Kronecker substitution is a mathematical technique used primarily in the context of polynomial approximations and numerical methods for solving differential equations, particularly when dealing with linear differential operators. It converts differential equations into algebraic equations by substituting certain variables or expressions, which can simplify the problem and make it more manageable.

Monomial representation is a mathematical expression used to represent polynomials, particularly in certain contexts like computer science, algebra, and optimization. A monomial is a single term that can consist of a coefficient (which is a constant) multiplied by one or more variables raised to non-negative integer powers.

An \( O^* \)-algebra is a mathematical structure that arises in the field of functional analysis, particularly in the study of operator algebras. Specifically, an \( O^* \)-algebra is a type of non-self-adjoint operator algebra that is equipped with a specific topological structure and certain algebraic properties.

In order theory, a branch of mathematics, the term "prime" can refer to a particular type of element within a partially ordered set (poset).

In the context of algebra and functional analysis, a **principal subalgebra** typically refers to a specific type of subalgebra that is generated by a single element, particularly in the study of operator algebras, such as von Neumann algebras or C*-algebras. To elaborate, let's consider the following definitions: 1. **Subalgebra**: A subalgebra of an algebra is a subset of that algebra that is itself an algebra under the same operations.

A *Quasi-Lie algebra* is a generalization of Lie algebras that relaxes some of the traditional properties that define a Lie algebra. While Lie algebras are defined by a bilinear operation (the Lie bracket) that is antisymmetric and satisfies the Jacobi identity, quasi-Lie algebras may abandon or modify some of these conditions.

A **simplicial Lie algebra** is a mathematical structure that arises in the study of algebraic topology and differentiable geometry, particularly in the context of generalized symmetries and homotopy theory. It combines concepts from both Lie algebras and simplicial sets.

The tensor product of quadratic forms is a mathematical operation that combines two quadratic forms into a new quadratic form. To understand this concept, we first need to clarify what a quadratic form is.

Tropical compactification is a mathematical technique used in algebraic geometry and related areas, particularly those involving tropical geometry. To understand tropical compactification, it's helpful to first grasp some concepts in both algebraic geometry and tropical geometry. ### Tropical Geometry: 1. **Tropical Semiring**: In tropical geometry, we typically work with a modified version of the arithmetic called the tropical semiring.

The Alon–Boppana bound is a result in the field of graph theory and spectral graph theory. It provides a lower bound on the largest eigenvalue (also known as the spectral radius) of a regular graph. More formally, let \( G \) be a \( d \)-regular graph on \( n \) vertices.

A distance-regular graph is a specific type of graph that has a high degree of regularity in the distances between pairs of vertices. Formally, a graph \( G \) is said to be distance-regular if it satisfies the following conditions: 1. **Regularity**: The graph is \( k \)-regular, meaning each vertex has exactly \( k \) neighbors.

Maude is a high-level programming language and system that is based on rewriting logic. It is designed for specifying, programming, and reasoning about systems in a formal and executable manner. Maude allows for the definition of systems in terms of algebraic specifications, and it can be used for a wide range of applications in formal methods, including model checking, theorem proving, and symbolic simulation.

Graph automorphism is a concept in graph theory that refers to a symmetry of a graph that preserves its structure. More specifically, an automorphism of a graph is a bijection (one-to-one and onto mapping) from the set of vertices of the graph to itself that preserves the adjacency relationship between vertices.

The Ihara zeta function is a mathematical object that arises in the study of finite graphs, particularly in the context of algebraic topology and number theory. It was introduced by Yoshio Ihara in the 1960s.

Mac Lane's planarity criterion, also known as the "Mac Lane's formation", is a combinatorial condition used to determine whether a graph can be embedded in the plane without any edges crossing. Specifically, the criterion states that a graph is planar if and only if it does not contain a specific type of subgraph as a "minor.

The Parry–Sullivan invariant is a concept in the field of dynamical systems and statistical mechanics, particularly related to the study of interval exchanges and translations. It is associated with the study of the dynamics of certain classes of transformations, particularly those that exhibit specific structural and statistical properties. The invariant itself is often connected to topological and measure-theoretic characteristics of systems that exhibit a certain type of symmetry or recurrence.

Sims' conjecture is a hypothesis in the field of algebraic topology and combinatorial group theory, specifically relating to the properties of certain types of groups. Named after mathematician Charles Sims, the conjecture primarily deals with the structure of finite groups and representation theory. While specific details or formulations may vary, Sims' conjecture is generally focused on establishing a relationship between the orders of groups and their representations or modules.

A strongly regular graph is a specific type of graph characterized by a regular structure that satisfies certain conditions regarding its vertices and edges. Formally, a strongly regular graph \( G \) is defined by three parameters \( (n, k, \lambda, \mu) \) where: - \( n \) is the total number of vertices in the graph.