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Levi-Civita Field
In mathematics, the Levi-Civita field, named after Tullio Levi-Civita, is a non-Archimedean ordered field; i.e., a system of numbers containing infinite and infinitesimal quantities. It is usually denoted \mathcal. Each member a can be constructed as a formal series of the form : a = \sum_ a_q\varepsilon^q , where \mathbb is the set of rational numbers, the coefficients a_q are real numbers, and \varepsilon is to be interpreted as a fixed positive infinitesimal. We require that for every rational number r, there are only finitely many q\in\mathbb less than r with a_q\neq 0; this restriction is necessary in order to make multiplication and division well defined and unique. Two such series are considered equal only if all their coefficients are equal. The ordering is defined according to the dictionary ordering of the list of coefficients, which is equivalent to the assumption that \varepsilon is an infinitesimal. The real numbers are embedded in this field as series in which a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tullio Levi-Civita
Tullio Levi-Civita, (; ; 29 March 1873 – 29 December 1941) was an Italian mathematician, most famous for his work on absolute differential calculus ( tensor calculus) and its applications to the theory of relativity, but who also made significant contributions in other areas. He was a pupil of Gregorio Ricci-Curbastro, the inventor of tensor calculus. His work included foundational papers in both pure and applied mathematics, celestial mechanics (notably on the three-body problem), analytic mechanics (the Levi-Civita separability conditions in the Hamilton–Jacobi equation) and hydrodynamics. Biography Born into an Italian Jewish family in Padua, Levi-Civita was the son of Giacomo Levi-Civita, a lawyer and former senator. He graduated in 1892 from the University of Padua Faculty of Mathematics. In 1894 he earned a teaching diploma after which he was appointed to the Faculty of Science teacher's college in Pavia. In 1898 he was appointed to the Padua Chair of Rational Me ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cauchy Complete
In mathematical analysis, a metric space is called complete (or a Cauchy space) if every Cauchy sequence of points in has a limit that is also in . Intuitively, a space is complete if there are no "points missing" from it (inside or at the boundary). For instance, the set of rational numbers is not complete, because e.g. \sqrt is "missing" from it, even though one can construct a Cauchy sequence of rational numbers that converges to it (see further examples below). It is always possible to "fill all the holes", leading to the ''completion'' of a given space, as explained below. Definition Cauchy sequence A sequence x_1, x_2, x_3, \ldots of elements from X of a metric space (X, d) is called Cauchy if for every positive real number r > 0 there is a positive integer N such that for all positive integers m, n > N, d(x_m, x_n) < r. Complete space A metric space is complete if any of the following equivalent conditions are satisfied: #Every Cauchy seq ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real Closed Field
In mathematics, a real closed field is a field F that has the same first-order properties as the field of real numbers. Some examples are the field of real numbers, the field of real algebraic numbers, and the field of hyperreal numbers. Definition A real closed field is a field ''F'' in which any of the following equivalent conditions is true: #''F'' is elementarily equivalent to the real numbers. In other words, it has the same first-order properties as the reals: any sentence in the first-order language of fields is true in ''F'' if and only if it is true in the reals. #There is a total order on ''F'' making it an ordered field such that, in this ordering, every positive element of ''F'' has a square root in ''F'' and any polynomial of odd degree with coefficients in ''F'' has at least one root in ''F''. #''F'' is a formally real field such that every polynomial of odd degree with coefficients in ''F'' has at least one root in ''F'', and for every element ''a'' o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Epsilon Numbers (mathematics)
In mathematics, the epsilon numbers are a collection of transfinite numbers whose defining property is that they are fixed points of an exponential map. Consequently, they are not reachable from 0 via a finite series of applications of the chosen exponential map and of "weaker" operations like addition and multiplication. The original epsilon numbers were introduced by Georg Cantor in the context of ordinal arithmetic; they are the ordinal numbers ''ε'' that satisfy the equation :\varepsilon = \omega^\varepsilon, \, in which ω is the smallest infinite ordinal. The least such ordinal is ''ε''0 (pronounced epsilon nought (chiefly British), epsilon naught (chiefly American), or epsilon zero), which can be viewed as the "limit" obtained by transfinite recursion from a sequence of smaller limit ordinals: :\varepsilon_0 = \omega^ = \sup \left\\,, where is the supremum, which is equivalent to set union in the case of the von Neumann representation of ordinals. Larger ordinal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Surreal Numbers
In mathematics, the surreal number system is a totally ordered proper class containing not only the real numbers but also infinite and infinitesimal numbers, respectively larger or smaller in absolute value than any positive real number. Research on the Go endgame by John Horton Conway led to the original definition and construction of surreal numbers. Conway's construction was introduced in Donald Knuth's 1974 book ''Surreal Numbers: How Two Ex-Students Turned On to Pure Mathematics and Found Total Happiness''. The surreals share many properties with the reals, including the usual arithmetic operations (addition, subtraction, multiplication, and division); as such, they form an ordered field. If formulated in von Neumann–Bernays–Gödel set theory, the surreal numbers are a universal ordered field in the sense that all other ordered fields, such as the rationals, the reals, the rational functions, the Levi-Civita field, the superreal numbers (including the hyperreal numbers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Transseries
