An activity-driven model is a framework used in various fields, including business process modeling, project management, and systems development, that emphasizes the activities or tasks that are necessary to achieve specific goals or outcomes. Rather than focusing primarily on resources (like people, tools, or capital) or outputs (like products or services), this model prioritizes the workflows and processes that drive success.

Loop-erased random walk (LERW) is a mathematical construct and a type of random walk that is particularly interesting in the fields of probability theory and statistical mechanics. It can be thought of as a model for exploring the behavior of paths in a random environment while avoiding certain obstacles (loops). Here's how it works: 1. **Random Walk**: Begin with a simple random walk on a lattice (for example, the integer grid in two dimensions), starting from an origin point.

Modelling British railway prototypes generally refers to the hobby of creating scale models that represent British railway vehicles, infrastructure, and scenes. This can include a variety of elements, such as locomotives, rolling stock (passenger and freight cars), stations, tracks, and the broader railway environment including scenery and buildings.

Normen Europäischer Modellbahnen (NEM) refers to a set of standards that govern the various aspects of model railroading in Europe. These standards were established to ensure compatibility and interoperability among different manufacturers' products, thus enhancing the overall experience of model railroad enthusiasts. The NEM standards cover a wide range of topics, including but not limited to: 1. **Scale and Gauge**: Definitions of various scales (e.g., H0, N) and respective track gauges.

The Railway of the Prince Imperial, also known as the "Chemin de fer du Prince Impérial," refers to a narrow-gauge railway located in the region of La Réunion, an island in the Indian Ocean that is overseas territory of France. This railway was built in the late 19th century, specifically between 1889 and 1890, to transport tourists to the scenic areas of the island.

Ramsey's theorem is a fundamental result in combinatorial mathematics and graph theory that addresses the conditions under which order must appear in a large enough structure. The theorem essentially states that in any sufficiently large graph, one can find certain types of complete subgraphs.

SuperTrain is an annual model train and hobby show that typically takes place in Calgary, Alberta, Canada. It showcases a variety of model railroads, trains, and related hobbies, attracting enthusiasts of all ages. The exhibition features large operating layouts, vendor booths selling supplies and merchandise, workshops, and family-friendly activities. It's an opportunity for model train enthusiasts to connect, share ideas, and participate in the vibrant model railroading community.

The Biggest Little Railway in the World is an anthology model railway layout located in the United Kingdom, specifically in the town of Bexhill-on-Sea, East Sussex. It is renowned for its intricate design and attention to detail, showcasing a miniature world complete with landscapes, buildings, and operational trains. The railway features various gauges and is designed to entertain both model railway enthusiasts and the general public.

A toy train is a miniature model train used primarily for play and entertainment. Toy trains come in various sizes, materials, and designs, and they can be operated in different ways, such as by hand, battery, or electricity. They can be part of a simple set designed for young children or more complex systems with tracks, scenery, and multiple cars designed for older hobbyists.

"Tracks Ahead" is a television series that focuses on railroads and railroading in the United States. The program, which began airing in the early 1990s, showcases various aspects of rail transport, including trains, rail systems, historical train journeys, and the impact of railroads on communities. It often features interviews with railroad enthusiasts, operators, and historians, as well as discussions about the technology and operations of railroads.

A **Cap set** is a specific configuration in the context of combinatorial geometry and number theory, specifically concerning subsets of integers or points in higher-dimensional spaces. The concept is particularly related to the study of sets that avoid certain geometric configurations or progressions.

Corners theorem, often referred to in the context of graph theory and combinatorial geometry, generally deals with conditions on the arrangement of points or vertices in a specific geometric or combinatorial setting. The theorem states that given a finite set of points in the plane, one can find a subset of these points such that certain geometric or combinatorial properties hold, often involving the vertices (or corners) of a configuration.

The Erdős–Dushnik–Miller theorem is a result in the field of graph theory, specifically in relation to the coloring of graphs. The theorem addresses the concept of coloring infinite graphs, particularly the problem of how many colors are needed to color an infinite graph such that no two adjacent vertices share the same color.

The Erdős–Hajnal conjecture is a famous conjecture in combinatorial set theory and graph theory, proposed by mathematicians Paul Erdős and András Hajnal in the early 1970s. It addresses the structure of graphs that do not contain certain types of subgraphs, specifically focusing on the clique and independent set sizes.

The Erdős–Szekeres theorem is a significant result in combinatorial geometry and discrete mathematics. It addresses the problem of monotone subsequences in sequences of points in the plane. The theorem states that for any integer \( n \), any sequence of \( n^2 \) distinct points in the plane, no three of which are collinear, contains either: 1. An increasing subsequence of length \( n \), or 2. A decreasing subsequence of length \( n \).

Folkman's theorem is a result in combinatorial mathematics, specifically in the area of Ramsey theory. It was proven by mathematician Frank P. Ramsey and is concerned with the coloring of edges in complete graphs.

The Green–Tao theorem is a significant result in additive combinatorics and number theory, established by mathematicians Ben Green and Terence Tao. It was proven in 2004 and states that the set of prime numbers contains arbitrarily long arithmetic progressions. More formally, the theorem asserts that for any integer \( k \), there exists a sequence of prime numbers that contains an arithmetic progression of length \( k \).

An **IP Set** is a data structure used primarily in the context of firewalls and network security systems to manage and store sets of IP addresses efficiently. IP sets allow network administrators to: 1. **Group IP Addresses**: Instead of creating individual rules for each IP address, administrators can create a single entry that represents a set of IPs. This is particularly useful for managing rules related to large numbers of IP addresses, such as those belonging to known malicious sources or trusted partners.

Milliken's tree theorem is a result in the field of combinatorial set theory, specifically in the area of Ramsey theory. It deals with properties of certain types of trees, which are hierarchical structures that can be thought of as branching diagrams. The theorem states that for any finite coloring of the nodes of a tree, one can find a subtree of a certain structure that is monochromatic (i.e., all nodes in that subtree have the same color) and satisfies certain conditions.