A repeated game is a standard concept in game theory where a specific game (usually a stage game) is played multiple times by the same players. This can involve either a fixed number of repetitions (finite repeated game) or an indefinite number of repetitions (infinite repeated game). ### Key Characteristics of Repeated Games: 1. **Stage Game**: The repeated game is based on a single, well-defined game, often involving decisions that players must make simultaneously.

"Sir Philip Sidney game" typically refers to a classic board game, often called "Sidney's Game" or "The Game of Sidney," which is centered around the literary contributions of Sir Philip Sidney, a prominent English poet, and courtier of the Elizabethan era. However, it’s worth noting that there isn't a well-known board game that directly bears his name in contemporary contexts.

The Banach game, also known as the Banach-Mazur game, is a two-player game that arises in the field of set-theoretical topology and functional analysis. The game is named after mathematicians Stefan Banach and Juliusz Mazur, who studied related concepts in the early 20th century.

A **biconnected graph** (or **bi-connected graph**) is a type of connected graph with a specific structural property related to its vertices and edges. In the context of graph theory, a biconnected graph is defined as follows: 1. **Connectivity**: A biconnected graph is a connected graph. This means there is a path between any two vertices in the graph.

Video game gameplay refers to the interactive experience provided by a video game, encompassing the mechanics, rules, challenges, and player actions within the game environment. It includes how players interact with the game, the objectives they must achieve, and the feedback they receive from the game in response to their actions. Here are some key elements that define gameplay: 1. **Mechanics**: These are the rules and systems that govern how the game operates.

The Impulse-based Turn System is a gaming mechanic often used in tabletop role-playing games (RPGs) and certain video games to manage turn order and actions during gameplay. This system emphasizes the spontaneity and dynamism of player actions rather than adhering strictly to a predetermined turn order. ### Key Features: 1. **Impulse Points**: Players may have a pool of points that they can spend to take actions in a turn.

Fractional Pareto efficiency is a concept that extends the traditional notion of Pareto efficiency in economics and optimization theory. While a traditional Pareto efficient allocation occurs when it is impossible to make one individual better off without making another individual worse off, fractional Pareto efficiency introduces a more nuanced approach. In fractional Pareto efficiency, one assesses configurations where some individuals may be partially made better off without entirely disadvantaging others.

Ordinal Pareto efficiency is a concept in economics and social choice theory that builds upon the idea of Pareto efficiency in a way that incorporates ordinal preferences rather than cardinal utility. ### Key Concepts: 1. **Pareto Efficiency**: A state is Pareto efficient if there is no other allocation of resources that can make at least one individual better off without making someone else worse off. In other words, a distribution cannot be improved for one individual without degrading the situation for another.

Asymptote can refer to two primary concepts: one in mathematics and the other as a programming language for technical graphics. 1. **Mathematical Concept**: In mathematics, an asymptote is a line that a curve approaches as it heads towards infinity. Asymptotes can be horizontal, vertical, or oblique (slant). They represent the behavior of a function as the input or output becomes very large or very small.

Line coordinates typically refer to the mathematical representation of a line in a coordinate system, such as a two-dimensional (2D) or three-dimensional (3D) space. The precise meaning can vary based on context, but here are some common interpretations: ### 1.

Hyperbolic geometry is a non-Euclidean geometry that arises from altering Euclid's fifth postulate, the parallel postulate. In hyperbolic geometry, the essential distinction is that, given a line and a point not on that line, there are infinitely many lines through that point that do not intersect the original line. This contrasts with Euclidean geometry, where there is exactly one parallel line that can be drawn through a point not on a line.

Inversive geometry is a branch of geometry that focuses on properties and relations of figures that are invariant under the process of inversion in a circle (or sphere in higher dimensions). This type of transformation maps points outside a given circle to points inside the circle and vice versa, while points on the circle itself remain unchanged. Key concepts and characteristics of inversive geometry include: 1. **Inversion**: The basic operation in inversive geometry is the inversion with respect to a circle.

Minhyong Kim is a notable mathematician specializing in number theory and arithmetic geometry. He is known for his work in several areas, including the study of Diophantine geometry, the arithmetic of abelian varieties, and various aspects of algebraic geometry and number theory. His research includes contributions to understanding rational points on algebraic varieties and connections between arithmetic and geometry. In addition to his research, Minhyong Kim is involved in mathematics education and outreach, promoting mathematics to a broader audience.

Dicaearchus was an ancient Greek philosopher and geographer, active in the 4th century BCE. He was a pupil of Aristotle and a member of the Peripatetic school. Dicaearchus is best known for his work in geography and for his attempts to systematically study the earth and its regions, as well as for his contributions to political theory and ethics. One of his notable contributions was his work on the division of the earth into regions and the description of various geography-related topics.

Menelaus of Alexandria was a Greek mathematician and astronomer who lived during the 1st century AD. He is best known for his work in geometry and spherical astronomy. One of his most significant contributions is the formulation of Menelaus' theorem, which relates to the geometry of triangles and is particularly important in the study of spherical triangles.

Claude Ambrose Rogers is not widely recognized as a public figure or a notable entity in historical or contemporary contexts, based on information available up to October 2023.

Eric Harold Neville was a British astronomer known for his contributions to the field of astronomy and astrophysics. He was particularly recognized for his work in photometry and the study of celestial objects. Neville's research helped enhance the understanding of star brightness variations and the physical properties of various astronomical bodies. Apart from his scientific contributions, he may also be remembered for his involvement in education and outreach within the astronomical community.

Thomas Willmore is associated with mathematics, specifically in the field of differential geometry. The term "Willmore" often refers to the Willmore energy or Willmore surfaces, which are concepts related to the study of surfaces in three-dimensional space. The Willmore energy of a surface is a measure of its bending and is defined as the integral of the square of the mean curvature over the surface. Willmore surfaces are those that minimize this energy.