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Dickson's Lemma
In mathematics, Dickson's lemma states that every set of n-tuples of natural numbers has finitely many minimal elements. This simple fact from combinatorics has become attributed to the American algebraist L. E. Dickson, who used it to prove a result in number theory about perfect numbers. However, the lemma was certainly known earlier, for example to Paul Gordan in his research on invariant theory.. Example Let K be a fixed number, and let S = \ be the set of pairs of numbers whose product is at least K. When defined over the positive real numbers, S has infinitely many minimal elements of the form (x,K/x), one for each positive number x; this set of points forms one of the branches of a hyperbola. The pairs on this hyperbola are minimal, because it is not possible for a different pair that belongs to S to be less than or equal to (x,K/x) in both of its coordinates. However, Dickson's lemma concerns only tuples of natural numbers, and over the natural numbers there are only fini ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orthant
In geometry, an orthant or hyperoctant is the analogue in ''n''-dimensional Euclidean space of a quadrant in the plane or an octant in three dimensions. In general an orthant in ''n''-dimensions can be considered the intersection of ''n'' mutually orthogonal half-spaces. By independent selections of half-space signs, there are 2''n'' orthants in ''n''-dimensional space. More specifically, a closed orthant in R''n'' is a subset defined by constraining each Cartesian coordinate to be nonnegative or nonpositive. Such a subset is defined by a system of inequalities: :ε1''x''1 ≥ 0 ε2''x''2 ≥ 0 · · · ε''n''''x''''n'' ≥ 0, where each ε''i'' is +1 or −1. Similarly, an open orthant in R''n'' is a subset defined by a system of strict inequalities :ε1''x''1 > 0 ε2''x''2 > 0 · ·&nb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Set
In mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which one could in principle count and finish counting. For example, :\ is a finite set with five elements. The number of elements of a finite set is a natural number (possibly zero) and is called the '' cardinality (or the cardinal number)'' of the set. A set that is not a finite set is called an ''infinite set''. For example, the set of all positive integers is infinite: :\. Finite sets are particularly important in combinatorics, the mathematical study of counting. Many arguments involving finite sets rely on the pigeonhole principle, which states that there cannot exist an injective function from a larger finite set to a smaller finite set. Definition and terminology Formally, a set is called finite if there exists a bijection :f\colon S\to\ for some natural number . The number is the set's cardinality, denoted as . The empty set o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Smooth Number
In number theory, an ''n''-smooth (or ''n''-friable) number is an integer whose prime factors are all less than or equal to ''n''. For example, a 7-smooth number is a number whose every prime factor is at most 7, so 49 = 72 and 15750 = 2 × 32 × 53 × 7 are both 7-smooth, while 11 and 702 = 2 × 33 × 13 are not 7-smooth. The term seems to have been coined by Leonard Adleman. Smooth numbers are especially important in cryptography, which relies on factorization of integers. The 2-smooth numbers are just the powers of 2, while 5-smooth numbers are known as regular numbers. Definition A positive integer is called B-smooth if none of its prime factors are greater than B. For example, 1,620 has prime factorization 22 × 34 × 5; therefore 1,620 is 5-smooth because none of its prime factors are greater than 5. This definition includes numbers that lack some of the smaller prime factors; for example, both 10 and 12 are 5-smooth, even though they miss out the prime factors 3 and 5, resp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Relation (mathematics)
In mathematics, a relation on a set may, or may not, hold between two given set members. For example, ''"is less than"'' is a relation on the set of natural numbers; it holds e.g. between 1 and 3 (denoted as 1 is an asymmetric relation, but ≥ is not. Again, the previous 3 alternatives are far from being exhaustive; as an example over the natural numbers, the relation defined by is neither symmetric nor antisymmetric, let alone asymmetric. ; : for all , if and then . A transitive relation is irreflexive if and only if it is asymmetric. For example, "is ancestor of" is a transitive relation, while "is parent of" is not. ; : for all , if then or . This property is sometimes called "total", which is distinct from the definitions of "total" given in the section . ; : for all , or . This property is sometimes called "total", which is distinct from the definitions of "total" given in the section . ; : every nonempty subset of contains a minimal element with respect to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Divisibility
In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by another integer m if m is a divisor of n; this implies dividing n by m leaves no remainder. Definition An integer is divisible by a nonzero integer if there exists an integer such that n=km. This is written as :m\mid n. Other ways of saying the same thing are that divides , is a divisor of , is a factor of , and is a multiple of . If does not divide , then the notation is m\not\mid n. Usually, is required to be nonzero, but is allowed to be zero. With this convention, m \mid 0 for every nonzero integer . Some definitions omit the requirement that m be nonzero. General Divisors can be negative as well as positive, although sometimes the term is restricted to positive divisors. For example, there are six divisors of 4; they are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Odd Perfect Number
