A degree-constrained spanning tree (DCST) is a specific type of spanning tree in a graph with the additional restriction that the degree (i.e., the number of edges connected) of each vertex must not exceed a specified limit. In other words, a DCST is a tree that spans all the vertices of a graph while ensuring that no vertex has a degree greater than a predefined upper bound.

The Kinetic Minimum Spanning Tree (KMST) is a concept derived from dynamic graph algorithms, specifically focusing on the minimum spanning tree (MST) in scenarios where the graph changes over time. In a typical minimum spanning tree problem, you have a weighted, undirected graph, and the goal is to find a tree that spans all vertices while minimizing the total edge weight. When the edges or weights of a graph change dynamically, maintaining an efficient representation of the minimum spanning tree becomes challenging.

Multiple Spanning Tree Protocol (MSTP) is a network protocol used in Ethernet networks to prevent loops in network topologies while allowing for the efficient redundancy and load balancing of the network. Specifically, MSTP is an extension of the Spanning Tree Protocol (STP) and Multiple Spanning Tree Protocol (MSTP) to work across multiple VLANs (Virtual Local Area Networks).

A Rectilinear Minimum Spanning Tree (RMST) is a specific type of minimum spanning tree that is defined in a rectilinear (or grid-like) space, where the coordinates are aligned with the axes of a Cartesian plane. In a rectilinear geometry, the distance between two points is measured using the Manhattan distance (also known as the L1 distance), which is calculated as the sum of the absolute differences of their Cartesian coordinates.

The term "Tornado family" can refer to a few different contexts, but it most commonly pertains to either: 1. **Meteorology**: In meteorological terms, a "Tornado family" often describes a series of tornadoes that occur within the same storm system or weather event. Tornadoes can sometimes form in succession or in the same geographic area during a severe weather outbreak, and these may be referred to as part of a "family" because they share similar characteristics and conditions.

An Apollonian network is a type of geometric network that is constructed using a recursive process based on the properties of triangular tiling. It begins with a single triangle, which is then subdivided into smaller triangles recursively. The network has a rich structure and exhibits fractal characteristics, making it interesting in the study of complex networks.

Kinetic triangulation is a concept from computational geometry that deals with the dynamic problem of maintaining the properties of a triangulation of a set of points in motion. Specifically, it refers to the process of efficiently updating the triangulation structure as the points in the plane change their positions over time.

A triangle mesh is a type of geometric representation commonly used in computer graphics, 3D modeling, and computational geometry. It consists of a collection of triangular faces that define a 3D shape or surface. Each triangle is typically defined by three vertices, which are points in 3D space, and the edges connecting these vertices.

The Even Circuit Theorem, often referred to in the context of graph theory and circuit design, primarily deals with the properties of circuits within graphs. While the term itself may not be universally defined across all disciplines, it is likely related to concepts in electrical engineering and theoretical computer science, where circuits can be represented as graphs. In general terms, in a graph: - A circuit (or cycle) is a closed path where no edges are repeated.

The Geiringer–Laman theorem is a result in the field of graph theory and combinatorial geometry, specifically concerning the rigidity of frameworks. The theorem provides a criterion for determining when a certain kind of graph, known as a "framework", can be considered rigid, meaning that its vertices cannot be moved without distorting the distances between them.

The Planar Separator Theorem is a concept in computational geometry and graph theory which states that for any planar graph, it is possible to partition the vertices of the graph into three disjoint sets: X, Y, and S. The sets have the following properties: 1. **Small Separator Size**: The size of the set S (the separator) is proportional to the square root of the number of vertices in the graph.

Wagner's theorem is a result in graph theory that provides a characterization of planar graphs. Specifically, it states that a graph is planar if and only if it does not contain a subgraph that is a subdivision of the complete graph \( K_{5} \) (the complete graph on five vertices) or a subdivision of the complete bipartite graph \( K_{3,3} \) (the complete bipartite graph with three vertices in each part).

Here’s a list of some notable star systems located within the distance range of 55 to 60 light-years from Earth: 1. **Luyten 726-8** (also known as **GJ 725**) - This binary system contains two red dwarf stars, Luyten 726-8A and Luyten 726-8B.

Basically it is a larger space such that there exists a surjection from the large space onto the smaller space, while still being compatible with the topology of the small space.

We can characterize the cover by how injective the function is. E.g. if two elements of the large space map to each element of the small space, then we have a double cover and so on.

The key concept of topology.

Argumentation theory is a multidisciplinary field that studies the structure, content, and dynamics of arguments, focusing on how they are constructed, understood, and evaluated. It draws from various fields, including philosophy, linguistics, communication, artificial intelligence, and law. Key aspects of argumentation theory include: 1. **Structure of Arguments**: Examination of the components that make up an argument, such as premises, conclusions, and inferential connections.

We map each point and a small enough neighbourhood of it to ${\mathbb{R}}^n$, so we can talk about the manifold points in terms of coordinates.

Does not require any further structure besides a consistent topological map. Notably, does not require metric nor an addition operation to make a vector space.

Manifolds are cool. Especially differentiable manifolds which we can do calculus on.

A notable example of a Non-Euclidean geometry manifold is the space of generalized coordinates of a Lagrangian. For example, in a problem such as the double pendulum, some of those generalized coordinates could be angles, which wrap around and thus are not euclidean.

A generalized definition of derivative that works on manifolds.

TODO: how does it maintain a single value even across different coordinate charts?