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Pompeiu Derivative
In mathematical analysis, a Pompeiu derivative is a real-valued function of one real variable that is the derivative of an everywhere differentiable function and that vanishes in a dense set. In particular, a Pompeiu derivative is discontinuous at any point where it is not 0. Whether non-identically zero such functions may exist was a problem that arose in the context of early-1900s research on functional differentiability and integrability. The question was affirmatively answered by Dimitrie Pompeiu by constructing an explicit example; these functions are therefore named after him. Pompeiu's construction Pompeiu's construction is described here. Let \sqrt /math> denote the real cube root of the real number . Let \_ be an enumeration of the rational numbers in the unit interval . Let \_ be positive real numbers with . Define g: , 1\rarr \mathbb by :g(x): = a_0+\sum_^\infty \,a_j \sqrt For any in , each term of the series is less than or equal to in absolute value, so the serie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (mathematics), series, and analytic functions. These theories are usually studied in the context of Real number, real and Complex number, complex numbers and Function (mathematics), functions. Analysis evolved from calculus, which involves the elementary concepts and techniques of analysis. Analysis may be distinguished from geometry; however, it can be applied to any Space (mathematics), space of mathematical objects that has a definition of nearness (a topological space) or specific distances between objects (a metric space). History Ancient Mathematical analysis formally developed in the 17th century during the Scientific Revolution, but many of its ideas can be traced back to earlier mathematicians. Early results in analysis were i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Surjective Function
In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of its domain. It is not required that be unique; the function may map one or more elements of to the same element of . The term ''surjective'' and the related terms ''injective'' and ''bijective'' were introduced by Nicolas Bourbaki, a group of mainly French 20th-century mathematicians who, under this pseudonym, wrote a series of books presenting an exposition of modern advanced mathematics, beginning in 1935. The French word '' sur'' means ''over'' or ''above'', and relates to the fact that the image of the domain of a surjective function completely covers the function's codomain. Any function induces a surjection by restricting its codomain to the image of its domain. Every surjective function has a right inverse assuming the axiom ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bounded Function
In mathematics, a function ''f'' defined on some set ''X'' with real or complex values is called bounded if the set of its values is bounded. In other words, there exists a real number ''M'' such that :, f(x), \le M for all ''x'' in ''X''. A function that is ''not'' bounded is said to be unbounded. If ''f'' is real-valued and ''f''(''x'') ≤ ''A'' for all ''x'' in ''X'', then the function is said to be bounded (from) above by ''A''. If ''f''(''x'') ≥ ''B'' for all ''x'' in ''X'', then the function is said to be bounded (from) below by ''B''. A real-valued function is bounded if and only if it is bounded from above and below. An important special case is a bounded sequence, where ''X'' is taken to be the set N of natural numbers. Thus a sequence ''f'' = (''a''0, ''a''1, ''a''2, ...) is bounded if there exists a real number ''M'' such that :, a_n, \le M for every natural number ''n''. The set of all bounded sequences forms the sequence space l^\infty. The definition of bound ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Class (set Theory)
In set theory and its applications throughout mathematics, a class is a collection of sets (or sometimes other mathematical objects) that can be unambiguously defined by a property that all its members share. Classes act as a way to have set-like collections while differing from sets so as to avoid Russell's paradox (see ). The precise definition of "class" depends on foundational context. In work on Zermelo–Fraenkel set theory, the notion of class is informal, whereas other set theories, such as von Neumann–Bernays–Gödel set theory, axiomatize the notion of "proper class", e.g., as entities that are not members of another entity. A class that is not a set (informally in Zermelo–Fraenkel) is called a proper class, and a class that is a set is sometimes called a small class. For instance, the class of all ordinal numbers, and the class of all sets, are proper classes in many formal systems. In Quine's set-theoretical writing, the phrase "ultimate class" is often used in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Intersection (set Theory)
