test for…
We study the problem of learning features through self-supervision that are generalisable to multiple graphs. State-of-the-art graph self-supervision restricts training to only one graph, resulting in graph-specific models that are…
Many real world networks contain a statistically surprising number of certain subgraphs, called network motifs. In the prevalent approach to motif analysis, network motifs are detected by comparing subgraph frequencies in the original…
Why is it that we can recognize object identity and 3D shape from line drawings, even though they do not exist in the natural world? This paper hypothesizes that the human visual system perceives line drawings as if they were approximately…
This is an innovative treatise on triangles, resting upon 1) 3-body problem techniques including mass-weighted relative Jacobi coordinates. 2) Part I's detailed layer-by-layer topological and geometrical study of Kendall-type shape spaces -…
Graph are a ubiquitous data representation, as they represent a flexible and compact representation. For instance, the 3D structure of RNA can be efficiently represented as $\textit{2.5D graphs}$, graphs whose nodes are nucleotides and…
Problems involving rotating systems analyzed from an inertial frame, without invoking fictitious forces, is something that freshman students find difficult to understand in an introductory mechanics course. One of the problems that I…
Real-world scenarios demand reasoning about process, more than final outcome prediction, to discover latent causal chains and better understand complex systems. It requires the learning algorithms to offer both accurate predictions and…