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Levmore–Cook Moving-knives Procedure
The Levmore–Cook moving-knives procedure is a procedure for envy-free cake-cutting among three partners. It is named after Saul X. Levmore and Elizabeth Early Cook who presented it in 1981. It assumes that the cake is two-dimensional. It requires a referee, two knives and four cuts, so some partners may receive disconnected pieces. Procedure We name the partners Alice, Bob and Carl. Initially, Alice cuts the cake to three pieces equal in her eyes. Bob and Carl each point to their favorite piece. Easy case: Bob and Carl point to different pieces. Each receives his favorite piece and Alice the remaining piece. Hard case: Bob and Carl point to the same piece. Say this is piece X and the other pieces are Y and Z. Now the referee (ideally it could be Alice) takes one knife and moves it horizontally over piece X, and Alice takes the other knife and moves it vertically over piece X: * Knife #1 is moved ''horizontally'' and ''continuously'' by the referee from the left of piece X to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Envy-free Cake-cutting
An envy-free cake-cutting is a kind of fair cake-cutting. It is a division of a heterogeneous resource ("cake") that satisfies the envy-free criterion, namely, that every partner feels that their allocated share is at least as good as any other share, according to their own subjective valuation. When there are only two partners, the problem is easy and was solved in antiquity by the divide and choose protocol. When there are three or more partners, the problem becomes much more challenging. Two major variants of the problem have been studied: * Connected pieces, e.g. if the cake is a 1-dimensional interval then each partner must receive a single sub-interval. If there are n partners, only n-1 cuts are needed. * General pieces, e.g. if the cake is a 1-dimensional interval then each partner can receive a union of disjoint sub-intervals. Short history Modern research into the fair cake-cutting problem started in the 1940s. The first fairness criterion studied was proportional divi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Saul Levmore
Saul Levmore (born 1953) is the William B. Graham Distinguished Service Professor of Law, and former Dean of the University of Chicago Law School. Career Levmore was the Brokaw Professor at the University of Virginia, and he has also been a visiting professor at Yale, Harvard, Toronto, Michigan, and Northwestern Universities. He earned a B.A. from Columbia University in 1973, where William Vickrey supervised his thesis; a PhD in Economics from Yale University in 1978; and a J.D. from Yale Law School in 1980. Levmore is an elected member of the American Academy of Arts and Sciences, and a former president of the American Law Deans Association. During his time at the University of Virginia, Levmore was a signatory to a letter of fifty-one individuals supporting Anita Hill during the confirmation hearing of Clarence Thomas. The letter was addressed to then-Senate Judiciary Committee Chairman Joe Biden and expressed the signatories' "complete confidence in ill'ssincerity and go ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
University Of Chicago
The University of Chicago (UChicago, Chicago, U of C, or UChi) is a private university, private research university in Chicago, Illinois. Its main campus is located in Chicago's Hyde Park, Chicago, Hyde Park neighborhood. The University of Chicago is consistently ranked among the best universities in the world and it is among the most selective in the United States. The university is composed of College of the University of Chicago, an undergraduate college and five graduate research divisions, which contain all of the university's graduate programs and interdisciplinary committees. Chicago has eight professional schools: the University of Chicago Law School, Law School, the Booth School of Business, the Pritzker School of Medicine, the Crown Family School of Social Work, Policy, and Practice, the Harris School of Public Policy, the University of Chicago Divinity School, Divinity School, the Graham School of Continuing Liberal and Professional Studies, and the Pritzker School of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cake
Cake is a flour confection made from flour, sugar, and other ingredients, and is usually baked. In their oldest forms, cakes were modifications of bread, but cakes now cover a wide range of preparations that can be simple or elaborate, and which share features with desserts such as pastries, meringues, custards, and pies. The most common ingredients include flour, sugar, eggs, fat (such as butter, oil or margarine), a liquid, and a leavening agent, such as baking soda or baking powder. Common additional ingredients include dried, candied, or fresh fruit, nuts, cocoa, and extracts such as vanilla, with numerous substitutions for the primary ingredients. Cakes can also be filled with fruit preserves, nuts or dessert sauces (like custard, jelly, cooked fruit, whipped cream or syrups), iced with buttercream or other icings, and decorated with marzipan, piped borders, or candied fruit. Cake is often served as a celebratory dish on ceremonial occasions, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Two-dimensional Space
In mathematics, a plane is a Euclidean ( flat), two-dimensional surface that extends indefinitely. A plane is the two-dimensional analogue of a point (zero dimensions), a line (one dimension) and three-dimensional space. Planes can arise as subspaces of some higher-dimensional space, as with one of a room's walls, infinitely extended, or they may enjoy an independent existence in their own right, as in the setting of two-dimensional Euclidean geometry. Sometimes the word ''plane'' is used more generally to describe a two-dimensional surface, for example the hyperbolic plane and elliptic plane. When working exclusively in two-dimensional Euclidean space, the definite article is used, so ''the'' plane refers to the whole space. Many fundamental tasks in mathematics, geometry, trigonometry, graph theory, and graphing are performed in a two-dimensional space, often in the plane. Euclidean geometry Euclid set forth the first great landmark of mathematical thought, an axioma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Knife
