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Ruzsa–Szemerédi Problem
In combinatorial mathematics and extremal graph theory, the Ruzsa–Szemerédi problem or (6,3)-problem asks for the maximum number of edges in a graph in which every edge belongs to a unique triangle. Equivalently it asks for the maximum number of edges in a balanced bipartite graph whose edges can be partitioned into a linear number of induced matchings, or the maximum number of triples one can choose from n points so that every six points contain at most two triples. The problem is named after Imre Z. Ruzsa and Endre Szemerédi, who first proved that its answer is smaller than n^2 by a slowly-growing (but still unknown) factor. Equivalence between formulations The following questions all have answers that are asymptotically equivalent: they differ by, at most, constant factors from each other. *What is the maximum possible number of edges in a graph with n vertices in which every edge belongs to a unique triangle? The graphs with this property are called locally linear graph ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jacob Fox
Jacob Fox (born Jacob Licht in 1984) is an American mathematician. He is a professor at Stanford University. His research interests are in Hungarian-style combinatorics, particularly Ramsey theory, extremal graph theory, combinatorial number theory, and probabilistic methods in combinatorics. Fox grew up in West Hartford, Connecticut and attended Hall High School. As a senior he won second place overall and first place in his category in the annual Intel Science Talent Search, also winning the Karl Menger Memorial Prize of the American Mathematical Society for his project. The project was titled "Rainbow Ramsey Theory: Rainbow Arithmetic Progressions and Anti-Ramsey Results" and was based on a research project he did at a six-week summer camp in mathematics at the Massachusetts Institute of Technology (MIT); he also participated in an earlier high school mathematics program at Ohio State University. Fox became an undergraduate at MIT, and was awarded the 2006 Morgan Prize fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tripod Packing
In combinatorics, tripod packing is a problem of finding many disjoint tripods in a three-dimensional grid, where a tripod is an infinite polycube, the union of the grid cubes along three positive axis-aligned rays with a shared apex. Several problems of tiling and packing tripods and related shapes were formulated in 1967 by Sherman K. Stein. Stein originally called the tripods of this problem "semicrosses", and they were also called Stein corners by Solomon W. Golomb. A collection of disjoint tripods can be represented compactly as a monotonic matrix, a square matrix whose nonzero entries increase along each row and column and whose equal nonzero entries are placed in a monotonic sequence of cells, and the problem can also be formulated in terms of finding sets of triples satisfying a compatibility condition called "2-comparability", or of finding compatible sets of triangles in a convex polygon. The best lower bound known for the number of tripods that can have their apexes p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cap Set
In affine geometry, a cap set is a subset of \mathbb_3^n (an n-dimensional affine space over a three-element field) with no three elements in a line. The cap set problem is the problem of finding the size of the largest possible cap set, as a function of n.. The first few cap set sizes are 1, 2, 4, 9, 20, 45, 112, ... . Cap sets may be defined more generally as subsets of finite affine or projective spaces with no three in line, where these objects are simply called caps. The "cap set" terminology should be distinguished from other unrelated mathematical objects with the same name, and in particular from sets with the compact absorption property in function spaces as well as from compact convex co-convex subsets of a convex set. Example An example of cap sets comes from the card game Set, a card game in which each card has four features (its number, symbol, shading, and color), each of which can take one of three values. The cards of this game can be interpreted as representing p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Approximation Ratio
An approximation is anything that is intentionally similar but not exactly equal to something else. Etymology and usage The word ''approximation'' is derived from Latin ''approximatus'', from ''proximus'' meaning ''very near'' and the prefix ''ad-'' (''ad-'' before ''p'' becomes ap- by assimilation) meaning ''to''. Words like ''approximate'', ''approximately'' and ''approximation'' are used especially in technical or scientific contexts. In everyday English, words such as ''roughly'' or ''around'' are used with a similar meaning. It is often found abbreviated as ''approx.'' The term can be applied to various properties (e.g., value, quantity, image, description) that are nearly, but not exactly correct; similar, but not exactly the same (e.g., the approximate time was 10 o'clock). Although approximation is most often applied to numbers, it is also frequently applied to such things as mathematical functions, shapes, and physical laws. In science, approximation can refer to u ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Streaming Algorithm
