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Riemann Integral
In the branch of mathematics known as real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval. It was presented to the faculty at the University of Göttingen in 1854, but not published in a journal until 1868. For many functions and practical applications, the Riemann integral can be evaluated by the fundamental theorem of calculus or approximated by numerical integration, or simulated using Monte Carlo integration. Overview Imagine you have a curve on a graph, and the curve stays above the x-axis between two points, a and b. The area under that curve, from a to b, is what we want to figure out. This area can be described as the set of all points (x, y) on the graph that follow these rules: a ≤ x ≤ b (the x-coordinate is between a and b) and 0 < y < f(x) (the y-coordinate is between 0 and the height of the curve f(x)). Mathematically, this region can be expressed in [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integral As Region Under Curve
In mathematics, an integral is the continuous analog of a sum, which is used to calculate areas, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental operations of calculus,Integral calculus is a very well established mathematical discipline for which there are many sources. See and , for example. the other being differentiation. Integration was initially used to solve problems in mathematics and physics, such as finding the area under a curve, or determining displacement from velocity. Usage of integration expanded to a wide variety of scientific fields thereafter. A definite integral computes the signed area of the region in the plane that is bounded by the graph of a given function between two points in the real line. Conventionally, areas above the horizontal axis of the plane are positive while areas below are negative. Integrals also refer to the concept of an '' antiderivative'', a function whose d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemann Sum
In mathematics, a Riemann sum is a certain kind of approximation of an integral by a finite sum. It is named after nineteenth century German mathematician Bernhard Riemann. One very common application is in numerical integration, i.e., approximating the area of functions or lines on a graph, where it is also known as the rectangle rule. It can also be applied for approximating the length of curves and other approximations. The sum is calculated by partitioning the region into shapes (rectangles, trapezoids, parabolas, or cubics—sometimes infinitesimally small) that together form a region that is similar to the region being measured, then calculating the area for each of these shapes, and finally adding all of these small areas together. This approach can be used to find a numerical approximation for a definite integral even if the fundamental theorem of calculus does not make it easy to find a closed-form solution. Because the region by the small shapes is usually not ... [...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 (its image) 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) \leq A for all x in X, then the function is said to be bounded (from) above by A. If f(x) \geq 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 \mathbb N of natural numbers. Thus a sequence f = (a_0, a_1, a_2, \ldots) 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 boundedness can be generalized to functions f: X \rightarrow Y taking ... [...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 (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called scalar (mathematics), ''scalars''. The operations of vector addition and scalar multiplication must satisfy certain requirements, called ''vector axioms''. Real vector spaces and complex vector spaces are kinds of vector spaces based on different kinds of scalars: real numbers and complex numbers. Scalars can also be, more generally, elements of any field (mathematics), field. Vector spaces generalize Euclidean vectors, which allow modeling of Physical quantity, physical quantities (such as forces and velocity) that have not only a Magnitude (mathematics), magnitude, but also a Orientation (geometry), direction. The concept of vector spaces is fundamental for linear algebra, together with the concept of matrix (mathematics), matrices, which ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Form
In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear mapIn some texts the roles are reversed and vectors are defined as linear maps from covectors to scalars from a vector space to its field (mathematics), field of scalar (mathematics), scalars (often, the real numbers or the complex numbers). If is a vector space over a field , the set of all linear functionals from to is itself a vector space over with addition and scalar multiplication defined pointwise. This space is called the dual space of , or sometimes the algebraic dual space, when a topological dual space is also considered. It is often denoted , p. 19, §3.1 or, when the field is understood, V^*; other notations are also used, such as V', V^ or V^. When vectors are represented by column vectors (as is common when a basis (linear algebra), basis is fixed), then linear functionals are represented as row vectors, and their values on specific vectors are given by matri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cofinal (mathematics)
