1.
Injective module
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This concept is dual to that of projective modules. Injective modules were introduced in and are discussed in detail in the textbook. Injective resolutions measure how far from injective a module is in terms of the injective dimension, injective hulls are maximal essential extensions, and turn out to be minimal injective extensions. Over a Noetherian ring, every module is uniquely a direct sum of indecomposable modules. An injective module over one ring, may not be injective over another, Rings which are themselves injective modules have a number of interesting properties and include rings such as group rings of finite groups over fields. Injective modules include divisible groups and are generalized by the notion of objects in category theory. e. Any short exact sequence 0 →Q → M → K →0 of left R-modules splits, injective right R-modules are defined in complete analogy. Trivially, the module is injective. Given a field k, every k-vector space Q is an injective k-module, reason, if Q is a subspace of V, we can find a basis of Q and extend it to a basis of V. The new extending basis vectors span a subspace K of V and V is the direct sum of Q and K. Note that the direct complement K of Q is not uniquely determined by Q, the rationals Q form an injective abelian group. The factor group Q/Z and the group are also injective Z-modules. The factor group Z/nZ for n >1 is injective as a Z/nZ-module, more generally, for any integral domain R with field of fractions K, the R-module K is an injective R-module, and indeed the smallest injective R-module containing R. For any Dedekind domain, the quotient module K/R is also injective, the zero ideal is also prime and corresponds to the injective K. In this way there is a 1-1 correspondence between ideals and indecomposable injective modules. A particularly rich theory is available for commutative noetherian rings due to Eben Matlis, the injective hull of R/P as an R-module is canonically an RP module, and is the RP-injective hull of R/P. In other words, it suffices to consider local rings, the endomorphism ring of the injective hull of R/P is the completion R ^ P of R at P. Two examples are the hull of the Z-module Z/pZ
2.
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
3.
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
4.
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
5.
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
6.
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
7.
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
8.
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
9.
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
10.
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
11.
Complex analysis
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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
12.
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
13.
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
14.
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
15.
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
16.
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
17.
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
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Smoothness
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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
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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
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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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Bijection
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In mathematical terms, a bijective function f, X → Y is a one-to-one and onto mapping of a set X to a set Y. A bijection from the set X to the set Y has a function from Y to X. If X and Y are finite sets, then the existence of a means they have the same number of elements. For infinite sets the picture is complicated, leading to the concept of cardinal number. A bijective function from a set to itself is called a permutation. Bijective functions are essential to many areas of including the definitions of isomorphism, homeomorphism, diffeomorphism, permutation group. Satisfying properties and means that a bijection is a function with domain X and it is more common to see properties and written as a single statement, Every element of X is paired with exactly one element of Y. Functions which satisfy property are said to be onto Y and are called surjections, Functions which satisfy property are said to be one-to-one functions and are called injections. With this terminology, a bijection is a function which is both a surjection and an injection, or using words, a bijection is a function which is both one-to-one and onto. Consider the batting line-up of a baseball or cricket team, the set X will be the players on the team and the set Y will be the positions in the batting order The pairing is given by which player is in what position in this order. Property is satisfied since each player is somewhere in the list, property is satisfied since no player bats in two positions in the order. Property says that for each position in the order, there is some player batting in that position, in a classroom there are a certain number of seats. A bunch of students enter the room and the instructor asks them all to be seated. After a quick look around the room, the instructor declares that there is a bijection between the set of students and the set of seats, where each student is paired with the seat they are sitting in. The instructor was able to conclude there were just as many seats as there were students. For any set X, the identity function 1X, X → X, the function f, R → R, f = 2x +1 is bijective, since for each y there is a unique x = /2 such that f = y. In more generality, any linear function over the reals, f, R → R, f = ax + b is a bijection, each real number y is obtained from the real number x = /a. The function f, R →, given by f = arctan is bijective since each real x is paired with exactly one angle y in the interval so that tan = x
