1.
Function (mathematics)
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In mathematics, a function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. An example is the function that each real number x to its square x2. The output of a function f corresponding to a x is denoted by f. In this example, if the input is −3, then the output is 9, likewise, if the input is 3, then the output is also 9, and we may write f =9. The input variable are sometimes referred to as the argument of the function, Functions of various kinds are the central objects of investigation in most fields of modern mathematics. There are many ways to describe or represent a function, some functions may be defined by a formula or algorithm that tells how to compute the output for a given input. Others are given by a picture, called the graph of the function, in science, functions are sometimes defined by a table that gives the outputs for selected inputs. A function could be described implicitly, for example as the inverse to another function or as a solution of a differential equation, sometimes the codomain is called the functions range, but more commonly the word range is used to mean, instead, specifically the set of outputs. For example, we could define a function using the rule f = x2 by saying that the domain and codomain are the numbers. The image of this function is the set of real numbers. In analogy with arithmetic, it is possible to define addition, subtraction, multiplication, another important operation defined on functions is function composition, where the output from one function becomes the input to another function. Linking each shape to its color is a function from X to Y, each shape is linked to a color, there is no shape that lacks a color and no shape that has more than one color. This function will be referred to as the color-of-the-shape function, the input to a function is called the argument and the output is called the value. The set of all permitted inputs to a function is called the domain of the function. Thus, the domain of the function is the set of the four shapes. The concept of a function does not require that every possible output is the value of some argument, a second example of a function is the following, the domain is chosen to be the set of natural numbers, and the codomain is the set of integers. The function associates to any number n the number 4−n. For example, to 1 it associates 3 and to 10 it associates −6, a third example of a function has the set of polygons as domain and the set of natural numbers as codomain
2.
Domain of a function
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In mathematics, and more specifically in naive set theory, the domain of definition of a function is the set of input or argument values for which the function is defined. That is, the function provides an output or value for each member of the domain, conversely, the set of values the function takes on as output is termed the image of the function, which is sometimes also referred to as the range of the function. For instance, the domain of cosine is the set of all real numbers, if the domain of a function is a subset of the real numbers and the function is represented in a Cartesian coordinate system, then the domain is represented on the X-axis. Given a function f, X→Y, the set X is the domain of f, in the expression f, x is the argument and f is the value. One can think of an argument as a member of the domain that is chosen as an input to the function, the image of f is the set of all values assumed by f for all possible x, this is the set. The image of f can be the set as the codomain or it can be a proper subset of it. It is, in general, smaller than the codomain, it is the whole codomain if, a well-defined function must map every element of its domain to an element of its codomain. For example, the function f defined by f =1 / x has no value for f, thus, the set of all real numbers, R, cannot be its domain. In cases like this, the function is defined on R\ or the gap is plugged by explicitly defining f. If we extend the definition of f to f = {1 / x x ≠00 x =0 then f is defined for all real numbers, any function can be restricted to a subset of its domain. The restriction of g, A → B to S, where S ⊆ A, is written g |S, S → B. The natural domain of a function is the set of values for which the function is defined, typically within the reals. For instance the natural domain of square root is the non-negative reals when considered as a real number function, when considering a natural domain, the set of possible values of the function is typically called its range. There are two meanings in current mathematical usage for the notion of the domain of a partial function from X to Y, i. e. a function from a subset X of X to Y. Most mathematicians, including recursion theorists, use the domain of f for the set X of all values x such that f is defined. But some, particularly category theorists, consider the domain to be X, in category theory one deals with morphisms instead of functions. Morphisms are arrows from one object to another, the domain of any morphism is the object from which an arrow starts. In this context, many set theoretic ideas about domains must be abandoned or at least formulated more abstractly, for example, the notion of restricting a morphism to a subset of its domain must be modified
3.
Codomain
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In mathematics, the codomain or target set of a function is the set Y into which all of the output of the function is constrained to fall. It is the set Y in the f, X → Y. The codomain is also referred to as the range but that term is ambiguous as it may also refer to the image. The set F is called the graph of the function, the set of all elements of the form f, where x ranges over the elements of the domain X, is called the image of f. In general, the image of a function is a subset of its codomain, thus, it may not coincide with its codomain. Namely, a function that is not surjective has elements y in its codomain for which the equation f = y does not have a solution. An alternative definition of function by Bourbaki, namely as just a functional graph, for example in set theory it is desirable to permit the domain of a function to be a proper class X, in which case there is formally no such thing as a triple. With such a definition functions do not have a codomain, although some still use it informally after introducing a function in the form f, X → Y. For a function f, R → R defined by f, x ↦ x 2, or equivalently f = x 2, the codomain of f is R, but f does not map to any negative number. Thus the image of f is the set R0 +, i. e. the interval [0, an alternative function g is defined thus, g, R → R0 + g, x ↦ x 2. While f and g map a given x to the number, they are not, in this view. A third function h can be defined to demonstrate why, h, x ↦ x, the domain of h must be defined to be R0 +, h, R0 + → R. The compositions are denoted h ∘ f, h ∘ g, on inspection, h ∘ f is not useful. The codomain affects whether a function is a surjection, in that the function is surjective if, in the example, g is a surjection while f is not. The codomain does not affect whether a function is an injection, each matrix represents a map with the domain R2 and codomain R2. Some transformations may have image equal to the codomain but many do not. Take for example the matrix T given by T = which represents a linear transformation that maps the point to, the point is not in the image of T, but is still in the codomain since linear transformations from R2 to R2 are of explicit relevance. Just like all 2×2 matrices, T represents a member of that set, examining the differences between the image and codomain can often be useful for discovering properties of the function in question
4.
Integer-valued function
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In mathematics, an integer-valued function is a function whose values are integers. In other words, it is a function that assigns an integer to each member of its domain, floor and ceiling functions are examples of an integer-valued function of a real variable, but on real numbers and generally, on topological spaces integer-valued functions are not especially useful. Any such function on a connected space either has discontinuities or is constant, on the other hand, on discrete and other totally disconnected spaces integer-valued functions have roughly the same importance as real-valued functions have on non-discrete spaces. Any function with natural, or non-negative integer values is a case of integer-valued function. Integer-valued functions defined on the domain of all real numbers include the floor and ceiling functions, the Dirichlet function, the sign function, integer-valued functions defined on the domain of non-negative real numbers include the integer square root function and the prime-counting function. On an arbitrary set X, integer-valued functions form a ring with pointwise operations of addition and multiplication, since the latter is an ordered ring, the functions form a partially ordered ring, f ≤ g ⟺ ∀ x, f ≤ g. Integer-valued functions are ubiquitous in graph theory and they also have similar uses in geometric group theory, where length function represents the concept of norm, and word metric represents the concept of metric. Integer-valued polynomials are important in ring theory, in mathematical logic such concepts as a primitive recursive function and a μ-recursive function represent integer-valued functions of several natural variables or, in other words, functions on Nn. Gödel numbering, defined on well-formed formulae of some language, is a natural-valued function. Computability theory is based on natural numbers and natural functions on them. In number theory, many functions are integer-valued. In computer programming many functions return values of integer type due to simplicity of implementation, integer-valued polynomial Semi-continuity Rank #Mathematics Grade #In mathematics
5.
Sequence
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In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed. Like a set, it contains members, the number of elements is called the length of the sequence. Unlike a set, order matters, and exactly the elements can appear multiple times at different positions in the sequence. Formally, a sequence can be defined as a function whose domain is either the set of the numbers or the set of the first n natural numbers. The position of an element in a sequence is its rank or index and it depends on the context or of a specific convention, if the first element has index 0 or 1. For example, is a sequence of letters with the letter M first, also, the sequence, which contains the number 1 at two different positions, is a valid sequence. Sequences can be finite, as in these examples, or infinite, the empty sequence is included in most notions of sequence, but may be excluded depending on the context. A sequence can be thought of as a list of elements with a particular order, Sequences are useful in a number of mathematical disciplines for studying functions, spaces, and other mathematical structures using the convergence properties of sequences. In particular, sequences are the basis for series, which are important in differential equations, Sequences are also of interest in their own right and can be studied as patterns or puzzles, such as in the study of prime numbers. There are a number of ways to denote a sequence, some of which are useful for specific types of sequences. One way to specify a sequence is to list the elements, for example, the first four odd numbers form the sequence. This notation can be used for sequences as well. For instance, the sequence of positive odd integers can be written. Listing is most useful for sequences with a pattern that can be easily discerned from the first few elements. Other ways to denote a sequence are discussed after the examples, the prime numbers are the natural numbers bigger than 1, that have no divisors but 1 and themselves. Taking these in their natural order gives the sequence, the prime numbers are widely used in mathematics and specifically in number theory. The Fibonacci numbers are the integer sequence whose elements are the sum of the two elements. The first two elements are either 0 and 1 or 1 and 1 so that the sequence is, for a large list of examples of integer sequences, see On-Line Encyclopedia of Integer Sequences
6.
