1.
Mathematics
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Mathematics is the study of topics such as quantity, structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope, Mathematicians seek out patterns and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof, when mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry, rigorous arguments first appeared in Greek mathematics, most notably in Euclids Elements. Galileo Galilei said, The universe cannot be read until we have learned the language and it is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth, carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. Benjamin Peirce called mathematics the science that draws necessary conclusions, David Hilbert said of mathematics, We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules, rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise. Albert Einstein stated that as far as the laws of mathematics refer to reality, they are not certain, Mathematics is essential in many fields, including natural science, engineering, medicine, finance and the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics, Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, the history of mathematics can be seen as an ever-increasing series of abstractions. The earliest uses of mathematics were in trading, land measurement, painting and weaving patterns, in Babylonian mathematics elementary arithmetic first appears in the archaeological record. Numeracy pre-dated writing and numeral systems have many and diverse. Between 600 and 300 BC the Ancient Greeks began a study of mathematics in its own right with Greek mathematics. Mathematics has since been extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made today, the overwhelming majority of works in this ocean contain new mathematical theorems and their proofs. The word máthēma is derived from μανθάνω, while the modern Greek equivalent is μαθαίνω, in Greece, the word for mathematics came to have the narrower and more technical meaning mathematical study even in Classical times
2.
Integral
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In mathematics, an integral assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining infinitesimal data. Integration is one of the two operations of calculus, with its inverse, differentiation, being the other. The area above the x-axis adds to the total and that below the x-axis subtracts from the total, roughly speaking, the operation of integration is the reverse of differentiation. For this reason, the integral may also refer to the related notion of the antiderivative. In this case, it is called an integral and is written. The integrals discussed in this article are those termed definite integrals, a rigorous mathematical definition of the integral was given by Bernhard Riemann. It is based on a procedure which approximates the area of a curvilinear region by breaking the region into thin vertical slabs. A line integral is defined for functions of two or three variables, and the interval of integration is replaced by a curve connecting two points on the plane or in the space. In a surface integral, the curve is replaced by a piece of a surface in the three-dimensional space and this method was further developed and employed by Archimedes in the 3rd century BC and used to calculate areas for parabolas and an approximation to the area of a circle. A similar method was developed in China around the 3rd century AD by Liu Hui. This method was used in the 5th century by Chinese father-and-son mathematicians Zu Chongzhi. The next significant advances in integral calculus did not begin to appear until the 17th century, further steps were made in the early 17th century by Barrow and Torricelli, who provided the first hints of a connection between integration and differentiation. Barrow provided the first proof of the theorem of calculus. Wallis generalized Cavalieris method, computing integrals of x to a power, including negative powers. The major advance in integration came in the 17th century with the independent discovery of the theorem of calculus by Newton. The theorem demonstrates a connection between integration and differentiation and this connection, combined with the comparative ease of differentiation, can be exploited to calculate integrals. In particular, the theorem of calculus allows one to solve a much broader class of problems. Equal in importance is the mathematical framework that both Newton and Leibniz developed
3.
Uniform norm
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In mathematical analysis, the uniform norm assigns to real- or complex-valued bounded functions f defined on a set S the non-negative number ∥ f ∥ ∞ = ∥ f ∥ ∞, S = sup. This norm is called the supremum norm, the Chebyshev norm. The name uniform norm derives from the fact that a sequence of functions converges to f under the derived from the uniform norm if. In this case, the norm is called the maximum norm. In particular, for the case of a vector x = in finite dimensional coordinate space, it takes the form ∥ x ∥ ∞ = max. The reason for the subscript ∞ is that f is continuous lim p → ∞ ∥ f ∥ p = ∥ f ∥ ∞. The binary function d = ∥ f − g ∥ ∞ is then a metric on the space of all bounded functions on a particular domain. A sequence converges uniformly to a function f if and only if lim n → ∞ ∥ f n − f ∥ ∞ =0. We can define closed sets and closures of sets with respect to this topology, closed sets in the uniform norm are sometimes called uniformly closed. The uniform closure of a set of functions A is the space of all functions that can be approximated by a sequence of uniformly-converging functions on A. For instance, one restatement of the Stone–Weierstrass theorem is that the set of all functions on is the uniform closure of the set of polynomials on. For complex continuous functions over a space, this turns it into a C* algebra. Chebyshev distance Uniform continuity Uniform space
4.
Step function
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In mathematics, a function on the real numbers is called a step function if it can be written as a finite linear combination of indicator functions of intervals. Informally speaking, a function is a piecewise constant function having only finitely many pieces. Indeed, if that is not the case to start with, a constant function is a trivial example of a step function. Then there is only one interval, A0 = R, the sign function sgn , which is −1 for negative numbers and +1 for positive numbers, and is the simplest non-constant step function. The Heaviside function H, which is 0 for negative numbers and 1 for positive numbers, is an important step function and it is the mathematical concept behind some test signals, such as those used to determine the step response of a dynamical system. The rectangular function, the boxcar function, is the next simplest step function. The integer part function is not a step function according to the definition of this article, however, some authors also define step functions with an infinite number of intervals. The sum and product of two functions is again a step function. The product of a function with a number is also a step function. As such, the functions form an algebra over the real numbers. A step function takes only a number of values. If the intervals A i, i =0,1, …, n, in the definition of the step function are disjoint and their union is the real line. The definite integral of a function is a piecewise linear function. In fact, this equality can be the first step in constructing the Lebesgue integral, unit step function Crenel function Simple function Piecewise defined function Sigmoid function Step detection
5.
