Children: Group theory
SL₂(ℝ) refers to the special linear group of degree 2 over the real numbers. It is defined as the group of all 2x2 real matrices with determinant equal to 1.
Schoof's algorithm is a polynomial-time algorithm used to compute the number of points on an elliptic curve defined over a finite field. The significance of this algorithm arises from its application in number theory and cryptography, particularly in elliptic curve cryptography (ECC).
The Schoof–Elkies–Atkin (SEA) algorithm is an efficient computational method for counting the number of points on an elliptic curve over a finite field. This is a key problem in number theory and is particularly important in the fields of cryptography and computational algebraic geometry.
A Schottky group is a specific type of group of isometries of hyperbolic space, particularly in the context of hyperbolic geometry. More formally, it can be defined as a free group of isometries of hyperbolic space, which acts on the hyperbolic plane or hyperbolic 3-space.
The Schur multiplier is an important concept in group theory, particularly in the study of algebraic topology and the classification of group extensions. It can be understood in the context of central extensions of groups. Given a group \( G \), the Schur multiplier \( M(G) \) is defined as the second homology group of \( G \) with coefficients in the integers, denoted as \( H_2(G, \mathbb{Z}) \).
A **spherical 3-manifold** is a type of three-dimensional manifold that is topologically equivalent to a quotient of the 3-dimensional sphere \( S^3 \) by a group of isometries (which preserve distances). More formally, a spherical 3-manifold can be described as a space of the form \( S^3 / G \), where \( G \) is a group of finite isometries of the 3-sphere.
A subgroup is a subset of a group that itself forms a group under the same operation as the original group. In mathematical group theory, a group is defined as a set equipped with a binary operation that satisfies four properties: closure, associativity, the existence of an identity element, and the existence of inverses.
The Subgroup Method is a technique used in various fields, notably in statistics, market research, and organizational analysis, to analyze data by dividing it into smaller, more manageable groups (or subgroups). This method allows for a more detailed understanding of trends, behaviors, or characteristics that may differ across different segments of the population or dataset.
The Suzuki Group refers to a collection of companies and entities that are associated with Suzuki Motor Corporation, a Japanese multinational corporation known primarily for its automobiles, motorcycles, all-terrain vehicles, and other products. Founded in 1909 by Michio Suzuki, the company originally started as a manufacturer of looms before transitioning into the automotive sector in the 1950s. Suzuki Motor Corporation is recognized for its small cars, compact vehicles, and motorcycle production.
In mathematics and set theory, a symmetric set is often defined in the context of a set of elements where certain symmetrical properties are present. 1. **General Symmetry**: A set \( S \) could be considered symmetric if for every element \( a \) in \( S \), there exists a corresponding element that reflects a specific symmetry property. This could be under operations such as reflection across a line, rotation in a plane, or an inversion in some metric space.
In group theory, "transfer" typically refers to a specific concept related to the behavior of groups under certain conditions, particularly in the context of transfer homomorphisms or transfer maps which can arise in the study of group cohomology and modular representations. However, in a more specific context, "transfer" often relates to the idea of transferring properties or structures from one group to another.
"Virtually" can refer to a few different concepts depending on the context: 1. **Adverbial Meaning**: In general usage, "virtually" means "almost" or "nearly," suggesting that something is true in effect but not in an absolute sense. For example, if someone says "I virtually finished the project," it implies that they are very close to finishing, but not quite there yet.
The Von Neumann paradox, also known as the "Von Neumann architecture paradox," is a concept in the field of game theory and economics, particularly in the context of decision-making and self-referential systems. However, there is another related concept often referred to as the "paradox of choice" in decision-making processes.
The Whitehead problem is a classic question in the field of algebraic topology, specifically in the area of group theory relating to homotopy theory. Formulated by the mathematician J.H.C. Whitehead in the 1940s, the problem asks whether a certain type of homomorphism between two groups can be lifted to a homotopy equivalence.
Wyckoff positions refer to a specific method of categorizing the arrangement of atoms in a crystal lattice, specifically within the field of crystallography. The concept is named after the American physicist and crystallographer Walter Bradford Wyckoff, who developed a systematic way to describe crystal structures. In the context of crystallography, Wyckoff positions describe the symmetry and positioning of atoms in a crystal symmetric group.