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Hill–Beck Land Division Problem
The following variant of the fair cake-cutting problem was introduced by Ted Hill in 1983. There is a territory ''D'' adjacent to ''n'' countries. Each country values the different subsets of ''D'' differently. The countries would like to divide ''D'' fairly among them, where "fair" means a proportional division. Additionally, ''the share allocated to each country must be connected and adjacent to that country''. This geographic constraint distinguishes this problem from classic fair cake-cutting. Formally, every country ''Ci'' must receive a disjoint piece of ''D'', marked ''Di'', such that a portion of the border between ''Ci'' and ''D'' is contained in the interior of ''Ci ∪ Di''. Impossibility and possibility There are cases in which the problem cannot be solved: # If there is a single point to which two countries assign all their value (e.g. a holy place), then obviously the territory cannot be divided proportionally. To prevent such situations, we assume that all count ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fair Cake-cutting
Fair cake-cutting is a kind of fair division problem. The problem involves a ''heterogeneous'' resource, such as a cake with different toppings, that is assumed to be ''divisible'' – it is possible to cut arbitrarily small pieces of it without destroying their value. The resource has to be divided among several partners who have different preferences over different parts of the cake, i.e., some people prefer the chocolate toppings, some prefer the cherries, some just want as large a piece as possible. The division should be ''unanimously'' fair - each person should receive a piece that he or she believes to be a fair share. The "cake" is only a metaphor; procedures for fair cake-cutting can be used to divide various kinds of resources, such as land estates, advertisement space or broadcast time. The prototypical procedure for fair cake-cutting is divide and choose, which is mentioned already in the book of Genesis. It solves the fair division problem for two people. The modern ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ted Hill (mathematician)
Theodore Preston Hill (born December 28, 1943) is an American mathematician specializing in probability theory. He is a professor emeritus at the Georgia Institute of Technology and a researcher at California Polytechnic State University, San Luis Obispo. Contributions Hill is known for his research on mathematical probability theory, in particular for his work on Benford's law, and for his work in the theories of optimal stopping ( secretary problems) and fair division, in particular the Hill-Beck land division problem. Hill has attracted attention for a paper on the theory that men exhibit greater variability than women in genetically controlled traits that he wrote with Sergei Tabachnikov. It was accepted but not published by ''The Mathematical Intelligencer''; a later version authored by Hill alone was peer reviewed and accepted by '' The New York Journal of Mathematics'' (''NYJM'') and retracted after publication. A revised version, again authored by Hill alone, was subs ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proportional Division
A proportional division is a kind of fair division in which a resource is divided among ''n'' partners with subjective valuations, giving each partner at least 1/''n'' of the resource by his/her own subjective valuation. Proportionality was the first fairness criterion studied in the literature; hence it is sometimes called "simple fair division". It was first conceived by Steinhaus. Example Consider a land asset that has to be divided among 3 heirs: Alice and Bob who think that it's worth 3 million dollars, and George who thinks that it's worth $4.5M. In a proportional division, Alice receives a land-plot that she believes to be worth at least $1M, Bob receives a land-plot that ''he'' believes to be worth at least $1M (even though Alice may think it is worth less), and George receives a land-plot that he believes to be worth at least $1.5M. Existence A proportional division does not always exist. For example, if the resource contains several indivisible items and the number ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Anatole Beck
Anatole Beck (19 March 1930 – 21 December 2014) was an American mathematician. Beck graduated from Stuyvesant High School in 1947, studied at Brooklyn College (Bachelor's degree 1951) and in 1956 received his PhD from Yale University under Shizuo Kakutani PhD (On the Random Ergodic theorem). In 1958 he became Assistant Professor and in 1966 Professor in thDepartment of Mathematicsat the University of Wisconsin–Madison. He married Jewish feminist writer Evelyn Torton Beck in 1954; they had two children before divorcing in 1974. He met his second wife Eve-Lynn Siegel Beck in 1998, she was cousins with Michael Bleicher, one of Anatole’s longtime friends. He was a visiting professor at the Technical University of Munich, the London School of Economics and a visiting scholar at the University of Göttingen, University of Warwick, University of London, and the Hebrew University. Beck's work dealt with ergodic theory, topological dynamics, Probability in Banach spaces, measure t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Last Diminisher
A last is a mechanical form shaped like a human foot. It is used by shoemakers and cordwainers in the manufacture and repair of shoes. Lasts typically come in pairs and have been made from various materials, including hardwoods, cast iron, and high-density plastics. The term is derived from the Proto-Germanic *''laistaz'' ("track, trace, footprint"); cognates include Swedish ''läst'', Danish ''læste'', German ''Leisten''. Production Lasts come in many styles and sizes, depending on the exact job they are designed for. Common variations include simple one-size lasts used for repairing soles and heels, durable lasts used in modern mass production, and custom-made lasts used in the making of bespoke footwear. Though a last is made approximately in the shape of a human foot, the precise shape is tailored to the kind of footwear being made. For example, a boot last would be designed to hug the instep for a close fit. Modern last shapes are typically designed using dedicated c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simply-connected