In mathematics, the field \mathbb^ of logarithmic-exponential transseries is a non-Archimedean ordered differential field which extends comparability of asymptotic growth rates of elementary nontrigonometric functions to a much broader class of objects. Each log-exp transseries represents a formal asymptotic behavior, and it can be manipulated formally, and when it converges (or in every case if using special semantics such as through infinite surreal numbers), corresponds to actual behavior. Transseries can also be convenient for representing functions. Through their inclusion of exponentiation and logarithms, transseries are a strong generalization of the power series at infinity (\sum_^\infty \frac) and other similar asymptotic expansions. The field \mathbb^ was introduced independently by Dahn-Göring and Ecalle in the respective contexts of model theory or exponential fields and of the study of analytic singularity and proof by Ecalle of the Dulac conjectures. It constitute ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Formal Power Series
In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subtraction, multiplication, division, partial sums, etc.). A formal power series is a special kind of formal series, of the form \sum_^\infty a_nx^n=a_0+a_1x+ a_2x^2+\cdots, where the a_n, called ''coefficients'', are numbers or, more generally, elements of some ring, and the x^n are formal powers of the symbol x that is called an indeterminate or, commonly, a variable. Hence, power series can be viewed as a generalization of polynomials where the number of terms is allowed to be infinite, and differ from usual power series by the absence of convergence requirements, which implies that a power series may not represent a function of its variable. Formal power series are in one to one correspondence with their sequences of coefficients, but the two concepts must not be confused, sin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rational Function
In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be rational numbers; they may be taken in any field . In this case, one speaks of a rational function and a rational fraction ''over ''. The values of the variables may be taken in any field containing . 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 . The set of rational functions over a field is a field, the field of fractions of the ring of the polynomial functions over . Definitions A function f is called a rational function if it can be written in the form : f(x) = \frac where P and Q are polynomial functions of x and Q is not the zero function. The domain of f is the set of all values of x for which the denominator Q(x) is not zero. How ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Puiseux Series
In mathematics, Puiseux series are a generalization of power series that allow for negative and fractional exponents of the indeterminate. For example, the series : \begin x^ &+ 2x^ + x^ + 2x^ + x^ + x^5 + \cdots\\ &=x^+ 2x^ + x^ + 2x^ + x^ + x^ + \cdots \end is a Puiseux series in the indeterminate . Puiseux series were first introduced by Isaac Newton in 1676 and rediscovered by Victor Puiseux in 1850.Puiseux (1850, 1851) The definition of a Puiseux series includes that the denominators of the exponents must be bounded. So, by reducing exponents to a common denominator , a Puiseux series becomes a Laurent series in an th root of the indeterminate. For example, the example above is a Laurent series in x^. Because a complex number has th roots, a convergent Puiseux series typically defines functions in a neighborhood of . Puiseux's theorem, sometimes also called the Newton–Puiseux theorem, asserts that, given a polynomial equation P(x,y)=0 with complex coefficie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hahn Series
In mathematics, Hahn series (sometimes also known as Hahn–Mal'cev–Neumann series) are a type of formal series, formal infinite series. They are a generalization of Puiseux series (themselves a generalization of formal power series) and were first introduced by Hans Hahn (mathematician), Hans Hahn in 1907 (and then further generalized by Anatoly Maltsev and Bernhard Neumann to a non-commutative setting). They allow for arbitrary exponents of the Indeterminate (variable), indeterminate so long as the set supporting them forms a well-ordered subset of the Valuation (algebra), value group (typically \mathbb or \mathbb). Hahn series were first introduced, as groups, in the course of the mathematical proof, proof of the Hahn embedding theorem and then studied by him in relation to Hilbert's second problem. Formulation The field (mathematics), field of Hahn series K\left[\left[T^\Gamma\right]\right] (in the indeterminate T) over a field K and with value group \Gamma (an ordered group) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spherically Complete
In mathematics, a field ''K'' with an absolute value is called spherically complete if the intersection of every decreasing sequence of balls (in the sense of the metric induced by the absolute value) is nonempty: :B_1\supseteq B_2\supseteq \cdots \Rightarrow\bigcap_ B_n\neq \empty. The definition can be adapted also to a field ''K'' with a valuation ''v'' taking values in an arbitrary ordered abelian group: (''K'',''v'') is spherically complete if every collection of balls that is totally ordered by inclusion has a nonempty intersection. Spherically complete fields are important in nonarchimedean functional analysis, since many results analogous to theorems of classical functional analysis require the base field to be spherically complete. Examples *Any locally compact field is spherically complete. This includes, in particular, the fields Q''p'' of p-adic numbers, and any of their finite extensions. *Every spherically complete field is complete. On the other hand, C''p'', ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
