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Sum-free Set
In additive combinatorics and number theory, a subset ''A'' of an abelian group ''G'' is said to be sum-free if the sumset ''A'' + ''A'' is disjoint from ''A''. In other words, ''A'' is sum-free if the equation a + b = c has no solution with a,b,c \in A. For example, the set of odd numbers is a sum-free subset of the integers, and the set forms a large sum-free subset of the set . Fermat's Last Theorem is the statement that, for a given integer ''n'' > 2, the set of all nonzero ''n''th powers of the integers is a sum-free set. Some basic questions that have been asked about sum-free sets are: * How many sum-free subsets of are there, for an integer ''N''? Ben Green has shown that the answer is O(2^), as predicted by the Cameron–Erdős conjecture. * How many sum-free sets does an abelian group ''G'' contain?Ben Green and Imre RuzsaSum-free sets in abelian groups 2005. * What is the size of the largest sum-free set that an abelian group ''G'' contains? A sum-free set is s ... [...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 differen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cameron–Erdős Conjecture
In combinatorics, the Cameron–Erdős conjecture (now a theorem) is the statement that the number of sum-free sets contained in = \ is O\big(\big). The sum of two odd numbers is even, so a set of odd numbers is always sum-free. There are \lceil N/2\rceil odd numbers in 'N'' and so 2^ subsets of odd numbers in 'N'' The Cameron–Erdős conjecture says that this counts a constant proportion of the sum-free sets. The conjecture was stated by Peter Cameron and Paul Erdős in 1988. It was proved by Ben Green and independently by Alexander Sapozhenko. in 2003. See also * Erdős conjecture Erdős, Erdos, or Erdoes is a Hungarian surname. People with the surname include: * Ágnes Erdős (born 1950), Hungarian politician * Brad Erdos (born 1990), Canadian football player * Éva Erdős (born 1964), Hungarian handball player * Józ ... Notes Additive number theory Combinatorics Theorems in discrete mathematics Paul Erdős Conjectures that have been ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sum-free Sequence
In mathematics, a sum-free sequence is an increasing sequence of positive integers, :a_1, a_2, a_3, \ldots, such that no term a_n can be represented as a sum of any subset of the preceding elements of the sequence. This differs from a sum-free set, where only pairs of sums must be avoided, but where those sums may come from the whole set rather than just the preceding terms. Example The powers of two, :1, 2, 4, 8, 16, ... form a sum-free sequence: each term in the sequence is one more than the sum of all preceding terms, and so cannot be represented as a sum of preceding terms. Sums of reciprocals A set of integers is said to be small if the sum of its reciprocals converges to a finite value. For instance, by the prime number theorem, the prime numbers are not small. proved that every sum-free sequence is small, and asked how large the sum of reciprocals could be. For instance, the sum of the reciprocals of the powers of two (a geometric series) is two. If R denotes the ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Erdős–Szemerédi Theorem
In arithmetic combinatorics, the Erdős–Szemerédi theorem states that for every finite set A of integers, at least one of A+A, the set of pairwise sums or A\cdot A, the set of pairwise products form a significantly larger set. More precisely, the Erdős–Szemerédi theorem states that there exist positive constants ''c'' and \varepsilon such that for any non-empty set A \subset \mathbb :\max( , A+A, , , A \cdot A, ) \geq c , A, ^ . It was proved by Paul Erdős and Endre Szemerédi in 1983.. The notation , A, denotes the cardinality of the set A. The set of pairwise sums is A+A = \ and is called sum set of A. The set of pairwise products is A \cdot A = \ and is called the product set of A. The theorem is a version of the maxim that additive structure and multiplicative structure cannot coexist. It can also be viewed as an assertion that the real line does not contain any set resembling a finite subring or finite subfield; it is the first example of what is now known as t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conjecture
In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis (still a conjecture) or Fermat's Last Theorem (a conjecture until proven in 1995 by Andrew Wiles), have shaped much of mathematical history as new areas of mathematics are developed in order to prove them. Important examples Fermat's Last Theorem In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers a, ''b'', and ''c'' can satisfy the equation ''a^n + b^n = c^n'' for any integer value of ''n'' greater than two. This theorem was first conjectured by Pierre de Fermat in 1637 in the margin of a copy of '' Arithmetica'', where he claimed that he had a proof that was too large to fit in the margin. The first successful proof was released in 1994 by Andrew Wiles, and formally published in 1995, after 358 years of effort by ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Proof
A mathematical proof is an inferential argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems; but every proof can, in principle, be constructed using only certain basic or original assumptions known as axioms, along with the accepted rules of inference. Proofs are examples of exhaustive deductive reasoning which establish logical certainty, to be distinguished from empirical arguments or non-exhaustive inductive reasoning which establish "reasonable expectation". Presenting many cases in which the statement holds is not enough for a proof, which must demonstrate that the statement is true in ''all'' possible cases. A proposition that has not been proved but is believed to be true is known as a conjecture, or a hypothesis if frequently used as an assumption for further mathematical work. Proofs employ logic expressed in mathematical symbols ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Erdős
