In the context of lattice theory, particularly in the fields of mathematics and physics, a dual lattice is a concept that arises in the study of periodic structures, such as crystals or in the theory of vector spaces. 1. **Lattice Definition**: A lattice typically refers to a discrete subgroup of Euclidean space that is generated by a finite set of basis vectors.

The concept of a "lattice of stable matchings" arises in the context of matching theory, which is often studied in economics, game theory, and computer science. It involves systems in which two groups (such as men and women, or job applicants and jobs) are matched based on preferences in such a way that no pair of individuals would prefer each other over their current matches. This idea is closely associated with the Gale-Shapley algorithm, which produces stable matchings.

A **modular lattice** is a type of lattice in order theory with a specific property regarding its elements. A lattice is an algebraic structure that consists of a set equipped with two binary operations, meet ( ∧ ) and join ( ∨ ), which satisfy certain axioms. Lattices can be visualized as a partially ordered set (poset) where every two elements have a unique supremum (join) and infimum (meet).

Harmonic analysis is a branch of mathematics that studies the representation of functions or signals as the superposition of basic waves, often referred to as harmonics. It encompasses a variety of techniques and theories used to analyze functions in terms of their frequency components. Key aspects of harmonic analysis include: 1. **Fourier Series**: This involves expressing periodic functions as sums of sines and cosines. The Fourier coefficients provide a way to compute how much of each harmonic is present in the original function.

Admissible representation is a concept that can refer to various contexts, such as mathematics, logic, and artificial intelligence. Generally, it pertains to a system of representing knowledge, information, or states in a way that adheres to specific criteria or constraints. For example: 1. **In Artificial Intelligence and Search Algorithms**: An admissible heuristic is one that never overestimates the cost to reach the goal from the current state.

An affine Lie algebra is a certain kind of Lie algebra that arises as an extension of finite-dimensional simple Lie algebras. It plays a significant role in various areas of mathematics and theoretical physics, including representation theory, vertex operator algebras, and integrable systems.

Cellular algebra is a type of algebraic structure that arises in the context of representation theory, particularly in the study of coherent and modular representations of certain algebraic objects. It provides a framework for understanding the representation theory of groups, algebras, and related structures using a combinatorial approach.

The Chang number is a concept from the field of mathematics, specifically in topology and combinatorics. It is named after the mathematician Chao-Chih Chang. In more detail, the Chang number is a cardinal number that arises in the context of certain properties of functions and transformations, particularly in the study of large cardinals and their relationships to set theory.

Clifford theory, named after the mathematician William Kingdon Clifford, is a concept in the field of group theory, specifically dealing with the representation of finite groups. It is particularly concerned with the relationship between representations of a group and its normal subgroups, as well as the way representations can be lifted to larger groups.

The Weil-Brezin map is a concept in the fields of mathematical physics and algebraic geometry. It pertains to the study of integrable systems and is notably related to the context of matrix models, specifically within the realm of random matrices and their connections to two-dimensional quantum gravity. In essence, the Weil-Brezin map provides a correspondence that links certain algebraic objects to geometric structures.

The Gelfand–Graev representation is a specific type of representation associated with the theory of finite groups, particularly in the context of group algebras and representation theory. Named after I. M. Gelfand and M. I. Graev, this representation is a construction that arises in the study of group characters and modular representations.

The term "Hecke algebra" can refer to several related but distinct concepts in mathematics, particularly in the fields of number theory, representation theory, and algebra. Here are a few notable interpretations: 1. **Hecke Algebras in Representation Theory**: In this context, Hecke algebras arise in the study of algebraic groups and their representations. They are associated with Coxeter groups and provide a way to study representations of symmetric groups and general linear groups.

The McKay graph is a type of graph used in the field of algebraic combinatorics, particularly in the study of group theory and representation theory. Specifically, it arises in the context of the representation theory of finite groups. For a given finite group \( G \), the McKay graph is constructed as follows: 1. **Vertices**: The vertices of the McKay graph correspond to the irreducible representations of the group \( G \).

A prehomogeneous vector space is a concept from the field of invariant theory and representation theory, particularly concerning vector spaces that admit a group action with certain properties.

The Steinberg formula is a mathematical expression used in the context of estimating the performance of a certain type of algorithm, specifically in areas such as numerical analysis, optimization, and machine learning.

Univel was a joint venture between IBM and Novell in the early 1990s, aimed at combining IBM's software and hardware expertise with Novell's networking and operating system capabilities. The goal of Univel was primarily to produce a version of the UNIX operating system that would be compatible with IBM's hardware and to enhance networking solutions, particularly in enterprise environments. The collaboration produced a UNIX variant known as "UnixWare," which was designed for performance on IBM's systems.

An associated graded ring is a construction in algebra that arises in the study of filtered rings, which are rings equipped with a specified filtration.

An **Azumaya algebra** is a specific type of algebra over a commutative ring that behaves like a matrix algebra over a field in a certain sense, while being more general. Formally, an Azumaya algebra is a sheaf of algebras that satisfies a particular condition related to the notion of being "frobenius". More precisely, let \( R \) be a commutative ring.

Division algebra is a type of algebraic structure where division is possible, except by zero. More formally, a division algebra is a vector space over a field \( F \) equipped with a bilinear multiplication operation that satisfies the following conditions: 1. **Non-Associativity or Associativity**: In a general division algebra, multiplication can be either associative or non-associative. If it is associative, the algebra is called an associative division algebra.