The Higman–Sims graph is a highly symmetric, 22-vertex graph that arises in the context of group theory and combinatorial design. It is named after mathematicians Graham Higman and Charles Sims, who studied its properties in relation to the Higman–Sims group, a specific group in group theory. Here are some important characteristics of the Higman–Sims graph: 1. **Vertices and Edges**: The graph has 22 vertices and 57 edges.

The Northern Ireland Statistics and Research Agency (NISRA) is the principal source of official statistics and social research in Northern Ireland. It operates under the Department of Finance and aims to provide accurate, reliable, and relevant statistical information to support decision-making, policy formulation, and public understanding of various issues in the region. NISRA is responsible for conducting the Census of Population in Northern Ireland and compiling various demographic, social, economic, and environmental statistics.

Northern resident orcas, also known as Northern Resident Killer Whales, are a ecotype of orcas (Orcinus orca) that inhabit the coastal waters of the northern Pacific Ocean, particularly around the waters of British Columbia, Canada, and the southeastern portion of Alaska. They are part of the larger population of orcas found in the North Pacific, but they exhibit specific social structures, behaviors, and feeding habits that distinguish them from other ecotypes.

A quartic graph is a graphical representation of a polynomial function of degree four.

A **random regular graph** is a type of graph in which each vertex has the same degree, a property known as **regularity**, and the graph is generated in a random manner. Specifically, a random \( d \)-regular graph is a graph where: 1. **Degree**: Every vertex has exactly \( d \) edges (or connections) to other vertices, meaning it has a degree of \( d \).

A supersingular isogeny graph is a mathematical structure used primarily in number theory and algebraic geometry, particularly in the study of elliptic curves and their isogenies (which are morphisms between elliptic curves that respect the group structure). These graphs have become increasingly important in the field of cryptography, especially in post-quantum cryptographic protocols.

The Block Wiedemann algorithm is an efficient method for solving large sparse linear systems, specifically those defined over finite fields or in the context of polynomial time computations in algebraic structures. It is particularly useful for solving systems of linear equations that can be represented in matrix form where the matrix may be very large and sparse.

The eigenvalue algorithm refers to a collection of methods used to compute the eigenvalues and eigenvectors of matrices. Eigenvalues and eigenvectors are fundamental concepts in linear algebra with applications in many areas such as stability analysis, vibrational analysis, and principal component analysis, among others.

The Minimal Residual Method, commonly referred to as the MinRes method, is an iterative algorithm used to solve linear systems of equations, especially those that are symmetric and positive definite. It is particularly useful for large-scale problems where direct methods (like Gaussian elimination) may be computationally expensive or infeasible due to memory constraints.

Nested dissection is an algorithmic technique used primarily in numerical linear algebra for solving large sparse systems of linear equations, particularly those arising from finite element methods and related applications. It efficiently exploits the sparse structure of matrices and is particularly suited for problems where the matrix can be partitioned into smaller submatrices.

Power iteration is a numerical method used to find the dominant eigenvalue and its corresponding eigenvector of a matrix. This technique is particularly effective for large, sparse matrices, where traditional methods like direct diagonalization may be computationally expensive or impractical. ### How Power Iteration Works: 1. **Initialization**: Start with a random vector \( \mathbf{b_0} \) (which should not be orthogonal to the eigenvector corresponding to the dominant eigenvalue).

Relaxation is an iterative method used to solve mathematical problems, particularly those involving linear or nonlinear equations, optimization problems, and differential equations. The technique involves making successive approximations to the solution of the problem until a desired level of accuracy is achieved. ### Key Concepts of Relaxation Methods: 1. **Iterative Process**: The relaxation method starts with an initial guess for the solution and improves this guess through a series of iterations. Each iteration updates the current estimate based on a specified rule.

Speakeasy is a computational environment designed for developing and executing code, particularly in the context of machine learning, data analysis, and similar disciplines. It provides an interactive platform where users can write, run, and test code in real-time. Some of the key features of Speakeasy include: 1. **Interactive Coding**: Users can write and execute code in a dynamic way, which is useful for exploratory data analysis and iterative development.

The Christoffel–Darboux formula is a significant result in the theory of orthogonal polynomials. It provides a way to express sums of products of orthogonal polynomials in a concise form. Typically, the formula relates the orthogonal polynomials defined on a specific interval with respect to a weight function.

The term "rates" can refer to various concepts depending on the context. Here are some common interpretations: 1. **Interest Rates**: The percentage charged on borrowed money or earned on investments, typically expressed on an annual basis. For example, a bank might offer a savings account with an interest rate of 2% per year. 2. **Exchange Rates**: The value of one currency in terms of another. For instance, if the exchange rate between the U.S.

Jacobi polynomials are a class of orthogonal polynomials that arise in various areas of mathematics, including approximation theory, numerical analysis, and the theory of special functions. They are named after the mathematician Carl Gustav Jacob Jacobi.

The Meixner–Pollaczek polynomials are a class of orthogonal polynomials that arise in various areas of mathematics, particularly in spectral theory, probability, and mathematical physics. They can be defined as a part of the broader family of Meixner polynomials, which are associated with certain types of stochastic processes, especially those arising in the context of random walks and queuing theory.

Q-Bessel polynomials, also known as Bessel polynomials of the first kind, are specific types of orthogonal polynomials that are related to Bessel functions. These polynomials arise in various areas of mathematics and applied sciences, particularly in solutions to differential equations, mathematical physics, and numerical analysis. Q-Bessel polynomials can be defined through their generating function or through a recurrence relation.

The Q-Meixner–Pollaczek polynomials are a family of orthogonal polynomials that arise in the context of certain special functions and quantum mechanics. They are a generalization of both the Meixner and Pollaczek polynomials and are associated with q-analogues, which are modifications of classic mathematical structures that depend on a parameter \( q \).

Elliptic rational functions are mathematical functions that arise in the study of elliptic curves and, more generally, in the theory of elliptic functions. They can be thought of as generalizations of rational functions that incorporate properties of elliptic functions. To understand elliptic rational functions, it's helpful to break down the components of the term: 1. **Elliptic Functions:** These are meromorphic functions that are periodic in two directions (often associated with the complex plane's lattice structure).