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Cantor Function
In mathematics, the Cantor function is an example of a function that is continuous, but not absolutely continuous. It is a notorious counterexample in analysis, because it challenges naive intuitions about continuity, derivative, and measure. Though it is continuous everywhere and has zero derivative almost everywhere, its value still goes from 0 to 1 as its argument reaches from 0 to 1. Thus, in one sense the function seems very much like a constant one which cannot grow, and in another, it does indeed monotonically grow. It is also called the Cantor ternary function, the Lebesgue function, Lebesgue's singular function, the Cantor–Vitali function, the Devil's staircase, the Cantor staircase function, and the Cantor–Lebesgue function. introduced the Cantor function and mentioned that Scheeffer pointed out that it was a counterexample to an extension of the fundamental theorem of calculus claimed by Harnack. The Cantor function was discussed and popularized by , and . Defin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cumulative Distribution Function
In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x. Every probability distribution supported on the real numbers, discrete or "mixed" as well as continuous, is uniquely identified by an ''upwards continuous'' ''monotonic increasing'' cumulative distribution function F : \mathbb R \rightarrow ,1/math> satisfying \lim_F(x)=0 and \lim_F(x)=1. In the case of a scalar continuous distribution, it gives the area under the probability density function from minus infinity to x. Cumulative distribution functions are also used to specify the distribution of multivariate random variables. Definition The cumulative distribution function of a real-valued random variable X is the function given by where the right-hand side represents the probability that the random variable X takes on a value less tha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fractal
In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as illustrated in successive magnifications of the Mandelbrot set. This exhibition of similar patterns at increasingly smaller scales is called self-similarity, also known as expanding symmetry or unfolding symmetry; if this replication is exactly the same at every scale, as in the Menger sponge, the shape is called affine self-similar. Fractal geometry lies within the mathematical branch of measure theory. One way that fractals are different from finite geometric figures is how they scale. Doubling the edge lengths of a filled polygon multiplies its area by four, which is two (the ratio of the new to the old side length) raised to the power of two (the conventional dimension of the filled polygon). Likewise, if the radius of a filled sphere i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Base (exponentiation)
In exponentiation, the base is the number b in an expression of the form bn. Related terms The number n is called the exponent and the expression is known formally as exponentiation of b by n or the exponential of n with base b. It is more commonly expressed as "the nth power of b", "b to the nth power" or "b to the power n". For example, the fourth power of 10 is 10,000 because . The term ''power'' strictly refers to the entire expression, but is sometimes used to refer to the exponent. Radix is the traditional term for ''base'', but usually refers then to one of the common bases: decimal (10), binary (2), hexadecimal (16), or sexagesimal (60). When the concepts of variable and constant came to be distinguished, the process of exponentiation was seen to transcend the algebraic functions. In his 1748 ''Introductio in analysin infinitorum'', Leonhard Euler referred to "base a = 10" in an example. He referred to ''a'' as a "constant number" in an extensive consideration of the ... [...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] |
Cantor Function Sequence
A cantor or chanter is a person who leads people in singing or sometimes in prayer. In formal Jewish worship, a cantor is a person who sings solo verses or passages to which the choir or congregation responds. In Judaism, a cantor sings and leads congregants in prayer in Jewish religious services; sometimes called a hazzan. A cantor in Reform and Conservative Judaism, just like in Orthodox Judaism, goes through years of extensive religious education, similar to that of a Rabbi, in order to become an officially recognized cantor. They often come from a long line of cantors in their family; born with a natural gift of singing with incredible vocal range. The term itself was shaped by the Latin term for "singer," but is not an inherently Latin word. It is frequently used to translate a range of equivalent terms in other languages, such as for the leader of singing on a traditional Kerala snake boat, a Chundan Vallam. A similar term is precentor, defined as a leader of the singing of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pairwise Disjoint
In mathematics, two sets are said to be disjoint sets if they have no element in common. Equivalently, two disjoint sets are sets whose intersection is the empty set.. For example, and are ''disjoint sets,'' while and are not disjoint. A collection of two or more sets is called disjoint if any two distinct sets of the collection are disjoint. Generalizations This definition of disjoint sets can be extended to a family of sets \left(A_i\right)_: the family is pairwise disjoint, or mutually disjoint if A_i \cap A_j = \varnothing whenever i \neq j. Alternatively, some authors use the term disjoint to refer to this notion as well. For families the notion of pairwise disjoint or mutually disjoint is sometimes defined in a subtly different manner, in that repeated identical members are allowed: the family is pairwise disjoint if A_i \cap A_j = \varnothing whenever A_i \neq A_j (every two ''distinct'' sets in the family are disjoint).. For example, the collection of sets is d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