Riemann integral
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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. For many functions and practical applications, the Riemann integral can be evaluated by the theorem of calculus or approximated by numerical integration. The Riemann integral is unsuitable for many theoretical purposes, some of the technical deficiencies in Riemann integration can be remedied with the Riemann–Stieltjes integral, and most disappear with the Lebesgue integral. Let f be a nonnegative real-valued function on the interval, and let S = be the region of the plane under the graph of the function f and we are interested in measuring the area of S. Once we have measured it, we denote the area by. The basic idea of the Riemann integral is to use very simple approximations for the area of S, by taking better and better approximations, we can say that in the limit we get exactly the area of S under the curve. A partition of an interval is a sequence of numbers of the form a = x 0 < x 1 < x 2 < ⋯ < x n = b Each is called a subinterval of the partition. The mesh or norm of a partition is defined to be the length of the longest subinterval, a tagged partition P of an interval is a partition together with a finite sequence of numbers t0. Tn −1 subject to the conditions that for each i, in other words, it is a partition together with a distinguished point of every subinterval. The mesh of a partition is the same as that of an ordinary partition. Suppose that two partitions P and Q are both partitions of the interval. We say that Q is a refinement of P if for each i, with i ∈, there exists an integer r such that xi = yr and such that ti = sj for some j with j ∈ [r. Said more simply, a refinement of a tagged partition breaks up some of the subintervals and adds tags to the partition where necessary, thus it refines the accuracy of the partition. We can define a partial order on the set of all tagged partitions by saying that one tagged partition is greater or equal to if the former is a refinement of the latter. Let f be a function defined on the interval. The Riemann sum of f with respect to the tagged partition x0, tn −1 is ∑ i =0 n −1 f. Each term in the sum is the product of the value of the function at a given point, consequently, each term represents the area of a rectangle with height f and width xi +1 − xi. The Riemann sum is the area of all the rectangles, a closely related concept are the lower and upper Darboux sums
6.
Nicolas Bourbaki
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With the goal of grounding all of mathematics on set theory, the group strove for rigour and generality. Their work led to the discovery of several concepts and terminologies still used, in 1934, young French mathematicians from various French universities felt the need to form a group to jointly produce textbooks that they could all use for teaching. André Weil organized the first meeting on 10 December 1934 in the basement of a Parisian grill room, Bourbakis main work is the Elements of Mathematics series. This series aims to be a completely self-contained treatment of the areas of modern mathematics. Assuming no special knowledge of mathematics, it takes up mathematics from the beginning, proceeds axiomatically. The volume on spectral theory from 1967 was for almost four decades the last new book to be added to the series, after that several new chapters to existing books as well as revised editions of existing chapters appeared until the publication of chapters 8-9 of Commutative Algebra in 1983. Then a long break in publishing activity occurred, leading many to suspect the end of the publishing project, however, chapter 10 of Commutative Algebra appeared in 1998, and after another long break a completely re-written and expanded chapter 8 of Algèbre was published in 2012. More importantly, the first four chapters of a new book on algebraic topology were published in 2016. Besides the Éléments de mathématique series, lectures from the Séminaire Bourbaki also have been published in monograph form since 1948. Notations introduced by Bourbaki include the symbol ∅ for the empty set and a dangerous bend symbol ☡, and the terms injective, surjective, and bijective. The emphasis on rigour may be seen as a reaction to the work of Henri Poincaré, the impact of Bourbakis work initially was great on many active research mathematicians world-wide. For example, Our time is witnessing the creation of a monumental work and it provoked some hostility, too, mostly on the side of classical analysts, they approved of rigour but not of high abstraction. This led to a gulf with the way theoretical physics was practiced, Bourbakis direct influence has decreased over time. This is partly because certain concepts which are now important, such as the machinery of category theory, are not covered in the treatise. It also mattered that, while especially algebraic structures can be defined in Bourbakis terms. On the other hand, the approach and rigour advocated by Bourbaki have permeated the current mathematical practices to such extent that the task undertaken was completed and this is particularly true for the less applied parts of mathematics. The Bourbaki seminar series founded in post-WWII Paris continues, it has going on since 1948. It is an important source of survey articles, with sketches of proofs, the topics range through all branches of mathematics, including sometimes theoretical physics
7.
Bounded set
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Bounded and boundary are distinct concepts, for the latter see boundary. A circle in isolation is a bounded set, while the half plane is unbounded yet has a boundary. In mathematical analysis and related areas of mathematics, a set is called bounded, if it is, in a certain sense, conversely, a set which is not bounded is called unbounded. The word bounded makes no sense in a topological space without a metric. A set S of real numbers is called bounded from above if there is a number k such that k ≥ s for all s in S. The number k is called a bound of S. The terms bounded from below and lower bound are similarly defined, a set S is bounded if it has both upper and lower bounds. Therefore, a set of numbers is bounded if it is contained in a finite interval. M is a metric space if M is bounded as a subset of itself. For subsets of Rn the two are equivalent, a metric space is compact if and only if it is complete and totally bounded. A subset of Euclidean space Rn is compact if and only if it is closed and bounded, in topological vector spaces, a different definition for bounded sets exists which is sometimes called von Neumann boundedness. If the topology of the vector space is induced by a metric which is homogeneous, as in the case of a metric induced by the norm of normed vector spaces. A set of numbers is bounded if and only if it has an upper and lower bound. This definition is extendable to subsets of any ordered set. Note that this general concept of boundedness does not correspond to a notion of size. A subset S of an ordered set P is called bounded above if there is an element k in P such that k ≥ s for all s in S. The element k is called a bound of S. The concepts of bounded below and lower bound are defined similarly, a subset S of a partially ordered set P is called bounded if it has both an upper and a lower bound, or equivalently, if it is contained in an interval
8.
