In mathematics, a group is a set equipped with a binary operation which combines any two elements to form a third element in such a way that four conditions called group axioms are satisfied, namely closure, associativity and invertibility. One of the most familiar examples of a group is the set of integers together with the addition operation, but groups are encountered in numerous areas within and outside mathematics, help focusing on essential structural aspects, by detaching them from the concrete nature of the subject of the study. Groups share a fundamental kinship with the notion of symmetry. For example, a symmetry group encodes symmetry features of a geometrical object: the group consists of the set of transformations that leave the object unchanged and the operation of combining two such transformations by performing one after the other. Lie groups are the symmetry groups used in the Standard Model of particle physics; the concept of a group arose from the study of polynomial equations, starting with Évariste Galois in the 1830s.
After contributions from other fields such as number theory and geometry, the group notion was generalized and established around 1870. Modern group theory—an active mathematical discipline—studies groups in their own right. To explore groups, mathematicians have devised various notions to break groups into smaller, better-understandable pieces, such as subgroups, quotient groups and simple groups. In addition to their abstract properties, group theorists study the different ways in which a group can be expressed concretely, both from a point of view of representation theory and of computational group theory. A theory has been developed for finite groups, which culminated with the classification of finite simple groups, completed in 2004. Since the mid-1980s, geometric group theory, which studies finitely generated groups as geometric objects, has become an active area in group theory; the modern concept of an abstract group developed out of several fields of mathematics. The original motivation for group theory was the quest for solutions of polynomial equations of degree higher than 4.
The 19th-century French mathematician Évariste Galois, extending prior work of Paolo Ruffini and Joseph-Louis Lagrange, gave a criterion for the solvability of a particular polynomial equation in terms of the symmetry group of its roots. The elements of such a Galois group correspond to certain permutations of the roots. At first, Galois' ideas were rejected by his contemporaries, published only posthumously. More general permutation groups were investigated in particular by Augustin Louis Cauchy. Arthur Cayley's On the theory of groups, as depending on the symbolic equation θn = 1 gives the first abstract definition of a finite group. Geometry was a second field in which groups were used systematically symmetry groups as part of Felix Klein's 1872 Erlangen program. After novel geometries such as hyperbolic and projective geometry had emerged, Klein used group theory to organize them in a more coherent way. Further advancing these ideas, Sophus Lie founded the study of Lie groups in 1884; the third field contributing to group theory was number theory.
Certain abelian group structures had been used implicitly in Carl Friedrich Gauss' number-theoretical work Disquisitiones Arithmeticae, more explicitly by Leopold Kronecker. In 1847, Ernst Kummer made early attempts to prove Fermat's Last Theorem by developing groups describing factorization into prime numbers; the convergence of these various sources into a uniform theory of groups started with Camille Jordan's Traité des substitutions et des équations algébriques. Walther von Dyck introduced the idea of specifying a group by means of generators and relations, was the first to give an axiomatic definition of an "abstract group", in the terminology of the time; as of the 20th century, groups gained wide recognition by the pioneering work of Ferdinand Georg Frobenius and William Burnside, who worked on representation theory of finite groups, Richard Brauer's modular representation theory and Issai Schur's papers. The theory of Lie groups, more locally compact groups was studied by Hermann Weyl, Élie Cartan and many others.
Its algebraic counterpart, the theory of algebraic groups, was first shaped by Claude Chevalley and by the work of Armand Borel and Jacques Tits. The University of Chicago's 1960–61 Group Theory Year brought together group theorists such as Daniel Gorenstein, John G. Thompson and Walter Feit, laying the foundation of a collaboration that, with input from numerous other mathematicians, led to the classification of finite simple groups, with the final step taken by Aschbacher and Smith in 2004; this project exceeded previous mathematical endeavours by its sheer size, in both length of proof and number of researchers. Research is ongoing to simplify the proof of this classification; these days, group theory is still a active mathematical branch, impacting many other fields. One of the most familiar groups is the set of integers Z which consists of the numbers... − 4, − 3, − − 1, 0, 1, 2, 3, 4... together with addition. The following properties of integer addition serve as a model for the group axioms given in the definition below.
For any two integers a and b, the sum a + b is an integer. That is, addition of integers always yields an integer; this property is known as closure under addition. For all integers a, b and c, + c = a +. Expressed in words
The method of least squares is a standard approach in regression analysis to approximate the solution of overdetermined systems, i.e. sets of equations in which there are more equations than unknowns. "Least squares" means that the overall solution minimizes the sum of the squares of the residuals made in the results of every single equation. The most important application is in data fitting; the best fit in the least-squares sense minimizes the sum of squared residuals. When the problem has substantial uncertainties in the independent variable simple regression and least-squares methods have problems. Least-squares problems fall into two categories: linear or ordinary least squares and nonlinear least squares, depending on whether or not the residuals are linear in all unknowns; the linear least-squares problem occurs in statistical regression analysis. The nonlinear problem is solved by iterative refinement. Polynomial least squares describes the variance in a prediction of the dependent variable as a function of the independent variable and the deviations from the fitted curve.
