The Tutte–Grothendieck invariant is an important concept in graph theory and combinatorics, associated with the study of matroids and graphs. This invariant is commonly denoted as \( T(G) \) for a graph \( G \) and is defined in terms of the graph's structure, specifically its connected components and edges.

Bounded expansion is a concept in graph theory that pertains to the behavior of certain classes of graphs, particularly in relation to their structure and properties. A family of graphs is said to have bounded expansion if, roughly speaking, the density of the graphs in the family does not grow too quickly as the size of the graphs increases.

In graph theory, the term "core" refers to a specific type of subgraph that captures some essential structural properties of the original graph. A **core** of a graph is often defined as a maximal subgraph in which every vertex has a degree (number of edges connected to it) of at least \( k \). This means that for a \( k \)-core, every vertex in the graph has at least \( k \) connections.

A citation graph is a directed graph that represents the relationship between academic papers, articles, patents, or other scholarly works based on citations. In a citation graph: - **Nodes**: Each node corresponds to a publication or scholarly work. - **Edges**: A directed edge from node A to node B indicates that publication A cites publication B. This means that A references or relies on B in its content.

A Program Dependence Graph (PDG) is a graphical representation of the dependencies within a program, specifically focusing on the relationships between different computations and data in the program. PDGs are useful for various analyses and optimizations in compiler design and software engineering. ### Key Components of a PDG: 1. **Nodes:** - **Statements or Instructions:** Each node in the graph represents a basic operation or statement in the program.

Confirmatory blockmodeling is a statistical technique used in social network analysis to test hypothesized structures within network data. It is concerned with identifying and validating specific patterns of connections (or relationships) among a set of actors (nodes) that belong to different groups (blocks). This method is useful in understanding how these groups interact within a network.

Catalan's conjecture, also known as Mihăilescu's theorem, states that the only solution in positive integers to the equation \( x^a - y^b = 1 \), where \( a \) and \( b \) are integers greater than 1, is the pair \( (3, 2) \) for the values \( (x, y) \) and \( (a, b) = (3, 2) \).

Harrison White can refer to a couple of different things depending on the context: 1. **Harrison C. White**: He is a sociologist known for his contributions to the fields of social theory and social networks. White has made significant contributions to understanding social structures and the dynamics of social relationships. 2. **Harrison White (Fictional Character)**: In some media, there may be fictional characters named Harrison White.

The Oberwolfach problem is a problem in combinatorial design and graph theory that involves the arrangement of pairs (or "couples") of items, typically represented as graphs or edges. It is named after the Oberwolfach Institute for Mathematics in Germany, where the problem was first studied. The classical statement of the problem can be described as follows: You have a finite group of \( n \) people (or vertices) who need to meet in pairs over a series of days (or rounds).

In statistics, **consistency** refers to a desirable property of an estimator. An estimator is said to be consistent if, as the sample size increases, it converges in probability to the true value of the parameter being estimated.

An Euler spiral, also known as a "spiral of constant curvature" or "clothoid," is a curve in which the curvature changes linearly with the arc length. This means that the radius of curvature of the spiral increases (or decreases) smoothly as you move along the curve. The curvature is a measure of how sharply a curve bends, and in an Euler spiral, the curvature increases from zero at the start of the spiral to a constant value at the end.

"Elementary Calculus: An Infinitesimal Approach" is a textbook authored by H. Edward Verhulst. It presents calculus using the concept of infinitesimals, which are quantities that are closer to zero than any standard real number yet are not zero themselves. This approach is different from the traditional epsilon-delta definitions commonly used in calculus classes. The book aims to provide a more intuitive understanding of calculus concepts by employing infinitesimals in the explanation of limits, derivatives, and integrals.

A quasi-continuous function is a type of function that is continuous on a dense subset of its domain.

Tensor calculus is a mathematical framework that extends the concepts of calculus to tensors, which are geometric entities that describe linear relationships between vectors, scalars, and other tensors. Tensors can be thought of as multi-dimensional arrays that generalize scalars (zero-order tensors), vectors (first-order tensors), and matrices (second-order tensors) to higher dimensions.