Real line
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In mathematics, the real line, or real number line is the line whose points are the real numbers. That is, the line is the set R of all real numbers, viewed as a geometric space. It can be thought of as a space, a metric space, a topological space. Just like the set of numbers, the real line is usually denoted by the symbol R. However. This article focuses on the aspects of R as a space in topology, geometry. The real numbers also play an important role in algebra as a field, for more information on R in all of its guises, see real number. The real line is a linear continuum under the standard < ordering, specifically, the real line is linearly ordered by <, and this ordering is dense and has the least-upper-bound property. In addition to the properties, the real line has no maximum or minimum element. It also has a dense subset, namely the set of rational numbers. It is a theorem that any linear continuum with a dense subset. The real line also satisfies the countable chain condition, every collection of mutually disjoint, in order theory, the famous Suslin problem asks whether every linear continuum satisfying the countable chain condition that has no maximum or minimum element is necessarily order-isomorphic to R. This statement has been shown to be independent of the axiomatic system of set theory known as ZFC. The real line forms a space, with the distance function given by absolute difference. The metric tensor is clearly the 1-dimensional Euclidean metric, since the n-dimensional Euclidean metric can be represented in matrix form as the n by n identity matrix, the metric on the real line is simply the 1 by 1 identity matrix, i. e.1. If p ∈ R and ε >0, then the ε-ball in R centered at p is simply the open interval. This real line has several important properties as a space, The real line is a complete metric space. The real line is path-connected, and is one of the simplest examples of a metric space The Hausdorff dimension of the real line is equal to one. The real line carries a standard topology which can be introduced in two different, equivalent ways, first, since the real numbers are totally ordered, they carry an order topology
9.
Partition of an interval
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In mathematics, a partition of an interval on the real line is a finite sequence x0, x1, x2. Xk of real numbers such that a = x0 < x1 < x2 <, in other terms, a partition of a compact interval I is a strictly increasing sequence of numbers starting from the initial point of I and arriving at the final point of I. Every interval of the form is referred to as a subinterval of the partition x, given two partitions, P and Q, one can always form their common refinement, denoted P ∨ Q, which consists of all the points of P and Q, re-numbered in order. The norm of the partition x0 < x1 < x2 <, < xn is the length of the longest of these subintervals, that is max. Partitions are used in the theory of the Riemann integral, the Riemann–Stieltjes integral, specifically, as finer partitions of a given interval are considered, their mesh approaches zero and the Riemann sum based on a given partition approaches the Riemann integral. A tagged partition is a partition of a given interval together with a sequence of numbers t0. Tn −1 subject to the conditions that for each i, in other words, a tagged partition is a partition together with a distinguished point of every subinterval, its mesh is defined in the same way as for an ordinary partition. It is possible to define a partial order on the set of all tagged partitions by saying that one tagged partition is bigger than if the bigger one is a refinement of the smaller one. Tn −1 is a partition of, and that y0. Sm −1 is another tagged partition of, sm −1 together is a refinement of a tagged partition x0. Tn −1 if for each i with 0 ≤ i ≤ n, there is an integer r such that xi = yr. Said more simply, a refinement of a tagged partition takes the starting partition and adds more tags, regulated integral Riemann integral Riemann–Stieltjes integral Partition of a set Gordon, Russell A. The integrals of Lebesgue, Denjoy, Perron, and Henstock
10.
Open set
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In topology, an open set is an abstract concept generalizing the idea of an open interval in the real line. These conditions are very loose, and they allow enormous flexibility in the choice of open sets, in the two extremes, every set can be open, or no set can be open but the space itself and the empty set. In practice, however, open sets are usually chosen to be similar to the intervals of the real line. The notion of an open set provides a way to speak of nearness of points in a topological space. Once a choice of open sets is made, the properties of continuity, connectedness, and compactness, each choice of open sets for a space is called a topology. Although open sets and the topologies that they comprise are of importance in point-set topology. Intuitively, an open set provides a method to distinguish two points, for example, if about one point in a topological space there exists an open set not containing another point, the two points are referred to as topologically distinguishable. In this manner, one may speak of two subsets of a topological space are near without concretely defining a metric on the topological space. Therefore, topological spaces may be seen as a generalization of metric spaces, in the set of all real numbers, one has the natural Euclidean metric, that is, a function which measures the distance between two real numbers, d = |x - y|. Therefore, given a number, one can speak of the set of all points close to that real number. In essence, points within ε of x approximate x to an accuracy of degree ε, note that ε >0 always but as ε becomes smaller and smaller, one obtains points that approximate x to a higher and higher degree of accuracy. For example, if x =0 and ε =1, the points within ε of x are precisely the points of the interval, that is, however, with ε =0.5, the points within ε of x are precisely the points of. Clearly, these points approximate x to a degree of accuracy compared to when ε =1. The previous discussion shows, for the case x =0, in particular, sets of the form give us a lot of information about points close to x =0. Thus, rather than speaking of a concrete Euclidean metric, one may use sets to describe points close to x, thus, we find that in some sense, every real number is distance 0 away from 0. It may help in case to think of the measure as being a binary condition, all things in R are equally close to 0. In general, one refers to the family of sets containing 0, used to approximate 0, as a neighborhood basis, in fact, one may generalize these notions to an arbitrary set, rather than just the real numbers. In this case, given a point of that set, one may define a collection of sets around x, of course, this collection would have to satisfy certain properties for otherwise we may not have a well-defined method to measure distance
11.
