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Gelfond–Schneider Theorem
In mathematics, the Gelfond–Schneider theorem establishes the transcendence of a large class of numbers. History It was originally proved independently in 1934 by Aleksandr Gelfond and Theodor Schneider. Statement Comments The values of ''a'' and ''b'' are not restricted to real numbers; complex numbers are allowed (here complex numbers are not regarded as rational when they have an imaginary part not equal to 0, even if both the real and imaginary parts are rational). In general, is multivalued, where log stands for the complex natural logarithm. (This is the multivalued inverse of the exponential function exp.) This accounts for the phrase "any value of" in the theorem's statement. An equivalent formulation of the theorem is the following: if ''α'' and ''γ'' are nonzero algebraic numbers, and we take any non-zero logarithm of ''α'', then is either rational or transcendental. This may be expressed as saying that if , are linearly independent over the rationals ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complete Metric Space
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] |
Dover Publications
Dover Publications, also known as Dover Books, is an American book publisher founded in 1941 by Hayward and Blanche Cirker. It primarily reissues books that are out of print from their original publishers. These are often, but not always, books in the public domain. The original published editions may be scarce or historically significant. Dover republishes these books, making them available at a significantly reduced cost. Classic reprints Dover reprints classic works of literature, classical sheet music, and public-domain images from the 18th and 19th centuries. Dover also publishes an extensive collection of mathematical, scientific, and engineering texts. It often targets its reprints at a niche market, such as woodworking. Starting in 2015, the company branched out into graphic novel reprints, overseen by Dover acquisitions editor and former comics writer and editor Drew Ford. Most Dover reprints are photo facsimiles of the originals, retaining the original pagination ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second-largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business internationally, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cambridge University Press
Cambridge University Press was the university press of the University of Cambridge. Granted a letters patent by King Henry VIII in 1534, it was the oldest university press in the world. Cambridge University Press merged with Cambridge Assessment to form Cambridge University Press and Assessment under Queen Elizabeth II's approval in August 2021. With a global sales presence, publishing hubs, and offices in more than 40 countries, it published over 50,000 titles by authors from over 100 countries. Its publications include more than 420 academic journals, monographs, reference works, school and university textbooks, and English language teaching and learning publications. It also published Bibles, runs a bookshop in Cambridge, sells through Amazon, and has a conference venues business in Cambridge at the Pitt Building and the Sir Geoffrey Cass Sports and Social Centre. It also served as the King's Printer. Cambridge University Press, as part of the University of Cambridge, was a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schanuel's Conjecture
In mathematics, specifically transcendental number theory, Schanuel's conjecture is a conjecture about the transcendence degree of certain field extensions of the rational numbers \mathbb, which would establish the transcendence of a large class of numbers, for which this is currently unknown. It is due to Stephen Schanuel and was published by Serge Lang in 1966. Statement Schanuel's conjecture can be given as follows: Consequences Schanuel's conjecture, if proven, would generalize most known results in transcendental number theory and establish a large class of numbers transcendental. Special cases of Schanuel's conjecture include: Lindemann-Weierstrass theorem Considering Schanuel's conjecture for only n=1 gives that for nonzero complex numbers z, at least one of the numbers z and e^z must be transcendental. This was proved by Ferdinand von Lindemann in 1882. If the numbers z_1,...,z_n are taken to be all algebraic and linearly independent over \mathbb Q then the e^,.. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Baker's Theorem
In transcendental number theory, a mathematical discipline, Baker's theorem gives a lower bound for the absolute value of linear combinations of logarithms of algebraic numbers. Nearly fifteen years earlier, Alexander Gelfond had considered the problem with only integer coefficients to be of "extraordinarily great significance". The result, proved by , subsumed many earlier results in transcendental number theory. Baker used this to prove the transcendence of many numbers, to derive effective bounds for the solutions of some Diophantine equations, and to solve the class number problem of finding all imaginary quadratic fields with class number 1. History To simplify notation, let \mathbb be the set of logarithms to the base ''e'' of nonzero algebraic numbers, that is \mathbb = \left \, where \Complex denotes the set of complex numbers and \overline denotes the algebraic numbers (the algebraic closure of the rational numbers \Q). Using this notation, several results in transcenden ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lindemann–Weierstrass Theorem
