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The Reeb space of a smooth function is a topological and combinatoric object and fundamental and important in understanding topological and geometric properties of the manifold of the domain. It is the graph and a topological space endowed…
Graphene, dubbed as a two-dimensional material represents the topological concept of "surface" embedded in a three-dimensional space. This regard enables to employ existing theories/tools in topology to understand different…
A special class of Jordan algebras over a field $F$ of characteristic zero is considered. Such an algebra consists of an $r$-dimensional subspace of the vector space of all square matrices of a fixed order $n$ over $F$. It contains the…
Surface bubbles are present in many industrial processes and in nature, as well as in CO$_2$ beverage. They have motivated many theoretical, numerical and experimental works. This paper presents the current knowledge on the physics of…
In this paper, we continue our study of the class of diagram groups. Simply speaking, a diagram is a labelled plane graph bounded by a pair of paths (the top path and the bottom path). To multiply two diagrams, one simply identifies the top…
In this paper we first review the covering space method with constrained BV functions for solving the classical Plateau's problem. Next, we carefully analyze some interesting examples of soap films compatible with the covering space method:…
Many real-world networks describe systems in which interactions decay with the distance between nodes. Examples include systems constrained in real space such as transportation and communication networks, as well as systems constrained in…
We describe the extension of a "viscous froth" model to the dynamics of a wet foam in a Hele-Shaw cell. The two-dimensional model includes the friction experienced by the soap films as they are dragged along the cell walls, while retaining…
A $d$-dimensional hypercube drawing of a graph represents the vertices by distinct points in $\{0,1\}^d$, such that the line-segments representing the edges do not cross. We study lower and upper bounds on the minimum number of dimensions…
Collaborative filtering based algorithms, including Recurrent Neural Networks (RNN), tend towards predicting a perpetuation of past observed behavior. In a recommendation context, this can lead to an overly narrow set of suggestions lacking…
Reasoning paths are reliable information in knowledge graph completion (KGC) in which algorithms can find strong clues of the actual relation between entities. However, in real-world applications, it is difficult to guarantee that…
"Bubble Ball" is a game built on a 2D physics engine, where a finite set of objects can modify the motion of a bubble-like ball. The objective is to choose the set and the initial configuration of the objects, in order to get the ball to…
Series-parallel network topologies generally exhibit simplified dynamical behavior and avoid high combinatorial complexity. A comprehensive analysis of how flow complexity emerges with a graph's deviation from series-parallel topology is…
In this paper we try to organize machine teaching as a coherent set of ideas. Each idea is presented as varying along a dimension. The collection of dimensions then form the problem space of machine teaching, such that existing teaching…
As an example of the concept of rulial space, we explore the case of simple Turing machines. We construct the rulial multiway graph which represents the behavior of all possible Turing machines with a certain class of rules. This graph…
Consider a configuration of pebbles distributed on the vertices of a connected graph of order $n$. A pebbling step consists of removing two pebbles from a given vertex and placing one pebble on an adjacent vertex. A distribution of pebbles…
Many real-world networks of interest are embedded in physical space. We present a new random graph model aiming to reflect the interplay between the geometries of the graph and of the underlying space. The model favors configurations with…
A cube is an 8-rep-tile: it is the union of eight smaller copies of itself. Is there a set with a hole which has this property? The computer found an interesting and complicated solution, which then could be simplified. We discuss some…
The connectivity structure of graphs is typically related to the attributes of the nodes. In social networks for example, the probability of a friendship between two people depends on their attributes, such as their age, address, and…
To humans, a robin seems more like a bird than a bird seems like a robin, but does this asymmetry also hold for machine vision? Humans and modern vision models can match each other in accuracy while making systematically different kinds of…