A **Dedekind domain** is a specific type of ring that plays a significant role in number theory, algebraic geometry, and algebraic number theory. A Dedekind domain is defined as an integral domain that satisfies certain properties. Here are the key characteristics of a Dedekind domain: 1. **Noetherian**: The ring is Noetherian, meaning that every ideal is finitely generated.

The Fundamental Theorem of Ideal Theory in number fields is a crucial result in algebraic number theory that connects ideals in the ring of integers of a number field to the arithmetic and structure of these numbers. Here's an overview of the key concepts involved: 1. **Number Fields**: A number field \( K \) is a finite degree field extension of the rational numbers \( \mathbb{Q} \).

A Unique Factorization Domain (UFD) is a specific type of integral domain in abstract algebra that has properties relating to the factorization of its elements. Specifically, a UFD is defined as an integral domain in which every nonzero element that is not a unit can be factored into irreducible elements (often called prime elements) in a way that is unique up to order and unit factors.

Apportionment methods are mathematical techniques used to allocate resources, representation, or seats among various groups or entities based on specific criteria, typically in a fair and equitable manner. These methods are commonly applied in various fields, including political science, economics, and statistics. ### Some Common Apportionment Methods: 1. **Hamilton's Method (Largest Remainders Method)**: - This method involves calculating a standard divisor to determine the initial number of representatives.

The Barbanel-Brams moving-knives procedure is a method used in fair division, particularly in the context of dividing a continuous resource among multiple participants. This procedure is designed to ensure that each participant receives a fair share of the resource according to their subjective valuations. Here's a simplified overview of how it works: 1. **Participants and Resource**: Assume there are \( n \) participants and a continuous resource (like a cake or an interval on a line) that they want to divide among themselves.

The Envy-graph procedure is a method used in the field of fair division, particularly in the context of allocating goods or resources among individuals. It aims to ensure that each participant in a division process feels they have received a fair share, thus reducing feelings of envy regarding others’ allocations. Here’s a brief overview of how the Envy-graph procedure typically works: 1. **Initial Allocation**: The process starts with an initial allocation of resources to participants.

Envy minimization is a concept that arises primarily in the context of fair division and allocation problems, particularly in economics and game theory. It refers to an approach or criterion for distributing resources or goods among multiple agents (such as people or entities) in a way that reduces the feelings of envy among those agents regarding what they receive. When a division is said to minimize envy, it implies that no individual would prefer the allocation received by another individual over their own allocation.

Lester Dubins is a notable figure in the field of mathematics, particularly known for his work in probability theory, statistics, and related areas. He has contributed to various topics, including the theory of random processes, statistical inference, and combinatorial problems. Dubins is also known for the "Dubins' problem," which deals with the optimal strategies in certain stochastic models.

The term "lone divider" is often used in the context of fair division and mathematical game theory, particularly in the study of dividing goods, resources, or values among multiple parties in a manner that is equitable. The lone divider method is a specific strategy used to achieve fair division. ### Lone Divider Method 1. **Participants**: Typically involves multiple parties—usually one "divider" and one or more "choosers.

"Permanence" is a science fiction novel by the author Dante D'Anthony. It explores themes related to memory, identity, and the nature of existence in a speculative future. The story revolves around a society where certain individuals can manipulate or alter their memories, raising questions about the implications of such powers on personal relationships and societal structures. The narrative often delves into the ethical dilemmas associated with memory modification, such as the authenticity of experiences and the impact on one's sense of self.

Edith Elkind is a prominent computer scientist known for her work in artificial intelligence, particularly in the areas of multi-agent systems, computational social choice, and algorithms. Her research often involves topics such as game theory, social choice theory, and the interaction of algorithms in social contexts. Elkind has contributed significantly to the understanding of how computational methods can be applied to problems in economics and social science.

Francis Su is a prominent mathematician known for his work in the fields of mathematical economics, applied mathematics, and education. He is a professor of mathematics at Harvey Mudd College, where he has been involved in various mathematical research and educational initiatives. Su is particularly recognized for his contributions to the study of fair division, game theory, and the mathematics of voting.

An **N-monoid** is a concept in the field of algebra, specifically in the study of algebraic structures known as monoids. A monoid is a set equipped with an associative binary operation and an identity element. 1. **Basic Definition of a Monoid**: - A set \( M \) along with a binary operation \( \cdot: M \times M \to M \) (often written simply as juxtaposition, i.e.

The Herzog–Schönheim conjecture is a conjecture in the field of algebraic geometry and commutative algebra. It concerns the properties of ideals in polynomial rings or local rings. Specifically, it relates to the asymptotic behavior of the growth of the lengths of certain graded components of ideals.

Tressy is likely a reference to a fashion doll character that was popular in the 1960s and 1970s. The Tressy doll was notable for its unique feature: it had a mechanism that allowed her hair to be lengthened or shortened, giving the illusion of having different hairstyles. This was achieved by a pull-string mechanism, allowing children to style Tressy's hair in various ways.

The "17-animal inheritance puzzle" is a classic genetic puzzle that involves determining the inheritance patterns of certain traits in a group of animals, often used as an educational tool in genetics or introductory biology courses. The puzzle typically outlines a scenario where a certain trait is passed down through generations of animals, and participants must use information about the traits of the parents and offspring to deduce which animals carry specific traits.

Envy-free item allocation is a concept in resource allocation and fair division, particularly in economics and game theory, where the goal is to distribute a set of items (or resources) among a group of individuals in such a way that no individual prefers the items allocated to someone else over their own. In other words, each participant feels satisfied with what they received and does not desire what others have, leading to a perception of fairness.

Envy-free matching is a concept often discussed in the context of fair allocation and matching theory, particularly in economics and game theory. It describes a situation where a set of agents or participants is matched to a set of items (or other agents) in such a way that no agent prefers another agent's allocation over their own. To break it down: 1. **Agents**: These are the participants who have preferences for various items or other participants.

Equitability generally refers to the quality of being fair and impartial. In various contexts, it emphasizes the importance of justice, fairness, and equal treatment, often within social, economic, and legal frameworks. The concept can be applied in different fields, including economics, education, and social justice.

Proportional division is a method of dividing something, such as resources, assets, or benefits, among different parties in a way that reflects their respective shares or entitlements. This concept is often applied in various contexts, including finance, law, business, and even social or political settings.