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Extended Natural Numbers
In mathematics, the extended natural numbers is a set which contains the values 0, 1, 2, \dots and \infty (infinity). That is, it is the result of adding a maximum element \infty to the natural numbers. Addition and multiplication work as normal for finite values, and are extended by the rules n+\infty=\infty+n=\infty (n\in\mathbb\cup \), 0\times \infty=\infty \times 0=0 and m\times \infty=\infty\times m=\infty for m\neq 0. With addition and multiplication, \mathbb\cup \ is a semiring but not a ring, as \infty lacks an additive inverse. The set can be denoted by \overline, \mathbb_\infty or \mathbb^\infty. It is a subset of the extended real number line, which extends the real numbers by adding -\infty and +\infty. Applications In graph theory, the extended natural numbers are used to define distances in graphs, with \infty being the distance between two unconnected vertices. They can be used to show the extension of some results, such as the max-flow min-cut theorem, to infini ... [...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 t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Category (mathematics)
In mathematics, a category (sometimes called an abstract category to distinguish it from a concrete category) is a collection of "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose objects are sets and whose arrows are functions. ''Category theory'' is a branch of mathematics that seeks to generalize all of mathematics in terms of categories, independent of what their objects and arrows represent. Virtually every branch of modern mathematics can be described in terms of categories, and doing so often reveals deep insights and similarities between seemingly different areas of mathematics. As such, category theory provides an alternative foundation for mathematics to set theory and other proposed axiomatic foundations. In general, the objects and arrows may be abstract entities of any kind, and the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cambridge University Press
Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by King Henry VIII in 1534, it is the oldest university press in the world. It is also the King's Printer. Cambridge University Press is a department of the University of Cambridge and is both an academic and educational publisher. It became part of Cambridge University Press & Assessment, following a merger with Cambridge Assessment in 2021. With a global sales presence, publishing hubs, and offices in more than 40 countries, it publishes over 50,000 titles by authors from over 100 countries. Its publishing includes more than 380 academic journals, monographs, reference works, school and university textbooks, and English language teaching and learning publications. It also publishes Bibles, runs a bookshop in Cambridge, sells through Amazon, and has a conference venues business in Cambridge at the Pitt Building and the Sir Geoffrey Cass Sports and Social Centre. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Communications In Algebra
''Communications in Algebra'' is a monthly peer-reviewed scientific journal covering algebra, including commutative algebra, ring theory, module theory, non-associative algebra (including Lie algebras and Jordan algebras), group theory, and algebraic geometry. It was established in 1974 and is published by Taylor & Francis. The editor-in-chief is Scott Chapman ( Sam Houston State University). Earl J. Taft (Rutgers University) was the founding editor. Abstracting and indexing The journal is abstracted and indexed in CompuMath Citation Index, Current Contents/Chemical, Earth, and Physical Sciences, Mathematical Reviews, MathSciNet, Science Citation Index Expanded (SCIE), and Zentralblatt MATH. According to the ''Journal Citation Reports'', the journal has a 2018 impact factor The impact factor (IF) or journal impact factor (JIF) of an academic journal is a scientometric index calculated by Clarivate that reflects the yearly mean number of citations of articles publishe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of Symbolic Logic
The '' Journal of Symbolic Logic'' is a peer-reviewed mathematics journal published quarterly by Association for Symbolic Logic. It was established in 1936 and covers mathematical logic. The journal is indexed by ''Mathematical Reviews'', Zentralblatt MATH, and Scopus. Its 2009 MCQ was 0.28, and its 2009 impact factor The impact factor (IF) or journal impact factor (JIF) of an academic journal is a scientometric index calculated by Clarivate that reflects the yearly mean number of citations of articles published in the last two years in a given journal, as ... was 0.631. External links * Mathematics journals Publications established in 1936 Multilingual journals Quarterly journals Association for Symbolic Logic academic journals Logic journals Cambridge University Press academic journals {{math-journal-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of Combinatorial Theory
The ''Journal of Combinatorial Theory'', Series A and Series B, are mathematical journals specializing in combinatorics and related areas. They are published by Elsevier. ''Series A'' is concerned primarily with structures, designs, and applications of combinatorics. ''Series B'' is concerned primarily with graph and matroid theory. The two series are two of the leading journals in the field and are widely known as ''JCTA'' and ''JCTB''. The journal was founded in 1966 by Frank Harary and Gian-Carlo Rota.They are acknowledged on the journals' title pages and Web sites. SeEditorial board of JCTA Originally there was only one journal, which was split into two parts in 1971 as the field grew rapidly. An electronic, |
Limited Principle Of Omniscience