In set theory, the intersection of two sets A and B, denoted by A \cap B, is the set containing all elements of A that also belong to B or equivalently, all elements of B that also belong to A. Notation and terminology Intersection is written using the symbol "\cap" between the terms; that is, in infix notation. For example: \\cap\=\ \\cap\=\varnothing \Z\cap\N=\N \\cap\N=\ The intersection of more than two sets (generalized intersection) can be written as: \bigcap_^n A_i which is similar to capital-sigma notation. For an explanation of the symbols used in this article, refer to the table of mathematical symbols. Definition The intersection of two sets A and B, denoted by A \cap B, is the set of all objects that are members of both the sets A and B. In symbols: A \cap B = \. That is, x is an element of the intersection A \cap B if and only if x is both an element of A and an element of B. For example: * The intersection of the sets and is . * The number 9 is in t ... [...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] |
Vector Space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can be complex numbers or, more generally, elements of any field. The operations of vector addition and scalar multiplication must satisfy certain requirements, called ''vector axioms''. The terms real vector space and complex vector space are often used to specify the nature of the scalars: real coordinate space or complex coordinate space. Vector spaces generalize Euclidean vectors, which allow modeling of physical quantities, such as forces and velocity, that have not only a magnitude, but also a direction. The concept of vector spaces is fundamental for linear algebra, together with the concept of matrix, which allows computing in vector spaces. This provides a concise and synthetic way for manipulating and studying systems of linear eq ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uncountably
In mathematics, an uncountable set (or uncountably infinite set) is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal number is larger than that of the set of all natural numbers. Characterizations There are many equivalent characterizations of uncountability. A set ''X'' is uncountable if and only if any of the following conditions hold: * There is no injective function (hence no bijection) from ''X'' to the set of natural numbers. * ''X'' is nonempty and for every ω-sequence of elements of ''X'', there exists at least one element of X not included in it. That is, ''X'' is nonempty and there is no surjective function from the natural numbers to ''X''. * The cardinality of ''X'' is neither finite nor equal to \aleph_0 (aleph-null, the cardinality of the natural numbers). * The set ''X'' has cardinality strictly greater than \aleph_0. The first three ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Residual Set
In the mathematical field of general topology, a meagre set (also called a meager set or a set of first category) is a subset of a topological space that is small or negligible in a precise sense detailed below. A set that is not meagre is called nonmeagre, or of the second category. See below for definitions of other related terms. The meagre subsets of a fixed space form a σ-ideal of subsets; that is, any subset of a meagre set is meagre, and the union of countably many meagre sets is meagre. Meagre sets play an important role in the formulation of the notion of Baire space and of the Baire category theorem, which is used in the proof of several fundamental results of functional analysis. Definitions Throughout, X will be a topological space. A subset of X is called X, a of X, or of the in X if it is a countable union of nowhere dense subsets of X (where a nowhere dense set is a set whose closure has empty interior). The qualifier "in X" can be omitted if the ambient ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
G Delta Set
G, or g, is the seventh Letter (alphabet), letter in the Latin alphabet, used in the English alphabet, modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is English alphabet#Letter names, ''gee'' (pronounced ), plural ''gees''. History The letter 'G' was introduced in the History of the Latin alphabet#Old Latin period, Old Latin period as a variant of 'C' to distinguish voiced from voiceless . The recorded originator of 'G' is freedman Spurius Carvilius Ruga, who added letter G to the teaching of the Roman alphabet during the 3rd century BC: he was the first Roman to open a fee-paying school, around 230 BCE. At this time, 'K' had fallen out of favor, and 'C', which had formerly represented both and before open vowels, had come to express in all environments. Ruga's positioning of 'G' shows that alphabetic order related to the letters' values as Greek numerals was a concern even in the 3rd century BC. Ac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Baire Function
In mathematics, Baire functions are function (mathematics), functions obtained from continuous functions by transfinite iteration of the operation of forming pointwise limits of sequences of functions. They were introduced by René-Louis Baire in 1899. A Baire set is a set whose indicator function, characteristic function is a Baire function. (There are other similar, but inequivalent definitions of Baire sets.) Classification of Baire functions Baire functions of class α, for any countable ordinal number α, form a vector space of Real number, real-valued functions defined on a topological space, as follows.T. Jech,The Brave New World of Determinacy (PDF download). Bulletin of the American Mathematical Society, vol. 5, number 3, November 1981 (pp.339--349). *The Baire class 0 functions are the Continuous function (topology), continuous functions. *The Baire class 1 functions are those functions which are the Pointwise convergence, pointwise limit of a sequence of Baire class 0 fun ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