Bounded function
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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 | ≤ M for all x in X, a function that is not bounded is said to be unbounded. Sometimes, if f ≤ A for all x in X, on the other hand, if f ≥ B for all x in X, then the function is said to be bounded below by B. The concept should not be confused with that of a bounded operator, an important special case is a bounded sequence, where X is taken to be the set N of natural numbers. Thus a sequence f = is bounded if there exists a real number M such that | a n | ≤ M for every natural number n, the set of all bounded sequences, equipped with a vector space structure, forms a sequence space. This definition can be extended to functions taking values in a metric space Y, if this is the case, there is also such an M for each other a, by the triangle inequality. The function f, R → R defined by f = sin is bounded, the sine function is no longer bounded if it is defined over the set of all complex numbers. The function f =1 x 2 −1 defined for all x except for −1 and 1 is unbounded. As x gets closer to −1 or to 1, the values of this function get larger and larger in magnitude and this function can be made bounded if one considers its domain to be, for example. The function f =1 x 2 +1 defined for all x is bounded. Every continuous function f, → R is bounded and this is really a special case of a more general fact, Every continuous function from a compact space into a metric space is bounded. The function f takes the value 0 for x rational number and 1 for x irrational number is bounded. Thus, a function does not need to be nice in order to be bounded, the set of all bounded functions defined on is much bigger than the set of continuous functions on that interval
12.
Linear subspace
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A linear subspace is usually called simply a subspace when the context serves to distinguish it from other kinds of subspaces. Let K be a field, V be a space over K. Then W is a if, The zero vector,0, is in W. If u and v are elements of W, then the sum u + v is an element of W, take W to be the set of all vectors in V whose last component is 0. Then W is a subspace of V. Proof, Given u and v in W, Thus, u + v is an element of W, too. Given u in W and a c in R, if u = again. Thus, cu is an element of W too, example II, Let the field be R again, but now let the vector space be the Cartesian plane R2. Take W to be the set of points of R2 such that x = y, then W is a subspace of R2. Proof, Let p = and q = be elements of W, then p + q =, since p1 = p2 and q1 = q2, then p1 + q1 = p2 + q2, so p + q is an element of W. Let p = be an element of W, that is, a point in the plane such that p1 = p2, then cp =, since p1 = p2, then cp1 = cp2, so cp is an element of W. In general, any subset of the coordinate space Rn that is defined by a system of homogeneous linear equations will yield a subspace. Geometrically, these subspaces are points, lines, planes, and so on, example III, Again take the field to be R, but now let the vector space V be the set RR of all functions from R to R. Let C be the subset consisting of continuous functions, then C is a subspace of RR. Proof, We know from calculus that 0 ∈ C ⊂ RR and we know from calculus that the sum of continuous functions is continuous. Again, we know from calculus that the product of a continuous function, example IV, Keep the same field and vector space as before, but now consider the set Diff of all differentiable functions. The same sort of argument as before shows that this is a subspace too, examples that extend these themes are common in functional analysis. A way to characterize subspaces is that they are closed under linear combinations, in a topological vector space X, a subspace W need not be closed in general, but a finite-dimensional subspace is always closed. The same is true for subspaces of finite codimension, i. e. determined by a number of continuous linear functionals
13.
Linear operator
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In mathematics, a linear map is a mapping V → W between two modules that preserves the operations of addition and scalar multiplication. An important special case is when V = W, in case the map is called a linear operator, or an endomorphism of V. Sometimes the term linear function has the meaning as linear map. A linear map always maps linear subspaces onto linear subspaces, for instance it maps a plane through the origin to a plane, Linear maps can often be represented as matrices, and simple examples include rotation and reflection linear transformations. In the language of algebra, a linear map is a module homomorphism. In the language of category theory it is a morphism in the category of modules over a given ring, let V and W be vector spaces over the same field K. e. that for any vectors x1. Am ∈ K, the equality holds, f = a 1 f + ⋯ + a m f. It is then necessary to specify which of these fields is being used in the definition of linear. If V and W are considered as spaces over the field K as above, for example, the conjugation of complex numbers is an R-linear map C → C, but it is not C-linear. A linear map from V to K is called a linear functional and these statements generalize to any left-module RM over a ring R without modification, and to any right-module upon reversing of the scalar multiplication. The zero map between two left-modules over the ring is always linear. The identity map on any module is a linear operator, any homothecy centered in the origin of a vector space, v ↦ c v where c is a scalar, is a linear operator. This does not hold in general for modules, where such a map might only be semilinear, for real numbers, the map x ↦ x2 is not linear. Conversely, any map between finite-dimensional vector spaces can be represented in this manner, see the following section. Differentiation defines a map from the space of all differentiable functions to the space of all functions. It also defines an operator on the space of all smooth functions. If V and W are finite-dimensional vector spaces over a field F, then functions that send linear maps f, V → W to dimF × dimF matrices in the way described in the sequel are themselves linear maps. The expected value of a variable is linear, as for random variables X and Y we have E = E + E and E = aE
14.