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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
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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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Image (mathematics)
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In mathematics, an image is the subset of a functions codomain which is the output of the function from a subset of its domain. Evaluating a function at each element of a subset X of the domain, the inverse image or preimage of a particular subset S of the codomain of a function is the set of all elements of the domain that map to the members of S. Image and inverse image may also be defined for binary relations. The word image is used in three related ways, in these definitions, f, X → Y is a function from the set X to the set Y. If x is a member of X, then f = y is the image of x under f, Y is alternatively known as the output of f for argument x. The image of a subset A ⊆ X under f is the subset f ⊆ Y defined by, f = When there is no risk of confusion and this convention is a common one, the intended meaning must be inferred from the context. This makes the image of f a function whose domain is the set of X. The image f of the entire domain X of f is called simply the image of f, let f be a function from X to Y. The set of all the fibers over the elements of Y is a family of sets indexed by Y, for example, for the function f = x2, the inverse image of would be. Again, if there is no risk of confusion, we may denote f −1 by f −1, the notation f −1 should not be confused with that for inverse function. The notation coincides with the one, though, for bijections. The traditional notations used in the section can be confusing. {\displaystyle f=\left\ The image of the set under f is f =, the image of the function f is. The preimage of a is f −1 =, the preimage of is the empty set. F, R → R defined by f = x2, the image of under f is f =, and the image of f is R+. The preimage of f is f −1 =. The preimage of set N = under f is the empty set, F, R2 → R defined by f = x2 + y2. The fibres f −1 are concentric circles about the origin, the origin itself, and the empty set, depending on whether a >0, a =0, or a <0, respectively
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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
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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
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Monomorphism
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In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from X to Y is often denoted with the notation X ↪ Y, monomorphisms are a categorical generalization of injective functions, in some categories the notions coincide, but monomorphisms are more general, as in the examples below. The categorical dual of a monomorphism is an epimorphism, i. e. a monomorphism in a category C is an epimorphism in the dual category Cop, every section is a monomorphism, and every retraction is an epimorphism. A left invertible morphism is called a split mono, however, a monomorphism need not be left-invertible. A morphism f, X → Y is monic if and only if the induced map f∗, Hom → Hom, in the category of sets the converse also holds, so the monomorphisms are exactly the injective morphisms. The converse also holds in most naturally occurring categories of algebras because of the existence of an object on one generator. In particular, it is true in the categories of all groups, of all rings and this is not an injective map, as for example every integer is mapped to 0. Nevertheless, it is a monomorphism in this category and this follows from the implication q ∘ h =0 ⇒ h =0, which we will now prove. If h, G → Q, where G is some divisible group, without loss of generality, we may assume that h ≥0. Then, letting n = h +1, since G is a group, there exists some y ∈ G such that x = ny. This says that h =0, as desired, to go from that implication to the fact that q is a monomorphism, assume that q ∘ f = q ∘ g for some morphisms f, g, G → Q, where G is some divisible group. Then q ∘ =0, where, x ↦ f − g, from the implication just proved, q ∘ =0 ⇒ f − g =0 ⇔ ∀ x ∈ G, f = g ⇔ f = g. Hence q is a monomorphism, as claimed, in a topos, every monic is an equalizer, and any map that is both monic and epic is an isomorphism. There are also useful concepts of regular monomorphism, strong monomorphism, a regular monomorphism equalizes some parallel pair of morphisms. An extremal monomorphism is a monomorphism that cannot be factored through an epimorphism, Precisely, if m = g ∘ e with e an epimorphism. A strong monomorphism satisfies a certain lifting property with respect to commutative squares involving an epimorphism, the companion terms monomorphism and epimorphism were originally introduced by Nicolas Bourbaki, Bourbaki uses monomorphism as shorthand for an injective function. Early category theorists believed that the generalization of injectivity to the context of categories was the cancellation property given above. While this is not exactly true for monic maps, it is close, so this has caused little trouble
