High Quality Content by WIKIPEDIA articles! In group theory, a branch of mathematics, the term order is used in two closely related senses: * the order of a group is its cardinality, i.e., the number of its elements; * the order, sometimes period, of an element a of a group is the smallest positive integer m such that am = e (where e denotes the identity element of the group, and am denotes the product of m copies of a). If no such m exists, we say that a has infinite order. All elements of finite groups have finite order.The order of a group and that of an element tend to speak about the structure of the group. Roughly speaking, the more complicated the factorization of the order the more complicated the group. If the order of group G...

ISBN: 978-6-1312-9561-4
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In mathematics an ordered vector space or partially ordered vector space is a vector space equipped with a partial order which is compatible with the vector space operations.A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied ("scaled") by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex numbers, rational numbers, or even more general fields instead. The operations of vector addition and scalar multiplication have to satisfy certain requirements, called axioms, listed below. An example of a vector space is...

ISBN: 978-6-1312-9912-4
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! High Quality Content by WIKIPEDIA articles! An order of magnitude is the class of scale or magnitude of any amount, where each class contains values of a fixed ratio to the class preceding it. In its most common usage, the amount being scaled is 10 and the scale is the (base 10) exponent being applied to this amount. Such differences in order of magnitude can be measured on the logarithmic scale in "factors of ten" or decades (as meaning "power of ten", not the term "10 years"). The entries in the table below lead to lists of items that are of the same order of magnitude in various units of measurement. This is useful for getting an intuitive sense of the comparative scale of familiar objects....

ISBN: 978-6-1310-2938-7
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In mathematics, the dimension of a partially ordered set (poset) is the smallest number of total orders the intersection of which gives rise to the partial order. This concept is also sometimes called the order dimension or the Dushnik–Miller dimension of the partial order. Dushnik & Miller (1941) first studied order dimension; for a more detailed treatment of this subject than provided here, see Trotter (1992).Let n be a positive integer, and let P be the partial order on the elements ai and bi (for 1 ? i ? n) in which ai ? bj whenever i ? j, but no other pairs are comparable. In particular, ai and bi are incomparable in P; P can be viewed as an oriented form of a crown graph. The...

ISBN: 978-6-1312-9647-5
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In mathematical order theory, an order-embedding is a special kind of monotone function, which provides a way to include one partially ordered set into another. Like Galois connections, order-embeddings constitute a notion which is strictly weaker than the concept of an order isomorphism. A way to understand the different nature of both of these weakenings by exploiting some category theory is discussed at the end of this article.An order isomorphism can be characterized as a surjective order-embedding. As a consequence, any order-embedding f restricts to an isomorphism between its domain S and its range f(S), which justifies the term "embedding". On the other hand, it might well be that two...

ISBN: 978-6-1312-9675-8
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In mathematics, an order in the sense of ring theory is a subring mathcal{O} of a ring R that satisfies the conditions.The third condition can be stated more accurately, in terms of the extension of scalars of R to the real numbers, embedding R in a real vector space (equivalently, taking the tensor product over mathbb{Q}). In less formal terms, additively mathcal{O} should be a free abelian group generated by a basis for R over mathbb{Q}. The leading example is the case where R is a number field K and mathcal{O} is its ring of integers. In algebraic number theory there are examples for any K other than the rational field of proper subrings of the ring of integers that are also orders. For...

ISBN: 978-6-1312-9621-5
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! This list compares various sizes of positive numbers, including counts of things, dimensionless quantity and probabilities. Each number is given a name in the so called short scale which is used in English speaking countries, as well as a name in the long scale which is used in a series of countries that do not have English as their national language. In physics, Planck units are physical units of measurement defined exclusively in terms of five universal physical constants listed below, in such a manner that these five physical constants take on the numerical value of one when expressed in terms of these units. Planck units elegantly simplify particular algebraic expressions appearing in...

ISBN: 978-6-1309-8090-0
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In statistics, the kth order statistic of a statistical sample is equal to its kth-smallest value. Together with rank statistics, order statistics are among the most fundamental tools in non-parametric statistics and inference. Important special cases of the order statistics are the minimum and maximum value of a sample, and (with some qualifications discussed below) the sample median and other sample quantiles. When using probability theory to analyze order statistics of random samples from a continuous distribution, the cumulative distribution function is used to reduce the analysis to the case of order statistics of the uniform distribution. Данное издание не является...

ISBN: 978-6-1303-3580-9
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In statistics, the kth order statistic of a statistical sample is equal to its kth-smallest value. Together with rank statistics, order statistics are among the most fundamental tools in non-parametric statistics and inference. Important special cases of the order statistics are the minimum and maximum value of a sample, and (with some qualifications discussed below) the sample median and other sample quantiles. When using probability theory to analyze order statistics of random samples from a continuous distribution, the cumulative distribution function is used to reduce the analysis to the case of order statistics of the uniform distribution. Данное издание не является...

