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Denjoy–Young–Saks Theorem
In mathematics, the Denjoy–Young–Saks theorem gives some possibilities for the Dini derivatives of a function that hold almost everywhere. proved the theorem for continuous functions, extended it to measurable function In mathematics and in particular measure theory, a measurable function is a function between the underlying sets of two measurable spaces that preserves the structure of the spaces: the preimage of any measurable set is measurable. This is in di ...s, and extended it to arbitrary functions. and give historical accounts of the theorem. Statement If ''f'' is a real valued function defined on an interval, then with the possible exception of a set of measure 0 on the interval, the Dini derivatives of ''f'' satisfy one of the following four conditions at each point: *''f'' has a finite derivative *''D''+''f'' = ''D''–''f'' is finite, ''D''−''f'' = ∞, ''D''+''f'' = –∞. *''D''−''f'' = ''D''+''f'' is finite, ''D''+''f'' = ∞, ''D''–''f'' = –∠... [...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] |
Dini Derivative
In mathematics and, specifically, real analysis, the Dini derivatives (or Dini derivates) are a class of generalizations of the derivative. They were introduced by Ulisse Dini, who studied continuous but nondifferentiable functions. The upper Dini derivative, which is also called an upper right-hand derivative, of a continuous function :f: \rightarrow , is denoted by and defined by :f'_+(t) = \limsup_ \frac, where is the supremum limit and the limit is a one-sided limit. The lower Dini derivative, , is defined by :f'_-(t) = \liminf_ \frac, where is the infimum limit. If is defined on a vector space, then the upper Dini derivative at in the direction is defined by :f'_+ (t,d) = \limsup_ \frac. If is locally Lipschitz, then is finite. If is differentiable at , then the Dini derivative at is the usual derivative at . Remarks * The functions are defined in terms of the infimum and supremum in order to make the Dini derivatives as "bullet proof" as possible, so that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Almost Everywhere
In measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of "almost everywhere" is a companion notion to the concept of measure zero, and is analogous to the notion of ''almost surely'' in probability theory. More specifically, a property holds almost everywhere if it holds for all elements in a set except a subset of measure zero, or equivalently, if the set of elements for which the property holds is conull. In cases where the measure is not complete, it is sufficient that the set be contained within a set of measure zero. When discussing sets of real numbers, the Lebesgue measure is usually assumed unless otherwise stated. The term ''almost everywhere'' is abbreviated ''a.e.''; in older literature ''p.p.'' is used, to stand for the equivalent French language phrase ''presque partout''. A set with full measure is one whose complement i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continuous Function
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value, known as '' discontinuities''. More precisely, a function is continuous if arbitrarily small changes in its value can be assured by restricting to sufficiently small changes of its argument. A discontinuous function is a function that is . Up until the 19th century, mathematicians largely relied on intuitive notions of continuity, and considered only continuous functions. The epsilon–delta definition of a limit was introduced to formalize the definition of continuity. Continuity is one of the core concepts of calculus and mathematical analysis, where arguments and values of functions are real and complex numbers. The concept has been generalized to functions between metric spaces and between topological spaces. The latter are the mo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Measurable Function
In mathematics and in particular measure theory, a measurable function is a function between the underlying sets of two measurable spaces that preserves the structure of the spaces: the preimage of any measurable set is measurable. This is in direct analogy to the definition that a continuous function between topological spaces preserves the topological structure: the preimage of any open set is open. In real analysis, measurable functions are used in the definition of the Lebesgue integral. In probability theory, a measurable function on a probability space is known as a random variable. Formal definition Let (X,\Sigma) and (Y,\Tau) be measurable spaces, meaning that X and Y are sets equipped with respective \sigma-algebras \Sigma and \Tau. A function f:X\to Y is said to be measurable if for every E\in \Tau the pre-image of E under f is in \Sigma; that is, for all E \in \Tau f^(E) := \ \in \Sigma. That is, \sigma (f)\subseteq\Sigma, where \sigma (f) is the σ-algebra gen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business internationally, o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Warszawa
Warsaw ( pl, Warszawa, ), officially the Capital City of Warsaw,, abbreviation: ''m.st. Warszawa'' is the capital and largest city of Poland. The metropolis stands on the River Vistula in east-central Poland, and its population is officially estimated at 1.86 million residents within a greater metropolitan area of 3.1 million residents, which makes Warsaw the 7th most-populous city in the European Union. The city area measures and comprises 18 districts, while the metropolitan area covers . Warsaw is an Alpha global city, a major cultural, political and economic hub, and the country's seat of government. Warsaw traces its origins to a small fishing town in Masovia. The city rose to prominence in the late 16th century, when Sigismund III decided to move the Polish capital and his royal court from Kraków. Warsaw served as the de facto capital of the Polish–Lithuanian Commonwealth until 1795, and subsequently as the seat of Napoleon's Duchy of Warsaw. The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lwów
Lviv ( uk, Львів) is the largest city in western Ukraine, and the seventh-largest in Ukraine, with a population of . It serves as the administrative centre of Lviv Oblast and Lviv Raion, and is one of the main cultural centres of Ukraine. It was named in honour of Leo, the eldest son of Daniel, King of Ruthenia. Lviv emerged as the centre of the historical regions of Red Ruthenia and Galicia in the 14th century, superseding Halych, Chełm, Belz and Przemyśl. It was the capital of the Kingdom of Galicia–Volhynia from 1272 to 1349, when it was conquered by King Casimir III the Great of Poland. From 1434, it was the regional capital of the Ruthenian Voivodeship in the Kingdom of Poland. In 1772, after the First Partition of Poland, the city became the capital of the Habsburg Kingdom of Galicia and Lodomeria. In 1918, for a short time, it was the capital of the West Ukrainian People's Republic. Between the wars, the city was the centre of the Lwów Voivodeship in the Se ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
