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Folkman's Theorem
Folkman's theorem is a theorem in mathematics, and more particularly in arithmetic combinatorics and Ramsey theory. According to this theorem, whenever the natural numbers are partitioned into finitely many subsets, there exist arbitrarily large sets of numbers all of whose sums belong to the same subset of the partition.. The theorem had been discovered and proved independently by several mathematicians,.. before it was named "Folkman's theorem", as a memorial to Jon Folkman, by Graham, Rothschild, and Spencer. Statement of the theorem Let N be the set of positive integers, and suppose that N is partitioned into ''k'' different subsets ''N''1, ''N''2, ... ''N''''k'', where ''k'' is any positive integer. Then Folkman's theorem states that, for every positive integer ''m'', there exists a set ''S''''m'' and an index ''i''''m'' such that ''S''''m'' has ''m'' elements and such that every sum of a nonempty subset of ''S''''m'' belongs to ''N''''i''''m''. Relation to Rado's the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theorem
In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of the axioms and previously proved theorems. In the mainstream of mathematics, the axioms and the inference rules are commonly left implicit, and, in this case, they are almost always those of Zermelo–Fraenkel set theory with the axiom of choice, or of a less powerful theory, such as Peano arithmetic. A notable exception is Wiles's proof of Fermat's Last Theorem, which involves the Grothendieck universes whose existence requires the addition of a new axiom to the set theory. Generally, an assertion that is explicitly called a theorem is a proved result that is not an immediate consequence of other known theorems. Moreover, many authors qualify as ''theorems'' only the most important results, and use the terms ''lemma'', ''proposition'' and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
System Of Linear Equations
In mathematics, a system of linear equations (or linear system) is a collection of one or more linear equations involving the same variable (math), variables. For example, :\begin 3x+2y-z=1\\ 2x-2y+4z=-2\\ -x+\fracy-z=0 \end is a system of three equations in the three variables . A solution to a linear system is an assignment of values to the variables such that all the equations are simultaneously satisfied. A Equation solving, solution to the system above is given by the Tuple, ordered triple :(x,y,z)=(1,-2,-2), since it makes all three equations valid. The word "system" indicates that the equations are to be considered collectively, rather than individually. In mathematics, the theory of linear systems is the basis and a fundamental part of linear algebra, a subject which is used in most parts of modern mathematics. Computational algorithms for finding the solutions are an important part of numerical linear algebra, and play a prominent role in engineering, physics, chemistry, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Additive Combinatorics
Additive combinatorics is an area of combinatorics in mathematics. One major area of study in additive combinatorics are ''inverse problems'': given the size of the sumset ''A'' + ''B'' is small, what can we say about the structures of A and B? In the case of the integers, the classical Freiman's theorem provides a partial answer to this question in terms of multi-dimensional arithmetic progressions. Another typical problem is to find a lower bound for , A + B, in terms of , A, and , B, . This can be viewed as an inverse problem with the given information that , A+B, is sufficiently small and the structural conclusion is then of the form that either A or B is the empty set; however, in literature, such problems are sometimes considered to be direct problems as well. Examples of this type include the Erdős–Heilbronn Conjecture (for a restricted sumset) and the Cauchy–Davenport Theorem. The methods used for tackling such questions often come from many different fi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theorems In Discrete Mathematics
In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of the axioms and previously proved theorems. In the mainstream of mathematics, the axioms and the inference rules are commonly left implicit, and, in this case, they are almost always those of Zermelo–Fraenkel set theory with the axiom of choice, or of a less powerful theory, such as Peano arithmetic. A notable exception is Wiles's proof of Fermat's Last Theorem, which involves the Grothendieck universes whose existence requires the addition of a new axiom to the set theory. Generally, an assertion that is explicitly called a theorem is a proved result that is not an immediate consequence of other known theorems. Moreover, many authors qualify as ''theorems'' only the most important results, and use the terms ''lemma'', ''proposition'' and '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ramsey theory
Ramsey theory, named after the British mathematician and philosopher Frank P. Ramsey, is a branch of the mathematical field of combinatorics that focuses on the appearance of order in a substructure given a structure of a known size. Problems in Ramsey theory typically ask a question of the form: "how big must some structure be to guarantee that a particular property holds?" Examples A typical result in Ramsey theory starts with some mathematical structure that is then cut into pieces. How big must the original structure be in order to ensure that at least one of the pieces has a given interesting property? This idea can be defined as partition regularity. For example, consider a complete graph of order ''n''; that is, there are ''n'' vertices and each vertex is connected to every other vertex by an edge. A complete graph of order 3 is called a triangle. Now colour each edge either red or blue. How large must ''n'' be in order to ensure that there is either a blue triangle or a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jon Folkman
