Stopped 2019 apparently. Shame. We need something to be upstreamed!

Persistent homology is a concept from computational topology, a branch of mathematics that studies the shape, structure, and features of data. It provides a way to analyze the topology of data sets, particularly those that vary with a parameter, by examining how the topological features of a shape persist across different scales. ### Key Concepts of Persistent Homology: 1. **Topological Spaces and Chains**: - Topological spaces are sets equipped with a structure that allows for the concept of continuity.

Computer Algebra Systems (CAS) are software programs designed to perform symbolic mathematics. They manipulate mathematical expressions in a way that is similar to how humans do algebra: by applying mathematical rules and properties symbolically rather than numerically. This allows users to perform complex calculations, simplifications, and transformations involving algebraic expressions, calculus, linear algebra, and other areas of mathematics.

"Coimage" can refer to different concepts depending on the context in which it's used, particularly in mathematics or computer science. Here are a couple of interpretations: 1. **In Mathematics (Category Theory):** The term "coimage" is often used in the context of category theory and algebraic topology. In this setting, the coimage of a morphism is related to the concept of the cokernel.

A glossary of commutative algebra is a collection of terms and definitions that are commonly used in the field of commutative algebra, which is a branch of mathematics that studies commutative rings, their ideals, and modules over those rings. Here are some key terms and concepts typically found in such a glossary: 1. **Ring**: A set equipped with two binary operations (addition and multiplication) that satisfy certain properties (associativity, distributivity, etc.).

Hilbert's Basis Theorem is a fundamental result in algebra, particularly in the theory of rings and ideals. It states that if \( R \) is a Noetherian ring (meaning that every ideal in \( R \) is finitely generated), then any ideal in the polynomial ring \( R[x] \) (the ring of polynomials in one variable \( x \) with coefficients in \( R \)) is also finitely generated.

The Hilbert–Samuel function is an important concept in commutative algebra and algebraic geometry, particularly in the study of the structure of space defined by ideals in rings and the geometry of schemes. It provides a way to measure the growth of the dimensions of the graded components of the quotient of a Noetherian ring by an ideal.

Their main innovation seems to be their 3D design which they call "Coaxmon".

Funding:

The \(I\)-adic topology is a concept from algebraic number theory and algebraic geometry that generalizes the notion of topology in the context of ideals in rings, specifically in relation to \(p\)-adic numbers.

The term "ideal norm" can have different meanings depending on the context. Here are a couple of interpretations based on various fields: 1. **Mathematics/Statistics**: In the context of mathematics, particularly in functional analysis and linear algebra, an "ideal norm" could refer to the notion of a norm that satisfies certain properties or conditions ideal for a given space.

Their status is a mess as of 2020s, with several systems ongoing. Long live the "original" collegiate university!

Ideal theory is a concept primarily associated with political philosophy and ethics, particularly in discussions surrounding justice, fairness, and the principles that should govern a well-ordered society. It can refer to the formulation of theoretical frameworks or principles that define what an ideal society should look like and how individuals within it should behave.

In the context of ring theory, an irreducible ring is typically referred to as a ring that cannot be factored into "simpler" rings in a specific way.

The J-2 ring, also known simply as a J-ring, refers to a particular type of ring in the study of algebraic structures in mathematics. Specifically, a J-2 ring is a ring where a certain condition related to Jacobson radical and nilpotent elements holds.

J-multiplicity is a concept that appears in the context of mathematical logic and model theory, particularly in the study of structures and their properties. It is often associated with the analysis of certain functions or relations over structures, and can be used to investigate how complex a particular model or theory is.

The Koszul–Tate resolution is a construction in algebraic geometry and homological algebra used to study certain algebraic structures, particularly those that involve differential forms or algebraic relations. It is named after Jean-Pierre Serre and William Tate, who contributed to the understanding of such resolutions. In simple terms, the Koszul-Tate resolution provides a way to resolve algebraic objects, such as modules or complexes associated with algebraic varieties, using tools from homological algebra.

The nilradical of a ring is an important concept in ring theory, a branch of abstract algebra. Specifically, the nilradical of a ring \( R \) is defined as the set of all nilpotent elements in \( R \). An element \( x \) of \( R \) is called nilpotent if there exists some positive integer \( n \) such that \( x^n = 0 \).