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Arithmetic Derivative
In number theory, the Lagarias arithmetic derivative or number derivative is a function defined for integers, based on prime factorization, by analogy with the product rule for the derivative of a function that is used in mathematical analysis. There are many versions of "arithmetic derivatives", including the one discussed in this article (the Lagarias arithmetic derivative), such as Ihara's arithmetic derivative and Buium's arithmetic derivatives. Early history The arithmetic derivative was introduced by Spanish mathematician Josè Mingot Shelly in 1911. The arithmetic derivative also appeared in the 1950 Putnam Competition. Definition For natural numbers , the arithmetic derivative In this article we use Oliver Heaviside's notation for the arithmetic derivative of . There are various other notations possible, such as ; a full discussion is available here for general differential operators, of which the arithmetic derivative can be considered one. Heaviside's notation is used ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Number Theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics."German original: "Die Mathematik ist die Königin der Wissenschaften, und die Arithmetik ist die Königin der Mathematik." Number theorists study prime numbers as well as the properties of mathematical objects made out of integers (for example, rational numbers) or defined as generalizations of the integers (for example, algebraic integers). Integers can be considered either in themselves or as solutions to equations (Diophantine geometry). Questions in number theory are often best understood through the study of Complex analysis, analytical objects (for example, the Riemann zeta function) that encode properties of the integers, primes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rational Number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all rational numbers, also referred to as "the rationals", the field of rationals or the field of rational numbers is usually denoted by boldface , or blackboard bold \mathbb. A rational number is a real number. The real numbers that are rational are those whose decimal expansion either terminates after a finite number of digits (example: ), or eventually begins to repeat the same finite sequence of digits over and over (example: ). This statement is true not only in base 10, but also in every other integer base, such as the binary and hexadecimal ones (see ). A real number that is not rational is called irrational. Irrational numbers include , , , and . Since the set of rational numbers is countable, and the set of real numbers is uncountable ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real Number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real number can be almost uniquely represented by an infinite decimal expansion. The real numbers are fundamental in calculus (and more generally in all mathematics), in particular by their role in the classical definitions of limits, continuity and derivatives. The set of real numbers is denoted or \mathbb and is sometimes called "the reals". The adjective ''real'' in this context was introduced in the 17th century by René Descartes to distinguish real numbers, associated with physical reality, from imaginary numbers (such as the square roots of ), which seemed like a theoretical contrivance unrelated to physical reality. The real numbers include the rational numbers, such as the integer and the fraction . The rest of the real number ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Univariate
In mathematics, a univariate object is an expression, equation, function or polynomial involving only one variable. Objects involving more than one variable are multivariate. In some cases the distinction between the univariate and multivariate cases is fundamental; for example, the fundamental theorem of algebra and Euclid's algorithm for polynomials are fundamental properties of univariate polynomials that cannot be generalized to multivariate polynomials. In statistics, a univariate distribution characterizes one variable, although it can be applied in other ways as well. For example, univariate data are composed of a single scalar component. In time series analysis, the whole time series is the "variable": a univariate time series is the series of values over time of a single quantity. Correspondingly, a "multivariate time series" characterizes the changing values over time of several quantities. In some cases, the terminology is ambiguous, since the values within a univariate t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Derivation (differential Algebra)
In mathematics, a derivation is a function on an algebra over a field, algebra which generalizes certain features of the derivative operator. Specifically, given an algebra ''A'' over a ring (mathematics), ring or a field (mathematics), field ''K'', a ''K''-derivation is a ''K''-linear map that satisfies Product rule, Leibniz's law: : D(ab) = a D(b) + D(a) b. More generally, if ''M'' is an ''A''-bimodule, a ''K''-linear map that satisfies the Leibniz law is also called a derivation. The collection of all ''K''-derivations of ''A'' to itself is denoted by Der''K''(''A''). The collection of ''K''-derivations of ''A'' into an ''A''-module ''M'' is denoted by . Derivations occur in many different contexts in diverse areas of mathematics. The partial derivative with respect to a variable is an R-derivation on the algebra of real-valued differentiable functions on R''n''. The Lie derivative with respect to a vector field is an R-derivation on the algebra of differentiable function ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polynomial Ring
In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables) with coefficients in another ring, often a field. Often, the term "polynomial ring" refers implicitly to the special case of a polynomial ring in one indeterminate over a field. The importance of such polynomial rings relies on the high number of properties that they have in common with the ring of the integers. Polynomial rings occur and are often fundamental in many parts of mathematics such as number theory, commutative algebra, and algebraic geometry. In ring theory, many classes of rings, such as unique factorization domains, regular rings, group rings, rings of formal power series, Ore polynomials, graded rings, have been introduced for generalizing some properties of polynomial rings. A closely related notion is that of the ring ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Field Of Fractions