Real-valued function
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In mathematics, a real-valued function or real function is a function whose values are real numbers. In other words, it is a function that assigns a number to each member of its domain. Many important function spaces are defined to consist of real functions, let X be an arbitrary set. Let F denote the set of all functions from X to real numbers R. F is an ordered ring. The σ-algebra of Borel sets is an important structure on real numbers, if X has its σ-algebra and a function f is such that the preimage f −1 of any Borel set B belongs to that σ-algebra, then f is said to be measurable. Measurable functions also form a space and an algebra as explained above. Moreover, a set of real-valued functions on X can actually define a σ-algebra on X generated by all preimages of all Borel sets and this is the way how σ-algebras arise in probability theory, where real-valued functions on the sample space Ω are real-valued random variables. Real numbers form a space and a complete metric space. Continuous real-valued functions are important in theories of topological spaces and of metric spaces, the extreme value theorem states that for any real continuous function on a compact space its global maximum and minimum exist. The concept of space itself is defined with a real-valued function of two variables, the metric, which is continuous. The space of functions on a compact Hausdorff space has a particular importance. Convergent sequences also can be considered as real-valued continuous functions on a topological space. Continuous functions also form a space and an algebra as explained above. Real numbers are used as the codomain to define smooth functions, a domain of a real smooth function can be the real coordinate space, a topological vector space, an open subset of them, or a smooth manifold. Spaces of smooth functions also are vector spaces and algebras as explained above, a measure on a set is a non-negative real-valued functional on a σ-algebra of subsets. Lp spaces on sets with a measure are defined from aforementioned real-valued measurable functions, though, real-valued Lp spaces still have some of the structure explicated above. For example, pointwise product of two L2 functions belongs to L1, Real analysis Partial differential equations, a major user of real-valued functions Norm Scalar Weisstein, Eric W. Real Function
7.
Function of a real variable
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The output, also called the value of the function, could be anything, simple examples include a single real number, or a vector of real numbers. Vector-valued functions of a real variable occur widely in applied mathematics and physics, particularly in classical mechanics of particles. But we could also have a matrix of numbers as the output. The output could also be other number fields, such as numbers, quaternions. For simplicity, in this article a real-valued function of a variable will be simply called a function. To avoid any ambiguity, the types of functions that may occur will be explicitly specified. In other words, a function of a real variable is a function f, X → R such that its domain X is a subset of ℝ that contains an open set. A simple example of a function in one variable could be, the image of a function f is the set of all values of f when the variable x runs in the whole domain of f. For a continuous real-valued function with a domain, the image is either an interval or a single value. In the latter case, the function is a constant function, the preimage of a given real number y is the set of the solutions of the equation y = f. The domain of a function of real variables is a subset of ℝ that is sometimes. In fact, if one restricts the domain X of a function f to a subset Y ⊂ X, one gets formally a different function, the restriction of f to Y, which is denoted f|Y. In practice, it is not harmful to identify f and f|Y. Conversely, it is possible to enlarge naturally the domain of a given function. This means that it is not worthy to explicitly define the domain of a function of a real variable, the arithmetic operations may be applied to the functions in the following way, For every real number r, the constant function ↦ r, is everywhere defined. For every real number r and every function f, the function r f, ↦ r f has the domain as f. One may similarly define 1 / f, ↦1 / f and this constraint implies that the above two algebras are not fields. Until the second part of 19th century, only continuous functions were considered by mathematicians, as continuous functions of a real variable are ubiquitous in mathematics, it is worth defining this notion without reference to the general notion of continuous maps between topological space
8.
Function of several real variables
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This concept extends the idea of a function of a real variable to several variables. The input variables take real values, while the output, also called the value of the function, the domain of a function of several variables is the subset of ℝn for which the function is defined. As usual, the domain of a function of real variables is supposed to contain an open subset of ℝn. A real-valued function of n variables is a function that takes as input n real numbers, commonly represented by the variables x1. Xn, for producing another number, the value of the function. For simplicity, in this article a real-valued function of real variables will be simply called a function. To avoid any ambiguity, the types of functions that may occur will be explicitly specified. In other words, a function of n real variables is a function f, X → R such that its domain X is a subset of ℝn that contains an open set. An element of X being an n-tuple, the notation for denoting functions would be f. The common usage, much older than the definition of functions between sets, it to not use double parentheses and to simply write f. It is also common to abbreviate the n-tuple by using a similar to that for vectors, like boldface x, underline x. The domain restricts all variables to be positive since lengths and areas must be positive, for an example of a function in two variables, z, R2 → R z = a x + b y where a and b are real non-zero constants. The function is well-defined at all points in ℝ2. The previous example can be extended easily to higher dimensions, z, R p → R z = a 1 x 1 + a 2 x 2 + ⋯ + a p x p for p non-zero real constants a1, ap, which describes a p-dimensional hyperplane. The Euclidean norm, f = ∥ x ∥ = x 12 + ⋯ + x n 2 is also a function of n variables which is everywhere defined, the function does not include the origin =, if it did then f would be ill-defined at that point. Using a 3d Cartesian coordinate system with the xy plane as the domain ℝ2, and the z axis the codomain ℝ, the image of a function f is the set of all values of f when the n-tuple runs in the whole domain of f. For a continuous real-valued function which has a domain, the image is either an interval or a single value. In the latter case, the function is a constant function, the preimage of a given real number y is called a level set
9.
Complex-valued function
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In mathematics, a complex-valued function is a function whose values are complex numbers. In other words, it is a function that assigns a number to each member of its domain. This domain does not necessarily have any structure related to complex numbers, most important uses of such functions in complex analysis and in functional analysis are explicated below. A vector space and an algebra of functions over complex numbers can be defined in the same way as for real-valued functions. Complex analysis considers holomorphic functions on manifolds, such as Riemann surfaces. The property of analytic continuation makes them very dissimilar from smooth functions, namely, if a function defined in a neighborhood can be continued to a wider domain, then this continuation is unique. As real functions, any function is infinitely smooth and analytic. But there is much freedom in construction of a holomorphic function than in one of a smooth function. Complex-valued L2 spaces on sets with a measure have a particular importance because they are Hilbert spaces and they often appear in functional analysis and operator theory. A major user of such spaces is quantum mechanics, as wave functions, the sets on which the complex-valued L2 is constructed have the potential to be more exotic than their real-valued analog. Also, complex-valued continuous functions are an important example in the theory of C*-algebras, Function of a complex variable, the dual concept Weisstein, Eric W. Complex Function
10.
Function of a complex variable
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Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. As a differentiable function of a variable is equal to the sum of its Taylor series. Complex analysis is one of the branches in mathematics, with roots in the 19th century. Important mathematicians associated with complex analysis include Euler, Gauss, Riemann, Cauchy, Weierstrass, Complex analysis, in particular the theory of conformal mappings, has many physical applications and is also used throughout analytic number theory. In modern times, it has very popular through a new boost from complex dynamics. Another important application of analysis is in string theory which studies conformal invariants in quantum field theory. A complex function is one in which the independent variable and the dependent variable are complex numbers. More precisely, a function is a function whose domain. In other words, the components of the f, u = u and v = v can be interpreted as real-valued functions of the two real variables, x and y. The basic concepts of complex analysis are often introduced by extending the elementary real functions into the complex domain, holomorphic functions are complex functions, defined on an open subset of the complex plane, that are differentiable. In the context of analysis, the derivative of f at z 0 is defined to be f ′ = lim z → z 0 f − f z − z 0, z ∈ C. Although superficially similar in form to the derivative of a real function, in particular, for this limit to exist, the value of the difference quotient must approach the same complex number, regardless of the manner in which we approach z 0 in the complex plane. Consequently, complex differentiability has much stronger consequences than usual differentiability, for instance, holomorphic functions are infinitely differentiable, whereas most real differentiable functions are not. For this reason, holomorphic functions are referred to as analytic functions. Such functions that are holomorphic everywhere except a set of isolated points are known as meromorphic functions. On the other hand, the functions z ↦ ℜ, z ↦ | z |, an important property that characterizes holomorphic functions is the relationship between the partial derivatives of their real and imaginary components, known as the Cauchy-Riemann conditions. If f, C → C, defined by f = f = u + i v, here, the differential operator ∂ / ∂ z ¯ is defined as. In terms of the real and imaginary parts of the function, u and v, this is equivalent to the pair of equations u x = v y and u y = − v x, where the subscripts indicate partial differentiation
11.
Constant function
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In mathematics, a constant function is a function whose value is the same for every input value. For example, the function y =4 is a constant function because the value of y is 4 regardless of the value x. As a real-valued function of an argument, a constant function has the general form y = c or just y = c. Example, The function y =2 or just y =2 is the constant function where the output value is c =2. The domain of function is the set of all real numbers ℝ. The codomain of this function is just, the independent variable x does not appear on the right side of the function expression and so its value is vacuously substituted. No matter what value of x is input, the output is 2, real-world example, A store where every item is sold for the price of 1 euro. The graph of the constant function y = c is a line in the plane that passes through the point. In the context of a polynomial in one variable x, the constant function is a polynomial of degree 0. This function has no point with the x-axis, that is. On the other hand, the polynomial f =0 is the zero function. It is the constant function and every x is a root and its graph is the x-axis in the plane. A constant function is a function, i. e. the graph of a constant function is symmetric with respect to the y-axis. In the context where it is defined, the derivative of a function is a measure of the rate of change of values with respect to change in input values. Because a constant function does not change, its derivative is 0 and this is often written, ′ =0. Namely, if y=0 for all numbers x, then y is a constant function. Example, Given the constant function y = −2, the derivative of y is the identically zero function y ′ = ′ =0. Every constant function whose domain and codomain are the same is idempotent, every constant function between topological spaces is continuous
12.
