Transcendental numbers and diophantine approximations
نویسندگان
چکیده
منابع مشابه
Algebraic and Transcendental Numbers
To begin with, recall that a complex number α is said to be a root of a polynomial P (X) if P (α) = 0. A complex number α is said to be algebraic if there is a nonzero polynomial P (X), with integer coefficients, of which α is a root. The set of algebraic numbers is denoted by Q̄. A complex number α which is not algebraic is said to be transcendental. The following numbers are obviously algebrai...
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This article centres around the contributions of the author and therefore, it is confined to topics where the author has worked. Between these topics there are connections and we explain them by a result of Liouville in 1844 that for an algebraic number α of degree n ≥ 2, there exists c > 0 depending only on α such that | α− p q |> c qn for all rational numbers p q with q > 0. This inequality i...
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A first example of a connection between transcendental numbers and complex dynamics is the following. Let p and q be polynomials with complex coefficients of the same degree. A classical result of Böttcher states that p and q are locally conjugates in a neighborhood of∞: there exists a function f , conformal in a neighborhood of infinity, such that f(p(z)) = q(f(z)). Under suitable assumptions,...
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The concept of “number” has formed the basis of civilzation since time immemorial. Looking back from our vantage point of the digital age, we can agree with Pythagoras that “all is number”. The study of numbers and their properties is the mathematical equivalent of the study of atoms and their structure. It is in fact more than that. The famous physicist and Nobel Laureate Eugene Wigner spoke o...
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Transcendental numbers play an important role in many areas of science. This paper contains a short survey on transcendental numbers and some relations among them. New inequalities for transcendental numbers are stated in Section 2 and proved in Section 4. Also, in relationship with these topics, we study the exponential function axioms related to the Yang-Baxter equation.
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ژورنال
عنوان ژورنال: Bulletin of the American Mathematical Society
سال: 1971
ISSN: 0002-9904
DOI: 10.1090/s0002-9904-1971-12761-1