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Darboux Integral
In real analysis, the Darboux integral is constructed using Darboux sums and is one possible definition of the integral of a function. Darboux integrals are equivalent to Riemann integrals, meaning that a function is Darboux-integrable if and only if it is Riemann-integrable, and the values of the two integrals, if they exist, are equal. The definition of the Darboux integral has the advantage of being easier to apply in computations or proofs than that of the Riemann integral. Consequently, introductory textbooks on calculus and real analysis often develop Riemann integration using the Darboux integral, rather than the true Riemann integral. Moreover, the definition is readily extended to defining Riemann–Stieltjes integration. Darboux integrals are named after their inventor, Gaston Darboux (1842–1917). Definition The definition of the Darboux integral considers upper and lower (Darboux) integrals, which exist for any bounded real-valued function f on the interval ,b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real Analysis
In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include convergence, limits, continuity, smoothness, differentiability and integrability. Real analysis is distinguished from complex analysis, which deals with the study of complex numbers and their functions. Scope Construction of the real numbers The theorems of real analysis rely on the properties of the (established) real number system. The real number system consists of an uncountable set (\mathbb), together with two binary operations denoted and \cdot, and a total order denoted . The operations make the real numbers a field, and, along with the order, an ordered field. The real number system is the unique '' complete ordered field'', in the sense that any other complete ordered field is isomorphic to it. Intuitively, completenes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partition Of An Interval
In mathematics, a partition of an interval on the real line is a finite sequence of real numbers such that :. In other terms, a partition of a compact interval is a strictly increasing sequence of numbers (belonging to the interval itself) starting from the initial point of and arriving at the final point of . Every interval of the form is referred to as a subinterval of the partition ''x''. Refinement of a partition Another partition of the given interval , bis defined as a refinement of the partition , if contains all the points of and possibly some other points as well; the partition is said to be “finer” than . Given two partitions, and , one can always form their common refinement, denoted , which consists of all the points of and , in increasing order. Norm of a partition The norm (or mesh) of the partition : is the length of the longest of these subintervals : . Applications Partitions are used in the theory of the Riemann integral, the Riemann� ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Minimum Bounding Rectangle
In computational geometry, the minimum bounding rectangle (MBR), also known as bounding box (BBOX) or envelope, is an expression of the maximum extents of a two-dimensional object (e.g. point, line, polygon) or set of objects within its coordinate system; in other words , , , {{math, max(''y''). The MBR is a 2-dimensional case of the minimum bounding box. MBRs are frequently used as an indication of the general position of a geographic feature or dataset, for either display, first-approximation spatial query, or spatial indexing purposes. The degree to which an "overlapping rectangles" query based on MBRs will be satisfactory (in other words, produce a low number of " false positive" hits) will depend on the extent to which individual spatial objects occupy (fill) their associated MBR. If the MBR is full or nearly so (for example, a mapsheet aligned with axes of latitude and longitude will normally entirely fill its associated MBR in the same coordinate space), then the " ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lebesgue Integration
In mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the axis. The Lebesgue integral, named after 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 generalized in a straightforward way to more gene ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regulated Integral
In mathematics, the regulated integral is a definition of integration for regulated functions, which are defined to be uniform limits of step functions. The use of the regulated integral instead of the Riemann integral has been advocated by Nicolas Bourbaki and Jean Dieudonné. Definition Definition on step functions Let 'a'', ''b''be a fixed closed, bounded interval in the real line R. A real-valued function ''φ'' : 'a'', ''b''→ R is called a step function if there exists a finite partition :\Pi = \ of 'a'', ''b''such that ''φ'' is constant on each open interval (''t''''i'', ''t''''i''+1) of Π; suppose that this constant value is ''c''''i'' ∈ R. Then, define the integral of a step function ''φ'' to be :\int_a^b \varphi(t) \, \mathrm t := \sum_^ c_i , t_ - t_i , . It can be shown that this definition is independent of the choice of partition, in that if Π1 is another partition of 'a'', ''b''such that ''φ'' is constant on the open intervals of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integer