In mathematics, a subset B \subseteq A of a preordered set (A, \leq) is said to be cofinal or frequent in A if for every a \in A, it is possible to find an element b in B that dominates a (formally, a\leq b). Cofinal subsets are very important in the theory of directed sets and nets, where “ cofinal subnet” is the appropriate generalization of " subsequence". They are also important in order theory, including the theory of cardinal numbers, where the minimum possible cardinality of a cofinal subset of A is referred to as the cofinality of A. Definitions Let \,\leq\, be a homogeneous binary relation on a set A. A subset B \subseteq A is said to be or with respect to \,\leq\, if it satisfies the following condition: :For every a \in A, there exists some b \in B that a \leq b. A subset that is not frequent is called . This definition is most commonly applied when (A, \leq) is a directed set, which is a preordered set with additional properties. ;Final functions ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jordan Measure
Jordan, officially the Hashemite Kingdom of Jordan, is a country in the Southern Levant region of West Asia. Jordan is bordered by Syria to the north, Iraq to the east, Saudi Arabia to the south, and Israel and the occupied Palestinian territories to the west. The Jordan River, flowing into the Dead Sea, is located along the country's western border within the Jordan Rift Valley. Jordan has a small coastline along the Red Sea in its southwest, separated by the Gulf of Aqaba from Egypt. Amman is the country's capital and largest city, as well as the most populous city in the Levant. Inhabited by humans since the Paleolithic period, three kingdoms developed in Transjordan during the Iron Age: Ammon, Moab and Edom. In the third century BC, the Arab Nabataeans established their kingdom centered in Petra. The Greco-Roman period saw the establishment of several cities in Transjordan that comprised the Decapolis. Later, after the end of Byzantine rule, the region beca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Smith–Volterra–Cantor Set
In mathematics, the Smith–Volterra–Cantor set (SVC), ε-Cantor set, or fat Cantor set is an example of a set of points on the real line that is nowhere dense (in particular it contains no intervals), yet has positive measure. The Smith–Volterra–Cantor set is named after the mathematicians Henry Smith, Vito Volterra and Georg Cantor. In an 1875 paper, Smith discussed a nowhere-dense set of positive measure on the real line, and Volterra introduced a similar example in 1881. The Cantor set as we know it today followed in 1883. The Smith–Volterra–Cantor set is topologically equivalent to the middle-thirds Cantor set. Construction Similar to the construction of the Cantor set, the Smith–Volterra–Cantor set is constructed by removing certain intervals from the unit interval , 1 The process begins by removing the middle 1/4 from the interval , 1/math> (the same as removing 1/8 on either side of the middle point at 1/2) so the remaining set is \left , \tf ... [...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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lebesgue Integral
In mathematics, the integral of a non-negative Function (mathematics), function of a single variable can be regarded, in the simplest case, as the area between the Graph of a function, graph of that function and the axis. The Lebesgue integral, named after france, French mathematician Henri Lebesgue, is one way to make this concept rigorous and to extend it to more general functions. The Lebesgue integral is more general than the Riemann integral, which it largely replaced in mathematical analysis since the first half of the 20th century. It can accommodate functions with discontinuities arising in many applications that are pathological from the perspective of the Riemann integral. The Lebesgue integral also has generally better analytical properties. For instance, under mild conditions, it is possible to exchange limits and Lebesgue integration, while the conditions for doing this with a Riemann integral are comparatively baroque. Furthermore, the Lebesgue integral can be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rational Point
In number theory and algebraic geometry, a rational point of an algebraic variety is a point whose coordinates belong to a given field. If the field is not mentioned, the field of rational numbers is generally understood. If the field is the field of real numbers, a rational point is more commonly called a real point. Understanding rational points is a central goal of number theory and Diophantine geometry. For example, Fermat's Last Theorem may be restated as: for , the Fermat curve of equation x^n+y^n=1 has no other rational points than , , and, if is even, and . Definition Given a field , and an algebraically closed extension of , an affine variety over is the set of common zeros in of a collection of polynomials with coefficients in : :\begin & f_1(x_1,\ldots,x_n)=0, \\ & \qquad \quad \vdots \\ & f_r(x_1,\dots,x_n)=0. \end These common zeros are called the ''points'' of . A -rational point (or -point) of is a point of that belongs to , that is, a sequence (a_1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Indicator Function
In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if is a subset of some set , then the indicator function of is the function \mathbf_A defined by \mathbf_\!(x) = 1 if x \in A, and \mathbf_\!(x) = 0 otherwise. Other common notations are and \chi_A. The indicator function of is the Iverson bracket of the property of belonging to ; that is, \mathbf_(x) = \left x\in A\ \right For example, the Dirichlet function is the indicator function of the rational numbers as a subset of the real numbers. Definition Given an arbitrary set , the indicator function of a subset of is the function \mathbf_A \colon X \mapsto \ defined by \operatorname\mathbf_A\!( x ) = \begin 1 & \text x \in A \\ 0 & \text x \notin A \,. \end The Iverson bracket provides the equivalent notation \left x\in A\ \right/math> or that can be used instead of \mathbf_\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
