Vincenty's formulae are a set of mathematical formulas used to calculate the distance between two points on the surface of an ellipsoidal model of the Earth, which takes into account the Earth's flattening and provides more accurate results than simpler spherical formulas. They are particularly useful for geodesic calculations in geodesy, cartography, and navigation.

A virtual globe is a three-dimensional computer-generated representation of Earth that allows users to visually explore geographic data and information. It typically provides a detailed view of the planet's surface, including terrain, bodies of water, political boundaries, and geographic features such as mountains and valleys. Virtual globes can also include layers of data that can be toggled on or off, displaying various information such as satellite imagery, weather patterns, population density, and more.

Anonymity in the context of social choice theory refers to a principle that focuses on the treatment of individuals in the decision-making process. Specifically, the anonymity principle states that all individuals should be treated equally and that the preferences of individuals should not be weighted differently based on their identity. In other words, if two individuals swap their preferences, the outcome of the social choice should remain unchanged.

An electoral system is a set of rules and processes that govern how votes are cast, counted, and translated into seats in a legislature or the outcome of an election. Electoral systems can significantly influence political processes, party systems, and voter behavior. They determine how representatives are elected, the methods by which votes are aggregated, and how those results translate into political power.

A. M. Cagle could refer to various individuals or entities, but without additional context, it's difficult to pinpoint a specific reference. It might refer to a person’s name, an author, a professional in a specific field, or even a business or organization. If you can provide more details about the context or topic related to A. M.

Fractional social choice is a concept in social choice theory that extends traditional voting and decision-making frameworks to incorporate scenarios where preferences can be expressed in a fractional or probabilistic manner. This approach recognizes that individual preferences might not be strictly ranked as in classical voting systems, allowing for a more nuanced representation of societal choices. In conventional social choice mechanisms, individuals provide their preferences in terms of complete rankings or simple majority votes.

Algebraic Normal Form (ANF) is a representation of Boolean functions that expresses them as a polynomial over the finite field \( \mathbb{F}_2 \) (the field with two elements, 0 and 1).

A **Boolean ring** is a specific type of ring in the field of abstract algebra that has particular properties.

Random algebra is a mathematical framework that generally deals with structures and operations involving randomness. It can have various interpretations depending on the context, including: 1. **Random Variables and Their Algebras**: In probability theory, random variables can be treated as elements of an algebra. This involves the study of functions that assign numerical values to outcomes of random processes, and the rules for combining these variables—such as addition and multiplication—can be defined in an algebraic manner.

A True Quantified Boolean Formula (TQBF) is a specific type of Boolean formula that is a decision problem in computational complexity theory. It extends the concept of Boolean formulas by incorporating quantifiers (universal and existential) over the variables involved. ### Definition: A TQBF is a prenex normal form Boolean formula that can be expressed as follows: - It consists of a sequence of quantifiers followed by a propositional formula.

A repeating decimal is a decimal representation of a number in which a digit or a group of digits repeats indefinitely. This means that after a certain point, the same sequence of digits appears over and over again, without end. Repeating decimals can result from the division of certain fractions. For example: - The fraction \( \frac{1}{3} \) is equal to \( 0.333...\), where the digit '3' repeats forever.

Alfred Tarski, a prominent logician and mathematician, developed an axiomatization of the real numbers based on first-order logic. Tarski's approach was notable for its focus on the completeness and consistency of the real number system, as well as its relationship to ordered fields.

Modulo, often represented by the symbol `%`, is a mathematical operation that finds the remainder of the division of one integer by another.

The multiplicative order of an integer \( a \) modulo \( n \) is defined as the smallest positive integer \( k \) such that \[ a^k \equiv 1 \mod n. \] In simpler terms, it is the smallest exponent \( k \) for which raising \( a \) to the power of \( k \) results in a value that, when divided by \( n \), leaves a remainder of 1.

A **root of unity** modulo \( n \) refers to an integer \( k \) such that \( k^m \equiv 1 \mod n \) for some positive integer \( m \). In other words, \( k \) is a root of unity if it raises to some integer power \( m \) and gives a result of 1 when taken modulo \( n \).