Lofting in 3D modeling is a technique used to create a three-dimensional object by defining a series of cross-sectional profiles and connecting them smoothly through a process called lofting. This technique is commonly used in various fields, including CAD (Computer-Aided Design), animation, and industrial design.

PLaSM, which stands for "Programming Language for Synthesis of Meshes," is a programming language designed for generating and manipulating geometric shapes and meshes, particularly in the context of computer graphics and computational geometry. PLaSM enables users to define complex shapes and models in a relatively straightforward way, allowing for the creation of intricate designs that can be used in various applications, such as 3D printing, computer-aided design (CAD), and animation.

MATLAB (Matrix Laboratory) is a high-level programming language and interactive environment designed primarily for numerical computing, data analysis, visualization, and algorithm development. Developed by MathWorks, MATLAB is widely used in academia, research, and industry for various applications, including mathematical modeling, simulation, signal processing, image processing, control systems, and machine learning.

British mathematicians have made significant contributions to the field throughout various centuries. Below are some notable mathematicians organized by century: ### 17th Century - **William Oughtred (1574–1660)**: Known for inventing the slide rule and for his work in the development of mathematical notation. - **John Wallis (1616–1703)**: A key figure in the development of calculus and the introduction of the concept of infinity.

TI InterActive! is an interactive software application developed by Texas Instruments specifically designed for education, particularly in mathematics and science. It serves as a digital learning platform that provides various tools and resources for students and teachers. Key features of TI InterActive! include: 1. **Graphing and Visualization**: Users can create graphs of mathematical functions, making it easier to visualize concepts like calculus and algebra.

KCNJ14, also known as the potassium voltage-gated channel subfamily J member 14, is a gene that encodes a protein that is part of the inwardly rectifying potassium channel family. These channels are essential for maintaining the resting membrane potential of cells and play significant roles in various physiological processes, including cardiac and neuronal excitability.

Brittleness is the property of a material that leads to fracture or failure with little to no plastic deformation under stress. In other words, brittle materials tend to break sharply without significant prior distortion or bending when they are subjected to strain. This characteristic is commonly observed in materials such as glass, ceramics, and some metals when they are cold, as they do not have the ability to absorb significant energy before breaking.

The term "congruence ideal" is primarily used in the context of algebra, particularly in the study of rings and ideals in ring theory. Although it's not as commonly referenced as some other concepts, the idea generally relates to how certain elements of a ring or algebraic structure can be used to define relationships and equivalences among elements. In the context of a ring \( R \), a congruence relation is an equivalence relation that is compatible with the ring operations.

Differential graded algebra (DGA) is a mathematical structure that combines concepts from algebra and topology, particularly in the context of homological algebra and algebraic topology. A DGA consists of a graded algebra equipped with a differential that satisfies certain properties. Here’s a more detailed breakdown of the components and properties: ### Components of a Differential Graded Algebra 1.

The term "G-ring" can refer to several different concepts depending on the context, such as mathematics, chemistry, or other specialized fields. However, it is most commonly known in the context of algebra, specifically in ring theory. In mathematics, a **G-ring** typically refers to a **generalized ring**, which is a structure that generalizes the concept of a ring by relaxing some of the usual requirements.

A Euclidean domain is a type of integral domain (a non-zero commutative ring with no zero divisors) that satisfies a certain property similar to the division algorithm in the integers.

A **Gorenstein ring** is a type of commutative ring that has particularly nice homological properties. More formally, a Noetherian ring \( R \) is called Gorenstein if it satisfies the following equivalent conditions: 1. **Dualizing Complex**: The singularity category of \( R \) has a dualizing complex which is concentrated in non-negative degrees, and the homological dimension of the ring is finite.

Hausdorff completion is a mathematical process used to construct a complete metric space from a given metric space that may not be complete. The idea is to extend the space in such a way that all Cauchy sequences converge within the new space. ### Overview of the Process: 1. **Metric Spaces and Completeness**: A metric space is a set equipped with a distance function (metric) that defines how far apart the points are.

Hironaka decomposition is a concept in the context of algebraic geometry and singularity theory, specifically related to the resolution of singularities. The term is often associated with the work of Heisuke Hironaka, who is well-known for his theorem on the resolution of singularities in higher-dimensional spaces.

Hodge algebra is a concept in mathematics that arises in the study of Hodge theory, which is a field connecting algebraic topology, differential geometry, and algebraic geometry. Hodge theory is centered on the decomposition of differential forms on a smooth manifold and the study of their topological and geometric properties. More formally, a Hodge algebra typically refers to a certain type of graded algebra that arises in the context of Hodge theory, particularly when considering cohomology and the Hodge decomposition theorem.

"Ideal reduction" can refer to different concepts depending on the context in which it is used. Here are a few interpretations based on various fields: 1. **Mathematics / Algebra**: In the context of algebraic structures, "ideal reduction" might refer to the process of simplifying algebraic expressions or problems using ideals in ring theory. An ideal is a special subset of a ring that can be used to create quotient rings, facilitating the study of various properties of the ring.

"Introduction to Commutative Algebra" is a well-known textbook written by David Eisenbud, which provides a comprehensive overview of the field of commutative algebra. It serves as an accessible entry point for students and researchers delving into the subject. Commutative algebra is a branch of algebra that studies commutative rings and their ideals, focusing on properties and structures that arise from these algebraic constructs.

In the context of abstract algebra, particularly in ring theory, an **irrelevant ideal** is typically discussed in relation to the properties of ideals in polynomial rings or local rings. While the term "irrelevant ideal" may not be universally defined across all mathematics literature, it's most commonly associated with certain ideals in the study of algebraic geometry and commutative algebra.

A **Krull ring** is a specific type of commutative ring that has certain ideal-theoretic properties. Named after Wolfgang Krull, these rings are important in algebraic geometry and commutative algebra due to their connection to the concept of dimension and the behavior of their prime ideals.

In the context of ring theory, a **minimal prime ideal** is a prime ideal \( P \) in a commutative ring \( R \) such that there are no other prime ideals contained within \( P \) except for \( P \) itself. In other words, \( P \) is a minimal element in the set of prime ideals of the ring with respect to inclusion.