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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] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Optimization Problem
In mathematics, engineering, computer science and economics Economics () is a behavioral science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and services. Economics focuses on the behaviour and interac ..., an optimization problem is the problem of finding the ''best'' solution from all feasible solutions. Optimization problems can be divided into two categories, depending on whether the variables are continuous or discrete: * An optimization problem with discrete variables is known as a '' discrete optimization'', in which an object such as an integer, permutation or graph must be found from a countable set. * A problem with continuous variables is known as a '' continuous optimization'', in which an optimal value from a continuous function must be found. They can include constrained problems and multimodal problems. Search space In the context of an optim ... [...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 believed 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 in the book of Book of Genesis, Genesis to resolve Abraham and Lot's conflict. This procedure solves the fa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ancient Greece
Ancient Greece () was a northeastern Mediterranean civilization, existing from the Greek Dark Ages of the 12th–9th centuries BC to the end of classical antiquity (), that comprised a loose collection of culturally and linguistically related city-states and communities. Prior to the Roman period, most of these regions were officially unified only once under the Kingdom of Macedon from 338 to 323 BC. In Western history, the era of classical antiquity was immediately followed by the Early Middle Ages and the Byzantine period. Three centuries after the decline of Mycenaean Greece during the Bronze Age collapse, Greek urban poleis began to form in the 8th century BC, ushering in the Archaic period and the colonization of the Mediterranean Basin. This was followed by the age of Classical Greece, from the Greco-Persian Wars to the death of Alexander the Great in 323 BC, and which included the Golden Age of Athens and the Peloponnesian War. The u ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convex Set
In geometry, a set of points is convex if it contains every line segment between two points in the set. For example, a solid cube (geometry), cube is a convex set, but anything that is hollow or has an indent, for example, a crescent shape, is not convex. The boundary (topology), boundary of a convex set in the plane is always a convex curve. The intersection of all the convex sets that contain a given subset of Euclidean space is called the convex hull of . It is the smallest convex set containing . A convex function is a real-valued function defined on an interval (mathematics), interval with the property that its epigraph (mathematics), epigraph (the set of points on or above the graph of a function, graph of the function) is a convex set. Convex minimization is a subfield of mathematical optimization, optimization that studies the problem of minimizing convex functions over convex sets. The branch of mathematics devoted to the study of properties of convex sets and convex f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Connected Set
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint non-empty open subsets. Connectedness is one of the principal topological properties that distinguish topological spaces. A subset of a topological space X is a if it is a connected space when viewed as a subspace of X. Some related but stronger conditions are path connected, simply connected, and n-connected. Another related notion is locally connected, which neither implies nor follows from connectedness. Formal definition A topological space X is said to be if it is the union of two disjoint non-empty open sets. Otherwise, X is said to be connected. A subset of a topological space is said to be connected if it is connected under its subspace topology. Some authors exclude the empty set (with its unique topology) as a connected space, but this article does not follow that practice. For a topological space X the fol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Measurable Set
In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as magnitude, mass, and probability of events. These seemingly distinct concepts have many similarities and can often be treated together in a single mathematical context. Measures are foundational in probability theory, integration theory, and can be generalized to assume negative values, as with electrical charge. Far-reaching generalizations (such as spectral measures and projection-valued measures) of measure are widely used in quantum physics and physics in general. The intuition behind this concept dates back to Ancient Greece, when Archimedes tried to calculate the area of a circle. But it was not until the late 19th and early 20th centuries that measure theory became a branch of mathematics. The foundations of modern measure theory were laid in the works of Émile Borel, Henri Lebesgue, Nikolai Luzin, Johann Rad ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Voronoi Diagram
In mathematics, a Voronoi diagram is a partition of a plane into regions close to each of a given set of objects. It can be classified also as a tessellation. In the simplest case, these objects are just finitely many points in the plane (called seeds, sites, or generators). For each seed there is a corresponding region, called a Voronoi cell, consisting of all points of the plane closer to that seed than to any other. The Voronoi diagram of a set of points is dual to that set's Delaunay triangulation. The Voronoi diagram is named after mathematician Georgy Voronoy, and is also called a Voronoi tessellation, a Voronoi decomposition, a Voronoi partition, or a Dirichlet tessellation (after Peter Gustav Lejeune Dirichlet). Voronoi cells are also known as Thiessen polygons, after Alfred H. Thiessen. Voronoi diagrams have practical and theoretical applications in many fields, mainly in science and technology, but also in visual art. Simplest case In the simplest case, shown in the ... [...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 believed 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 in the book of Book of Genesis, Genesis to resolve Abraham and Lot's conflict. This procedure solves the fa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 cou ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stone–Tukey Theorem
In mathematical measure theory, for every positive integer the ham sandwich theorem states that given measurable "objects" in -dimensional Euclidean space, it is possible to divide each one of them in half (with respect to their measure, e.g. volume) with a single -dimensional hyperplane. This is possible even if the objects overlap. It was proposed by Hugo Steinhaus and proved by Stefan Banach (explicitly in dimension 3, without stating the theorem in the -dimensional case), and also years later called the Stone–Tukey theorem after Arthur H. Stone and John Tukey. Naming The ham sandwich theorem takes its name from the case when and the three objects to be bisected are the ingredients of a ham sandwich. Sources differ on whether these three ingredients are two slices of bread and a piece of ham , bread and cheese and ham , or bread and butter and ham . In two dimensions, the theorem is known as the pancake theorem to refer to the flat nature of the two objects to be bise ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fair Division
Fair division is the problem in game theory of dividing a set of resources among several people who have an Entitlement (fair division), entitlement to them so that each person receives their due share. The central tenet of fair division is that such a division should be performed by the players themselves, without the need for external arbitration, as only the players themselves really know how they value the goods. There are many different kinds of fair division problems, depending on the nature of goods to divide, the criteria for fairness, the nature of the players and their preferences, and other criteria for evaluating the quality of the division. The archetypal fair division algorithm is divide and choose. The research in fair division can be seen as an extension of this procedure to various more complex settings. Description In game theory, fair division is the problem of dividing a set of resources among several people who have an Entitlement (fair division), entitlem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