Identity function
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In mathematics, an identity function, also called an identity relation or identity map or identity transformation, is a function that always returns the same value that was used as its argument. In equations, the function is given by f = x, formally, if M is a set, the identity function f on M is defined to be that function with domain and codomain M which satisfies f = x for all elements x in M. In other words, the value f in M is always the same input element x of M. The identity function on M is clearly a function as well as a surjective function. The identity function f on M is often denoted by idM, in set theory, where a function is defined as a particular kind of binary relation, the identity function is given by the identity relation, or diagonal of M. If f, M → N is any function, then we have f ∘ idM = f = idN ∘ f, in particular, idM is the identity element of the monoid of all functions from M to M. Since the identity element of a monoid is unique, one can define the identity function on M to be this identity element. Such a definition generalizes to the concept of an identity morphism in category theory, the identity function is a linear operator, when applied to vector spaces. The identity function on the integers is a completely multiplicative function. In an n-dimensional vector space the identity function is represented by the identity matrix In, in a metric space the identity is trivially an isometry. An object without any symmetry has as symmetry group the group only containing this isometry. In a topological space, the identity function is always continuous
13.
Linear map
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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.
Polynomial
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In mathematics, a polynomial is an expression consisting of variables and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents. An example of a polynomial of a single indeterminate x is x2 − 4x +7, an example in three variables is x3 + 2xyz2 − yz +1. Polynomials appear in a variety of areas of mathematics and science. In advanced mathematics, polynomials are used to construct polynomial rings and algebraic varieties, central concepts in algebra, the word polynomial joins two diverse roots, the Greek poly, meaning many, and the Latin nomen, or name. It was derived from the binomial by replacing the Latin root bi- with the Greek poly-. The word polynomial was first used in the 17th century, the x occurring in a polynomial is commonly called either a variable or an indeterminate. When the polynomial is considered as an expression, x is a symbol which does not have any value. It is thus correct to call it an indeterminate. However, when one considers the function defined by the polynomial, then x represents the argument of the function, many authors use these two words interchangeably. It is a convention to use uppercase letters for the indeterminates. However one may use it over any domain where addition and multiplication are defined, in particular, when a is the indeterminate x, then the image of x by this function is the polynomial P itself. This equality allows writing let P be a polynomial as a shorthand for let P be a polynomial in the indeterminate x. A polynomial is an expression that can be built from constants, the word indeterminate means that x represents no particular value, although any value may be substituted for it. The mapping that associates the result of substitution to the substituted value is a function. This can be expressed concisely by using summation notation, ∑ k =0 n a k x k That is. Each term consists of the product of a number—called the coefficient of the term—and a finite number of indeterminates, because x = x1, the degree of an indeterminate without a written exponent is one. A term and a polynomial with no indeterminates are called, respectively, a constant term, the degree of a constant term and of a nonzero constant polynomial is 0. The degree of the polynomial,0, is generally treated as not defined
15.
Rational function
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In mathematics, a rational function is any function which can be defined by a rational fraction, i. e. an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be numbers, they may be taken in any field K. In this case, one speaks of a function and a rational fraction over K. The values of the variables may be taken in any field L containing K, then the domain of the function is the set of the values of the variables for which the denominator is not zero and the codomain is L. The set of functions over a field K is a field. A function f is called a function if and only if it can be written in the form f = P Q where P and Q are polynomials in x and Q is not the zero polynomial. The domain of f is the set of all points x for which the denominator Q is not zero and it is a common usage to identify f and f 1, that is to extend by continuity the domain of f to that of f 1. Indeed, one can define a rational fraction as a class of fractions of polynomials. In this case P Q is equivalent to P1 Q1, a proper rational function is a rational function in which the degree of P is no greater than the degree of Q and both are real polynomials. The rational function f = x 3 −2 x 2 is not defined at x 2 =5 ⇔ x = ±5 and it is asymptotic to x 2 as x approaches infinity. A constant function such as f = π is a function since constants are polynomials. Note that the function itself is rational, even though the value of f is irrational for all x, every polynomial function f = P is a rational function with Q =1. A function that cannot be written in form, such as f = sin , is not a rational function. The adjective irrational is not generally used for functions, the rational function f = x x is equal to 1 for all x except 0, where there is a removable singularity. The sum, product, or quotient of two functions is itself a rational function. However, the process of reduction to standard form may result in the removal of such singularities unless care is taken. Using the definition of functions as equivalence classes gets around this. For example,1 x 2 − x +2 = ∑ k =0 ∞ a k x k, combining like terms gives 1 =2 a 0 + x + ∑ k =2 ∞ x k
16.
Algebraic function
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In mathematics, an algebraic function is a function that can be defined as the root of a polynomial equation. Examples of such functions are, f =1 / x f = x f =1 + x 3 x 3 /7 −7 x 1 /3 Some algebraic functions, however, cannot be expressed by such finite expressions. This is the case, for example, of the Bring radical, quite often, S = Q, and one then talks about function algebraic over Q, and the evaluation at a given rational value of such an algebraic function gives an algebraic number. A function which is not algebraic is called a function, as it is for example the case of exp , tan . A composition of functions can give an algebraic function, f = cos =1 − x 2. As an equation of degree n has n roots, a polynomial equation does not implicitly define a single function, consider for example the equation of the unit circle, y 2 + x 2 =1. This determines y, except only up to a sign, accordingly. An algebraic function in m variables is similarly defined as a function y which solves an equation in m +1 variables. It is normally assumed that p should be an irreducible polynomial, the existence of an algebraic function is then guaranteed by the implicit function theorem. Formally, a function in m variables over the field K is an element of the algebraic closure of the field of rational functions K. The informal definition of an algebraic function provides a number of clues about their properties and this is something of an oversimplification, because of the fundamental theorem of Galois theory, algebraic functions need not be expressible by radicals. First, note that any polynomial function y = p is an algebraic function, more generally, any rational function y = p q is algebraic, being the solution to q y − p =0. Moreover, the nth root of any polynomial y = p n is an algebraic function, surprisingly, the inverse function of an algebraic function is an algebraic function. For supposing that y is a solution to a n y n + ⋯ + a 0 =0, for each value of x, then x is also a solution of this equation for each value of y. Indeed, interchanging the roles of x and y and gathering terms, writing x as a function of y gives the inverse function, also an algebraic function. However, not every function has an inverse, for example, y = x2 fails the horizontal line test, it fails to be one-to-one. The inverse is the function x = ± y. Another way to understand this, is that the set of branches of the equation defining our algebraic function is the graph of an algebraic curve
17.
Smooth function
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In mathematical analysis, the smoothness of a function is a property measured by the number of derivatives it has which are continuous. A smooth function is a function that has derivatives of all orders everywhere in its domain, differentiability class is a classification of functions according to the properties of their derivatives. Higher order differentiability classes correspond to the existence of more derivatives, consider an open set on the real line and a function f defined on that set with real values. Let k be a non-negative integer, the function f is said to be of class Ck if the derivatives f′, f′′. The function f is said to be of class C∞, or smooth, if it has derivatives of all orders. The function f is said to be of class Cω, or analytic, if f is smooth, Cω is thus strictly contained in C∞. Bump functions are examples of functions in C∞ but not in Cω, to put it differently, the class C0 consists of all continuous functions. The class C1 consists of all differentiable functions whose derivative is continuous, thus, a C1 function is exactly a function whose derivative exists and is of class C0. In particular, Ck is contained in Ck−1 for every k, C∞, the class of infinitely differentiable functions, is the intersection of the sets Ck as k varies over the non-negative integers. The function f = { x if x ≥0,0 if x <0 is continuous, because cos oscillates as x →0, f ’ is not continuous at zero. Therefore, this function is differentiable but not of class C1, the functions f = | x | k +1 where k is even, are continuous and k times differentiable at all x. But at x =0 they are not times differentiable, so they are of class Ck, the exponential function is analytic, so, of class Cω. The trigonometric functions are also analytic wherever they are defined, the function f is an example of a smooth function with compact support. Let n and m be some positive integers, if f is a function from an open subset of Rn with values in Rm, then f has component functions f1. Each of these may or may not have partial derivatives, the classes C∞ and Cω are defined as before. These criteria of differentiability can be applied to the functions of a differential structure. The resulting space is called a Ck manifold, if one wishes to start with a coordinate-independent definition of the class Ck, one may start by considering maps between Banach spaces. A map from one Banach space to another is differentiable at a point if there is a map which approximates it at that point
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Continuous function
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In mathematics, a continuous function is a function for which sufficiently small changes in the input result in arbitrarily small changes in the output. Otherwise, a function is said to be a discontinuous function, a continuous function with a continuous inverse function is called a homeomorphism. Continuity of functions is one of the concepts of topology. The introductory portion of this focuses on the special case where the inputs and outputs of functions are real numbers. In addition, this article discusses the definition for the general case of functions between two metric spaces. In order theory, especially in theory, one considers a notion of continuity known as Scott continuity. Other forms of continuity do exist but they are not discussed in this article, as an example, consider the function h, which describes the height of a growing flower at time t. By contrast, if M denotes the amount of money in an account at time t, then the function jumps at each point in time when money is deposited or withdrawn. A form of the definition of continuity was first given by Bernard Bolzano in 1817. Cauchy defined infinitely small quantities in terms of quantities. The formal definition and the distinction between pointwise continuity and uniform continuity were first given by Bolzano in the 1830s but the work wasnt published until the 1930s, all three of those nonequivalent definitions of pointwise continuity are still in use. Eduard Heine provided the first published definition of continuity in 1872. This is not a definition of continuity since the function f =1 x is continuous on its whole domain of R ∖ A function is continuous at a point if it does not have a hole or jump. A “hole” or “jump” in the graph of a function if the value of the function at a point c differs from its limiting value along points that are nearby. Such a point is called a discontinuity, a function is then continuous if it has no holes or jumps, that is, if it is continuous at every point of its domain. Otherwise, a function is discontinuous, at the points where the value of the function differs from its limiting value, there are several ways to make this definition mathematically rigorous. These definitions are equivalent to one another, so the most convenient definition can be used to determine whether a function is continuous or not. In the definitions below, f, I → R. is a function defined on a subset I of the set R of real numbers and this subset I is referred to as the domain of f
19.