Discrete set
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In mathematics, a point x is called an isolated point of a subset S if x is an element of S but there exists a neighborhood of x which does not contain any other points of S. This is equivalent to saying that the singleton is a set in the topological space S. If the space X is a Euclidean space, then x is a point of S if there exists an open ball around x which contains no other points of S. A set that is made up only of isolated points is called a discrete set, however, not every countable set is discrete, of which the rational numbers under the usual Euclidean metric are the canonical example. A set with no isolated point is said to be dense-in-itself, a closed set with no isolated point is called a perfect set. The number of isolated points is an invariant, i. e. if two topological spaces X and Y are homeomorphic, the number of isolated points in each is equal. Topological spaces in the examples are considered as subspaces of the real line with the standard topology. For the set S = ∪, the point 0 is an isolated point. For the set S = ∪, each of the points 1/k is an isolated point, the set N = of natural numbers is a discrete set. The Morse lemma states that non-degenerate critical points of functions are isolated. Let us consider the set F of points x in the interval such that every digit of their binary representation x i fulfills the following conditions. X i =1 only for finitely many indexes i, if m denotes the biggest index such that x m =1, then x m −1 =0. If x i =1 and i < m, then one of the following two condition holds, x i −1 =1, x i +1 =1. Informally, this means that every digit of the binary representation of x equals to one, has a consecutive pair. Now, F is a set consisting entirely of isolated points. Besides, F has the property that its closure is an uncountable set. Another set F with the properties can be obtained as follows. Let C be the middle-thirds Cantor set, let I1, I2, I3, … be the component intervals of − C, since each I k contains only one point from F, every point of F is an isolated point
15.
Limit point
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Note that x does not have to be an element of S. This concept profitably generalizes the notion of a limit and is the underpinning of concepts such as closed set, Let S be a subset of a topological space X. A point x in X is a point of S if every neighbourhood of x contains at least one point of S different from x itself. Note that it doesnt make a difference if we restrict the condition to open neighbourhoods only and this is equivalent, in a T1 space, to requiring that every neighbourhood of x contains infinitely many points of S. If every open set containing x contains infinitely many points of S then x is a type of limit point called an ω-accumulation point of S. If every open set containing x contains uncountably many points of S then x is a type of limit point called a condensation point of S. If every open set U containing x satisfies |U ∩ S| = |S| then x is a type of limit point called a complete accumulation point of S. A point x ∈ X is a point or accumulation point of a sequence n ∈ N if, for every neighbourhood V of x. If the space is Fréchet–Urysohn, this is equivalent to the assertion that x is a limit of some subsequence of the sequence n ∈ N, the set of all cluster points of a sequence is sometimes called a limit set. The concept of a net generalizes the idea of a sequence, Let n, → X be a net, where is a directed set. Cluster points in nets encompass the idea of both points and ω-accumulation points. Clustering and limit points are defined for the related topic of filters. We have the following characterisation of limit points, x is a point of S if. Proof, We use the fact that a point is in the closure of a set if, If x is in S, we are done. If x is not in S, then every neighbourhood of x contains a point of S, in other words, x is a limit point of S and x is in L. If x is in S, then every neighbourhood of x clearly meets S, If x is in L, then every neighbourhood of x contains a point of S, so x is again in the closure of S. A corollary of this gives us a characterisation of closed sets, A set S is closed if. Proof, S is closed if and only if S is equal to its closure if and only if S = S ∪ L if, another proof, Let S be a closed set and x a limit point of S
16.
Compact space
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In mathematics, and more specifically in general topology, compactness is a property that generalizes the notion of a subset of Euclidean space being closed and bounded. Examples include a closed interval, a rectangle, or a set of points. This notion is defined for general topological spaces than Euclidean space in various ways. One such generalization is that a space is compact if any infinite sequence of points sampled from the space must frequently get arbitrarily close to some point of the space. An equivalent definition is that every sequence of points must have an infinite subsequence that converges to some point of the space, the Heine-Borel theorem states that a subset of Euclidean space is compact in this sequential sense if and only if it is closed and bounded. Thus, if one chooses a number of points in the closed unit interval some of those points must get arbitrarily close to some real number in that space. For instance, some of the numbers 1/2, 4/5, 1/3, 5/6, 1/4, 6/7, the same set of points would not accumulate to any point of the open unit interval, so the open unit interval is not compact. Euclidean space itself is not compact since it is not bounded, in particular, the sequence of points 0, 1, 2, 3, … has no subsequence that converges to any given real number. Apart from closed and bounded subsets of Euclidean space, typical examples of compact spaces include spaces consisting not of geometrical points, the term compact was introduced into mathematics by Maurice Fréchet in 1904 as a distillation of this concept. Various equivalent notions of compactness, including sequential compactness and limit point compactness, in general topological spaces, however, different notions of compactness are not necessarily equivalent. This more subtle notion, introduced by Pavel Alexandrov and Pavel Urysohn in 1929, the term compact set is sometimes a synonym for compact space, but usually refers to a compact subspace of a topological space. In the 19th century, several disparate mathematical properties were understood that would later be seen as consequences of compactness. On the one hand, Bernard Bolzano had been aware that any bounded sequence of points has a subsequence that must eventually get close to some other point. The process could then be repeated by dividing the resulting smaller interval into smaller and smaller parts until it closes down on the limit point. The full significance of Bolzanos theorem, and its method of proof, in the 1880s, it became clear that results similar to the Bolzano–Weierstrass theorem could be formulated for spaces of functions rather than just numbers or geometrical points. The idea of regarding functions as points of a generalized space dates back to the investigations of Giulio Ascoli. The uniform limit of this sequence then played precisely the same role as Bolzanos limit point and this ultimately led to the notion of a compact operator as an offshoot of the general notion of a compact space. It was Maurice Fréchet who, in 1906, had distilled the essence of the Bolzano–Weierstrass property, in 1870, Eduard Heine showed that a continuous function defined on a closed and bounded interval was in fact uniformly continuous
17.