In transcendental number theory, the Lindemann–Weierstrass theorem is a result that is very useful in establishing the transcendence of numbers. It states the following: In other words, the extension field \mathbb(e^, \dots, e^) has transcendence degree over \mathbb. An equivalent formulation from , is the following: This equivalence transforms a linear relation over the algebraic numbers into an algebraic relation over \mathbb by using the fact that a symmetric polynomial whose arguments are all conjugates of one another gives a rational number. The theorem is named for Ferdinand von Lindemann and Karl Weierstrass. Lindemann proved in 1882 that is transcendental for every non-zero algebraic number thereby establishing that is transcendental (see below). Weierstrass proved the above more general statement in 1885. The theorem, along with the Gelfond–Schneider theorem, is extended by Baker's theorem, and all of these would be further generalized by Schanuel's ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert's Seventh Problem
Hilbert's seventh problem is one of David Hilbert's list of open mathematical problems posed in 1900. It concerns the irrationality and transcendence of certain numbers (''Irrationalität und Transzendenz bestimmter Zahlen''). Statement of the problem Two specific equivalent questions are asked: #In an isosceles triangle, if the ratio of the base angle to the angle at the vertex is algebraic but not rational, is then the ratio between base and side always transcendental? #Is a^b always transcendental, for algebraic a \not\in \ and irrational algebraic b? Solution The question (in the second form) was answered in the affirmative by Aleksandr Gelfond in 1934, and refined by Theodor Schneider in 1935. This result is known as Gelfond's theorem or the Gelfond–Schneider theorem. (The restriction to irrational ''b'' is important, since it is easy to see that a^b is algebraic for algebraic ''a'' and rational ''b''.) From the point of view of generalizations, this is the case : ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gelfond's Constant
In mathematics, the exponential of pi , also called Gelfond's constant, is the real number raised to the power . Its decimal expansion is given by: :' = ... Like both and , this constant is both irrational and transcendental. This follows from the Gelfond–Schneider theorem, which establishes to be transcendental, given that is algebraic and not equal to zero or one and is algebraic but not rational. We have e^\pi = (e^)^ = (-1)^,where is the imaginary unit. Since is algebraic but not rational, is transcendental. The numbers and are also known to be algebraically independent over the rational numbers, as demonstrated by Yuri Nesterenko. It is not known whether is a Liouville number. The constant was mentioned in Hilbert's seventh problem alongside the Gelfond–Schneider constant and the name "Gelfond's constant" stems from Soviet mathematician Alexander Gelfond. Occurrences The constant appears in relation to the volumes of hyperspheres: The volume of an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gelfond–Schneider Constant
The Gelfond–Schneider constant or Hilbert number is two to the power of the square root of two: :2 ≈ ... which was proved to be a transcendental number by Rodion Kuzmin in 1930. In 1934, Aleksandr Gelfond and Theodor Schneider independently proved the more general '' Gelfond–Schneider theorem'', which solved the part of Hilbert's seventh problem described below. Properties The square root of the Gelfond–Schneider constant is the transcendental number :\sqrt=\sqrt^ \approx .... This same constant can be used to prove that "an irrational elevated to an irrational power may be rational", even without first proving its transcendence. The proof proceeds as follows: either \sqrt^\sqrt is a rational which proves the theorem, or it is irrational (as it turns out to be) and then :\left(\sqrt^\right)^=\sqrt^=\sqrt^2=2 is an irrational to an irrational power that is a rational which proves the theorem. The proof is not constructive, as it does not say which of the two cas ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
P-adic Exponential Function
In mathematics, particularly ''p''-adic analysis, the ''p''-adic exponential function is a ''p''-adic analogue of the usual exponential function on the complex numbers. As in the complex case, it has an inverse function, named the ''p''-adic logarithm. Definition The usual exponential function on C is defined by the infinite series :\exp(z)=\sum_^\infty \frac. Entirely analogously, one defines the exponential function on C''p'', the completion of the algebraic closure of Q''p'', by :\exp_p(z)=\sum_^\infty\frac. However, unlike exp which converges on all of C, exp''p'' only converges on the disc :, z, _p [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
