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Fueter–Pólya Theorem
The Fueter–Pólya theorem, first proved by Rudolf Fueter and George Pólya, states that the only quadratic polynomial pairing functions are the Cantor polynomials. Introduction In 1873, Georg Cantor showed that the so-called Cantor polynomial : P(x,y) := \frac ((x+y)^2+3x+y) = \frac2 = x + \frac2 = \binom1 + \binom is a bijective mapping from \N^2 to \N. The polynomial given by swapping the variables is also a pairing function. Fueter was investigating whether there are other quadratic polynomials with this property, and concluded that this is not the case assuming P(0,0)=0. He then wrote to Pólya, who showed the theorem does not require this condition. Statement If P is a real quadratic polynomial in two variables whose restriction to \N^2 is a bijection from \N^2 to \N then it is : P(x,y) := \frac ((x+y)^2+3x+y) or :P(x,y) := \frac ((y+x)^2+3y+x). Proof The original proof is surprisingly difficult, using the Lindemann–Weierstrass theorem to prove the t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rudolf Fueter
Karl Rudolf Fueter (30 June 1880 – 9 August 1950) was a Swiss mathematician, known for his work on number theory. Biography After a year of graduate study of mathematics in Basel, Fueter began study in 1899 at the University of Göttingen and completed his Promotion in 1903 with dissertation ''Der Klassenkörper der quadratischen Körper und die komplexe Multiplikation'' under David Hilbert. After his Promotion, Fueter studied for 1 year in Paris, 3 months in Vienna, and 6 months in London. In 1905 he completed his Habilitierung at the University of Marburg. Fueter worked as a docent in 1907/1908 at Marburg and in the winter of 1907/1908 at the Bergakademie Clausthal. He was called to positions as professor ordinarius in 1908 at Basel, in 1913 at the Technische Hochschule Karlsruhe, and in 1916 at the University of Zurich. From 1920 to 1922 he was the rector of the University of Zurich. Fueter did research on algebraic number theory and quaternion analysis p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
George Pólya
George Pólya (; ; December 13, 1887 – September 7, 1985) was a Hungarian-American mathematician. He was a professor of mathematics from 1914 to 1940 at ETH Zürich and from 1940 to 1953 at Stanford University. He made fundamental contributions to combinatorics, number theory, numerical analysis and probability theory. He is also noted for his work in heuristics and mathematics education. He has been described as one of The Martians (scientists), The Martians, an informal category which included one of his most famous students at ETH Zurich, John von Neumann. Life and works Pólya was born in Budapest, Austria-Hungary, to Anna Deutsch and Jakab Pólya, History of the Jews in Hungary, Hungarian Jews who had converted to Christianity in 1886. Although his parents were religious and he was baptized into the Catholic Church upon birth, George eventually grew up to be an agnostic. He received a PhD under Lipót Fejér in 1912, at Eötvös Loránd University. He was a professor o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quadratic Function
In mathematics, a quadratic function of a single variable (mathematics), variable is a function (mathematics), function of the form :f(x)=ax^2+bx+c,\quad a \ne 0, where is its variable, and , , and are coefficients. The mathematical expression, expression , especially when treated as an mathematical object, object in itself rather than as a function, is a quadratic polynomial, a polynomial of degree two. In elementary mathematics a polynomial and its associated polynomial function are rarely distinguished and the terms ''quadratic function'' and ''quadratic polynomial'' are nearly synonymous and often abbreviated as ''quadratic''. The graph of a function, graph of a function of a real variable, real single-variable quadratic function is a parabola. If a quadratic function is equation, equated with zero, then the result is a quadratic equation. The solutions of a quadratic equation are the zero of a function, zeros (or ''roots'') of the corresponding quadratic function, of which ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pairing Function
In mathematics, a pairing function is a process to uniquely encode two natural numbers into a single natural number. Any pairing function can be used in set theory to prove that integers and rational numbers have the same cardinality as natural numbers. Definition A pairing function is a bijection :\pi:\mathbb \times \mathbb \to \mathbb. Generalization More generally, a pairing function on a set A is a function that maps each pair of elements from A into an element of A, such that any two pairs of elements of A are associated with different elements of A, or a bijection from A^2 to A. Instead of abstracting from the domain, the arity of the pairing function can also be generalized: there exists an n-ary generalized Cantor pairing function on \mathbb. Cantor pairing function The Cantor pairing function is a primitive recursive pairing function :\pi:\mathbb \times \mathbb \to \mathbb defined by :\pi(k_1,k_2) := \frac(k_1 + k_2)(k_1 + k_2 + 1)+k_2 where k_1, k_2\in\. It ca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polynomial