Lebesgue integration
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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 x-axis. The Lebesgue integral extends the integral to a class of functions. It also extends the domains on which these functions can be defined, however, as the need to consider more irregular functions arose—e. g. Also, one wish to integrate on spaces more general than the real line. The Lebesgue integral provides the right abstractions needed to do this important job, the Lebesgue integral plays an important role in probability theory, in the branch of mathematics called real analysis and in many other fields in the mathematical sciences. It is named after Henri Lebesgue, who introduced the integral and it is also a pivotal part of the axiomatic theory of probability. The integral of a function f between limits a and b can be interpreted as the area under the graph of f and this is easy to understand for familiar functions such as polynomials, but what does it mean for more exotic functions. In general, for class of functions does area under the curve make sense. The answer to this question has great theoretical and practical importance, as part of a general movement toward rigor in mathematics in the nineteenth century, mathematicians attempted to put integral calculus on a firm foundation. The Riemann integral—proposed by Bernhard Riemann —is a broadly successful attempt to provide such a foundation, riemanns definition starts with the construction of a sequence of easily calculated areas that converge to the integral of a given function. This definition is successful in the sense that it gives the answer for many already-solved problems. However, Riemann integration does not interact well with taking limits of sequences of functions and this is important, for instance, in the study of Fourier series, Fourier transforms, and other topics. The Lebesgue integral is able to describe how and when it is possible to take limits under the integral sign. For this reason, the Lebesgue definition makes it possible to calculate integrals for a class of functions. For example, the Dirichlet function, which is 0 where its argument is irrational and 1 otherwise, has a Lebesgue integral, but does not have a Riemann integral. Lebesgue summarized his approach to integration in a letter to Paul Montel, I have to pay a certain sum, I take the bills and coins out of my pocket and give them to the creditor in the order I find them until I have reached the total sum. After I have taken all the out of my pocket I order the bills and coins according to identical values. The insight is that one should be able to rearrange the values of a function freely and this process of rearrangement can convert a very pathological function into one that is nice from the point of view of integration, and thus let such pathological functions be integrated
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JSTOR
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JSTOR is a digital library founded in 1995. Originally containing digitized back issues of journals, it now also includes books and primary sources. It provides full-text searches of almost 2,000 journals, more than 8,000 institutions in more than 160 countries have access to JSTOR, most access is by subscription, but some older public domain content is freely available to anyone. William G. Bowen, president of Princeton University from 1972 to 1988, JSTOR originally was conceived as a solution to one of the problems faced by libraries, especially research and university libraries, due to the increasing number of academic journals in existence. Most libraries found it prohibitively expensive in terms of cost and space to maintain a collection of journals. By digitizing many journal titles, JSTOR allowed libraries to outsource the storage of journals with the confidence that they would remain available long-term, online access and full-text search ability improved access dramatically. Bowen initially considered using CD-ROMs for distribution, JSTOR was initiated in 1995 at seven different library sites, and originally encompassed ten economics and history journals. JSTOR access improved based on feedback from its sites. Special software was put in place to make pictures and graphs clear, with the success of this limited project, Bowen and Kevin Guthrie, then-president of JSTOR, wanted to expand the number of participating journals. They met with representatives of the Royal Society of London and an agreement was made to digitize the Philosophical Transactions of the Royal Society dating from its beginning in 1665, the work of adding these volumes to JSTOR was completed by December 2000. The Andrew W. Mellon Foundation funded JSTOR initially, until January 2009 JSTOR operated as an independent, self-sustaining nonprofit organization with offices in New York City and in Ann Arbor, Michigan. JSTOR content is provided by more than 900 publishers, the database contains more than 1,900 journal titles, in more than 50 disciplines. Each object is identified by an integer value, starting at 1. In addition to the site, the JSTOR labs group operates an open service that allows access to the contents of the archives for the purposes of corpus analysis at its Data for Research service. This site offers a facility with graphical indication of the article coverage. Users may create focused sets of articles and then request a dataset containing word and n-gram frequencies and they are notified when the dataset is ready and may download it in either XML or CSV formats. The service does not offer full-text, although academics may request that from JSTOR, JSTOR Plant Science is available in addition to the main site. The materials on JSTOR Plant Science are contributed through the Global Plants Initiative and are only to JSTOR
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International Standard Book Number
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The International Standard Book Number is a unique numeric commercial book identifier. An ISBN is assigned to each edition and variation of a book, for example, an e-book, a paperback and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, the method of assigning an ISBN is nation-based and varies from country to country, often depending on how large the publishing industry is within a country. The initial ISBN configuration of recognition was generated in 1967 based upon the 9-digit Standard Book Numbering created in 1966, the 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108. Occasionally, a book may appear without a printed ISBN if it is printed privately or the author does not follow the usual ISBN procedure, however, this can be rectified later. Another identifier, the International Standard Serial Number, identifies periodical publications such as magazines, the ISBN configuration of recognition was generated in 1967 in the United Kingdom by David Whitaker and in 1968 in the US by Emery Koltay. The 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108, the