Complexification of a Lie group is a process that involves taking a real Lie group and extending it to a complex Lie group. This technique is useful in many areas of mathematics and theoretical physics because it allows for the application of complex analysis techniques to problems originally framed in the context of real manifolds.

Idempotence is a property of certain operations in mathematics and computer science where applying the operation multiple times has the same effect as applying it just once. In other words, performing an operation a number of times doesn't change the result beyond the initial application. ### Mathematical Definition In mathematics, a function \( f \) is considered idempotent if: \[ f(f(x)) = f(x) \quad \text{for all } x \] ### Examples 1.

Quadratic irrational numbers are a type of irrational number that can be expressed in the form \( \frac{a + b\sqrt{d}}{c} \), where \( a \), \( b \), and \( c \) are integers, \( d \) is a non-square positive integer, and \( c \) is a positive integer. In simpler terms, they can be represented as a root of a quadratic equation with integer coefficients.

A Severi–Brauer variety, named after the mathematicians Francesco Severi and Hans von Brauer, is a specific type of algebraic variety that is related to the study of division algebras and central simple algebras in algebraic geometry.

In the context of mathematics, particularly in category theory and algebra, an epimorphism is a morphism (or map) between two objects that generalizes the notion of an "onto" function in set theory.

Lie groups are mathematical structures that combine concepts from algebra and geometry. They are named after the Norwegian mathematician Sophus Lie, who studied them in the context of continuous transformation groups. ### Basic Definition A **Lie group** is a group that is also a smooth manifold, meaning that the group operations (multiplication and inversion) are smooth (infinitely differentiable) functions. This combination allows for the study of algebraic structures (like groups) with the tools of calculus and differential geometry.

A Hardy field is a type of mathematical structure used in the field of real analysis and model theory, specifically in the study of asymptotic behaviors of functions. It is named after the mathematician G. H. Hardy. A Hardy field is essentially a field of functions that satisfies certain algebraic and order properties.

Kleene algebra is a mathematical structure used in theoretical computer science, formal language theory, and algebra. It is named after the mathematician Stephen Kleene, who made significant contributions to the foundations of automata theory and formal languages. Kleene algebra consists of a set equipped with certain operations and axioms that support reasoning about the properties of regular languages and automata.

In algebra, particularly in the context of ring theory, the term "rng" (pronounced "ring") is an abbreviation that refers to a mathematical structure that is similar to a ring but does not necessarily require the existence of a multiplicative identity (i.e., an element that acts as 1 in multiplication).

A **semigroupoid** is an algebraic structure that generalizes the notion of a semigroup to a situation where the elements can be thought of as processes or mappings rather than simple algebraic objects. More formally, a semigroupoid can be defined as a category in which every morphism (or arrow) is invertible, but it has a single object, or it can be thought of as a partially defined operation among elements.

As of my last update in October 2023, Yasmine Amhis does not appear to be a widely recognized public figure, concept, or term in popular culture, politics, science, or other notable fields. It's possible that she could be a private individual or a rising figure not widely documented in mainstream sources up to that time.

Minimum Description Length (MDL) is a principle from information theory and statistics that provides a method for model selection. It is based on the idea that the best model for a given set of data is the one that leads to the shortest overall description of both the model and the data when encoded. In essence, it seeks to balance the complexity of the model against how well the model fits or explains the data.

Proxima Centauri is a red dwarf star located in the Alpha Centauri star system, which is the closest star system to Earth, situated about 4.24 light-years away. Proxima Centauri is the smallest and faintest of the three stars in this system, the other two being Alpha Centauri A and Alpha Centauri B, which are more similar to our Sun.

Sequence alignment is a bioinformatics method used to arrange sequences of DNA, RNA, or proteins to identify regions of similarity and difference. This process is crucial for understanding evolutionary relationships, functional similarities, and structural characteristics among biological sequences. There are two primary types of sequence alignment: 1. **Global Alignment**: This method aligns sequences from start to finish, ensuring that every residue in the sequences is aligned. It is typically used when comparing sequences that are of similar length and contain many conserved regions.

Agnes Giberne was a British author, best known for her works in the genre of children's literature and science fiction in the late 19th and early 20th centuries. She was born in 1857 and gained recognition for her ability to blend scientific themes with adventure and educational elements in her stories. One of her notable works is "The World of the Future," which explores themes of future societies and technological advancements.

Canadian amber is a type of fossilized tree resin that is found in Canada, particularly in areas like the province of Alberta. It is primarily derived from ancient coniferous trees, such as those from the family Pinaceae. Canadian amber is notable for its age and unique characteristics, which can include a range of colors from yellow to deep orange and even shades of green or brown.

Arthur Butler Phillips Mee (1875–1943) was a British writer, editor, and journalist, best known for his work in the field of popular history and encyclopedias. He is particularly famous for his contributions to children's literature and educational material. One of his notable works is the "Children's Encyclopedia," which aimed to present historical and scientific knowledge in an engaging format for young readers.

Hugh Percy Wilkins (1896–1960) was a British geologist and paleontologist notable for his work in the field of paleontology and for his contributions to the understanding of fossils, particularly those from the Mesozoic era. He is perhaps best known for his theories regarding dinosaurs and the classification of prehistoric reptiles. Wilkins also worked on various geological surveys and made significant contributions to the field of stratigraphy.

Mostafa Roustaei is an Iranian mathematician notable for his contributions to various fields in mathematics, particularly in functional analysis, operator theory, and numerical analysis. He has published works in these areas and has been involved in academic research and teaching.

Patrick Moore was a notable British astronomer, journalist, and television presenter, best known for his work on the long-running BBC television program "The Sky at Night," which he began hosting in 1957. He was widely recognized for his contributions to popularizing astronomy and science communication. In addition to his television work, Moore authored numerous books on astronomy and was involved in various astronomical societies and organizations. He had a distinctive persona characterized by his monocle and enthusiasm for the night sky.