Erdős, Erdos, or Erdoes is a Hungarian surname. People with the surname include: * Ágnes Erdős (born 1950), Hungarian politician * Brad Erdos (born 1990), Canadian football player * Éva Erdős (born 1964), Hungarian handball player * József Erdős (born 1977), Hungarian entomologist * Mary Callahan Erdoes (born 1967), American banker * Paul Erdős (1913–1996), Hungarian mathematician * Richárd Erdős (1881–1912), Jewish Hungarian bass opera singer, father of Richard Erdoes * Richard Erdoes (1912–2008), Hungarian-Austrian born American artist * Sándor Erdős (born 1947), Hungarian fencer * Thomas Erdos (born 1965), Brazilian auto racing driver * Todd Erdos (born 1973), American middle-relief pitcher * Viktor Erdős (born 1987), Hungarian chess grandmaster See also * Erdő * Erdődy The House of Erdődy de Monyorókerék et Monoszló (also House of Erdödy) is the name of an old Hungarian- Croatian noble family with possessions in Hungary and Croatia. Elevat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subadditivity
In mathematics, subadditivity is a property of a function that states, roughly, that evaluating the function for the sum of two elements of the domain always returns something less than or equal to the sum of the function's values at each element. There are numerous examples of subadditive functions in various areas of mathematics, particularly norms and square roots. Additive maps are special cases of subadditive functions. Definitions A subadditive function is a function f \colon A \to B, having a domain ''A'' and an ordered codomain ''B'' that are both closed under addition, with the following property: \forall x, y \in A, f(x+y)\leq f(x)+f(y). An example is the square root function, having the non-negative real numbers as domain and codomain, since \forall x, y \geq 0 we have: \sqrt\leq \sqrt+\sqrt. A sequence \left \, n \geq 1, is called subadditive if it satisfies the inequality a_\leq a_n+a_m for all ''m'' and ''n''. This is a special case of subadditive function, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subadditive
In mathematics, subadditivity is a property of a function that states, roughly, that evaluating the function for the sum of two elements of the domain always returns something less than or equal to the sum of the function's values at each element. There are numerous examples of subadditive functions in various areas of mathematics, particularly norms and square roots. Additive maps are special cases of subadditive functions. Definitions A subadditive function is a function f \colon A \to B, having a domain ''A'' and an ordered codomain ''B'' that are both closed under addition, with the following property: \forall x, y \in A, f(x+y)\leq f(x)+f(y). An example is the square root function, having the non-negative real numbers as domain and codomain, since \forall x, y \geq 0 we have: \sqrt\leq \sqrt+\sqrt. A sequence \left \, n \geq 1, is called subadditive if it satisfies the inequality a_\leq a_n+a_m for all ''m'' and ''n''. This is a special case of subadditive function, i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Function (mathematics)
In mathematics, a function from a set to a set assigns to each element of exactly one element of .; the words map, mapping, transformation, correspondence, and operator are often used synonymously. The set is called the domain of the function and the set is called the codomain of the function.Codomain ''Encyclopedia of Mathematics'Codomain. ''Encyclopedia of Mathematics''/ref> The earliest known approach to the notion of function can be traced back to works of Persian mathematicians Al-Biruni and Sharaf al-Din al-Tusi. Functions were originally the idealization of how a varying quantity depends on another quantity. For example, the position of a planet is a ''function'' of time. Historically, the concept was elaborated with the infinitesimal calculus at the end of the 17th century, and, until the 19th century, the functions that were considered were differentiable (that is, they had a high degree of regularity). The concept of a function was formalized at the end of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proper Subset
In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ''B''. The relationship of one set being a subset of another is called inclusion (or sometimes containment). ''A'' is a subset of ''B'' may also be expressed as ''B'' includes (or contains) ''A'' or ''A'' is included (or contained) in ''B''. A ''k''-subset is a subset with ''k'' elements. The subset relation defines a partial order on sets. In fact, the subsets of a given set form a Boolean algebra under the subset relation, in which the join and meet are given by intersection and union, and the subset relation itself is the Boolean inclusion relation. Definition If ''A'' and ''B'' are sets and every element of ''A'' is also an element of ''B'', then: :*''A'' is a subset of ''B'', denoted by A \subseteq B, or equivalently, :* ''B'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
London Mathematical Society
The London Mathematical Society (LMS) is one of the United Kingdom's learned societies for mathematics (the others being the Royal Statistical Society (RSS), the Institute of Mathematics and its Applications (IMA), the Edinburgh Mathematical Society and the Operational Research Society (ORS). History The Society was established on 16 January 1865, the first president being Augustus De Morgan. The earliest meetings were held in University College, but the Society soon moved into Burlington House, Piccadilly. The initial activities of the Society included talks and publication of a journal. The LMS was used as a model for the establishment of the American Mathematical Society in 1888. Mary Cartwright was the first woman to be President of the LMS (in 1961–62). The Society was granted a royal charter in 1965, a century after its foundation. In 1998 the Society moved from rooms in Burlington House into De Morgan House (named after the society's first president), at 57 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