United Kingdom continued to use the 9-digit SBN code until 1974. The ISO on-line facility only refers back to 1978, an SBN may be converted to an ISBN by prefixing the digit 0. For example, the edition of Mr. J. G. Reeder Returns, published by Hodder in 1965, has SBN340013818 -340 indicating the publisher,01381 their serial number. This can be converted to ISBN 0-340-01381-8, the check digit does not need to be re-calculated, since 1 January 2007, ISBNs have contained 13 digits, a format that is compatible with Bookland European Article Number EAN-13s. An ISBN is assigned to each edition and variation of a book, for example, an ebook, a paperback, and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, a 13-digit ISBN can be separated into its parts, and when this is done it is customary to separate the parts with hyphens or spaces. Separating the parts of a 10-digit ISBN is also done with either hyphens or spaces, figuring out how to correctly separate a given ISBN number is complicated, because most of the parts do not use a fixed number of digits. ISBN issuance is country-specific, in that ISBNs are issued by the ISBN registration agency that is responsible for country or territory regardless of the publication language. Some ISBN registration agencies are based in national libraries or within ministries of culture, in other cases, the ISBN registration service is provided by organisations such as bibliographic data providers that are not government funded. In Canada, ISBNs are issued at no cost with the purpose of encouraging Canadian culture. In the United Kingdom, United States, and some countries, where the service is provided by non-government-funded organisations. Australia, ISBNs are issued by the library services agency Thorpe-Bowker
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Numerical integration
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This article focuses on calculation of definite integrals. The term numerical quadrature is more or less a synonym for numerical integration, Some authors refer to numerical integration over more than one dimension as cubature, others take quadrature to include higher-dimensional integration. The basic problem in numerical integration is to compute an approximate solution to a definite integral ∫ a b f d x to a degree of accuracy. If f is a smooth function integrated over a number of dimensions. The term numerical integration first appears in 1915 in the publication A Course in Interpolation, Quadrature is a historical mathematical term that means calculating area. Quadrature problems have served as one of the sources of mathematical analysis. Mathematicians of Ancient Greece, according to the Pythagorean doctrine, understood calculation of area as the process of constructing geometrically a square having the same area and that is why the process was named quadrature. For example, a quadrature of the circle, Lune of Hippocrates and this construction must be performed only by means of compass and straightedge. The ancient Babylonians used the trapezoidal rule to integrate the motion of Jupiter along the ecliptic, for a quadrature of a rectangle with the sides a and b it is necessary to construct a square with the side x = a b. For this purpose it is possible to use the fact, if we draw the circle with the sum of a and b as the diameter. The similar geometrical construction solves a problem of a quadrature for a parallelogram, problems of quadrature for curvilinear figures are much more difficult. The quadrature of the circle with compass and straightedge had been proved in the 19th century to be impossible, nevertheless, for some figures a quadrature can be performed. The quadratures of a surface and a parabola segment done by Archimedes became the highest achievement of the antique analysis. The area of the surface of a sphere is equal to quadruple the area of a circle of this sphere. The area of a segment of the cut from it by a straight line is 4/3 the area of the triangle inscribed in this segment. For the proof of the results Archimedes used the Method of exhaustion of Eudoxus, in medieval Europe the quadrature meant calculation of area by any method. More often the Method of indivisibles was used, it was less rigorous, john Wallis algebrised this method, he wrote in his Arithmetica Infinitorum series that we now call the definite integral, and he calculated their values. Isaac Barrow and James Gregory made further progress, quadratures for some algebraic curves, christiaan Huygens successfully performed a quadrature of some Solids of revolution
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Darboux integral
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In real analysis, a branch of mathematics, the Darboux integral is constructed using Darboux sums and is one possible definition of the integral of a function. 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 extended to defining Riemann-Stieltjes integration. Darboux integrals are named after their inventor, Gaston Darboux, the definition of the Darboux integral considers upper and lower integrals, which exist for any bounded real-valued function f on the interval. The Darboux integral exists if and only if the upper and lower integrals are equal, the upper and lower integrals are in turn the infimum and supremum, respectively, of upper and lower sums which over- and underestimate, respectively, the area under the curve. These ideas are made precise below, A partition of an interval is a sequence of values xi such that a = x 0 < x 1 < ⋯ < x n = b. Each interval is called a subinterval of the partition, let ƒ, →R be a bounded function, and let P = be a partition of. Let M i = sup x ∈ f, m i = inf x ∈ f, the upper Darboux sum of ƒ with respect to P is U f, P = ∑ i =1 n M i. The lower Darboux sum of ƒ with respect to P is L f, P = ∑ i =1 n m i, the lower and upper Darboux sums are often called the lower and upper sums. The upper Darboux integral of ƒ is U f = inf, the lower Darboux integral of ƒ is L f = sup. In some literature an integral symbol with an underline and overline represent the lower and upper Darboux integrals respectively. L f ≡ ∫ a b _ f d x U f ≡ ∫ a b ¯ f d x And like Darboux sums they are simply called the lower and upper integrals. If Uƒ = Lƒ, then we call the value the Darboux Integral. Furthermore, the lower Darboux sum is bounded below by the rectangle of width, likewise, the upper sum is bounded above by the rectangle of width and height sup. Suppose that g, →R is also a function, then the upper and lower integrals satisfy the following inequalities. An identical result holds if F is defined using an upper Darboux integral, suppose we want to show that the function f = x is Darboux-integrable on the interval and determine its value. To do this we partition into n equally sized subintervals each of length 1/n and we denote a partition of n equally sized subintervals as Pn. Now since f = x is increasing on, the infimum on any particular subinterval is given by its starting point
22.