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Category theory
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Category theory formalizes mathematical structure and its concepts in terms of a collection of objects and of arrows. A category has two properties, the ability to compose the arrows associatively and the existence of an identity arrow for each object. The language of category theory has been used to formalize concepts of other high-level abstractions such as sets, rings, several terms used in category theory, including the term morphism, are used differently from their uses in the rest of mathematics. In category theory, morphisms obey conditions specific to category theory itself, Category theory has practical applications in programming language theory, in particular for the study of monads in functional programming. Categories represent abstraction of other mathematical concepts, many areas of mathematics can be formalised by category theory as categories. Hence category theory uses abstraction to make it possible to state and prove many intricate, a basic example of a category is the category of sets, where the objects are sets and the arrows are functions from one set to another. However, the objects of a category need not be sets, any way of formalising a mathematical concept such that it meets the basic conditions on the behaviour of objects and arrows is a valid category—and all the results of category theory apply to it. The arrows of category theory are said to represent a process connecting two objects, or in many cases a structure-preserving transformation connecting two objects. There are, however, many applications where more abstract concepts are represented by objects. The most important property of the arrows is that they can be composed, in other words, linear algebra can also be expressed in terms of categories of matrices. A systematic study of category theory allows us to prove general results about any of these types of mathematical structures from the axioms of a category. The class Grp of groups consists of all objects having a group structure, one can proceed to prove theorems about groups by making logical deductions from the set of axioms. For example, it is immediately proven from the axioms that the identity element of a group is unique, in the case of groups, the morphisms are the group homomorphisms. The study of group homomorphisms then provides a tool for studying properties of groups. Not all categories arise as structure preserving functions, however, the example is the category of homotopies between pointed topological spaces. If one axiomatizes relations instead of functions, one obtains the theory of allegories, a category is itself a type of mathematical structure, so we can look for processes which preserve this structure in some sense, such a process is called a functor. Diagram chasing is a method of arguing with abstract arrows joined in diagrams. Functors are represented by arrows between categories, subject to specific defining commutativity conditions, functors can define categorical diagrams and sequences
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Contraposition
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In logic, contraposition is an inference that says that a conditional statement is logically equivalent to its contrapositive. The contrapositive of the statement has its antecedent and consequent inverted and flipped, for instance, the proposition All bats are mammals can be restated as the conditional If something is a bat, then it is a mammal. Now, the law says that statement is identical to the contrapositive If something is not a mammal, then it is not a bat. The contrapositive can be compared with three other relationships between conditional statements, Inversion, ¬ P → ¬ Q If something is not a bat, then it is not a mammal. Unlike the contrapositive, the truth value is not at all dependent on whether or not the original proposition was true. The inverse here is not true. Conversion, Q → P If something is a mammal, then it is a bat, the converse is actually the contrapositive of the inverse and so always has the same truth value as the inverse, which is not necessarily the same as that of the original proposition. Negation, ¬ There exists a bat that is not a mammal, If the negation is true, the original proposition is false. Here, of course, the negation is false, note that if P → Q is true and we are given that Q is false, ¬ Q, it can logically be concluded that P must be false, ¬ P. This is often called the law of contrapositive, or the modus tollens rule of inference, according to this diagram, if something is in A, it must be in B as well. So we can all of A is in B as, A → B It is also clear that anything that is not within B cannot be within A. This statement, ¬ B → ¬ A is the contrapositive, therefore, we can say that →. Practically speaking, this may make much easier when trying to prove something. Alternatively, we can try to prove ¬ B → ¬ A by checking all girls without brown hair to see if they are all outside the US. This means that if we find at least one girl without brown hair within the US, we will have disproved ¬ B → ¬ A, to conclude, for any statement where A implies B, then not B always implies not A. Proving or disproving either one of these statements automatically proves or disproves the other. A proposition Q is implicated by a proposition P when the relationship holds, This states that, if P, then Q, or, if Socrates is a man. In a conditional such as this, P is the antecedent, one statement is the contrapositive of the other only when its antecedent is the negated consequent of the other, and vice versa