ISBN: 978-6-1303-3820-6
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! The ordered exponential (also called the path-ordered exponential) is a mathematical object, defined in non-commutative algebras, which is equivalent to the exponential function of the integral in the commutative algebras. Therefore it is a function, defined by means of a function from real numbers to a real or complex associative algebra. In practice the values lie in matrix and operator algebras.In mathematics, commutativity is the property that changing the order of something does not change the end result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. The commutativity of simple operations, such as multiplication and addition of numbers,...

ISBN: 978-6-1312-9734-2
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In mathematics and computer programming, an expression or string of symbols intended to represent a numerical value must follow commonly accepted and unambiguous rules. For example, the rule for evaluating 2 + 3 * 4 in mathematics and in most computer languages is to do the multiplication first, so the correct answer is 14. Sometimes parentheses, which have their own rules, may be used to avoid confusion, thus: 2 + (3 * 4). When a term in an expression is both preceded and followed by an operator such as minus or times, the convention needed to clarify which operator should be applied first is known as a precedence rule or, more informally, order of operation. From the earliest use of...

ISBN: 978-6-1303-3807-7
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! An order of magnitude is the class of scale or magnitude of any amount, where each class contains values of a fixed ratio to the class preceding it. In its most common usage, the amount being scaled is 10 and the scale is the exponent being applied to this amount. Such differences in order of magnitude can be measured on the logarithmic scale in "factors of ten" or decades. The entries in the table below lead to lists of items that are of the same order of magnitude in various units of measurement. This is useful for getting an intuitive sense of the comparative scale of familiar objects. Данное издание не является оригинальным. Книга...

ISBN: 978-6-1303-0476-8
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In mathematics, an ordered pair is a collection of objects having two coordinates (or entries or projections), such that one can always uniquely determine the object, which is the first coordinate (or first entry or left projection) of the pair as well as the second coordinate (or second entry or right projection). If the first coordinate is a and the second is b, the usual notation for an ordered pair is (a, b). The pair is "ordered" in that (a, b) differs from (b, a) unless a = b. Данное издание не является оригинальным. Книга печатается по технологии принт-он-деманд после получения заказа.

ISBN: 978-6-1303-3835-0
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In proof theory, ordinal analysis assigns ordinals (often large countable ordinals) to mathematical theories as a measure of their strength. The field was formed when Gerhard Gentzen in 1934 used cut elimination to prove, in modern terms, that the proof theoretic ordinal of Peano arithmetic is ?0.Ordinal analysis concerns true, effective (recursive) theories that can interpret a sufficient portion of arithmetic to make statements about ordinal notations. The proof theoretic ordinal of such a theory is the smallest recursive ordinal that the theory cannot prove is well founded — the supremum of all ordinals ? for which there exists a notation o in Kleene's sense such that T proves that o is an...

ISBN: 978-6-1312-9947-6
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In mathematics, an ordinary differential equation (or ODE) is a relation that contains functions of only one independent variable, and one or more of its derivatives with respect to that variable. A simple example is Newton's second law of motion, which leads to the differential equation m frac{d^2 x(t)}{dt^2} = F(x(t)),, for the motion of a particle of constant mass m. In general, the force F depends upon the position of the particle x(t) at time t, and thus the unknown function x(t) appears on both sides of the differential equation, as is indicated in the notation F(x(t)). Данное издание не является оригинальным. Книга печатается по...

ISBN: 978-6-1303-3862-6
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In the philosophy of mathematics, ordinary mathematics is an inexact term, used to distinguish the body of most mathematical work from that of, for example, constructivist, intuitionist, or finitist mathematics. Ordinary mathematics is usually studied within the universe SN, or sometimes V?+? (see Von Neumann universe). Contrast with finitist mathematics, which limits to the study of V? (see hereditarily finite sets), or with metamathematics and the study of large cardinals, which study objects contained in a larger universe. Ordinary mathematicians generally assume the axiom of choice (at least, because it makes their work easier), whereas constructivists reject it on the grounds that it is...

ISBN: 978-6-1313-0292-3
Издательство: Книга по требованию
Дата выхода: июль 2011

ISBN: 978-6-1300-9063-0
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In mathematical logic and set theory, an ordinal collapsing function (or projection function) is a technique for defining (notations for) certain recursive large countable ordinals, whose principle is to give names to certain ordinals much larger than the one being defined, perhaps even large cardinals (though they can be replaced with recursively large ordinals at the cost of extra technical difficulty), and then “collapse” them down to a system of notations for the sought-after ordinal. For this reason, ordinal collapsing functions are described as an impredicative manner of naming ordinals. The details of the definition of ordinal collapsing functions vary, and get more complicated as...

