منابع مشابه
The Hurwitz Theorem on Sums of Squares
This was discovered by Euler in the 18th century, forgotten, and then rediscovered in the 19th century by Hamilton in his work on quaternions. Shortly after Hamilton’s rediscovery of (1.2) Cayley discovered a similar 8-square identity. In all of these sum-of-squares identities, the terms being squared on the right side are all bilinear expressions in the x’s and y’s: each such expression, like ...
متن کاملThe Hurwitz Theorem on Sums of Squares by Linear Algebra
(1.1) (x1 + x 2 2)(y 2 1 + y 2 2) = (x1y1 − x2y2) + (x1y2 + x2y1). There is also an identity like this for a sum of four squares: (x1 + x 2 2 + x 2 3 + x 2 4)(y 2 1 + y 2 2 + y 2 3 + y 2 4) = (x1y1 − x2y2 − x3y3 − x4y4) + (1.2) (x1y2 + x2y1 + x3y4 − x4y3) + (x1y3 + x3y1 − x2y4 + x4y2) + (x1y4 + x4y1 + x2y3 − x3y2). This was discovered by Euler in the 18th century, forgotten, and then rediscover...
متن کاملThe Hurwitz Theorem on Sums of Squares by Representation Theory
This was discovered by Euler in the 18th century, forgotten, and then rediscovered in the 19th century by Hamilton in his work on quaternions. Shortly after Hamilton’s rediscovery of (1.2) Cayley discovered a similar 8-square identity. In all of these sum-of-squares identities, the terms being squared on the right side are all bilinear expressions in the x’s and y’s: each such expression, like ...
متن کاملHurwitz’ Theorem
In this article we describe several results based on the paper [Hur98] and which we will refer to as Hurwitz’ theorem. There are several related results: the classification of real normed division algebras, the classification of complex composition algebras and the classification of real composition algebras. The classification of real division algebras is an open problem. Our interest is in co...
متن کاملLatin Squares: Transversals and counting of Latin squares
Author: Jenny Zhang First, let’s preview what mutually orthogonal Latin squares are. Two Latin squares L1 = [aij ] and L2 = [bij ] on symbols {1, 2, ...n}, are said to be orthogonal if every ordered pair of symbols occurs exactly once among the n2 pairs (aij , bij), 1 ≤ i ≤ n, 1 ≤ j ≤ n. Now, let me introduce a related concept which is called transversal. A transversal of a Latin square is a se...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Expositiones Mathematicae
سال: 2015
ISSN: 0723-0869
DOI: 10.1016/j.exmath.2014.12.002