Measurable function
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In mathematics, particularly in measure theory, a measurable function is a structure-preserving function between measurable spaces. For example, the notion of integrability can be defined for a real-valued measurable function on a measurable space, a measurable function is said to be bimeasurable if it is bijective and its inverse is also measurable. For example, in probability theory, a function on a probability space is known as a random variable. In contrast, functions that are not Lebesgue measurable are generally considered pathological, let and be measurable spaces, meaning that X and Y are sets equipped with respective σ -algebras Σ and T. A function f, X → Y is said to be if the preimage of E under f is in Σ for every E ∈ T, i. e. f −1, = ∈ Σ, ∀ E ∈ T. The notion of measurability depends on the sigma algebras Σ and T, to emphasize this dependency, if f, X → Y is a measurable function, we will write f, →. This definition can be simple, however, as special care must be taken regarding the σ -algebras involved. Here, L is the σ -algebra of Lebesgue measurable sets, and B is the Borel algebra on R, as a result, the composition of Lebesgue-measurable functions need not be Lebesgue-measurable. By convention a topological space is assumed to be equipped with the Borel algebra unless otherwise specified, most commonly this space will be the real or complex numbers. For instance, a measurable function is a function for which the preimage of each Borel set is measurable. A complex-valued measurable function is defined analogously, in practice, some authors use measurable functions to refer only to real-valued measurable functions with respect to the Borel algebra. If the values of the lie in an infinite-dimensional vector space instead of R or C, usually other definitions of measurability are used, such as weak measurability. If and are Borel spaces, a function f, → is also called a Borel function. Continuous functions are Borel functions but not all Borel functions are continuous, however, a measurable function is nearly a continuous function, see Luzins theorem. If a Borel function happens to be a section of some map Y → π X, it is called a Borel section. A Lebesgue measurable function is a function f, →, where L is the σ -algebra of Lebesgue measurable sets. Lebesgue measurable functions are of interest in analysis because they can be integrated. In the case f, X → R, f is Lebesgue measurable iff = is measurable for all α ∈ R and this is also equivalent to any of, being measurable for all α
20.
Injective function
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In mathematics, an injective function or injection or one-to-one function is a function that preserves distinctness, it never maps distinct elements of its domain to the same element of its codomain. In other words, every element of the codomain is the image of at most one element of its domain. The term one-to-one function must not be confused with one-to-one correspondence, occasionally, an injective function from X to Y is denoted f, X ↣ Y, using an arrow with a barbed tail. A function f that is not injective is sometimes called many-to-one, however, the injective terminology is also sometimes used to mean single-valued, i. e. each argument is mapped to at most one value. A monomorphism is a generalization of a function in category theory. Let f be a function whose domain is a set X, the function f is said to be injective provided that for all a and b in X, whenever f = f, then a = b, that is, f = f implies a = b. Equivalently, if a ≠ b, then f ≠ f, in particular the identity function X → X is always injective. If the domain X = ∅ or X has only one element, the function f, R → R defined by f = 2x +1 is injective. The function g, R → R defined by g = x2 is not injective, however, if g is redefined so that its domain is the non-negative real numbers [0, +∞), then g is injective. The exponential function exp, R → R defined by exp = ex is injective, the natural logarithm function ln, → R defined by x ↦ ln x is injective. The function g, R → R defined by g = xn − x is not injective, since, for example, g = g =0. More generally, when X and Y are both the real line R, then a function f, R → R is one whose graph is never intersected by any horizontal line more than once. This principle is referred to as the line test. Functions with left inverses are always injections and that is, given f, X → Y, if there is a function g, Y → X such that, for every x ∈ X g = x then f is injective. In this case, g is called a retraction of f, conversely, f is called a section of g. Conversely, every injection f with non-empty domain has an inverse g. Note that g may not be an inverse of f because the composition in the other order, f o g. In other words, a function that can be undone or reversed, injections are reversible but not always invertible
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Surjective function
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It is not required that x is unique, the function f may map one or more elements of X to the same element of Y. The French prefix sur means over or above and relates to the fact that the image of the domain of a surjective function completely covers the functions codomain, any function induces a surjection by restricting its codomain to its range. Every surjective function has an inverse, and every function with a right inverse is necessarily a surjection. The composite of surjective functions is always surjective, any function can be decomposed into a surjection and an injection. A surjective function is a function whose image is equal to its codomain, equivalently, a function f with domain X and codomain Y is surjective if for every y in Y there exists at least one x in X with f = y. Surjections are sometimes denoted by a two-headed rightwards arrow, as in f, X ↠ Y, symbolically, If f, X → Y, then f is said to be surjective if ∀ y ∈ Y, ∃ x ∈ X, f = y. For any set X, the identity function idX on X is surjective, the function f, Z → defined by f = n mod 2 is surjective. The function f, R → R defined by f = 2x +1 is surjective, because for every real number y we have an x such that f = y, an appropriate x is /2. However, this function is not injective since e. g. the pre-image of y =2 is, the function g, R → R defined by g = x2 is not surjective, because there is no real number x such that x2 = −1. However, the g, R → R0+ defined by g = x2 is surjective because for every y in the nonnegative real codomain Y there is at least one x in the real domain X such that x2 = y. The natural logarithm ln, → R is a surjective. Its inverse, the function, is not surjective as its range is the set of positive real numbers. The matrix exponential is not surjective when seen as a map from the space of all n×n matrices to itself. It is, however, usually defined as a map from the space of all n×n matrices to the linear group of degree n, i. e. the group of all n×n invertible matrices. Under this definition the matrix exponential is surjective for complex matrices, the projection from a cartesian product A × B to one of its factors is surjective unless the other factor is empty. In a 3D video game vectors are projected onto a 2D flat screen by means of a surjective function, a function is bijective if and only if it is both surjective and injective. If a function is identified with its graph, then surjectivity is not a property of the function itself, unlike injectivity, surjectivity cannot be read off of the graph of the function alone. The function g, Y → X is said to be an inverse of the function f, X → Y if f = y for every y in Y
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Restriction (mathematics)
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In mathematics, the restriction of a function f is a new function f|A obtained by choosing a smaller domain A for the original function f. The notation f ↾ A is also used, let f, E → F be a function from a set E to a set F, so that the domain of f is in E. The restriction of the function f, R → R, x ↦ x 2 to R + = [0, ∞ ) is the injection f, R + → R, x ↦ x 2. The factorial function is the restriction of the function to the integers. Restricting a function f, X → Y to its entire domain X gives back the original function, i. e. f | X = f. Restricting a function twice is the same as restricting it once, i. e. if A ⊆ B ⊆ d o m f, the restriction of the identity function on a space X to a subset A of X is just the inclusion map of A into X. The restriction of a function is continuous. For a function to have an inverse, it must be one-to-one, if a function f is not one-to-one, it may be possible to define a partial inverse of f by restricting the domain. For example, the function f = x 2 is not one-to-one, however, the function becomes one-to-one if we restrict to the domain x ≥0, in which case f −1 = y. The selection σ a θ v selects all those tuples in R for which θ holds between the a attribute and the value v. Thus, the selection operator restricts to a subset of the entire database. The pasting lemma is a result in topology that relates the continuity of a function with the continuity of its restrictions to subsets. Let X, Y be both closed subsets of a topological space A such that A = X ∪ Y, if f, A → B is continuous when restricted to both X and Y, then f is continuous. This result allows one to take two continuous functions defined on closed subsets of a space and create a new one. Sheaves provide a way of generalizing restrictions to objects besides functions, in sheaf theory, one assigns an object F in a category to each open set U of a topological space, and requires that the objects satisfy certain conditions. The most important condition is that there are restriction morphisms between every pair of objects associated to nested sets, i. e. If we have three open sets W ⊆ V ⊆ U, then the composite resW, V o resV, U = resW, the collection of all such objects is called a sheaf. If only the first two properties are satisfied, it is a pre-sheaf, more generally, the restriction A ◁ R of a binary relation R between E and F may be defined as a relation having domain A, codomain F and graph G = . Similarly, one can define a right-restriction or range restriction R ▷ B, indeed, one could define a restriction to n-ary relations, as well as to subsets understood as relations, such as ones of E × F for binary relations
23.
