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Pfister's Sixteen-square Identity
In algebra, Pfister's sixteen-square identity is a non- bilinear identity of form \left(x_1^2+x_2^2+x_3^2+\cdots+x_^2\right)\left(y_1^2+y_2^2+y_3^2+\cdots+y_^2\right) = z_1^2+z_2^2+z_3^2+\cdots+z_^2 It was first proven to exist by H. Zassenhaus and W. Eichhorn in the 1960s, and independently by Albrecht Pfister around the same time. There are several versions, a concise one of which is \begin &\scriptstyle \\ &\scriptstyle \\ &\scriptstyle \\ &\scriptstyle \\ &\scriptstyle \\ &\scriptstyle \\ &\scriptstyle \\ &\scriptstyle \\ &\scriptstyle \\ &\scriptstyle \\ &\scriptstyle \\ &\scriptstyle \\ &\scriptstyle \\ &\scriptstyle \\ &\scriptstyle \\ &\scriptstyle \end If all x_i and y_i with i>8 are set equal to zero, then it reduces to Degen's eight-square identity (in blue). The u_i are \begin &u_1 = \tfrac \\ &u_2 = \tfrac \\ &u_3 = \tfrac \\ &u_4 = \tfrac \\ &u_5 = \tfrac \\ &u_6 = \tfrac \\ &u_7 = \tfrac \\ &u_8 = \tfrac \end and, a=-1,\;\;b=0,\;\;c=x_1^2+x_2^2+x_3^2+x_4^2+x_5^ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary algebra deals with the manipulation of variables (commonly represented by Roman letters) as if they were numbers and is therefore essential in all applications of mathematics. Abstract algebra is the name given, mostly in education, to the study of algebraic structures such as groups, rings, and fields (the term is no more in common use outside educational context). Linear algebra, which deals with linear equations and linear mappings, is used for modern presentations of geometry, and has many practical applications (in weather forecasting, for example). There are many areas of mathematics that belong to algebra, some having "algebra" in their name, such as commutative algebra, and some not, such as Galois theory. The word ''algebra'' is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bilinear Map
In mathematics, a bilinear map is a function combining elements of two vector spaces to yield an element of a third vector space, and is linear in each of its arguments. Matrix multiplication is an example. Definition Vector spaces Let V, W and X be three vector spaces over the same base field F. A bilinear map is a function B : V \times W \to X such that for all w \in W, the map B_w v \mapsto B(v, w) is a linear map from V to X, and for all v \in V, the map B_v w \mapsto B(v, w) is a linear map from W to X. In other words, when we hold the first entry of the bilinear map fixed while letting the second entry vary, the result is a linear operator, and similarly for when we hold the second entry fixed. Such a map B satisfies the following properties. * For any \lambda \in F, B(\lambda v,w) = B(v, \lambda w) = \lambda B(v, w). * The map B is additive in both components: if v_1, v_2 \in V and w_1, w_2 \in W, then B(v_1 + v_2, w) = B(v_1, w) + B(v_2, w) and B(v, w_1 + w_2) = B(v ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hans Zassenhaus
Hans Julius Zassenhaus (28 May 1912 – 21 November 1991) was a German mathematician, known for work in many parts of abstract algebra, and as a pioneer of computer algebra. Biography He was born in Koblenz in 1912. His father was a historian and advocate for Reverence for Life as expressed by Albert Schweitzer. Hans had two brothers, Guenther and Wilfred, and sister Hiltgunt, who wrote an autobiography in 1974. According to her, their father lost his position as school principal due to his philosophy. She wrote:Hiltgunt Zassenhaus (1974) ''Walls: Resisting the Third Reich'', Beacon Press :Hans, my eldest brother, studied mathematics. My brothers Guenther and Wilfred were in medical school. ... only students who participated in Nazi activities would get scholarships. That left us out. Together we made an all-out effort. ... soon our house became a beehive. Day in and day out for the next four years a small army of children of all ages would arrive to be tutored. At the University ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Albrecht Pfister (mathematician)
Albrecht Pfister (born July 30, 1934) is a German mathematician specializing in algebra and in particular quadratic forms. Pfister received his doctoral degree in 1961 at the Ludwig Maximilian University of Munich. The title of his doctoral thesis was ''Über das Koeffizientenproblem der beschränkten Funktionen von zwei Veränderlichen'' ("On the coefficient problem of the bounded functions of two variables"). His thesis advisors were Martin Kneser and Karl Stein. In 1966 he received his habilitation at the Georg August University of Göttingen. From 1970 until his retirement he was professor at the Johannes Gutenberg University of Mainz. In the theory of quadratic forms over fields, the Pfister forms that he introduced in 1965 bear his name. In 1970, he was an invited speaker on the topic ''Sums of squares in real function fields'' at the International Congress of Mathematicians in Nice Nice ( , ; Niçard dialect, Niçard: , classical norm, or , nonstandard, ; it, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Degen's Eight-square Identity
In mathematics, Degen's eight-square identity establishes that the product of two numbers, each of which is a sum of eight squares, is itself the sum of eight squares. Namely: \begin & \left(a_1^2+a_2^2+a_3^2+a_4^2+a_5^2+a_6^2+a_7^2+a_8^2\right)\left(b_1^2+b_2^2+b_3^2+b_4^2+b_5^2+b_6^2+b_7^2+b_8^2\right) = \\ ex & \quad \left(a_1 b_1 - a_2 b_2 - a_3 b_3 - a_4 b_4 - a_5 b_5 - a_6 b_6 - a_7 b_7 - a_8 b_8\right)^2+ \\ & \quad \left(a_1 b_2 + a_2 b_1 + a_3 b_4 - a_4 b_3 + a_5 b_6 - a_6 b_5 - a_7 b_8 + a_8 b_7\right)^2+ \\ & \quad \left(a_1 b_3 - a_2 b_4 + a_3 b_1 + a_4 b_2 + a_5 b_7 + a_6 b_8 - a_7 b_5 - a_8 b_6\right)^2+ \\ & \quad \left(a_1 b_4 + a_2 b_3 - a_3 b_2 + a_4 b_1 + a_5 b_8 - a_6 b_7 + a_7 b_6 - a_8 b_5\right)^2+ \\ & \quad \left(a_1 b_5 - a_2 b_6 - a_3 b_7 - a_4 b_8 + a_5 b_1 + a_6 b_2 + a_7 b_3 + a_8 b_4\right)^2+ \\ & \quad \left(a_1 b_6 + a_2 b_5 - a_3 b_8 + a_4 b_7 - a_5 b_2 + a_6 b_1 - a_7 b_4 + a_8 b_3\right)^2+ \\ & \quad \left(a_1 b_7 + a_2 b_8 + a_3 b_5 - a_4 b_6 - a ... [...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] |
Hurwitz's Theorem (normed Division Algebras)
{{mathdab ...