When the observations come from an exponential family and mild conditions are satisfied, least-squares estimates and maximum-likelihood estimates are identical. The method of least squares can be derived as a method of moments estimator; the following discussion is presented in terms of linear functions but the use of least squares is valid and practical for more general families of functions. By iteratively applying local quadratic approximation to the likelihood, the least-squares method may be used to fit a generalized linear model; the least-squares method is credited to Carl Friedrich Gauss, but it was first published by Adrien-Marie Legendre. The method of least squares grew out of the fields of astronomy and geodesy, as scientists and mathematicians sought to provide solutions to the challenges of navigating the Earth's oceans during the Age of Exploration; the accurate description of the behavior of celestial bodies was the key to enabling ships to sail in open seas, where sailors could no longer rely on land sightings for navigation.
The method was the culmination of several advances that took place during the course of the eighteenth century: The combination of different observations as being the best estimate of the true value. The combination of different observations taken under the same conditions contrary to trying one's best to observe and record a single observation accurately; the approach was known as the method of averages. This approach was notably used by Tobias Mayer while studying the librations of the moon in 1750, by Pierre-Simon Laplace in his work in explaining the differences in motion of Jupiter and Saturn in 1788; the combination of different observations taken under different conditions. The method came to be known as the method of least absolute deviation, it was notably performed by Roger Joseph Boscovich in his work on the shape of the earth in 1757 and by Pierre-Simon Laplace for the same problem in 1799. The development of a criterion that can be evaluated to determine when the solution with the minimum error has been achieved.
Laplace tried to specify a mathematical form of the probability density for the errors and define a method of estimation that minimizes the error of estimation. For this purpose, Laplace used a symmetric two-sided exponential distribution we now call Laplace distribution to model the error distribution, used the sum of absolute deviation as error of estimation, he felt these to be the simplest assumptions he could make, he had hoped to obtain the arithmetic mean as the best estimate. Instead, his estimator was the posterior median; the first clear and concise exposition of the method of least squares was published by Legendre in 1805. The technique is described as an algebraic procedure for fitting linear equations to data and Legendre demonstrates the new method by analyzing the same data as Laplace for the shape of the earth; the value of Legendre's method of least squares was recognized by leading astronomers and geodesists of the time. In 1809 Carl Friedrich Gauss published his method of calculating the orbits of celestial bodies.
In that work he claimed to have been in possession of the method of least squares since 1795. This led to a priority dispute with Legendre. However, to Gauss's credit, he went beyond Legendre and succeeded in connecting the method of least squares with the principles of probability and to the normal distribution, he had managed to complete Laplace's program of specifying a mathematical form of the probability density for the observations, depending on a finite number of unknown parameters, define a method of estimation that minimizes the error of estimation. Gauss showed that the arithmetic mean is indeed the best estimate of the location parameter by changing both the probability density and the method of estimation, he turned the problem around by asking what form the density should have and what method of estimation should be used to get the arithmetic mean as estimate of the location parameter. In this attempt, he invented the normal distribution. An early demonstration of the strength of Gauss's method came when it was used to predict the future location of the newly discovered astero
Idempotence is the property of certain operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of places in abstract algebra and functional programming; the term was introduced by Benjamin Peirce in the context of elements of algebras that remain invariant when raised to a positive integer power, means " the same power", from idem + potence. An element x of a magma is said to be idempotent if: x • x = x. If all elements are idempotent with respect to • • is called idempotent; the formula ∀x, x • x = x is called the idempotency law for •. The natural number 1 is an idempotent element with respect to multiplication, so is 0, but no other natural number is. For the latter reason, multiplication of natural numbers is not an idempotent operation. More formally, in the monoid, idempotent elements are just 0 and 1. In a magma, an identity element e or an absorbing element a, if it exists, is idempotent.