In abstract algebra, the field of fractions of an integral domain is the smallest field in which it can be embedded. The construction of the field of fractions is modeled on the relationship between the integral domain of integers and the field of rational numbers. Intuitively, it consists of ratios between integral domain elements. The field of fractions of R is sometimes denoted by \operatorname(R) or \operatorname(R), and the construction is sometimes also called the fraction field, field of quotients, or quotient field of R. All four are in common usage, but are not to be confused with the quotient of a ring by an ideal, which is a quite different concept. For a commutative ring which is not an integral domain, the analogous construction is called the localization or ring of quotients. Definition Given an integral domain and letting R^* = R \setminus \, we define an equivalence relation on R \times R^* by letting (n,d) \sim (m,b) whenever nb = md. We denote the equivale ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eisenstein Integer
In mathematics, the Eisenstein integers (named after Gotthold Eisenstein), occasionally also known as Eulerian integers (after Leonhard Euler), are the complex numbers of the form :z = a + b\omega , where and are integers and :\omega = \frac = e^ is a primitive (hence non-real) cube root of unity. The Eisenstein integers form a triangular lattice in the complex plane, in contrast with the Gaussian integers, which form a square lattice in the complex plane. The Eisenstein integers are a countably infinite set. Properties The Eisenstein integers form a commutative ring of algebraic integers in the algebraic number field \mathbb(\omega) — the third cyclotomic field. To see that the Eisenstein integers are algebraic integers note that each is a root of the monic polynomial :z^2 - (2a - b)\;\!z + \left(a^2 - ab + b^2\right)~. In particular, satisfies the equation :\omega^2 + \omega + 1 = 0~. The product of two Eisenstein integers and is given explicitly by :(a + ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gaussian Integer
In number theory, a Gaussian integer is a complex number whose real and imaginary parts are both integers. The Gaussian integers, with ordinary addition and multiplication of complex numbers, form an integral domain, usually written as \mathbf /math> or \Z Gaussian integers share many properties with integers: they form a Euclidean domain, and have thus a Euclidean division and a Euclidean algorithm; this implies unique factorization and many related properties. However, Gaussian integers do not have a total ordering that respects arithmetic. Gaussian integers are algebraic integers and form the simplest ring of quadratic integers. Gaussian integers are named after the German mathematician Carl Friedrich Gauss. Basic definitions The Gaussian integers are the set :\mathbf \, \qquad \text i^2 = -1. In other words, a Gaussian integer is a complex number such that its real and imaginary parts are both integers. Since the Gaussian integers are closed under addition and multip ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unique Factorization Domain
In mathematics, a unique factorization domain (UFD) (also sometimes called a factorial ring following the terminology of Bourbaki) is a ring in which a statement analogous to the fundamental theorem of arithmetic holds. Specifically, a UFD is an integral domain (a nontrivial commutative ring in which the product of any two non-zero elements is non-zero) in which every non-zero non-unit element can be written as a product of prime elements (or irreducible elements), uniquely up to order and units. Important examples of UFDs are the integers and polynomial rings in one or more variables with coefficients coming from the integers or from a field. Unique factorization domains appear in the following chain of class inclusions: Definition Formally, a unique factorization domain is defined to be an integral domain ''R'' in which every non-zero element ''x'' of ''R'' can be written as a product (an empty product if ''x'' is a unit) of irreducible elements ''p''i of ''R'' and a uni ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Irrational Number
In mathematics, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two integers. When the ratio of lengths of two line segments is an irrational number, the line segments are also described as being '' incommensurable'', meaning that they share no "measure" in common, that is, there is no length ("the measure"), no matter how short, that could be used to express the lengths of both of the two given segments as integer multiples of itself. Among irrational numbers are the ratio of a circle's circumference to its diameter, Euler's number ''e'', the golden ratio ''φ'', and the square root of two. In fact, all square roots of natural numbers, other than of perfect squares, are irrational. Like all real numbers, irrational numbers can be expressed in positional notation, notably as a decimal number. In the cas ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bo Åhlander
Bo or BO may refer to Arts and entertainment Film, television, and theatre *Box office, where tickets to an event are sold, and by extension, the amount of business a production receives *'' BA:BO'', 2008 South Korean film * ''Bo'' (film), a Belgian film starring Ella-June Henrard and directed by Hans Herbots Gaming *'' Call of Duty: Black Ops'', a first-person shooter video game *'' Blood Omen: Legacy of Kain'', first in the Legacy of Kain video game series Music *Bo (instrument), a Chinese cymbal * Bo, a Greek rapper. Religion *Bo or Bodhi Tree *Bo (parsha), fifteenth weekly Torah reading Ethnic groups *Bo people (China), a nearly extinct minority population in Southern China *Bo people of Laos, see List of ethnic groups in Laos *Bo people (Andaman), a recently extinct group in the Andaman Islands Human names *Bo (given name), name origin, plus a list of people and fictional characters with the name or nickname *Bo (surname), name origin, plus a list of people with the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