Function composition
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In mathematics, function composition is the pointwise application of one function to the result of another to produce a third function. The resulting composite function is denoted g ∘ f, X → Z, the notation g ∘ f is read as g circle f, or g round f, or g composed with f, g after f, g following f, or g of f, or g on f. Intuitively, composing two functions is a process in which the output of the inner function becomes the input of the outer function. The composition of functions is a case of the composition of relations. The composition of functions has some additional properties, Composition of functions on a finite set, If f =, and g =, then g ∘ f =. The composition of functions is always associative—a property inherited from the composition of relations, since there is no distinction between the choices of placement of parentheses, they may be left off without causing any ambiguity. In a strict sense, the composition g ∘ f can be only if fs codomain equals gs domain, in a wider sense it is sufficient that the former is a subset of the latter. The functions g and f are said to commute with each other if g ∘ f = f ∘ g, commutativity is a special property, attained only by particular functions, and often in special circumstances. For example, | x | +3 = | x + 3 | only when x ≥0, the composition of one-to-one functions is always one-to-one. Similarly, the composition of two functions is always onto. It follows that composition of two bijections is also a bijection, the inverse function of a composition has the property that −1 =. Derivatives of compositions involving differentiable functions can be using the chain rule. Higher derivatives of functions are given by Faà di Brunos formula. Suppose one has two functions f, X → X, g, X → X having the domain and codomain. Then one can form chains of transformations composed together, such as f ∘ f ∘ g ∘ f, such chains have the algebraic structure of a monoid, called a transformation monoid or composition monoid. In general, transformation monoids can have remarkably complicated structure, one particular notable example is the de Rham curve. The set of all functions f, X → X is called the transformation semigroup or symmetric semigroup on X. If the transformation are bijective, then the set of all combinations of these functions forms a transformation group
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Inverse function
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I. e. f = y if and only if g = x. As a simple example, consider the function of a real variable given by f = 5x −7. Thinking of this as a procedure, to reverse this and get x back from some output value, say y. In this case means that we should add 7 to y. In functional notation this inverse function would be given by, g = y +75, with y = 5x −7 we have that f = y and g = x. Not all functions have inverse functions, in order for a function f, X → Y to have an inverse, it must have the property that for every y in Y there must be one, and only one x in X so that f = y. This property ensures that a function g, Y → X will exist having the necessary relationship with f, let f be a function whose domain is the set X, and whose image is the set Y. Then f is invertible if there exists a g with domain Y and image X, with the property. If f is invertible, the g is unique, which means that there is exactly one function g satisfying this property. That function g is called the inverse of f, and is usually denoted as f −1. Stated otherwise, a function is invertible if and only if its inverse relation is a function on the range Y, not all functions have an inverse. For a function to have an inverse, each element y ∈ Y must correspond to no more than one x ∈ X, a function f with this property is called one-to-one or an injection. If f −1 is to be a function on Y, then each element y ∈ Y must correspond to some x ∈ X. Functions with this property are called surjections. This property is satisfied by definition if Y is the image of f, to be invertible a function must be both an injection and a surjection. If a function f is invertible, then both it and its inverse function f−1 are bijections, there is another convention used in the definition of functions. This can be referred to as the set-theoretic or graph definition using ordered pairs in which a codomain is never referred to, under this convention all functions are surjections, and so, being a bijection simply means being an injection. Authors using this convention may use the phrasing that a function is invertible if, the two conventions need not cause confusion as long as it is remembered that in this alternate convention the codomain of a function is always taken to be the range of the function. With this type of function it is impossible to deduce an input from its output, such a function is called non-injective or, in some applications, information-losing
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Partial function
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In mathematics, a partial function from X to Y is a function f, X ′ → Y, for some subset X ′ of X. It generalizes the concept of an f, X → Y by not forcing f to map every element of X to an element of Y. If X ′ = X, then f is called a function and is equivalent to a function. Partial functions are used when the exact domain, X, is not known. Specifically, we say that for any x ∈ X, either. For example, we can consider the square root function restricted to the g, Z → Z g = n. Thus g is defined for n that are perfect squares. So, g =5, but g is undefined, there are two distinct meanings in current mathematical usage for the notion of the domain of a partial function. Most mathematicians, including recursion theorists, use the domain of f for the set of all values x such that f is defined. But some, particularly category theorists, consider the domain of a function f, X → Y to be X. Similarly, the range can refer to either the codomain or the image of a function. Occasionally, a function with domain X and codomain Y is written as f, X ⇸ Y. A partial function is said to be injective or surjective when the function given by the restriction of the partial function to its domain of definition is. A partial function may be both injective and surjective, because a function is trivially surjective when restricted to its image, the term partial bijection denotes a partial function which is injective. An injective partial function may be inverted to a partial function. Furthermore, a function which is injective may be inverted to an injective partial function. The notion of transformation can be generalized to functions as well. A partial transformation is a function f, A → B, total function is a synonym for function
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Multivalued function
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In mathematics, a multivalued function is a left-total relation. In the strict sense, a well-defined function associates one, and only one, the term multivalued function is, therefore, a misnomer because functions are single-valued. Multivalued functions often arise as inverses of functions that are not injective, such functions do not have an inverse function, but they do have an inverse relation. The multivalued function corresponds to this inverse relation, every real number greater than zero has two real square roots. The square roots of 4 are in the set, the square root of 0 is 0. Each complex number except zero has two roots, three cube roots, and in general n nth roots. The complex logarithm function is multiple-valued, the values assumed by log for real numbers a and b are log a 2 + b 2 + i arg +2 π n i for all integers n. Inverse trigonometric functions are multiple-valued because trigonometric functions are periodic and we have tan = tan = tan = tan = ⋯ =1. As a consequence, arctan is intuitively related to several values, π/4, 5π/4, −3π/4 and we can treat arctan as a single-valued function by restricting the domain of tan x to −π/2 < x < π/2 – a domain over which tan x is monotonically increasing. Thus, the range of arctan becomes −π/2 < y < π/2 and these values from a restricted domain are called principal values. The indefinite integral can be considered as a multivalued function, the indefinite integral of a function is the set of functions whose derivative is that function. The constant of integration follows from the fact that the derivative of a constant function is 0 and these are all examples of multivalued functions that come about from non-injective functions. Since the original functions do not preserve all the information of their inputs, often, the restriction of a multivalued function is a partial inverse of the original function. Multivalued functions of a variable have branch points. For example, for the nth root and logarithm functions,0 is a point, for the arctangent function. Using the branch points, these functions may be redefined to be single-valued functions, by restricting the range. A suitable interval may be found through use of a branch cut, as in the case with real functions, the restricted range may be called principal branch of the function. Set-valued analysis is the study of sets in the spirit of mathematical analysis, instead of considering collections of only points, set-valued analysis considers collections of sets
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Implicit function
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In mathematics, an implicit equation is a relation of the form R =0, where R is a function of several variables. For example, the equation of the unit circle is x 2 + y 2 −1 =0. An implicit function is a function that is defined implicitly by an implicit equation, thus, an implicit function for y in the context of the unit circle is defined implicitly by x 2 + f 2 −1 =0. This implicit equation defines f as a function of x only if −1 ≤ x ≤1, the implicit function theorem provides conditions under which a relation defines an implicit function. A common type of function is an inverse function. If f is a function of x, then the function of f. This solution is x = f −1, intuitively, an inverse function is obtained from f by interchanging the roles of the dependent and independent variables. Stated another way, the function gives the solution for x of the equation R = y − f =0. Example The product log is a function giving the solution for x of the equation y − x ex =0. An algebraic function is a function satisfies a polynomial equation whose coefficients are themselves polynomials. Algebraic functions play an important role in analysis and algebraic geometry. A simple example of a function is given by the left side of the unit circle equation. Solving for y gives a solution, y = ±1 − x 2. But even without specifying this explicit solution, it is possible to refer to the solution of the unit circle equation. Nevertheless, one can refer to the implicit solution y = g involving the multi-valued implicit function g. Not every equation R =0 implies a graph of a single-valued function, another example is an implicit function given by x − C =0 where C is a cubic polynomial having a hump in its graph. Thus, for a function to be a true function it might be necessary to use just part of the graph. An implicit function can sometimes be successfully defined as a function only after zooming in on some part of the x-axis
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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
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Set (mathematics)
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In mathematics, a set is a well-defined collection of distinct objects, considered as an object in its own right. For example, the numbers 2,4, and 6 are distinct objects when considered separately, Sets are one of the most fundamental concepts in mathematics. Developed at the end of the 19th century, set theory is now a part of mathematics. In mathematics education, elementary topics such as Venn diagrams are taught at a young age, the German word Menge, rendered as set in English, was coined by Bernard Bolzano in his work The Paradoxes of the Infinite. A set is a collection of distinct objects. The objects that make up a set can be anything, numbers, people, letters of the alphabet, other sets, Sets are conventionally denoted with capital letters. Sets A and B are equal if and only if they have precisely the same elements. Cantors definition turned out to be inadequate, instead, the notion of a set is taken as a notion in axiomatic set theory. There are two ways of describing, or specifying the members of, a set, one way is by intensional definition, using a rule or semantic description, A is the set whose members are the first four positive integers. B is the set of colors of the French flag, the second way is by extension – that is, listing each member of the set. An extensional definition is denoted by enclosing the list of members in curly brackets, one often has the choice of specifying a set either intensionally or extensionally. In the examples above, for instance, A = C and B = D, there are two important points to note about sets. First, in a definition, a set member can be listed two or more times, for example. However, per extensionality, two definitions of sets which differ only in one of the definitions lists set members multiple times, define, in fact. Hence, the set is identical to the set. The second important point is that the order in which the elements of a set are listed is irrelevant and we can illustrate these two important points with an example, = =. For sets with many elements, the enumeration of members can be abbreviated, for instance, the set of the first thousand positive integers may be specified extensionally as, where the ellipsis indicates that the list continues in the obvious way. Ellipses may also be used where sets have infinitely many members, thus the set of positive even numbers can be written as
30.