Indeed, e • e = e and a • a = a. In a group, the identity element e is the only idempotent element. Indeed, if x is an element of G such that x • x = x x • x = x • e and x = e by multiplying on the left by the inverse element of x. Taking the intersection x∩y of two sets x and y is an idempotent operation, since x∩x always equals x; this means that the idempotency law ∀ x ∩ x = x is true. Taking the union of two sets is an idempotent operation. Formally, in the monoids and of the power set of the set E with the set union ∪ and set intersection ∩ all elements are idempotent. In the monoids and of the Boolean domain with the logical disjunction ∨ and the logical conjunction ∧ all elements are idempotent. In a Boolean ring, multiplication is idempotent. In the monoid of the functions from a set E to a subset F of E with the function composition ∘, idempotent elements are the functions f: E → F such that f ∘ f = f, in other words such that for all x in E, f = f. For example: Taking the absolute value abs of an integer number x is an idempotent function for the following reason: abs = abs is true for each integer number x.
This means that abs ∘ abs = abs holds, that is, abs is an idempotent element in the set of all functions with respect to function composition. Therefore, abs satisfies the above definition of an idempotent function. Other examples include: the identity function is idempotent. If the set E has n elements, we can partition it into k chosen fixed points and n − k non-fixed points under f, kn−k is the number of different idempotent functions. Hence, taking into account all possible partitions, ∑ k = 0 n k n − k is the total number of possible idempotent functions on the set; the integer sequence of the number of idempotent functions as given by the sum above for n = 0, 1, 2, 3, 4, 5, 6, 7, 8, … starts with 1, 1, 3, 10, 41, 196, 1057, 6322, 41393, …. Neither the property of being idempotent nor that of being not is preserved under function composition; as an example for the former, f = x mod 3 and g = max are both idempotent, but f ∘ g is not, although g ∘ f happens to be. As an example for the latter, the negation function ¬ on the Boolean domain is not idempotent, but ¬ ∘ ¬ is.
Unary negation − of real numbers is not idempotent, but − ∘ − is. In computer science, the term idempotence may have a different meaning depending on the context in which it is applied: in imperative programming, a subroutine with side effects is idempotent if the system state remains the same after one or several calls, in other words if the function from the system state space to itself associated to the subroutine is idempotent in the mathematical sense given in the definition; this is a useful property in many situations, as it means that an operation can be repeated or retried as as necessary without causing unintended effects. With non-idempotent operations, the algorithm may have to keep track of whether the operation was performed or not. A function looking up a customer's name and address in a database is idempotent, since this will not cause the database to change. Changing a customer's address is idempotent, because the final address will be the same no matter how many times it is submitted.
However, placing an order for a cart for the customer is not idempotent, since running the call several t
Multiplication is one of the four elementary mathematical operations of arithmetic. The multiplication of whole numbers may be thought as a repeated addition; the multiplier can be multiplicand second. A × b = b + ⋯ + b ⏟ a For example, 4 multiplied by 3 can be calculated by adding 3 copies of 4 together: 3 × 4 = 4 + 4 + 4 = 12 Here 3 and 4 are the factors and 12 is the product. One of the main properties of multiplication is the commutative property: adding 3 copies of 4 gives the same result as adding 4 copies of 3: 4 × 3 = 3 + 3 + 3 + 3 = 12 Thus the designation of multiplier and multiplicand does not affect the result of the multiplication; the multiplication of integers, rational numbers and real numbers is defined by a systematic generalization of this basic definition. Multiplication can be visualized as counting objects arranged in a rectangle or as finding the area of a rectangle whose sides have given lengths; the area of a rectangle does not depend on which side is measured first, which illustrates the commutative property.
The product of two measurements is a new type of measurement, for instance multiplying the lengths of the two sides of a rectangle gives its area, this is the subject of dimensional analysis. The inverse operation of multiplication is division. For example, since 4 multiplied by 3 equals 12 12 divided by 3 equals 4. Multiplication by 3, followed by division by 3, yields the original number. Multiplication is defined for other types of numbers, such as complex numbers, more abstract constructs, like matrices. For some of these more abstract constructs, the order in which the operands are multiplied together matters. A listing of the many different kinds of products that are used in mathematics is given in the product page. In arithmetic, multiplication is written using the sign "×" between the terms. For example, 2 × 3 = 6 3 × 4 = 12 2 × 3 × 5 = 6 × 5 = 30 2 × 2 × 2 × 2 × 2 = 32 The sign is encoded in Unicode at U+00D7 × MULTIPLICATION SIGN. There are other mathematical notations for multiplication: Multiplication is denoted by dot signs a middle-position dot:5 ⋅ 2 or 5.