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Inclusion map
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In mathematics, if A is a subset of B, then the inclusion map is the function ι that sends each element, x, of A to x, treated as an element of B, ι, A → B, ι = x. A hooked arrow ↪ is sometimes used in place of the function arrow above to denote an inclusion map and this and other analogous injective functions from substructures are sometimes called natural injections. Given any morphism f between objects X and Y, if there is a map into the domain ι, A → X. In many instances, one can construct a canonical inclusion into the codomain R→Y known as the range of f. Inclusion maps tend to be homomorphisms of algebraic structures, thus, more precisely, given a sub-structure closed under some operations, the inclusion map will be an embedding for tautological reasons. For example, for a binary operation ⋆, to require that ι = ι ⋆ ι is simply to say that ⋆ is consistently computed in the sub-structure and the large structure. The case of an operation is similar, but one should also look at nullary operations. Here the point is that closure means such constants must already be given in the substructure, inclusion maps are seen in algebraic topology where if A is a strong deformation retract of X, the inclusion map yields an isomorphism between all homotopy groups. Inclusion maps in geometry come in different kinds, for example embeddings of submanifolds, contravariant objects such as differential forms restrict to submanifolds, giving a mapping in the other direction. Another example, more sophisticated, is that of affine schemes, for which the inclusions Spec → Spec and Spec → Spec may be different morphisms, fundamental Concepts of Algebra, Academic Press, New York, ISBN 0-12-172050-0. Mac Lane, S. Birkhoff, G. Algebra, AMS Chelsea Publishing, Providence, Rhode Island, ISBN 0-8218-1646-2
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Empty set
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In mathematics, and more specifically set theory, the empty set is the unique set having no elements, its size or cardinality is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, in other theories, many possible properties of sets are vacuously true for the empty set. Null set was once a synonym for empty set, but is now a technical term in measure theory. The empty set may also be called the void set, common notations for the empty set include, ∅, and ∅. The latter two symbols were introduced by the Bourbaki group in 1939, inspired by the letter Ø in the Norwegian, although now considered an improper use of notation, in the past,0 was occasionally used as a symbol for the empty set. The empty-set symbol ∅ is found at Unicode point U+2205, in LaTeX, it is coded as \emptyset for ∅ or \varnothing for ∅. In standard axiomatic set theory, by the principle of extensionality, hence there is but one empty set, and we speak of the empty set rather than an empty set. The mathematical symbols employed below are explained here, in this context, zero is modelled by the empty set. For any property, For every element of ∅ the property holds, There is no element of ∅ for which the property holds. Conversely, if for some property and some set V, the two statements hold, For every element of V the property holds, There is no element of V for which the property holds. By the definition of subset, the empty set is a subset of any set A. That is, every element x of ∅ belongs to A. Indeed, since there are no elements of ∅ at all, there is no element of ∅ that is not in A. Any statement that begins for every element of ∅ is not making any substantive claim and this is often paraphrased as everything is true of the elements of the empty set. When speaking of the sum of the elements of a finite set, the reason for this is that zero is the identity element for addition. Similarly, the product of the elements of the empty set should be considered to be one, a disarrangement of a set is a permutation of the set that leaves no element in the same position. The empty set is a disarrangment of itself as no element can be found that retains its original position. Since the empty set has no members, when it is considered as a subset of any ordered set, then member of that set will be an upper bound. For example, when considered as a subset of the numbers, with its usual ordering, represented by the real number line
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Exponential function