Cardinal number
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In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality of sets. The cardinality of a set is a natural number, the number of elements in the set. The transfinite cardinal numbers describe the sizes of infinite sets, cardinality is defined in terms of bijective functions. Two sets have the same cardinality if, and only if, in the case of finite sets, this agrees with the intuitive notion of size. In the case of sets, the behavior is more complex. It is also possible for a subset of an infinite set to have the same cardinality as the original set. There is a sequence of cardinal numbers,0,1,2,3, …, n, …, ℵ0, ℵ1, ℵ2, …, ℵ α, …. This sequence starts with the natural numbers including zero, which are followed by the aleph numbers, the aleph numbers are indexed by ordinal numbers. Under the assumption of the axiom of choice, this transfinite sequence includes every cardinal number, If one rejects that axiom, the situation is more complicated, with additional infinite cardinals that are not alephs. Cardinality is studied for its own sake as part of set theory and it is also a tool used in branches of mathematics including model theory, combinatorics, abstract algebra, and mathematical analysis. In category theory, the numbers form a skeleton of the category of sets. The notion of cardinality, as now understood, was formulated by Georg Cantor, cardinality can be used to compare an aspect of finite sets, e. g. the sets and are not equal, but have the same cardinality, namely three. Cantor applied his concept of bijection to infinite sets, e. g. the set of natural numbers N =, thus, all sets having a bijection with N he called denumerable sets and they all have the same cardinal number. This cardinal number is called ℵ0, aleph-null and he called the cardinal numbers of these infinite sets transfinite cardinal numbers. Cantor proved that any unbounded subset of N has the same cardinality as N and he later proved that the set of all real algebraic numbers is also denumerable. His proof used an argument with nested intervals, but in an 1891 paper he proved the result using his ingenious. The new cardinal number of the set of numbers is called the cardinality of the continuum. His continuum hypothesis is the proposition that c is the same as ℵ1 and this hypothesis has been found to be independent of the standard axioms of mathematical set theory, it can neither be proved nor disproved from the standard assumptions
31.
Permutation
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These differ from combinations, which are selections of some members of a set where order is disregarded. For example, written as tuples, there are six permutations of the set, namely and these are all the possible orderings of this three element set. As another example, an anagram of a word, all of whose letters are different, is a permutation of its letters, in this example, the letters are already ordered in the original word and the anagram is a reordering of the letters. The study of permutations of finite sets is a topic in the field of combinatorics, Permutations occur, in more or less prominent ways, in almost every area of mathematics. For similar reasons permutations arise in the study of sorting algorithms in computer science, the number of permutations of n distinct objects is n factorial, usually written as n. which means the product of all positive integers less than or equal to n. In algebra and particularly in group theory, a permutation of a set S is defined as a bijection from S to itself and that is, it is a function from S to S for which every element occurs exactly once as an image value. This is related to the rearrangement of the elements of S in which each element s is replaced by the corresponding f, the collection of such permutations form a group called the symmetric group of S. The key to this structure is the fact that the composition of two permutations results in another rearrangement. Permutations may act on structured objects by rearranging their components, or by certain replacements of symbols, in elementary combinatorics, the k-permutations, or partial permutations, are the ordered arrangements of k distinct elements selected from a set. When k is equal to the size of the set, these are the permutations of the set, fabian Stedman in 1677 described factorials when explaining the number of permutations of bells in change ringing. Starting from two bells, first, two must be admitted to be varied in two ways which he illustrates by showing 12 and 21 and he then explains that with three bells there are three times two figures to be produced out of three which again is illustrated. His explanation involves cast away 3, and 1.2 will remain, cast away 2, and 1.3 will remain, cast away 1, and 2.3 will remain. He then moves on to four bells and repeats the casting away argument showing that there will be four different sets of three, effectively this is an recursive process. He continues with five bells using the casting method and tabulates the resulting 120 combinations. At this point he gives up and remarks, Now the nature of these methods is such, in modern mathematics there are many similar situations in which understanding a problem requires studying certain permutations related to it. There are two equivalent common ways of regarding permutations, sometimes called the active and passive forms, or in older terminology substitutions and permutations, which form is preferable depends on the type of questions being asked in a given discipline. The active way to regard permutations of a set S is to them as the bijections from S to itself. Thus, the permutations are thought of as functions which can be composed with each other, forming groups of permutations
32.
Isomorphism
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In mathematics, an isomorphism is a homomorphism or morphism that admits an inverse. Two mathematical objects are isomorphic if an isomorphism exists between them, an automorphism is an isomorphism whose source and target coincide. For most algebraic structures, including groups and rings, a homomorphism is an isomorphism if, in topology, where the morphisms are continuous functions, isomorphisms are also called homeomorphisms or bicontinuous functions. In mathematical analysis, where the morphisms are functions, isomorphisms are also called diffeomorphisms. A canonical isomorphism is a map that is an isomorphism. Two objects are said to be isomorphic if there is a canonical isomorphism between them. Isomorphisms are formalized using category theory, let R + be the multiplicative group of positive real numbers, and let R be the additive group of real numbers. The logarithm function log, R + → R satisfies log = log x + log y for all x, y ∈ R +, so it is a group homomorphism. The exponential function exp, R → R + satisfies exp = for all x, y ∈ R, the identities log exp x = x and exp log y = y show that log and exp are inverses of each other. Since log is a homomorphism that has an inverse that is also a homomorphism, because log is an isomorphism, it translates multiplication of positive real numbers into addition of real numbers. This facility makes it possible to real numbers using a ruler. Consider the group, the integers from 0 to 5 with addition modulo 6 and these structures are isomorphic under addition, if you identify them using the following scheme, ↦0 ↦1 ↦2 ↦3 ↦4 ↦5 or in general ↦ mod 6. For example, + =, which translates in the system as 1 +3 =4. Even though these two groups look different in that the sets contain different elements, they are indeed isomorphic, more generally, the direct product of two cyclic groups Z m and Z n is isomorphic to if and only if m and n are coprime. For example, R is an ordering ≤ and S an ordering ⊑, such an isomorphism is called an order isomorphism or an isotone isomorphism. If X = Y, then this is a relation-preserving automorphism, in a concrete category, such as the category of topological spaces or categories of algebraic objects like groups, rings, and modules, an isomorphism must be bijective on the underlying sets. In algebraic categories, an isomorphism is the same as a homomorphism which is bijective on underlying sets, in abstract algebra, two basic isomorphisms are defined, Group isomorphism, an isomorphism between groups Ring isomorphism, an isomorphism between rings. Just as the automorphisms of an algebraic structure form a group, letting a particular isomorphism identify the two structures turns this heap into a group
33.