In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addition, subtraction, multiplication and exponentiation to nonnegative integer powers, and has a finite number of terms. An example of a polynomial of a single indeterminate is . An example with three indeterminates is . Polynomials appear in many areas of mathematics and science. For example, they are used to form polynomial equations, which encode a wide range of problems, from elementary word problem (mathematics education), word problems to complicated scientific problems; they are used to define polynomial functions, which appear in settings ranging from basic chemistry and physics to economics and social science; and they are used in calculus and numerical analysis to approximate other functions. In advanced mathematics, polynomials are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Georg Cantor
Georg Ferdinand Ludwig Philipp Cantor ( ; ; – 6 January 1918) was a mathematician who played a pivotal role in the creation of set theory, which has become a foundations of mathematics, fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence between the members of two sets, defined infinite set, infinite and well-order, well-ordered sets, and proved that the real numbers are more numerous than the natural numbers. Cantor's method of proof of this theorem implies the existence of an infinity of infinities. He defined the cardinal number, cardinal and ordinal number, ordinal numbers and their arithmetic. Cantor's work is of great philosophical interest, a fact he was well aware of. Originally, Cantor's theory of transfinite numbers was regarded as counter-intuitive – even shocking. This caused it to encounter resistance from mathematical contemporaries such as Leopold Kronecker and Henri Poincaré and later from Hermann Wey ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bijective
In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Equivalently, a bijection is a relation between two sets such that each element of either set is paired with exactly one element of the other set. A function is bijective if it is invertible; that is, a function f:X\to Y is bijective if and only if there is a function g:Y\to X, the ''inverse'' of , such that each of the two ways for composing the two functions produces an identity function: g(f(x)) = x for each x in X and f(g(y)) = y for each y in Y. For example, the ''multiplication by two'' defines a bijection from the integers to the even numbers, which has the ''division by two'' as its inverse function. A function is bijective if and only if it is both injective (or ''one-to-one'')—meaning that each element in the codomain is mappe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Restriction (mathematics)
In mathematics, the restriction of a function f is a new function, denoted f\vert_A or f , obtained by choosing a smaller domain A for the original function f. The function f is then said to extend f\vert_A. Formal definition Let f : E \to F be a function from a set E to a set F. If a set A is a subset of E, then the restriction of f to A is the function _A : A \to F given by _A(x) = f(x) for x \in A. Informally, the restriction of f to A is the same function as f, but is only defined on A. If the function f is thought of as a relation (x,f(x)) on the Cartesian product E \times F, then the restriction of f to A can be represented by its graph, :G(_A) = \ = G(f)\cap (A\times F), where the pairs (x,f(x)) represent ordered pairs in the graph G. Extensions A function F is said to be an ' of another function f if whenever x is in the domain of f then x is also in the domain of F and f(x) = F(x). That is, if \operatorname f \subseteq \operatorname F and F\big\vert_ = f. ... [...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] |
Binomial Coefficients
In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the term in the polynomial expansion of the binomial power ; this coefficient can be computed by the multiplicative formula : \binom nk = \frac, which using factorial notation can be compactly expressed as : \binom = \frac. For example, the fourth power of is : \begin (1 + x)^4 &= \tbinom x^0 + \tbinom x^1 + \tbinom x^2 + \tbinom x^3 + \tbinom x^4 \\ &= 1 + 4x + 6 x^2 + 4x^3 + x^4, \end and the binomial coefficient \tbinom =\tfrac = \tfrac = 6 is the coefficient of the term. Arranging the numbers \tbinom, \tbinom, \ldots, \tbinom in successive rows for gives a triangular array called Pascal's triangle, satisfying the recurrence relation : \binom = \binom + \binom . The binomial coefficients occur in many areas of mathematics, and especia ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Theorems
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] |
