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.

As of my last knowledge update in October 2021, VORTEX projects refer to various initiatives and programs across different fields, including scientific research, technology, and computing, among others. The term "VORTEX" itself can be associated with different contexts, so without a specific reference or area, it's challenging to provide a precise answer.

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 \( n \).

The Petrie dual is a concept in the field of geometry and topology, particularly in the study of polyhedra and regular polytopes. It is a specific type of duality that applies to certain polyhedra. In essence, each polyhedron can be associated with a dual polyhedron where the vertices, edges, and faces are transformed in a systematic way.

Tropical cyclone meteorology is the study of tropical cyclones, which are powerful storm systems characterized by low-pressure centers, organized convection, and sustained winds that can exceed 74 miles per hour (119 kilometers per hour). These storms, which include hurricanes and typhoons depending on their location, are significant meteorological phenomena due to their potential for severe weather, including heavy rainfall, strong winds, and storm surges.

The term "Tropical cyclone stubs" typically refers to a specific type of article or entry on platforms like Wikipedia that provides only basic or minimal information about a tropical cyclone. These stubs are often very brief and lack comprehensive details, serving as placeholders for more extensive articles that can be expanded upon by contributors over time. On Wikipedia, stubs are designated with a template that encourages users to edit and add information to improve the article.

A "hurricane party" is a social gathering that takes place in the days leading up to or during a hurricane, particularly when people are in areas that are expected to be affected by the storm. The term is often used somewhat humorously, suggesting that some people choose to host a party at their home to ride out the storm with friends and family, rather than evacuating.

The term "Genoa lows" refers to a type of weather system associated with the Mediterranean region, particularly in areas influenced by the city of Genoa, Italy. These lows are characterized by the development of low-pressure systems that can bring significant precipitation and strong winds to the surrounding areas. They typically form in the fall and winter months and are often linked to the Adriatic and Ligurian Seas.

A Nor'easter is a type of storm that primarily affects the northeastern United States, characterized by strong winds coming from the northeast. These storms typically occur in the fall, winter, and early spring and can bring heavy snow, rain, and coastal flooding. Nor'easters develop when warm, moist air from the Gulf of Mexico meets cold air from Canada. As the warm air rises over the cold air, it can create low-pressure systems that can intensify rapidly.

Wall's finiteness obstruction is a concept from algebraic topology, particularly in the study of finite group actions on spaces and the homology of groups. It arises in the context of understanding when a group can be represented by a finite-dimensional space or manifold.

The term "E-quadratic form" appears to refer to a type of quadratic form characterized by a specific kind of structure or properties, particularly in the context of mathematics. While there isn't a universally recognized definition for "E-quadratic form" specifically, the term might relate to concepts in algebra, geometry, or particularly in number theory. In general, a **quadratic form** is a homogeneous polynomial of degree two in a number of variables.

A proxy bid is a bidding method used primarily in auctions, where a bidder allows an auction house or a bidding platform to place bids on their behalf up to a specified maximum amount. The purpose of a proxy bid is to automate the bidding process and ensure that the bidder doesn't have to continuously monitor the auction or manually place each bid.

Turán's brick factory problem is a classic problem in combinatorial optimization, particularly in the field of graph theory. It is named after the Hungarian mathematician Paul Erdős and his colleague László Turán, who studied problems involving extremal graph theory. The problem can be described as follows: Imagine a brick factory that produces bricks of various colors.

The Brauer group is a fundamental concept in algebraic geometry and algebra, particularly in the study of central simple algebras. It encodes information about dividing algebras and Galois cohomology. In more precise terms, the Brauer group of a field \( K \), denoted \( \text{Br}(K) \), is defined as the group of equivalence classes of central simple algebras over \( K \) under the operation of tensor product.

The Lefschetz hyperplane theorem is a fundamental result in algebraic geometry and topology that relates the topology of a projective variety to that of its hyperplane sections. Specifically, it provides information about the cohomology groups of a projective variety and its hyperplane sections. To state the theorem more formally: Let \(X\) be a smooth projective variety of dimension \(n\) defined over an algebraically closed field.