Integration by parts
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It is frequently used to transform the antiderivative of a product of functions into an antiderivative for which a solution can be more easily found. The rule can be derived in one line simply by integrating the product rule of differentiation, more general formulations of integration by parts exist for the Riemann–Stieltjes and Lebesgue–Stieltjes integrals. The discrete analogue for sequences is called summation by parts, the theorem can be derived as follows. Suppose u and v are two differentiable functions. The product rule states, d d x = v d d x + u d d x and it is not actually necessary for u and v to be continuously differentiable. Integration by parts works if u is continuous and the function designated v is Lebesgue integrable. This is only if we choose v = − exp . One can also come up with similar examples in which u and v are not continuously differentiable. This visualisation also explains why integration by parts may help find the integral of an inverse function f−1 when the integral of the f is known. Indeed, the x and y are inverses, and the integral ∫x dy may be calculated as above from knowing the integral ∫y dx. The following form is useful in illustrating the best strategy to take, as a simple example, consider, ∫ ln x 2 d x. Since the derivative of ln is 1/x, one makes part u, since the antiderivative of 1/x2 is -1/x, the formula now yields, ∫ ln x 2 d x = − ln x − ∫ d x. The antiderivative of −1/x2 can be found with the rule and is 1/x. Alternatively, one may choose u and v such that the product u simplifies due to cancellation, for example, suppose one wishes to integrate, ∫ sec 2 ⋅ ln d x. The integrand simplifies to 1, so the antiderivative is x, finding a simplifying combination frequently involves experimentation. Some other special techniques are demonstrated in the examples below, exponentials and trigonometric functions An example commonly used to examine the workings of integration by parts is I = ∫ e x cos d x. Here, integration by parts is performed twice, then, ∫ e x sin d x = e x sin − ∫ e x cos d x. Putting these together, ∫ e x cos d x = e x cos + e x sin − ∫ e x cos d x, the same integral shows up on both sides of this equation
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Inverse function integration
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This formula was published in 1905 by Charles-Ange Laisant. Let I1 and I2 be two intervals of R, assume that f, I1 → I2 is a continuous and invertible function, and let f −1 denote its inverse I2 → I1. In his 1905 article, Laisant gave three proofs, first, under the additional hypothesis that f −1 is differentiable, one may differentiate the above formula, which completes the proof immediately. If f = c and f = d, the theorem can be written, the figure on the right is a proof without words of this formula. Laisant does not discuss the necessary to make this proof rigorous. Laisants third proof uses the hypothesis that f is differentiable. Beginning with f −1 = x, one multiplies by f ′, the right-hand side is calculated using integration by parts to be x f − ∫ f d x, and the formula follows. Nevertheless, it can be shown that this holds even if f or f −1 is not differentiable, it suffices, for example. On the other hand, even though general monotonic functions are differentiable almost everywhere and it is also possible to check that for every y in I2, the derivative of the function y → y f −1 − F is equal to f −1. In other words, ∀ x ∈ I1 lim h →0 f − x f − f − f = x. To this end, it suffices to apply the mean value theorem to F between x and x + h, taking into account that f is monotonic, assume that f = exp , hence f −1 = ln . The formula above gives immediately ∫ ln d y = y ln − y + C, similarly, with f = cos and f −1 = arccos , ∫ arccos d y = y arccos − sin + C. With f = tan and f −1 = arctan , ∫ arctan d y = y arctan + ln | cos | + C. This result was published independently in 1912 by an Italian engineer, Alberto Caprilli and it was rediscovered in 1955 by Parker, and by a number of mathematicians following him. Nevertheless, they all assume that f or f−1 is differentiable and this proof relies on the very definition of the Darboux integral, and consists in showing that the upper Darboux sums of the function f are in 1-1 correspondence with the lower Darboux sums of f−1. The above theorem generalizes in the way to holomorphic functions. Then f and f −1 have antiderivatives, and if F is an antiderivative of f, because all holomorphic functions are differentiable, the proof is immediate by complex differentiation
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Order of integration (calculus)
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In some cases, the order of integration can be validly interchanged, in others it cannot. The problem for examination is evaluation of an integral of the form ∬ D f d x d y, for some functions f straightforward integration is feasible, but where that is not true, the integral can sometimes be reduced to simpler form by changing the order of integration. The difficulty with this interchange is determining the change in description of the domain D, the method also is applicable to other multiple integrals. Sometimes, even though a full evaluation is difficult, or perhaps requires a numerical integration, reduction to a single integration makes a numerical evaluation much easier and more efficient. Consider the iterated integral ∫ a z ∫ a x h d y d x and this forms a three dimensional slice dx wide along the x-axis, from y=a to y=x along the y axis, and in the z direction z=f. Notice that if the thickness dx is infinitesimal, x varies only infinitesimally on the slice and we can assume that x is constant. This integration is as shown in the panel of Figure 1. The integral can be reduced to an integration by reversing the order of integration as shown in the right panel of the figure. For application to principal-value integrals, see Whittaker and Watson, Gakhov, Lu, see also the discussion of the Poincaré-Bertrand transformation in Obolashvili. The second form is evaluated using a partial fraction expansion and an evaluation using the Sokhotski–Plemelj formula, the notation ∫ L ∗ indicates a Cauchy principal value. A good discussion of the basis for reversing the order of integration is found in the book Fourier Analysis by T. W. Körner. He introduces his discussion with an example where interchange of integration leads to two different answers because the conditions of Theorem II below are not satisfied. Here is the example, ∫1 ∞ x 2 − y 22 d y =1 ∞ = −11 + x 2, ron Miechs UCLA Calculus Problems More complex examples of changing the order of integration Duane Nykamps University of Minnesota website A general introduction