In number theory, a perfect number is a positive integer that is equal to the sum of its positive divisors, excluding the number itself. For instance, 6 has divisors 1, 2 and 3 (excluding itself), and 1 + 2 + 3 = 6, so 6 is a perfect number. The sum of divisors of a number, excluding the number itself, is called its aliquot sum, so a perfect number is one that is equal to its aliquot sum. Equivalently, a perfect number is a number that is half the sum of all of its positive divisors including itself; in symbols, \sigma_1(n)=2n where \sigma_1 is the sum-of-divisors function. For instance, 28 is perfect as 1 + 2 + 4 + 7 + 14 = 28. This definition is ancient, appearing as early as Euclid's ''Elements'' (VII.22) where it is called (''perfect'', ''ideal'', or ''complete number''). Euclid also proved a formation rule (IX.36) whereby q(q+1)/2 is an even perfect number whenever q is a prime of the form 2^p-1 for positive integer p—what is now called a Mersenne prime. Two millennia l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime Number
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, or , involve 5 itself. However, 4 is composite because it is a product (2 × 2) in which both numbers are smaller than 4. Primes are central in number theory because of the fundamental theorem of arithmetic: every natural number greater than 1 is either a prime itself or can be factorized as a product of primes that is unique up to their order. The property of being prime is called primality. A simple but slow method of checking the primality of a given number n, called trial division, tests whether n is a multiple of any integer between 2 and \sqrt. Faster algorithms include the Miller–Rabin primality test, which is fast but has a small chance of error, and the AKS primality test, which always pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Odd Number
In mathematics, parity is the property of an integer of whether it is even or odd. An integer is even if it is a multiple of two, and odd if it is not.. For example, −4, 0, 82 are even because \begin -2 \cdot 2 &= -4 \\ 0 \cdot 2 &= 0 \\ 41 \cdot 2 &= 82 \end By contrast, −3, 5, 7, 21 are odd numbers. The above definition of parity applies only to integer numbers, hence it cannot be applied to numbers like 1/2 or 4.201. See the section "Higher mathematics" below for some extensions of the notion of parity to a larger class of "numbers" or in other more general settings. Even and odd numbers have opposite parities, e.g., 22 (even number) and 13 (odd number) have opposite parities. In particular, the parity of zero is even. Any two consecutive integers have opposite parity. A number (i.e., integer) expressed in the decimal numeral system is even or odd according to whether its last digit is even or odd. That is, if the last digit is 1, 3, 5, 7, or 9, then it is odd; otherwis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Well-quasi-ordering
In mathematics, specifically order theory, a well-quasi-ordering or wqo is a quasi-ordering such that any infinite sequence of elements x_0, x_1, x_2, \ldots from X contains an increasing pair x_i \leq x_j with i x_2> \cdots) nor infinite sequences of ''pairwise incomparable'' elements. Hence a quasi-order (''X'', ≤) is wqo if and only if (''X'', <) is and has no infinite s. Examples 
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Descending Chain Condition
In mathematics, the ascending chain condition (ACC) and descending chain condition (DCC) are finiteness properties satisfied by some algebraic structures, most importantly ideals in certain commutative rings.Jacobson (2009), p. 142 and 147 These conditions played an important role in the development of the structure theory of commutative rings in the works of David Hilbert, Emmy Noether, and Emil Artin. The conditions themselves can be stated in an abstract form, so that they make sense for any partially ordered set. This point of view is useful in abstract algebraic dimension theory due to Gabriel and Rentschler. Definition A partially ordered set (poset) ''P'' is said to satisfy the ascending chain condition (ACC) if no infinite strictly ascending sequence :a_1 < a_2 < a_3 < \cdots of elements of ''P'' exists. Equivalently,Proof: first, a strictly increasing sequence cannot stabilize, obviously. Conversely, suppose there is an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Antichain
In mathematics, in the area of order theory, an antichain is a subset of a partially ordered set such that any two distinct elements in the subset are incomparable. The size of the largest antichain in a partially ordered set is known as its width. By Dilworth's theorem, this also equals the minimum number of chains (totally ordered subsets) into which the set can be partitioned. Dually, the height of the partially ordered set (the length of its longest chain) equals by Mirsky's theorem the minimum number of antichains into which the set can be partitioned. The family of all antichains in a finite partially ordered set can be given join and meet operations, making them into a distributive lattice. For the partially ordered system of all subsets of a finite set, ordered by set inclusion, the antichains are called Sperner families and their lattice is a free distributive lattice, with a Dedekind number of elements. More generally, counting the number of antichains of a finite pa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
