Arithmetic varieties, in the context of algebraic geometry, refer to varieties defined over number fields or more general arithmetic fields, and they can be studied using both algebraic techniques and number theoretic methods. These varieties are often associated with Diophantine equations, which seek integer or rational solutions to polynomial equations. More formally, an arithmetic variety is an algebraic variety defined over the field of rational numbers \( \mathbb{Q} \) or over more general number fields.

Cyclotomic units are a special class of elements in the field of algebraic number theory, particularly within the context of cyclotomic fields. Cyclotomic fields are extensions of the rational numbers obtained by adjoining a primitive \( n \)-th root of unity, denoted as \( \zeta_n \), to the rationals \( \mathbb{Q} \).

Fricke involution is a concept found in the context of modular forms and algebraic geometry, particularly in relation to the study of modular curves. It is a specific type of involution—meaning it is an operation that can be applied twice to return to the original state—defined on the upper half-plane or on modular forms.

The Graß conjecture, also known as the Graß problem, is a problem in number theory related to prime numbers. Specifically, it posits a certain property of the primes in relation to their distribution. The conjecture asserts that for any integer \( n \), there exist infinitely many primes that can be expressed in the form \( n^2 + k \), for \( k \) being a positive integer that is not a perfect square.

A Gregory number refers to a specific type of number that is related to the Gregory series or Gregory-Leibniz series, which is an infinite series that can be used to estimate the value of π (pi).

The "Hexagonal Tortoise Problem" is a common conceptual or computational exercise often found in recreational mathematics or programming challenges. It involves a tortoise that moves on a hexagonal grid, typically starting from a specific point and moving in various directions based on certain rules. The problem usually requires finding a path, counting the number of distinct cells visited, or calculating possible movements. In a more specific context, the problem may involve defining how the tortoise moves (e.g.

The Local Trace Formula is a significant result in the fields of number theory and representation theory, particularly in the study of automorphic forms and L-functions. It relates the trace of an operator on a space of functions to geometric and number-theoretic data associated with a locally symmetric space. In more specific terms, the Local Trace Formula often appears in the context of the theory of L-functions and automorphic representations.

A monogenic field is a concept that arises in the context of algebraic number theory and field theory. The term generally refers to a field extension that is generated by a single element, also known as a primitive element.

Overconvergent modular forms are a special class of modular forms that arise in the context of p-adic analysis and arithmetic geometry, particularly in relation to the theory of p-adic modular forms and overconvergent systems of forms. In classical terms, a modular form is a complex analytic function on the upper half-plane that satisfies specific transformation properties under the action of a congruence subgroup of \( SL(2, \mathbb{Z}) \).

The term "millieme" refers to a fractional currency unit that is used in some countries, particularly in the Arab world and parts of the Ottoman Empire's legacy. A millieme is typically 1/1000 of a dinar or other primary currency unit, although the specific relationship can vary by country. For example, in Iraq, the millieme was historically used as a subdivision of the dinar.

Large numbers are often named using a system that builds upon powers of ten. Here are some names for various large numbers, primarily based on the short scale, which is more commonly used in the United States and modern English-speaking countries: 1. **Thousand**: \(10^3\) (1,000) 2. **Million**: \(10^6\) (1,000,000) 3.

A googolplex is a very large number defined as \(10^{\text{googol}}\), where a googol is equal to \(10^{100}\). In other words, a googolplex is \(10^{10^{100}}\).

An ideal number is a concept that appears in various mathematical contexts, but it is perhaps most commonly associated with the field of algebraic number theory, where it is linked to the notion of ideals in ring theory. In ring theory, an *ideal* is a special subset of a ring that has certain properties, making it a useful structure for generalizing concepts such as divisibility. An ideal allows for the definition of quotient rings, which are fundamental in many areas of mathematics.

Argentina primarily uses the metric system as its standard system of measurement.

Indian logicians refer to scholars and philosophers from India who have contributed to the field of logic, particularly in the ancient and medieval periods. Indian logic has a rich tradition that is distinct from Western logic and has developed through a variety of philosophical schools, particularly within the broader context of Indian philosophy.

Navya-Nyāya, often referred to simply as Nyāya, is a school of Indian philosophy that emerged in the later part of the Indian philosophical tradition, around the 14th to 16th centuries. It builds upon and refines the earlier Nyāya system, which is primarily known for its focus on logic, epistemology (the study of knowledge), and the process of reasoning. **Key Features of Navya-Nyāya:** 1.

A quadratic form is a specific type of polynomial expression that involves variables raised to the second power, usually in the context of multiple variables.

Fringe physics refers to theories, ideas, and research that exist outside of mainstream scientific consensus and often lack empirical support or rigorous validation. This domain includes speculative concepts that may challenge established scientific principles or explore phenomena that are not fully understood by current scientific frameworks. Examples of fringe physics include theories related to free energy devices, perpetual motion machines, and various forms of alternative physics that propose new interpretations of fundamental concepts like gravity, time, and space.

The Antarctic Benthic Deep-Sea Biodiversity Project (ABDDBP) aims to gather comprehensive data on the biodiversity, distribution, and ecological functions of benthic (seafloor) organisms in the deep-sea regions of Antarctica. The project is part of larger efforts to understand marine ecosystems, particularly in extreme environments like the Southern Ocean.