In constructive mathematics, the limited principle of omniscience (LPO) and the lesser limited principle of omniscience (LLPO) are axioms that are nonconstructive but are weaker than the full law of the excluded middle . The LPO and LLPO axioms are used to gauge the amount of nonconstructivity required for an argument, as in constructive reverse mathematics. They are also related to weak counterexamples in the sense of Brouwer Brouwer (also Brouwers and de Brouwer) is a Dutch and Flemish surname. The word ''brouwer'' means 'beer brewer'. Brouwer * Adriaen Brouwer (1605–1638), Flemish painter * Alexander Brouwer (b. 1989), Dutch beach volleyball player * Andries Brou .... Definitions The limited principle of omniscience states : :LPO: For any sequence a_0, a_1, ... such that each a_i is either 0 or 1, the following holds: either a_i=0 for all i, or there is a k with a_k=1. The lesser limited principle of omniscience states: :LLPO: For any sequence a_0, a_1, ... such tha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Retraction (topology)
In topology, a branch of mathematics, a retraction is a continuous mapping from a topological space into a subspace that preserves the position of all points in that subspace. The subspace is then called a retract of the original space. A deformation retraction is a mapping that captures the idea of ''continuously shrinking'' a space into a subspace. An absolute neighborhood retract (ANR) is a particularly well-behaved type of topological space. For example, every topological manifold is an ANR. Every ANR has the homotopy type of a very simple topological space, a CW complex. Definitions Retract Let ''X'' be a topological space and ''A'' a subspace of ''X''. Then a continuous map :r\colon X \to A is a retraction if the restriction of ''r'' to ''A'' is the identity map on ''A''; that is, r(a) = a for all ''a'' in ''A''. Equivalently, denoting by :\iota\colon A \hookrightarrow X the inclusion, a retraction is a continuous map ''r'' such that :r \circ \iota = \operatorname ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binary Sequence
A bitstream (or bit stream), also known as binary sequence, is a sequence of bits. A bytestream is a sequence of bytes. Typically, each byte is an 8-bit quantity, and so the term octet stream is sometimes used interchangeably. An octet may be encoded as a sequence of 8 bits in multiple different ways (see bit numbering) so there is no unique and direct translation between bytestreams and bitstreams. Bitstreams and bytestreams are used extensively in telecommunications and computing. For example, synchronous bitstreams are carried by SONET, and Transmission Control Protocol transports an asynchronous bytestream. Relationship between bitstreams and bytestreams In practice, bitstreams are not used directly to encode bytestreams; a communication channel may use a signalling method that does not directly translate to bits (for instance, by transmitting signals of multiple frequencies) and typically also encodes other information such as framing and error correction together w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called the ''length'' of the sequence. Unlike a set, the same elements can appear multiple times at different positions in a sequence, and unlike a set, the order does matter. Formally, a sequence can be defined as a function from natural numbers (the positions of elements in the sequence) to the elements at each position. The notion of a sequence can be generalized to an indexed family, defined as a function from an ''arbitrary'' index set. For example, (M, A, R, Y) is a sequence of letters with the letter 'M' first and 'Y' last. This sequence differs from (A, R, M, Y). Also, the sequence (1, 1, 2, 3, 5, 8), which contains the number 1 at two different positions, is a valid sequence. Sequences can be ''finite'', as in these examples, or ''infi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
One-point Compactification
In the mathematical field of topology, the Alexandroff extension is a way to extend a noncompact topological space by adjoining a single point in such a way that the resulting space is compact. It is named after the Russian mathematician Pavel Alexandroff. More precisely, let ''X'' be a topological space. Then the Alexandroff extension of ''X'' is a certain compact space ''X''* together with an open embedding ''c'' : ''X'' → ''X''* such that the complement of ''X'' in ''X''* consists of a single point, typically denoted ∞. The map ''c'' is a Hausdorff compactification if and only if ''X'' is a locally compact, noncompact Hausdorff space. For such spaces the Alexandroff extension is called the one-point compactification or Alexandroff compactification. The advantages of the Alexandroff compactification lie in its simple, often geometrically meaningful structure and the fact that it is in a precise sense minimal among all compactifications; the disadvanta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Constructive Mathematics
In the philosophy of mathematics, constructivism asserts that it is necessary to find (or "construct") a specific example of a mathematical object in order to prove that an example exists. Contrastingly, in classical mathematics, one can prove the existence of a mathematical object without "finding" that object explicitly, by assuming its non-existence and then deriving a contradiction from that assumption. Such a proof by contradiction might be called non-constructive, and a constructivist might reject it. The constructive viewpoint involves a verificational interpretation of the existential quantifier, which is at odds with its classical interpretation. There are many forms of constructivism. These include the program of intuitionism founded by Brouwer, the finitism of Hilbert and Bernays, the constructive recursive mathematics of Shanin and Markov, and Bishop's program of constructive analysis. Constructivism also includes the study of constructive set theories such as CZF ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