In computer science, streaming algorithms are algorithms for processing data streams in which the input is presented as a sequence of items and can be examined in only a few passes (typically just one). In most models, these algorithms have access to limited memory (generally logarithmic in the size of and/or the maximum value in the stream). They may also have limited processing time per item. These constraints may mean that an algorithm produces an approximate answer based on a summary or "sketch" of the data stream. History Though streaming algorithms had already been studied by Munro and Paterson as early as 1978, as well as Philippe Flajolet and G. Nigel Martin in 1982/83, the field of streaming algorithms was first formalized and popularized in a 1996 paper by Noga Alon, Yossi Matias, and Mario Szegedy. For this paper, the authors later won the Gödel Prize in 2005 "for their foundational contribution to streaming algorithms." There has since been a large body of work ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Property Testing
In computer science, a property testing algorithm for a decision problem is an algorithm whose query complexity to its input is much smaller than the instance size of the problem. Typically property testing algorithms are used to distinguish if some combinatorial structure ''S'' (such as a graph or a boolean function) satisfies some property ''P'', or is "far" from having this property (meaning an ε-fraction of the representation of ''S'' need be modified in order to make ''S'' satisfy ''P''), using only a small number of "local" queries to the object. For example, the following promise problem admits an algorithm whose query complexity is independent of the instance size (for an arbitrary constant ε > 0): :"Given a graph ''G'' on ''n'' vertices, decide if ''G'' is bipartite, or ''G'' cannot be made bipartite even after removing an arbitrary subset of at most \epsilon\tbinom n2 edges of ''G''." Property testing algorithms are central to the definition of probabilistically ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computational Complexity Theory
In theoretical computer science and mathematics, computational complexity theory focuses on classifying computational problems according to their resource usage, and relating these classes to each other. A computational problem is a task solved by a computer. A computation problem is solvable by mechanical application of mathematical steps, such as an algorithm. A problem is regarded as inherently difficult if its solution requires significant resources, whatever the algorithm used. The theory formalizes this intuition, by introducing mathematical models of computation to study these problems and quantifying their computational complexity, i.e., the amount of resources needed to solve them, such as time and storage. Other measures of complexity are also used, such as the amount of communication (used in communication complexity), the number of gates in a circuit (used in circuit complexity) and the number of processors (used in parallel computing). One of the roles of compu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tripod Packing
In combinatorics, tripod packing is a problem of finding many disjoint tripods in a three-dimensional grid, where a tripod is an infinite polycube, the union of the grid cubes along three positive axis-aligned rays with a shared apex. Several problems of tiling and packing tripods and related shapes were formulated in 1967 by Sherman K. Stein. Stein originally called the tripods of this problem "semicrosses", and they were also called Stein corners by Solomon W. Golomb. A collection of disjoint tripods can be represented compactly as a monotonic matrix, a square matrix whose nonzero entries increase along each row and column and whose equal nonzero entries are placed in a monotonic sequence of cells, and the problem can also be formulated in terms of finding sets of triples satisfying a compatibility condition called "2-comparability", or of finding compatible sets of triangles in a convex polygon. The best lower bound known for the number of tripods that can have their apexes p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vera T
Vera may refer to: Names *Vera (surname), a surname (including a list of people with the name) * Vera (given name), a given name (including a list of people and fictional characters with the name) **Vera (), archbishop of the archdiocese of Tarragona Places Spain * Vera, Almería, a municipality in the province of Almería, Andalusia * Vera de Bidasoa, a municipality in the autonomous community of Navarra *La Vera, a comarca in the province of Cáceres, Extremadura United States *Vera, Illinois, an unincorporated community * Vera, Kansas, a ghost town * Vera, Missouri, an unincorporated community * Vera, Oklahoma, a town * Vera, Texas, an unincorporated community * Vera, Virginia, an unincorporated community *Veradale, Washington, originally known as Vera, CDP Elsewhere * Vera, Santa Fe, a city in the province of Santa Fe, Argentina * Vera Department, an administrative subdivision (departamento) of the province of Santa Fe * Vera, Mato Grosso, Brazil, a municipality * Cape Ve ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Paul Erdős
Paul Erdős ( hu, Erdős Pál ; 26 March 1913 – 20 September 1996) was a Hungarian mathematician. He was one of the most prolific mathematicians and producers of mathematical conjectures of the 20th century. pursued and proposed problems in discrete mathematics, graph theory, number theory, mathematical analysis, approximation theory, set theory, and probability theory. Much of his work centered around discrete mathematics, cracking many previously unsolved problems in the field. He championed and contributed to Ramsey theory, which studies the conditions in which order necessarily appears. Overall, his work leaned towards solving previously open problems, rather than developing or exploring new areas of mathematics. Erdős published around 1,500 mathematical papers during his lifetime, a figure that remains unsurpassed. He firmly believed mathematics to be a social activity, living an itinerant lifestyle with the sole purpose of writing mathematical papers with other mat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
