A *factor-critical graph* is a type of graph in which the removal of any single vertex results in a graph that has a perfect matching. In other words, a graph \( G \) is called factor-critical if for every vertex \( v \) in \( G \), the graph \( G - v \) (the graph obtained by removing vertex \( v \) and its incident edges) has a perfect matching.

Induced matching is a concept used primarily in the fields of psychiatry and social sciences, particularly in the context of observational studies and nonrandomized trials. The idea behind induced matching is to reduce bias in estimates of treatment effects by matching subjects in a way that accounts for certain covariates that could influence both treatment assignment and outcomes. In induced matching, subjects who receive a particular treatment are matched with subjects who do not, on the basis of observable characteristics.

Rank-maximal allocation is a concept that arises in the context of resource allocation problems, particularly in matching markets and auctions. The idea is to allocate resources (such as goods or services) to agents (such as individuals or organizations) in a way that maximizes the rank of the allocated outcomes according to each agent's preferences. In simpler terms, rank-maximal allocation attempts to ensure that each agent receives an allocation that is as high as possible on their personal preference list.

The Ruzsa–Szemerédi problem is a question in the field of combinatorial number theory, particularly concerning sets of integers and their structure. It was posed by Hungarian mathematicians Imre Ruzsa and Endre Szemerédi. The problem revolves around the concept of progressions in subsets of integers. Specifically, it asks how large a subset of integers can be if it avoids certain arithmetic progressions of a given length.

The Top Trading Cycle (TTC) is a notable algorithm used in the field of resource allocation and matching theory. It was primarily developed by economists to allocate resources or items efficiently among a group of agents based on their preferences. The basic idea of the Top Trading Cycle algorithm is as follows: 1. **Initial Setup**: Each participant (agent) has a list of preferences, indicating which items they would like to receive.

The Digital Differential Analyzer (DDA) is an algorithm used in computer graphics for line drawing. It is particularly important for rendering straight lines in raster graphics. The DDA algorithm is an incremental method that utilizes floating-point arithmetic to determine the points that lie on a straight line between two specified endpoints. ### Key Concepts of DDA 1.

The hit-or-miss transform is a morphological operation used in image processing and computer vision, particularly for shape matching and pattern recognition. It is a fundamental operation that allows one to detect specific shapes or patterns within a binary image. The hit-or-miss transform involves two sets: a structuring element (or template) and a binary image. The structuring element can be thought of as a defined shape or pattern that you want to detect in the image.

Reeve tetrahedra refer to a particular type of geometric structure in the field of topology and computational geometry. Specifically, a Reeve tetrahedron is a tetrahedron that is formed through a specific type of triangulation of a polytope or a higher-dimensional manifold. The term is named after mathematician J. M. Reeve, who contributed to the understanding of geometric structures and properties in higher dimensions.

A summed-area table (also known as an integral image) is a data structure used primarily in computer vision and image processing. It allows for rapid computation of the sum of pixel values in a rectangular subset of a grid or image. The key benefits of using a summed-area table include significantly reduced computation time and efficient querying for sums over image regions. ### How it Works: 1. **Construction**: A summed-area table is constructed from the original image by computing cumulative sums.

"Computing the Continuous Discretely" is a phrase commonly associated with the work and ideas of mathematician and computer scientist Steven Strogatz, particularly in the context of dynamical systems and complex systems. It highlights the interplay between continuous and discrete systems, illustrating how phenomena that are inherently continuous can be modeled, analyzed, or approximated using discrete computational methods.

The divisor summatory function, often denoted as \( \sum_{n \leq x} d(n) \), is a function that counts the total number of divisors of natural numbers up to \( x \). Specifically, \( d(n) \) represents the number of positive divisors of an integer \( n \).

Euclid's orchard is a mathematical concept that relates to the study of geometric configurations and properties, particularly in the context of number theory and combinatorial geometry. The term is not widely used in all mathematical contexts, but it can refer to a specific arrangement of points in a Euclidean space or an exploration of how to organize or distribute points according to certain rules or properties.

In mathematics, particularly in the field of topology and functional analysis, a Meyer set refers to a specific type of set associated with the theory of distributions and certain properties of functions in Sobolev spaces. More generally, the term can also refer to concepts in PDEs (partial differential equations) and harmonic analysis, but there isn't a universally accepted definition specifically for "Meyer set" across all mathematical disciplines.

The Poisson summation formula is a powerful and essential result in analytic number theory and Fourier analysis, connecting sums of a function at integer points to sums of its Fourier transform. Specifically, it relates a sum over a lattice (for example, the integers) to a sum over the dual lattice.

A regular grid is a structured arrangement of points or cells that are uniformly spaced along one or more dimensions. This type of grid is characterized by its consistent intervals in both the x and y (and possibly z) directions, forming a predictable pattern. Regular grids are commonly used in various fields such as: 1. **Geography and GIS**: In geographical information systems (GIS), regular grids help in spatial analysis and representation of spatial data.

Cauchy's theorem in geometry is a result concerning the properties of polygons, specifically convex polygons. The most well-known version pertains to the following statement: If two simple (non-intersecting) polygons are such that one can be continuously transformed into the other without self-intersection (while preserving the vertices and edges), then the two polygons have the same area.

A flexible polyhedron is a type of polyhedron that can change its shape without altering the lengths of its edges. In other words, the vertices of a flexible polyhedron can move while keeping the distance between connected vertices constant, allowing the polyhedron to "flex" or deform. This characteristic distinguishes flexible polyhedra from rigid polyhedra, which cannot change shape without changing the lengths of their edges.

A pseudotriangle is a geometric shape that resembles a triangle but does not necessarily meet all the criteria of a traditional triangle. The specific definition can vary depending on the context in which the term is used, such as in computational geometry or other mathematical fields. In some contexts, a pseudotriangle can refer to a polygon with three vertices that might not satisfy the requirements of having straight edges (i.e., it can contain curved segments) or other characteristics typically associated with standard triangles.

Barred galaxies are a specific type of spiral galaxy that feature a distinct elongated structure, or "bar," made up of stars that extends from the central region of the galaxy. This bar typically contains a higher density of stars compared to the surrounding regions and often influences the dynamics and structure of the galaxy. The bar structure can affect the motion of stars and gas within the galaxy, facilitating the transfer of material toward the center.

Fiction about galaxies often explores themes of space exploration, alien civilizations, the nature of humanity, and the vastness of the universe. It can take various forms, including novels, short stories, movies, and television series. Here are some common elements and themes found in galactic fiction: 1. **Space Exploration**: Many stories focus on human or alien endeavors to explore distant galaxies. This can involve interstellar travel, advanced spacecraft, and the challenges and adventures of navigating unknown worlds.