Hurwitz's theorem can refer to several theorems named after Adolf Hurwitz: * Hurwitz's theorem (complex analysis) * Riemann–Hurwitz formula in algebraic geometry * Hurwitz's theorem (composition algebras) on quadratic forms and nonassociative algebras * Hurwitz's automorphisms theorem on Riemann surfaces * Hurwitz's theorem (number theory) In number theory, Hurwitz's theorem, named after Adolf Hurwitz, gives a bound on a Diophantine approximation. The theorem states that for every irrational number ''ξ'' there are infinitely many relatively prime integers ''m'', ''n'' such that \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rational Functions
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 ''K''. In this case, one speaks of a rational function and a rational fraction ''over K''. The values of the variables may be taken in any field ''L'' containing ''K''. 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 ''L''. The set of rational functions over a field ''K'' is a field, the field of fractions of the ring of the polynomial functions over ''K''. Definitions A function f(x) is called a rational function if and only 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\, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Denominator
A fraction (from la, fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight-fifths, three-quarters. A ''common'', ''vulgar'', or ''simple'' fraction (examples: \tfrac and \tfrac) consists of a numerator, displayed above a line (or before a slash like ), and a non-zero denominator, displayed below (or after) that line. Numerators and denominators are also used in fractions that are not ''common'', including compound fractions, complex fractions, and mixed numerals. In positive common fractions, the numerator and denominator are natural numbers. The numerator represents a number of equal parts, and the denominator indicates how many of those parts make up a unit or a whole. The denominator cannot be zero, because zero parts can never make up a whole. For example, in the fraction , the numerator 3 indicates that the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euler's Four-square Identity
In mathematics, Euler's four-square identity says that the product of two numbers, each of which is a sum of four square (algebra), squares, is itself a sum of four squares. Algebraic identity For any pair of quadruples from a commutative ring, the following expressions are equal: \begin \left(a_1^2+a_2^2+a_3^2+a_4^2\right)\left(b_1^2+b_2^2+b_3^2+b_4^2\right) = & \left(a_1 b_1 - a_2 b_2 - a_3 b_3 - a_4 b_4\right)^2 \\ &+ \left(a_1 b_2 + a_2 b_1 + a_3 b_4 - a_4 b_3\right)^2 \\ &+ \left(a_1 b_3 - a_2 b_4 + a_3 b_1 + a_4 b_2\right)^2 \\ &+ \left(a_1 b_4 + a_2 b_3 - a_3 b_2 + a_4 b_1\right)^2. \end Leonhard Euler, Euler wrote about this identity in a letter dated May 4, 1748 to Christian Goldbach, Goldbach (but he used a different sign convention from the above). It can be verified with elementary algebra. The identity was used by Joseph Louis Lagrange, Lagrange to prove his Lagrange's four-square theorem, four square theorem. More specifically, it implies that it is sufficient ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Brahmagupta–Fibonacci Identity
In algebra, the Brahmagupta–Fibonacci identity expresses the product of two sums of two squares as a sum of two squares in two different ways. Hence the set of all sums of two squares is closed under multiplication. Specifically, the identity says :\begin \left(a^2 + b^2\right)\left(c^2 + d^2\right) & = \left(ac-bd\right)^2 + \left(ad+bc\right)^2 & & (1) \\ & = \left(ac+bd\right)^2 + \left(ad-bc\right)^2. & & (2) \end For example, :(1^2 + 4^2)(2^2 + 7^2) = 26^2 + 15^2 = 30^2 + 1^2. The identity is also known as the Diophantus identity,Daniel Shanks, Solved and unsolved problems in number theory, p.209, American Mathematical Society, Fourth edition 1993. as it was first proved by Diophantus of Alexandria. It is a special case of Euler's four-square identity, and also of Lagrange's identity. Brahmagupta proved and used a more general identity (the Brahmagupta identity), equivalent to :\begin \left(a^2 + nb^2\right)\left(c^2 + nd^2\r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