In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the space) into any other such path while preserving the two endpoints in question. The fundamental group of a topological space is an indicator of the failure for the space to be simply connected: a path-connected topological space is simply connected if and only if its fundamental group is trivial. Definition and equivalent formulations A topological space X is called if it is path-connected and any loop in X defined by f : S^1 \to X can be contracted to a point: there exists a continuous map F : D^2 \to X such that F restricted to S^1 is f. Here, S^1 and D^2 denotes the unit circle and closed unit disk in the Euclidean plane respectively. An equivalent formulation is this: X is simply connected if and only if it is path-connected, and wheneve ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemann Mapping
In complex analysis, the Riemann mapping theorem states that if ''U'' is a non-empty simply connected open subset of the complex number plane C which is not all of C, then there exists a biholomorphic mapping ''f'' (i.e. a bijective holomorphic mapping whose inverse is also holomorphic) from ''U'' onto the open unit disk :D = \. This mapping is known as a Riemann mapping. Intuitively, the condition that ''U'' be simply connected means that ''U'' does not contain any “holes”. The fact that ''f'' is biholomorphic implies that it is a conformal map and therefore angle-preserving. Such a map may be interpreted as preserving the shape of any sufficiently small figure, while possibly rotating and scaling (but not reflecting) it. Henri Poincaré proved that the map ''f'' is essentially unique: if ''z''0 is an element of ''U'' and φ is an arbitrary angle, then there exists precisely one ''f'' as above such that ''f''(''z''0) = 0 and such that the argument of the derivative of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unit Disc
In mathematics, the open unit disk (or disc) around ''P'' (where ''P'' is a given point in the plane), is the set of points whose distance from ''P'' is less than 1: :D_1(P) = \.\, The closed unit disk around ''P'' is the set of points whose distance from ''P'' is less than or equal to one: :\bar D_1(P)=\.\, Unit disks are special cases of disks and unit balls; as such, they contain the interior of the unit circle and, in the case of the closed unit disk, the unit circle itself. Without further specifications, the term ''unit disk'' is used for the open unit disk about the origin, D_1(0), with respect to the standard Euclidean metric. It is the interior of a circle of radius 1, centered at the origin. This set can be identified with the set of all complex numbers of absolute value less than one. When viewed as a subset of the complex plane (C), the unit disk is often denoted \mathbb. The open unit disk, the plane, and the upper half-plane The function :f(z)=\frac is an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
N-connected
In algebraic topology, homotopical connectivity is a property describing a topological space based on the dimension of its holes. In general, low homotopical connectivity indicates that the space has at least one low-dimensional hole. The concept of ''n''-connectedness generalizes the concepts of path-connectedness and simple connectedness. An equivalent definition of homotopical connectivity is based on the homotopy groups of the space. A space is ''n''-connected (or ''n''-simple connected) if its first ''n'' homotopy groups are trivial. Homotopical connectivity is defined for maps, too. A map is ''n''-connected if it is an isomorphism "up to dimension ''n,'' in homotopy". Definition using holes All definitions below consider a topological space ''X''. A hole in ''X'' is, informally, a thing that prevents some suitably-placed sphere from continuously shrinking to a point., Section 4.3 Equivalently, it is a sphere that cannot be continuously extended to a ball. Formally, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fair Cake-cutting
Fair cake-cutting is a kind of fair division problem. The problem involves a ''heterogeneous'' resource, such as a cake with different toppings, that is assumed to be ''divisible'' – it is possible to cut arbitrarily small pieces of it without destroying their value. The resource has to be divided among several partners who have different preferences over different parts of the cake, i.e., some people prefer the chocolate toppings, some prefer the cherries, some just want as large a piece as possible. The division should be ''unanimously'' fair - each person should receive a piece that he or she believes to be a fair share. The "cake" is only a metaphor; procedures for fair cake-cutting can be used to divide various kinds of resources, such as land estates, advertisement space or broadcast time. The prototypical procedure for fair cake-cutting is divide and choose, which is mentioned already in the book of Genesis. It solves the fair division problem for two people. The modern ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Map Segmentation
In mathematics, the map segmentation problem is a kind of optimization problem. It involves a certain geographic region that has to be partitioned into smaller sub-regions in order to achieve a certain goal. Typical optimization objectives include: * Minimizing the workload of a fleet of vehicles assigned to the sub-regions; * Balancing the consumption of a resource, as in fair cake-cutting. * Determining the optimal locations of supply depots; * Maximizing the surveillance coverage. Fair division of land has been an important issue since ancient times, e.g. in ancient Greece. Notation There is a geographic region denoted by C ("cake"). A partition of C, denoted by X, is a list of disjoint subregions whose union is C: :C = X_1\sqcup\cdots\sqcup X_n There is a certain set of additional parameters (such as: obstacles, fixed points or probability density functions), denoted by P. There is a real-valued function denoted by G ("goal") on the set of all partitions. The map segmentat ... [...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, an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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