3 The middle dot notation, encoded in Unicode as U+22C5 ⋅ DOT OPERATOR, is standard in the United States, the United Kingdom, other countries where the period is used as a decimal point. When the dot operator character is not accessible, the interpunct is used. In other countries that use a comma as a decimal mark, either the period or a middle dot is used for multiplication. In algebra, multiplication involving variables is written as a juxtaposition called implied multiplication; the notation can be used for quantities that are surrounded by parentheses. This implicit usage of multiplication can cause ambiguity when the concatenated variables happen to match the name of another variable, when a variable name in front of a parenthesis can be confused with a function name, or in the correct determination of the order of operations. In vector multiplication, there is a distinction between the dot symbols; the cross symbol denotes the taking a cross product of two vectors, yielding a vector as the result, while the dot denotes taking the dot product of two vectors, resulting in a scalar.
In computer programming, the asterisk is still the most common notation. This is due to the fact that most computers were limited to small character sets that lacked a multiplication sign, while the asterisk appeared on every keyboard; this usage originated in the FORTRAN programming language. The numbers to be multiplied are called the "factors"; the number to be multiplied is the "multiplicand", the number by which it is multiplied is the "multiplier". The multiplier is placed first and the multiplicand is placed second; as the result of a multiplication does not depend on the order of the factors, the distinction between "multiplicand" and "multiplier" is useful only at a elementary level and
Addition is one of the four basic operations of arithmetic. The addition of two whole numbers is the total amount of those values combined. For example, in the adjacent picture, there is a combination of three apples and two apples together, making a total of five apples; this observation is equivalent to the mathematical expression "3 + 2 = 5" i.e. "3 add 2 is equal to 5". Besides counting items, addition can be defined on other types of numbers, such as integers, real numbers and complex numbers; this is part of a branch of mathematics. In algebra, another area of mathematics, addition can be performed on abstract objects such as vectors and matrices. Addition has several important properties, it is commutative, meaning that order does not matter, it is associative, meaning that when one adds more than two numbers, the order in which addition is performed does not matter. Repeated addition of 1 is the same as counting. Addition obeys predictable rules concerning related operations such as subtraction and multiplication.
Performing addition is one of the simplest numerical tasks. Addition of small numbers is accessible to toddlers. In primary education, students are taught to add numbers in the decimal system, starting with single digits and progressively tackling more difficult problems. Mechanical aids range from the ancient abacus to the modern computer, where research on the most efficient implementations of addition continues to this day. Addition is written using the plus sign "+" between the terms; the result is expressed with an equals sign. For example, 1 + 1 = 2 2 + 2 = 4 1 + 2 = 3 5 + 4 + 2 = 11 3 + 3 + 3 + 3 = 12 There are situations where addition is "understood" though no symbol appears: A whole number followed by a fraction indicates the sum of the two, called a mixed number. For example, 3½ = 3 + ½ = 3.5. This notation can cause confusion since in most other contexts juxtaposition denotes multiplication instead; the sum of a series of related numbers can be expressed through capital sigma notation, which compactly denotes iteration.
For example, ∑ k = 1 5 k 2 = 1 2 + 2 2 + 3 2 + 4 2 + 5 2 = 55. The numbers or the objects to be added in general addition are collectively referred to as the terms, the addends or the summands; this is to be distinguished from factors. Some authors call. In fact, during the Renaissance, many authors did not consider the first addend an "addend" at all. Today, due to the commutative property of addition, "augend" is used, both terms are called addends. All of the above terminology derives from Latin. "Addition" and "add" are English words derived from the Latin verb addere, in turn a compound of ad "to" and dare "to give", from the Proto-Indo-European root *deh₃- "to give". Using the gerundive suffix -nd results in "addend", "thing to be added". From augere "to increase", one gets "augend", "thing to be increased". "Sum" and "summand" derive from the Latin noun summa "the highest, the top" and associated verb summare. This is appropriate not only because the sum of two positive numbers is greater than either, but because it was common for the ancient Greeks and Romans to add upward, contrary to the modern practice of adding downward, so that a sum was higher than the addends.
Addere and summare date back at least to Boethius, if not to earlier Roman writers such as Vitruvius and Frontinus. The Middle English terms "adden" and "adding" were popularized by Chaucer; the plus sign "+" is an abbreviation of the Latin word et, meaning "and". It appears in mathematical works dating back to at least 1489. Addition is used to model many physical processes. For the simple case of adding natural numbers, there are many possible interpretations and more visual representations; the most fundamental interpretation of addition lies in combining sets: When two or more disjoint collections are combined into a single collection, the number of objects in the single collection is the sum of the number of objects in the original collections. This interpretation is easy to visualize, with little danger of ambiguity, it is useful in higher mathematics. However, it is not obvious how one should extend this version of addition to include fractional numbers or negative numbers. One possible fix is to consider collections of objects that can be divided, such as pies or, still bet