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Order (algebra)
Order in mathematics may refer to: Set theory * Total order and partial order, a binary relation generalizing the usual ordering of numbers and of words in a dictionary * Ordered set * Order in Ramsey theory, uniform structures in consequence to critical set cardinality Algebra * Order (group theory), the cardinality of a group or period of an element * Order of a polynomial (other) * Order of a square matrix, its dimension *Order (ring theory), an algebraic structure *Ordered group *Ordered field Analysis *Order (differential equation) or order of highest derivative, of a differential equation *Leading-order terms *NURBS order, a number one greater than the degree of the polynomial representation of a non-uniform rational B-spline *Order of convergence, a measurement of convergence *Order of derivation *Order of an entire function * Order of a power series, the lowest degree of its terms *Ordered list, a sequence or tuple *Orders of approximation in Big O notation *Z-o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Total Order
In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X: # a \leq a ( reflexive). # If a \leq b and b \leq c then a \leq c ( transitive). # If a \leq b and b \leq a then a = b ( antisymmetric). # a \leq b or b \leq a (strongly connected, formerly called total). Total orders are sometimes also called simple, connex, or full orders. A set equipped with a total order is a totally ordered set; the terms simply ordered set, linearly ordered set, and loset are also used. The term ''chain'' is sometimes defined as a synonym of ''totally ordered set'', but refers generally to some sort of totally ordered subsets of a given partially ordered set. An extension of a given partial order to a total order is called a linear extension of that partial order. Strict and non-strict total orders A on a set X is a strict partia ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Power Series
In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots where ''an'' represents the coefficient of the ''n''th term and ''c'' is a constant. Power series are useful in mathematical analysis, where they arise as Taylor series of infinitely differentiable functions. In fact, Borel's theorem implies that every power series is the Taylor series of some smooth function. In many situations, ''c'' (the ''center'' of the series) is equal to zero, for instance when considering a Maclaurin series. In such cases, the power series takes the simpler form \sum_^\infty a_n x^n = a_0 + a_1 x + a_2 x^2 + \dots. Beyond their role in mathematical analysis, power series also occur in combinatorics as generating functions (a kind of formal power series) and in electronic engineering (under the name of the Z-transform). The familiar decimal notation for real numbers can also be viewed ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Permutation
In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or process of changing the linear order of an ordered set. Permutations differ from combinations, which are selections of some members of a set regardless of order. For example, written as tuples, there are six permutations of the set , namely (1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2), and (3, 2, 1). These are all the possible orderings of this three-element set. Anagrams of words whose letters are different are also permutations: the letters are already ordered in the original word, and the anagram is a reordering of the letters. The study of permutations of finite sets is an important topic in the fields of combinatorics and group theory. Permutations are used in almost every branch of mathematics, and in many other fields of s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cyclic Order
In mathematics, a cyclic order is a way to arrange a set of objects in a circle. Unlike most structures in order theory, a cyclic order is not modeled as a binary relation, such as "". One does not say that east is "more clockwise" than west. Instead, a cyclic order is defined as a ternary relation , meaning "after , one reaches before ". For example, une, October, February but not une, February, October cf. picture. A ternary relation is called a cyclic order if it is cyclic, asymmetric, transitive, and connected. Dropping the "connected" requirement results in a partial cyclic order. A set with a cyclic order is called a cyclically ordered set or simply a cycle. Some familiar cycles are discrete, having only a finite number of elements: there are seven days of the week, four cardinal directions, twelve notes in the chromatic scale, and three plays in rock-paper-scissors. In a finite cycle, each element has a "next element" and a "previous element". There are also continu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bijection
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other set, and each element of the other set is paired with exactly one element of the first set. There are no unpaired elements. In mathematical terms, a bijective function is a one-to-one (injective) and onto (surjective) mapping of a set ''X'' to a set ''Y''. The term ''one-to-one correspondence'' must not be confused with ''one-to-one function'' (an injective function; see figures). A bijection from the set ''X'' to the set ''Y'' has an inverse function from ''Y'' to ''X''. If ''X'' and ''Y'' are finite sets, then the existence of a bijection means they have the same number of elements. For infinite sets, the picture is more complicated, leading to the concept of cardinal number—a way to distinguish the various sizes of infinite se ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Twelvefold Way