Jon Hal Folkman (December 8, 1938 – January 23, 1969) was an American mathematician, a student of John Milnor, and a researcher at the RAND Corporation. Schooling Folkman was a Putnam Fellow in 1960. He received his Ph.D. in 1964 from Princeton University, under the supervision of Milnor, with a thesis entitled ''Equivariant Maps of Spheres into the Classical Groups''. Research Jon Folkman contributed important theorems in many areas of combinatorics. In geometric combinatorics, Folkman is known for his pioneering and posthumously-published studies of oriented matroids; in particular, the Folkman–Lawrence topological representation theorem is "one of the cornerstones of the theory of oriented matroids". In lattice theory, Folkman solved an open problem on the foundations of combinatorics by proving a conjecture of Gian–Carlo Rota; in proving Rota's conjecture, Folkman characterized the structure of the homology groups of "geometric lattices" in terms of the free ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Richard Rado
Richard Rado FRS (28 April 1906 – 23 December 1989) was a German-born British mathematician whose research concerned combinatorics and graph theory. He was Jewish and left Germany to escape Nazi persecution. He earned two PhDs: in 1933 from the University of Berlin, and in 1935 from the University of Cambridge. He was interviewed in Berlin by Lord Cherwell for a scholarship given by the chemist Sir Robert Mond which provided financial support to study at Cambridge. After he was awarded the scholarship, Rado and his wife left for the UK in 1933. He was appointed Professor of Mathematics at the University of Reading in 1954 and remained there until he retired in 1971. Contributions Rado made contributions in combinatorics and graph theory including 18 papers with Paul Erdős. In graph theory, the Rado graph, a countably infinite graph containing all countably infinite graphs as induced subgraphs, is named after Rado. He rediscovered it in 1964 after previous works on the sam ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Power Of Two
A power of two is a number of the form where is an integer, that is, the result of exponentiation with number two as the base and integer as the exponent. In a context where only integers are considered, is restricted to non-negative values, so there are 1, 2, and 2 multiplied by itself a certain number of times. The first ten powers of 2 for non-negative values of are: : 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, ... Because two is the base of the binary numeral system, powers of two are common in computer science. Written in binary, a power of two always has the form 100...000 or 0.00...001, just like a power of 10 in the decimal system. Computer science Two to the exponent of , written as , is the number of ways the bits in a binary word of length can be arranged. A word, interpreted as an unsigned integer, can represent values from 0 () to () inclusively. Corresponding signed integer values can be positive, negative and zero; see signed n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rado's Theorem (Ramsey Theory)
Rado's theorem is a theorem from the branch of mathematics known as Ramsey theory. It is named for the German mathematician Richard Rado. It was proved in his thesis, ''Studien zur Kombinatorik''. Statement Let A \mathbf = \mathbf be a system of linear equations, where A is a matrix with integer entries. This system is said to be r''-regular'' if, for every r-coloring of the natural numbers 1, 2, 3, ..., the system has a monochromatic solution. A system is ''regular'' if it is ''r-regular'' for all ''r'' ≥ 1. Rado's theorem states that a system A \mathbf = \mathbf is regular if and only if the matrix ''A'' satisfies the ''columns condition''. Let ''ci'' denote the ''i''-th column of ''A''. The matrix ''A'' satisfies the columns condition provided that there exists a partition ''C''1, ''C''2, ..., ''C''''n'' of the column indices such that if s_i = \Sigma_c_j, then # ''s''1 = 0 # for all ''i'' ≥ 2, ''si'' can be written as a ration ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arithmetic Combinatorics
In mathematics, arithmetic combinatorics is a field in the intersection of number theory, combinatorics, ergodic theory and harmonic analysis. Scope Arithmetic combinatorics is about combinatorial estimates associated with arithmetic operations (addition, subtraction, multiplication, and division). Additive combinatorics is the special case when only the operations of addition and subtraction are involved. Ben Green explains arithmetic combinatorics in his review of "Additive Combinatorics" by Tao and Vu. Important results Szemerédi's theorem Szemerédi's theorem is a result in arithmetic combinatorics concerning arithmetic progressions in subsets of the integers. In 1936, Erdős and Turán conjectured. that every set of integers ''A'' with positive natural density contains a ''k'' term arithmetic progression for every ''k''. This conjecture, which became Szemerédi's theorem, generalizes the statement of van der Waerden's theorem. Green–Tao theorem and extensions The Gre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schur's Theorem
In discrete mathematics, Schur's theorem is any of several theorems of the mathematician Issai Schur. In differential geometry, Schur's theorem is a theorem of Axel Schur. In functional analysis, Schur's theorem is often called Schur's property, also due to Issai Schur. Ramsey theory In Ramsey theory, Schur's theorem states that for any partition of the positive integers into a finite number of parts, one of the parts contains three integers ''x'', ''y'', ''z'' with :x + y = z. For every positive integer ''c'', ''S''(''c'') denotes the smallest number ''S'' such that for every partition of the integers \ into ''c'' parts, one of the parts contains integers ''x'', ''y'', and ''z'' with x + y = z. Schur's theorem ensures that ''S''(''c'') is well-defined for every positive integer ''c''. The numbers of the form ''S''(''c'') are called Schur's number. Folkman's theorem generalizes Schur's theorem by stating that there exist arbitrarily large sets of integers, all of whose ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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