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In mathematics, an exponential function is a function of the form in which the input variable x occurs as an exponent. A function of the form f = b x + c, as functions of a real variable, exponential functions are uniquely characterized by the fact that the growth rate of such a function is directly proportional to the value of the function. The constant of proportionality of this relationship is the logarithm of the base b. The argument of the function can be any real or complex number or even an entirely different kind of mathematical object. Its ubiquitous occurrence in pure and applied mathematics has led mathematician W. Rudin to opine that the function is the most important function in mathematics. In applied settings, exponential functions model a relationship in which a constant change in the independent variable gives the same change in the dependent variable. The graph of y = e x is upward-sloping, and increases faster as x increases, the graph always lies above the x -axis but can get arbitrarily close to it for negative x, thus, the x -axis is a horizontal asymptote. The slope of the tangent to the graph at each point is equal to its y -coordinate at that point, as implied by its derivative function. Its inverse function is the logarithm, denoted log, ln, or log e, because of this. The exponential function exp, C → C can be characterized in a variety of equivalent ways, the constant e is then defined as e = exp = ∑ k =0 ∞. The exponential function arises whenever a quantity grows or decays at a proportional to its current value. One such situation is continuously compounded interest, and in fact it was this observation that led Jacob Bernoulli in 1683 to the number lim n → ∞ n now known as e, later, in 1697, Johann Bernoulli studied the calculus of the exponential function. If instead interest is compounded daily, this becomes 365, letting the number of time intervals per year grow without bound leads to the limit definition of the exponential function, exp = lim n → ∞ n first given by Euler. This is one of a number of characterizations of the exponential function, from any of these definitions it can be shown that the exponential function obeys the basic exponentiation identity, exp = exp ⋅ exp which is why it can be written as ex. The derivative of the function is the exponential function itself. More generally, a function with a rate of change proportional to the function itself is expressible in terms of the exponential function and this function property leads to exponential growth and exponential decay. The exponential function extends to a function on the complex plane. Eulers formula relates its values at purely imaginary arguments to trigonometric functions, the exponential function also has analogues for which the argument is a matrix, or even an element of a Banach algebra or a Lie algebra
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Natural logarithm
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The natural logarithm of a number is its logarithm to the base of the mathematical constant e, where e is an irrational and transcendental number approximately equal to 2.718281828459. The natural logarithm of x is written as ln x, loge x, or sometimes, if the base e is implicit. Parentheses are sometimes added for clarity, giving ln, loge or log and this is done in particular when the argument to the logarithm is not a single symbol, to prevent ambiguity. The natural logarithm of x is the power to which e would have to be raised to equal x. The natural log of e itself, ln, is 1, because e1 = e, while the natural logarithm of 1, ln, is 0, since e0 =1. The natural logarithm can be defined for any real number a as the area under the curve y = 1/x from 1 to a. The simplicity of this definition, which is matched in many other formulas involving the natural logarithm, like all logarithms, the natural logarithm maps multiplication into addition, ln = ln + ln . However, logarithms in other bases differ only by a constant multiplier from the natural logarithm, for instance, the binary logarithm is the natural logarithm divided by ln, the natural logarithm of 2. Logarithms are useful for solving equations in which the unknown appears as the exponent of some other quantity, for example, logarithms are used to solve for the half-life, decay constant, or unknown time in exponential decay problems. They are important in many branches of mathematics and the sciences and are used in finance to solve problems involving compound interest, by Lindemann–Weierstrass theorem, the natural logarithm of any positive algebraic number other than 1 is a transcendental number. The concept of the natural logarithm was worked out by Gregoire de Saint-Vincent and their work involved quadrature of the hyperbola xy =1 by determination of the area of hyperbolic sectors. Their solution generated the requisite hyperbolic logarithm function having properties now associated with the natural logarithm, the notations ln x and loge x both refer unambiguously to the natural logarithm of x. log x without an explicit base may also refer to the natural logarithm. This usage is common in mathematics and some scientific contexts as well as in many programming languages, in some other contexts, however, log x can be used to denote the common logarithm. Historically, the notations l. and l were in use at least since the 1730s, finally, in the twentieth century, the notations Log and logh are attested. The graph of the logarithm function shown earlier on the right side of the page enables one to glean some of the basic characteristics that logarithms to any base have in common. Chief among them are, the logarithm of the one is zero. What makes natural logarithms unique is to be found at the point where all logarithms are zero. At that specific point the slope of the curve of the graph of the logarithm is also precisely one