A knife ( : knives; from Old Norse 'knife, dirk') is a tool or weapon with a cutting edge or blade, usually attached to a handle or hilt. One of the earliest tools used by humanity, knives appeared at least 2.5 million years ago, as evidenced by the Oldowan tools. Originally made of wood, bone, and stone (such as flint and obsidian), over the centuries, in step with improvements in both metallurgy and manufacturing, knife blades have been made from copper, bronze, iron, steel, ceramic, and titanium. Most modern knives have either fixed or folding blades; blade patterns and styles vary by maker and country of origin. Knives can serve various purposes. Hunters use a hunting knife, soldiers use the combat knife, scouts, campers, and hikers carry a pocket knife; there are kitchen knives for preparing foods (the chef's knife, the paring knife, bread knife, cleaver), table knives (butter knives and steak knives), weapons ( daggers or switchblades), knives for throwing or juggl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equality (mathematics)
In mathematics, equality is a relationship between two quantities or, more generally two mathematical expressions, asserting that the quantities have the same value, or that the expressions represent the same mathematical object. The equality between and is written , and pronounced equals . The symbol "" is called an "equals sign". Two objects that are not equal are said to be distinct. For example: * x=y means that and denote the same object. * The identity (x+1)^2=x^2+2x+1 means that if is any number, then the two expressions have the same value. This may also be interpreted as saying that the two sides of the equals sign represent the same function. * \ = \ if and only if P(x) \Leftrightarrow Q(x). This assertion, which uses set-builder notation, means that if the elements satisfying the property P(x) are the same as the elements satisfying Q(x), then the two uses of the set-builder notation define the same set. This property is often expressed as "two sets that have t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Analogy
Analogy (from Greek ''analogia'', "proportion", from ''ana-'' "upon, according to" lso "against", "anew"+ ''logos'' "ratio" lso "word, speech, reckoning" is a cognitive process of transferring information or meaning from a particular subject (the analog, or source) to another (the target), or a linguistic expression corresponding to such a process. In a narrower sense, analogy is an inference or an argument from one particular to another particular, as opposed to deduction, induction, and abduction, in which at least one of the premises, or the conclusion, is general rather than particular in nature. The term analogy can also refer to the relation between the source and the target themselves, which is often (though not always) a similarity, as in the biological notion of analogy. Analogy plays a significant role in problem solving, as well as decision making, argumentation, perception, generalization, memory, creativity, invention, prediction, emotion, explanation, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generalization
A generalization is a form of abstraction whereby common properties of specific instances are formulated as general concepts or claims. Generalizations posit the existence of a domain or set of elements, as well as one or more common characteristics shared by those elements (thus creating a conceptual model). As such, they are the essential basis of all valid deductive inferences (particularly in logic, mathematics and science), where the process of verification is necessary to determine whether a generalization holds true for any given situation. Generalization can also be used to refer to the process of identifying the parts of a whole, as belonging to the whole. The parts, which might be unrelated when left on their own, may be brought together as a group, hence belonging to the whole by establishing a common relation between them. However, the parts cannot be generalized into a whole—until a common relation is established among ''all'' parts. This does not mean that th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stromquist Moving-knives Procedure
The Stromquist moving-knives procedure is a procedure for envy-free cake-cutting among three players. It is named after Walter Stromquist who presented it in 1980. This procedure was the first envy-free moving knife procedure devised for three players. It requires four knives but only two cuts, so each player receives a single connected piece. There is no natural generalization to more than three players which divides the cake without extra cuts. The resulting partition is not necessarily efficient. Procedure A referee moves a sword from left to right over the cake, hypothetically dividing it into small left piece and a large right piece. Each player moves a knife over the right piece, always keeping it parallel to the sword. The players must move their knives in a continuous manner, without making any "jumps".The importance of this continuity is explained here: When any player shouts "cut", the cake is cut by the sword and by whichever of the players' knives happens to be th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fair Division Protocols
A fair (archaic: faire or fayre) is a gathering of people for a variety of entertainment or commercial activities. Fairs are typically temporary with scheduled times lasting from an afternoon to several weeks. Types Variations of fairs include: * Art fairs, including art exhibitions and arts festivals * County fair (USA) or county show (UK), a public agricultural show exhibiting the equipment, animals, sports and recreation associated with agriculture and animal husbandry. * Festival, an event ordinarily coordinated with a theme e.g. music, art, season, tradition, history, ethnicity, religion, or a national holiday. * Health fair, an event designed for outreach to provide basic preventive medicine and medical screening * Historical reenactments, including Renaissance fairs and Dickens fairs * Horse fair, an event where people buy and sell horses. * Job fair, event in which employers, recruiters, and schools give information to potential employees. * Regional or state fair ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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