In combinatorics, the twelvefold way is a systematic classification of 12 related enumerative problems concerning two finite sets, which include the classical problems of counting permutations, combinations, multisets, and partitions either of a set or of a number. The idea of the classification is credited to Gian-Carlo Rota, and the name was suggested by Joel Spencer. Overview Let and be finite sets. Let n=, N, and x=, X, be the cardinality of the sets. Thus is an -set, and is an -set. The general problem we consider is the enumeration of equivalence classes of functions f: N \to X. The functions are subject to one of the three following restrictions: # No condition: each in may be sent by to any in , and each may occur multiple times. # is injective: each value f(a) for in must be distinct from every other, and so each in may occur at most once in the image of . # is surjective: for each in there must be at least one in such that f(a) = b, thus e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Josephus Permutation
In computer science and mathematics, the Josephus problem (or Josephus permutation) is a theoretical problem related to a certain counting-out game. A number of people are standing in a circle waiting to be executed. Counting begins at a specified point in the circle and proceeds around the circle in a specified direction. After a specified number of people are skipped, the next person is executed. The procedure is repeated with the remaining people, starting with the next person, going in the same direction and skipping the same number of people, until only one person remains, and is freed. The problem—given the number of people, starting point, direction, and number to be skipped—is to choose the position in the initial circle to avoid execution. History The problem is named after Flavius Josephus, a Jewish historian living in the 1st century. According to Josephus' account of the siege of Yodfat, he and his 40 soldiers were trapped in a cave by Roman soldiers. They chose ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orders Of Magnitude
An order of magnitude is an approximation of the logarithm of a value relative to some contextually understood reference value, usually 10, interpreted as the base of the logarithm and the representative of values of magnitude one. Logarithmic distributions are common in nature and considering the order of magnitude of values sampled from such a distribution can be more intuitive. When the reference value is 10, the order of magnitude can be understood as the number of digits in the base-10 representation of the value. Similarly, if the reference value is one of some powers of 2, since computers store data in a binary format, the magnitude can be understood in terms of the amount of computer memory needed to store that value. Differences in order of magnitude can be measured on a base-10 logarithmic scale in “ decades” (i.e., factors of ten). Examples of numbers of different magnitudes can be found at Orders of magnitude (numbers). Definition Generally, the order of magnitu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Order Of Operations
In mathematics and computer programming, the order of operations (or operator precedence) is a collection of rules that reflect conventions about which procedures to perform first in order to evaluate a given mathematical expression. For example, in mathematics and most computer languages, multiplication is granted a higher precedence than addition, and it has been this way since the introduction of modern algebraic notation. Thus, the expression is interpreted to have the value , and not . When exponents were introduced in the 16th and 17th centuries, they were given precedence over both addition and multiplication, and could be placed only as a superscript to the right of their base. Thus and . These conventions exist to eliminate notational ambiguity, while allowing notation to be as brief as possible. Where it is desired to override the precedence conventions, or even simply to emphasize them, parentheses ( ) can be used. For example, forces addition to precede multipl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multiplicative Order
In number theory, given a positive integer ''n'' and an integer ''a'' coprime to ''n'', the multiplicative order of ''a'' modulo ''n'' is the smallest positive integer ''k'' such that a^k\ \equiv\ 1 \pmod n. In other words, the multiplicative order of ''a'' modulo ''n'' is the order of ''a'' in the multiplicative group of the units in the ring of the integers modulo ''n''. The order of ''a'' modulo ''n'' is sometimes written as \operatorname_n(a). Example The powers of 4 modulo 7 are as follows: : \begin 4^0 &= 1 &=0 \times 7 + 1 &\equiv 1\pmod7 \\ 4^1 &= 4 &=0 \times 7 + 4 &\equiv 4\pmod7 \\ 4^2 &= 16 &=2 \times 7 + 2 &\equiv 2\pmod7 \\ 4^3 &= 64 &=9 \times 7 + 1 &\equiv 1\pmod7 \\ 4^4 &= 256 &=36 \times 7 + 4 &\equiv 4\pmod7 \\ 4^5 &= 1024 &=146 \times 7 + 2 &\equiv 2\pmod7 \\ \vdots\end The smallest positive integer ''k'' such that 4''k'' ≡ 1 (mod 7) is 3, so the order of 4 (mod 7) is 3. Properties Even without knowledge that we are working in the multiplicative gr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Z-order (curve)
In mathematical analysis and computer science, functions which are Z-order, Lebesgue curve, Morton space-filling curve, Morton order or Morton code map multidimensional data to one dimension while preserving locality of the data points. It is named in France after Henri Lebesgue, who studied it in 1904, and named in US after Guy Macdonald Morton, who first applied the order to file sequencing in 1966. The z-value of a point in multidimensions is simply calculated by interleaving the binary representations of its coordinate values. Once the data are sorted into this ordering, any one-dimensional data structure can be used such as binary search trees, B-trees, skip lists or (with low significant bits truncated) hash tables. The resulting ordering can equivalently be described as the order one would get from a depth-first traversal of a quadtree or octree. Coordinate values The figure below shows the Z-values for the two dimensional case with integer coordinates 0 ≤& ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
