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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] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...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] |
Mathematical Identities
In mathematics, an identity is an equality relating one mathematical expression ''A'' to another mathematical expression ''B'', such that ''A'' and ''B'' (which might contain some variables) produce the same value for all values of the variables within a certain range of validity. In other words, ''A'' = ''B'' is an identity if ''A'' and ''B'' define the same functions, and an identity is an equality between functions that are differently defined. For example, (a+b)^2 = a^2 + 2ab + b^2 and \cos^2\theta + \sin^2\theta =1 are identities. Identities are sometimes indicated by the triple bar symbol instead of , the equals sign. Common identities Algebraic identities Certain identities, such as a+0=a and a+(-a)=0, form the basis of algebra, while other identities, such as (a+b)^2 = a^2 + 2ab +b^2 and a^2 - b^2 = (a+b)(a-b), can be useful in simplifying algebraic expressions and expanding them. Trigonometric identities Geometrically, trigonometric ide ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elementary Number Theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics."German original: "Die Mathematik ist die Königin der Wissenschaften, und die Arithmetik ist die Königin der Mathematik." Number theorists study prime numbers as well as the properties of mathematical objects made out of integers (for example, rational numbers) or defined as generalizations of the integers (for example, algebraic integers). Integers can be considered either in themselves or as solutions to equations (Diophantine geometry). Questions in number theory are often best understood through the study of analytical objects (for example, the Riemann zeta function) that encode properties of the integers, primes or other number-theoretic objects i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elementary Algebra
Elementary algebra encompasses the basic concepts of algebra. It is often contrasted with arithmetic: arithmetic deals with specified numbers, whilst algebra introduces variables (quantities without fixed values). This use of variables entails use of algebraic notation and an understanding of the general rules of the operations introduced in arithmetic. Unlike abstract algebra, elementary algebra is not concerned with algebraic structures outside the realm of real and complex numbers. It is typically taught to secondary school students and builds on their understanding of arithmetic. The use of variables to denote quantities allows general relationships between quantities to be formally and concisely expressed, and thus enables solving a broader scope of problems. Many quantitative relationships in science and mathematics are expressed as algebraic equations. Algebraic notation Algebraic notation describes the rules and conventions for writing mathematical expressio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Latin Square
In combinatorics and in experimental design, a Latin square is an ''n'' × ''n'' array filled with ''n'' different symbols, each occurring exactly once in each row and exactly once in each column. An example of a 3×3 Latin square is The name "Latin square" was inspired by mathematical papers by Leonhard Euler (1707–1783), who used Latin characters as symbols, but any set of symbols can be used: in the above example, the alphabetic sequence A, B, C can be replaced by the integer sequence 1, 2, 3. Euler began the general theory of Latin squares. History The Korean mathematician Choi Seok-jeong was the first to publish an example of Latin squares of order nine, in order to construct a magic square in 1700, predating Leonhard Euler by 67 years. Reduced form A Latin square is said to be ''reduced'' (also, ''normalized'' or ''in standard form'') if both its first row and its first column are in their natural order. For example, the La ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
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] |
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] |
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] |
University Of Connecticut
The University of Connecticut (UConn) is a public land-grant research university in Storrs, Connecticut, a village in the town of Mansfield. The primary 4,400-acre (17.8 km2) campus is in Storrs, approximately a half hour's drive from Hartford and 90 minutes from Boston. UConn was founded in 1881 as the Storrs Agricultural School, named after two brothers who donated the land for the school. In 1893, the school became a public land grant college, becoming the University of Connecticut in 1939. Over the following decade, social work, nursing and graduate programs were established, while the schools of law and pharmacy were also absorbed into the university. During the 1960s, UConn Health was established for new medical and dental schools. John Dempsey Hospital opened in Farmington in 1975. The university is classified among "R1: Doctoral Universities – Very high research activity". The university has been considered a Public Ivy. UConn is one of the founding institution ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quaternion
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quaternion as the quotient of two '' directed lines'' in a three-dimensional space, or, equivalently, as the quotient of two vectors. Multiplication of quaternions is noncommutative. Quaternions are generally represented in the form :a + b\ \mathbf i + c\ \mathbf j +d\ \mathbf k where , and are real numbers; and , and are the ''basic quaternions''. Quaternions are used in pure mathematics, but also have practical uses in applied mathematics, particularly for calculations involving three-dimensional rotations, such as in three-dimensional computer graphics, computer vision, and crystallographic texture analysis. They can be used alongside other methods of rotation, such as Euler angles and rotation matrices, or as an alternative to them ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