An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative integers. The set (mathematics), set of all integers is often denoted by the boldface or blackboard bold The set of natural numbers \mathbb is a subset of \mathbb, which in turn is a subset of the set of all rational numbers \mathbb, itself a subset of the real numbers \mathbb. Like the set of natural numbers, the set of integers \mathbb is Countable set, countably infinite. An integer may be regarded as a real number that can be written without a fraction, fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, , 5/4, and Square root of 2, are not. The integers form the smallest Group (mathematics), group and the smallest ring (mathematics), ring containing the natural numbers. In algebraic number theory, the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Darboux Refinement
Darboux is a surname. Notable people with the surname include: * Jean Gaston Darboux (1842–1917), French mathematician * Lauriane Doumbouya (née Darboux), the current First Lady of Guinea since 5 September 2021 * Paul Darboux (1919–1982), Beninese politician {{surname ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dense Subset
In topology and related areas of mathematics, a subset ''A'' of a topological space ''X'' is said to be dense in ''X'' if every point of ''X'' either belongs to ''A'' or else is arbitrarily "close" to a member of ''A'' — for instance, the rational numbers are a dense subset of the real numbers because every real number either is a rational number or has a rational number arbitrarily close to it (see Diophantine approximation). Formally, A is dense in X if the smallest closed subset of X containing A is X itself. The of a topological space X is the least cardinality of a dense subset of X. Definition A subset A of a topological space X is said to be a of X if any of the following equivalent conditions are satisfied: The smallest closed subset of X containing A is X itself. The closure of A in X is equal to X. That is, \operatorname_X A = X. The interior of the complement of A is empty. That is, \operatorname_X (X \setminus A) = \varnothing. Every point in X either ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dirichlet Function
In mathematics, the Dirichlet function is the indicator function \mathbf_\Q of the set of rational numbers \Q, i.e. \mathbf_\Q(x) = 1 if is a rational number and \mathbf_\Q(x) = 0 if is not a rational number (i.e. is an irrational number). \mathbf 1_\Q(x) = \begin 1 & x \in \Q \\ 0 & x \notin \Q \end It is named after the mathematician Peter Gustav Lejeune Dirichlet. It is an example of a pathological function which provides counterexamples to many situations. Topological properties Periodicity For any real number and any positive rational number , \mathbf_\Q(x + T) = \mathbf_\Q(x). The Dirichlet function is therefore an example of a real periodic function which is not constant but whose set of periods, the set of rational numbers, is a dense subset of \R. Integration properties See also * Thomae's function, a variation that is discontinuous only at the rational numbers References {{Reflist Dirichlet Johann Peter Gustav Lejeune Dirichlet (; ; 13 February ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lipschitz Continuous
In mathematical analysis, Lipschitz continuity, named after Germany, German mathematician Rudolf Lipschitz, is a strong form of uniform continuity for function (mathematics), functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: there exists a real number such that, for every pair of points on the graph of this function, the absolute value of the slope of the line connecting them is not greater than this real number; the smallest such bound is called the ''Lipschitz constant'' of the function (and is related to the ''modulus of continuity, modulus of uniform continuity''). For instance, every function that is defined on an interval and has a bounded first derivative is Lipschitz continuous. In the theory of differential equations, Lipschitz continuity is the central condition of the Picard–Lindelöf theorem which guarantees the existence and uniqueness of the solution to an initial value problem. A special type of Lipschitz continuity, cal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Darboux
Darboux is a surname. Notable people with the surname include: * Jean Gaston Darboux (1842–1917), French mathematician * Lauriane Doumbouya (née Darboux), the current First Lady of Guinea since 5 September 2021 * Paul Darboux (1919–1982), Beninese politician {{surname ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integral
In mathematics, an integral is the continuous analog of a Summation, sum, which is used to calculate area, areas, volume, 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 Derivative, 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 function, graph of a given Function (mathematics), function between two points in the real line. Conventionally, areas above the horizontal Coordinate axis, axis of the plane are positive while areas below are n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