ISBN: 978-6-1313-0296-1
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In mathematics, an ordinary differential equation (or ODE) is a relation that contains functions of only one independent variable, and one or more of its derivatives with respect to that variable.Much study has been devoted to the solution of ordinary differential equations. In the case where the equation is linear, it can be solved by analytical methods. Unfortunately, most of the interesting differential equations are non-linear and, with a few exceptions, cannot be solved exactly. Approximate solutions are arrived at using computer approximations. Данное издание не является оригинальным. Книга печатается по технологии...

ISBN: 978-6-1303-3931-9
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In mathematical logic and set theory, an ordinal notation is a finite sequence of symbols from a finite alphabet which names an ordinal number according to some scheme which gives meaning to the language. There are many such schemes of ordinal notations, including schemes by Wilhelm Ackermann, Heinz Bachmann, Buchholz, Georg Cantor, Solomon Feferman, Gerhard Jager, Isles, Pfeiffer, Wolfram Pohlers, Kurt Schutte, Gaisi Takeuti (called ordinal diagrams), Oswald Veblen. Given such a scheme, one should be able to define a recursive well-ordering of a subset of the natural numbers by associating a natural number with each finite sequence of symbols via a Godel numbering. Stephen Cole Kleene has a...

ISBN: 978-6-1303-3855-8
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In the mathematical field of set theory, ordinal arithmetic describes the three usual operations on ordinal numbers: addition, multiplication, and exponentiation. Each can be defined in essentially two different ways: either by constructing an explicit well-ordered set which represents the operation or by using transfinite recursion. Cantor normal form provides a standardized way of writing ordinals. The so-called "natural" arithmetical operations retain commutativity at the expense of continuity.The union of two disjoint well-ordered sets S and T can be well-ordered. The order-type of that union is the ordinal which results from adding the order-types of S and T. If two well-ordered sets are...

ISBN: 978-6-1312-9964-3
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! Ordnance Survey is an executive agency of the United Kingdom government. It is the national mapping agency for Great Britain, and one of the world's largest producers of maps. The name reflects the original military purpose of the organisation (see ordnance and surveying) in mapping Britain during the Napoleonic Wars when there was a threat of invasion from France, and its logo includes the War Department's broad arrow heraldic mark. Данное издание не является оригинальным. Книга печатается по технологии принт-он-деманд после получения заказа.

ISBN: 978-6-1303-2075-1
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! En mathematiques, la priorite des operations ou ordre des operations precise l'ordre dans lequel les calculs doivent etre effectues dans une expression complexe. Les regles de priorite sont : Si un calcul ne contient que des multiplications et divisions melees ou bien que additions et soustractions melees, alors en ce cas le calcul est effectue selon l'ordre naturel de lecture, de gauche a droite; les multiplications et divisions sont prioritaires sur les additions et soustractions; les calculs contenus entre parentheses (ou crochets) sont prioritaires sur les calculs situes en dehors de ces parentheses; les exposants sont prioritaires sur toute autre operation. Данное издание...

ISBN: 978-6-1340-7475-9
Издательство: Книга по требованию
Дата выхода: июль 2011

Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In mathematics, especially in the area of algebra known as ring theory, an Ore extension, named after Oystein Ore, is a special type of a ring extension whose properties are relatively well understood. Ore extensions appear in several natural contexts, including skew and differential polynomial rings, group algebras of polycyclic groups, enveloping algebras of solvable Lie algebras, and coordinate rings of quantum groups.The Weyl algebras are Ore extensions, with R any a commutative polynomial ring, the identity ring endomorphism, and the polynomial derivative. Данное издание не является...

ISBN: 978-6-1313-0287-9
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! Ore's theorem is a result in graph theory proved in 1960 by Norwegian mathematician ystein Ore. It gives a sufficient condition for a graph to be Hamiltonian, essentially stating that a graph with "sufficiently many edges" must contain a Hamilton cycle. Specifically, the theorem considers the sum of the degrees of any two non-adjacent vertices. Ore's theorem is a generalization of Dirac's theorem and further generalization leads to the Bondy-Chvatal theorem.Suppose for a contradiction that the result fails. It is therefore possible to pick a non-Hamiltonian maximal (i.e. with the most possible edges) graph G on n ? 3 vertices satisfying property. Данное издание не...

ISBN: 978-6-1313-0286-2
Издательство: Книга по требованию
Дата выхода: июль 2011