Homeomorphism
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In the mathematical field of topology, a homeomorphism or topological isomorphism or bi continuous function is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, two spaces with a homeomorphism between them are called homeomorphic, and from a topological viewpoint they are the same. The word homeomorphism comes from the Greek words ὅμοιος = similar and μορφή = shape, roughly speaking, a topological space is a geometric object, and the homeomorphism is a continuous stretching and bending of the object into a new shape. Thus, a square and a circle are homeomorphic to each other, but a sphere and a torus are not. A function f, X → Y between two spaces and is called a homeomorphism if it has the following properties, f is a bijection, f is continuous. A function with three properties is sometimes called bicontinuous. If such a function exists, we say X and Y are homeomorphic, a self-homeomorphism is a homeomorphism of a topological space and itself. The homeomorphisms form a relation on the class of all topological spaces. The resulting equivalence classes are called homeomorphism classes, the open interval is homeomorphic to the real numbers R for any a < b. The unit 2-disc D2 and the square in R2 are homeomorphic. An example of a mapping from the square to the disc is, in polar coordinates. The graph of a function is homeomorphic to the domain of the function. A differentiable parametrization of a curve is an homeomorphism between the domain of the parametrization and the curve, a chart of a manifold is an homeomorphism between an open subset of the manifold and an open subset of a Euclidean space. The stereographic projection is a homeomorphism between the sphere in R3 with a single point removed and the set of all points in R2. If G is a group, its inversion map x ↦ x −1 is a homeomorphism. Also, for any x ∈ G, the left translation y ↦ x y, the right translation y ↦ y x, rm and Rn are not homeomorphic for m ≠ n. The Euclidean real line is not homeomorphic to the circle as a subspace of R2, since the unit circle is compact as a subspace of Euclidean R2. The third requirement, that f −1 be continuous, is essential, consider for instance the function f, [0, 2π) → S1 defined by f =
34.
Diffeomorphism
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In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is a function that maps one differentiable manifold to another such that both the function and its inverse are smooth. Given two manifolds M and N, a map f, M → N is called a diffeomorphism if it is a bijection and its inverse f−1. If these functions are r times continuously differentiable, f is called a Cr-diffeomorphism, two manifolds M and N are diffeomorphic if there is a diffeomorphism f from M to N. They are Cr diffeomorphic if there is an r times continuously differentiable bijective map between them whose inverse is also r times continuously differentiable, F is said to be a diffeomorphism if it is bijective, smooth and its inverse is smooth. First remark It is essential for V to be connected for the function f to be globally invertible. g. Second remark Since the differential at a point D f x, T x U → T f V is a map, it has a well-defined inverse if. The matrix representation of Dfx is the n × n matrix of partial derivatives whose entry in the i-th row. This so-called Jacobian matrix is used for explicit computations. Third remark Diffeomorphisms are necessarily between manifolds of the same dimension, imagine f going from dimension n to dimension k. If n < k then Dfx could never be surjective, in both cases, therefore, Dfx fails to be a bijection. Fourth remark If Dfx is a bijection at x then f is said to be a local diffeomorphism. Fifth remark Given a smooth map from dimension n to k, if Df is surjective, f is said to be a submersion. Sixth remark A differentiable bijection is not necessarily a diffeomorphism, F = x3, for example, is not a diffeomorphism from R to itself because its derivative vanishes at 0. This is an example of a homeomorphism that is not a diffeomorphism, seventh remark When f is a map between differentiable manifolds, a diffeomorphic f is a stronger condition than a homeomorphic f. For a diffeomorphism, f and its inverse need to be differentiable, for a homeomorphism, f, every diffeomorphism is a homeomorphism, but not every homeomorphism is a diffeomorphism. F, M → N is called a diffeomorphism if, in coordinate charts, more precisely, Pick any cover of M by compatible coordinate charts and do the same for N. Let φ and ψ be charts on, respectively, M and N, with U and V as, respectively, the map ψfφ−1, U → V is then a diffeomorphism as in the definition above, whenever f ⊂ ψ−1
35.
Permutation group
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In mathematics, a permutation group is a group G whose elements are permutations of a given set M and whose group operation is the composition of permutations in G. The group of all permutations of a set M is the group of M. The term permutation group thus means a subgroup of the symmetric group, if M = then, Sym, the symmetric group on n letters is usually denoted by Sn. The way in which the elements of a permutation group permute the elements of the set is called its group action, group actions have applications in the study of symmetries, combinatorics and many other branches of mathematics, physics and chemistry. A general property of finite groups implies that a finite nonempty subset of a group is again a group if. The degree of a group of permutations of a set is the number of elements in the set. The order of a group is the number of elements in the group, by Lagranges theorem, the order of any finite permutation group of degree n must divide n. Since permutations are bijections of a set, they can be represented by Cauchys two-line notation and this notation lists each of the elements of M in the first row, and for each element, its image under the permutation below it in the second row. If σ is a permutation of the set M = then, for instance, a particular permutation of the set can be written as, σ =, this means that σ satisfies σ=2, σ=5, σ=4, σ=3, and σ=1. The elements of M need not appear in any order in the first row. This permutation could also be written as, σ =, the permutation written above in 2-line notation would be written in cyclic notation as σ =. The product of two permutations is defined as their composition as functions, in other words σ·π is the function maps any element x of the set to σ. Note that the rightmost permutation is applied to the argument first, with this convention, the product is given by xσ·π = π. However, this gives a different rule for multiplying permutations and this convention is commonly used in the permutation group literature, but this article uses the convention where the rightmost permutation is applied first. Since the composition of two bijections always gives another bijection, the product of two permutations is again a permutation. In two-line notation, the product of two permutations is obtained by rearranging the columns of the second permutation so that its first row is identical with the row of the first permutation. The product can then be written as the first row of the first permutation over the row of the modified second permutation. For example, given the permutations, P = and Q =, the composition of permutations, when they are written in cyclic form, is obtained by juxtaposing the two permutations and then simplifying to a disjoint cycle form if desired
36.
Projective map
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In projective geometry, a homography is an isomorphism of projective spaces, induced by an isomorphism of the vector spaces from which the projective spaces derive. It is a bijection that maps lines to lines, and thus a collineation, in general, some collineations are not homographies, but the fundamental theorem of projective geometry asserts that is not so in the case of real projective spaces of dimension at least two. Synonyms include projectivity, projective transformation, and projective collineation, at the end of 19th century, formal definitions of projective spaces were introduced, which differed from extending Euclidean or affine spaces by adding points at infinity. The term projective transformation originated in these abstract constructions and these constructions divide into two classes that have been shown to be equivalent. For sake of simplicity, unless stated, the projective spaces considered in this article are supposed to be defined over a field. Equivalently Pappuss hexagon theorem and Desargues theorem are supposed to be true, a large part of the results remain true, or may be generalized to projective geometries for which these theorems do not hold. The projection is not defined if the point A belongs to the passing through O. The notion of space was originally introduced by extending the Euclidean space. Given another plane Q, which does not contain O, the restriction to Q of the projection is called a perspectivity. With these definitions, a perspectivity is only a partial function, therefore, this notion is normally defined for projective spaces. Originally, a homography was defined as the composition of a number of perspectivities. It is a part of the theorem of projective geometry that this definition coincides with the more algebraic definition sketched in the introduction. A projective space P of dimension n over a field K may be defined as the set of the lines through the origin in a K-vector space V of dimension n +1. If a basis of V has been fixed, a point of V may be represented by a point of Kn+1. A point of P, being a line in V, may thus be represented by the coordinates of any point of this line. Given two projective spaces P and P of the dimension, an homography is a mapping from P to P. Such an isomorphism induces a bijection from P to P, because of the linearity of f, two such isomorphisms, f and g, define the same homography if and only if there is a nonzero element a of K such that g = af. This may be written in terms of coordinates in the following way, A homography φ may be defined by a nonsingular n+1 × n+1 matrix
37.
Onto
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In mathematics, a function f from a set X to a set Y is surjective, or a surjection, if every element y in Y has a corresponding element x in X such that f = y. The French prefix sur means over or above and relates to the fact that the image of the domain of a surjective function completely covers the functions codomain, a surjective function is a function whose image is equal to its codomain. Equivalently, a function f with domain X and codomain Y is surjective if for every y in Y there exists at least one x in X with f = y, surjections are sometimes denoted by a two-headed rightwards arrow, as in f, X ↠ Y. Symbolically, If f, X → Y, then f is said to be surjective if ∀ y ∈ Y, ∃ x ∈ X, f = y, for any set X, the identity function idX on X is surjective. The function f, Z → defined by f = n mod 2 is surjective. The function f, R → R defined by f = 2x +1 is surjective, because for every real number y we have an x such that f = y, an appropriate x is /2. However, this function is not injective since e. g. the pre-image of y =2 is, the function g, R → R defined by g = x2 is not surjective, because there is no real number x such that x2 = −1. However, the g, R → R0+ defined by g = x2 is surjective because for every y in the nonnegative real codomain Y there is at least one x in the real domain X such that x2 = y. The natural logarithm ln, → R is a surjective. Its inverse, the function, is not surjective as its range is the set of positive real numbers. The matrix exponential is not surjective when seen as a map from the space of all n×n matrices to itself. It is, however, usually defined as a map from the space of all n×n matrices to the linear group of degree n, i. e. the group of all n×n invertible matrices. Under this definition the matrix exponential is surjective for complex matrices, the projection from a cartesian product A × B to one of its factors is surjective unless the other factor is empty. In a 3D video game vectors are projected onto a 2D flat screen by means of a surjective function, a function is bijective if and only if it is both surjective and injective. If a function is identified with its graph, then surjectivity is not a property of the function itself, unlike injectivity, surjectivity cannot be read off of the graph of the function alone. The function g, Y → X is said to be an inverse of the function f, X → Y if f = y for every y in Y. In other words, g is an inverse of f if the composition f o g of g and f in that order is the identity function on the domain Y of g. The function g need not be an inverse of f because the composition in the other order, g o f
38.
One-to-one function
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In mathematics, an injective function or injection or one-to-one function is a function that preserves distinctness, it never maps distinct elements of its domain to the same element of its codomain. In other words, every element of the codomain is the image of at most one element of its domain. The term one-to-one function must not be confused with one-to-one correspondence, occasionally, an injective function from X to Y is denoted f, X ↣ Y, using an arrow with a barbed tail. A function f that is not injective is sometimes called many-to-one, however, the injective terminology is also sometimes used to mean single-valued, i. e. each argument is mapped to at most one value. A monomorphism is a generalization of a function in category theory. Let f be a function whose domain is a set X, the function f is said to be injective provided that for all a and b in X, whenever f = f, then a = b, that is, f = f implies a = b. Equivalently, if a ≠ b, then f ≠ f, in particular the identity function X → X is always injective. If the domain X = ∅ or X has only one element, the function f, R → R defined by f = 2x +1 is injective. The function g, R → R defined by g = x2 is not injective, however, if g is redefined so that its domain is the non-negative real numbers [0, +∞), then g is injective. The exponential function exp, R → R defined by exp = ex is injective, the natural logarithm function ln, → R defined by x ↦ ln x is injective. The function g, R → R defined by g = xn − x is not injective, since, for example, g = g =0. More generally, when X and Y are both the real line R, then a function f, R → R is one whose graph is never intersected by any horizontal line more than once. This principle is referred to as the line test. Functions with left inverses are always injections and that is, given f, X → Y, if there is a function g, Y → X such that, for every x ∈ X g = x then f is injective. In this case, g is called a retraction of f, conversely, f is called a section of g. Conversely, every injection f with non-empty domain has an inverse g. Note that g may not be an inverse of f because the composition in the other order, f o g. In other words, a function that can be undone or reversed, injections are reversible but not always invertible
39.
Unicode
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Unicode is a computing industry standard for the consistent encoding, representation, and handling of text expressed in most of the worlds writing systems. As of June 2016, the most recent version is Unicode 9.0, the standard is maintained by the Unicode Consortium. Unicodes success at unifying character sets has led to its widespread, the standard has been implemented in many recent technologies, including modern operating systems, XML, Java, and the. NET Framework. Unicode can be implemented by different character encodings, the most commonly used encodings are UTF-8, UTF-16 and the now-obsolete UCS-2. UTF-8 uses one byte for any ASCII character, all of which have the same values in both UTF-8 and ASCII encoding, and up to four bytes for other characters. UCS-2 uses a 16-bit code unit for each character but cannot encode every character in the current Unicode standard, UTF-16 extends UCS-2, using one 16-bit unit for the characters that were representable in UCS-2 and two 16-bit units to handle each of the additional characters. Many traditional character encodings share a common problem in that they allow bilingual computer processing, Unicode, in intent, encodes the underlying characters—graphemes and grapheme-like units—rather than the variant glyphs for such characters. In the case of Chinese characters, this leads to controversies over distinguishing the underlying character from its variant glyphs. In text processing, Unicode takes the role of providing a unique code point—a number, in other words, Unicode represents a character in an abstract way and leaves the visual rendering to other software, such as a web browser or word processor. This simple aim becomes complicated, however, because of concessions made by Unicodes designers in the hope of encouraging a more rapid adoption of Unicode, the first 256 code points were made identical to the content of ISO-8859-1 so as to make it trivial to convert existing western text. For other examples, see duplicate characters in Unicode and he explained that he name Unicode is intended to suggest a unique, unified, universal encoding. In this document, entitled Unicode 88, Becker outlined a 16-bit character model, Unicode could be roughly described as wide-body ASCII that has been stretched to 16 bits to encompass the characters of all the worlds living languages. In a properly engineered design,16 bits per character are more than sufficient for this purpose, Unicode aims in the first instance at the characters published in modern text, whose number is undoubtedly far below 214 =16,384. By the end of 1990, most of the work on mapping existing character encoding standards had been completed, the Unicode Consortium was incorporated in California on January 3,1991, and in October 1991, the first volume of the Unicode standard was published. The second volume, covering Han ideographs, was published in June 1992, in 1996, a surrogate character mechanism was implemented in Unicode 2.0, so that Unicode was no longer restricted to 16 bits. The Microsoft TrueType specification version 1.0 from 1992 used the name Apple Unicode instead of Unicode for the Platform ID in the naming table, Unicode defines a codespace of 1,114,112 code points in the range 0hex to 10FFFFhex. Normally a Unicode code point is referred to by writing U+ followed by its hexadecimal number, for code points in the Basic Multilingual Plane, four digits are used, for code points outside the BMP, five or six digits are used, as required. Code points in Planes 1 through 16 are accessed as surrogate pairs in UTF-16, within each plane, characters are allocated within named blocks of related characters
40.
Batting order (baseball)
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In baseball, the batting order or batting lineup is the sequence in which the members of the offense take their turns in batting against the pitcher. The batting order is the component of a teams offensive strategy. The home plate umpire keeps one copy of the card of each team. Once the home plate umpire gives the lineup cards to the managers, the batting lineup is final. If a team out of order, it is a violation of baseballs rules. Dictionary. com, however, defines bat around as to have every player in the lineup take a turn at bat during a single inning and it is not an official statistic. Opinions differ as to whether nine batters must get an at-bat, in modern American baseball, some batting positions have nicknames, leadoff for first, cleanup for fourth, and last for ninth. Others are known by the numbers or the term #-hole. At the start of each inning, the batting order resumes where it left off in the previous inning, early forms of baseball or rounders from the mid 19th-Century did not require a fixed batting order, any player who was not on base could be called upon to bat. The concept of a set batting order is said to have been invented by Alexander Cartwright, who also instituted rules such as the ball and tagging the runner. In the early days of baseball, the rules did not require that the order be announced before game time. This permitted strategic decisions regarding batting order to occur while the game was in progress, for example, Cap Anson was known to wait to see if the first two men got on base in the first inning. If they did not, he would wait and hit in the next inning, however, in the 1880s, organized baseball began mandating that the batting order be disclosed before the first pitch. For example, Rule 36 in The Playing Rules of Professional Base Ball Clubs of 1896 stated the following, after the first inning the first striker in each inning shall be the batsman whose name follows that of the last man who completed his turn. In cricket, the order is generally fixed so that players are sure of their role within the team. A batsman can be promoted to a spot in the batting order according to the teams wishes. The idea of a batting order, in which the on-deck batter at the time the final out is made in one inning becomes the lead-off batter in the next inning, is unique to baseball. In the shorter form of cricket, there is only one innings per side, in a typical innings of this latter form, all eleven players on the team will have a chance to bat, and the innings finishes when 10 players are out
41.
Baseball
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Baseball is a bat-and-ball game played between two teams of nine players each, who take turns batting and fielding. A run is scored when a player advances around the bases, Players on the batting team take turns hitting against the pitcher of the fielding team, which tries to prevent runs by getting hitters out in any of several ways. A player on the team who reaches a base safely can later attempt to advance to subsequent bases during teammates turns batting. The teams switch between batting and fielding whenever the team records three outs. One turn batting for both teams, beginning with the team, constitutes an inning. A game is composed of nine innings, and the team with the number of runs at the end of the game wins. Baseball has no clock, although almost all games end in the ninth inning. Baseball evolved from older bat-and-ball games already being played in England by the mid-18th century and this game was brought by immigrants to North America, where the modern version developed. By the late 19th century, baseball was widely recognized as the sport of the United States. Baseball is now popular in North America and parts of Central and South America, the Caribbean, in the United States and Canada, professional Major League Baseball teams are divided into the National League and American League, each with three divisions, East, West, and Central. The major league champion is determined by playoffs that culminate in the World Series, the top level of play is similarly split in Japan between the Central and Pacific Leagues and in Cuba between the West League and East League. The evolution of baseball from older bat-and-ball games is difficult to trace with precision, a French manuscript from 1344 contains an illustration of clerics playing a game, possibly la soule, with similarities to baseball. Other old French games such as thèque, la balle au bâton, consensus once held that todays baseball is a North American development from the older game rounders, popular in Great Britain and Ireland. Baseball Before We Knew It, A Search for the Roots of the Game, by David Block, suggests that the game originated in England, recently uncovered historical evidence supports this position. Block argues that rounders and early baseball were actually regional variants of other. It has long believed that cricket also descended from such games. The earliest known reference to baseball is in a 1744 British publication, A Little Pretty Pocket-Book, David Block discovered that the first recorded game of Bass-Ball took place in 1749 in Surrey, and featured the Prince of Wales as a player. William Bray, an English lawyer, recorded a game of baseball on Easter Monday 1755 in Guildford and this early form of the game was apparently brought to Canada by English immigrants