scholarly journals Appendix B: An Algebraic Identity

2015 ◽  
pp. 121-122
Keyword(s):  
Algebraic Identity

On structure theory of pre-Hilbert algebras

2009 ◽  
Vol 139 (2) ◽  
pp. 303-319 ◽  
Author(s):  
José Antonio Cuenca Mira
Keyword(s):  
Vector Space ◽  
Associative Algebra ◽  
Real Vector Space ◽  
Inner Product ◽  
Structure Theory ◽  
Classical Theorem ◽  
Real Vector ◽  
Algebraic Identity ◽  
Hilbert Algebras

Let A be a real (non-associative) algebra which is normed as real vector space, with a norm ‖·‖ deriving from an inner product and satisfying ‖ac‖ ≤ ‖a‖‖c‖ for any a,c ∈ A. We prove that if the algebraic identity (a((ac)a))a = (a2c)a2 holds in A, then the existence of an idempotent e such that ‖e‖ = 1 and ‖ea‖ = ‖a‖ = ‖ae‖, a ∈ A, implies that A is isometrically isomorphic to ℝ, ℂ, ℍ, $\mathbb{O}$,\, $\stackrel{\raisebox{4.5pt}[0pt][0pt]{\fontsize{4pt}{4pt}\selectfont$\star$}}{\smash{\CC}}$,\, $\stackrel{\raisebox{4.5pt}[0pt][0pt]{\fontsize{4pt}{4pt}\selectfont$\star$}}{\smash{\mathbb{H}}}$,\, $\stackrel{\raisebox{4.5pt}[0pt][0pt]{\fontsize{4pt}{4pt}\selectfont$\star$}}{\smash{\mathbb{O}}}$ or ℙ. This is a non-associative extension of a classical theorem by Ingelstam. Finally, we give some applications of our main result.

A serendipitous path to a famous inequality

2008 ◽  
Vol 92 (523) ◽  
pp. 50-54
Author(s):  
Robert M. Young
Keyword(s):  
Real Number ◽  
Geometric Means ◽  
Algebraic Identity ◽  
Binomial Expansion

The mysterious path of discovery – the tireless experimentation in search of patterns, the veiled connections that suddenly unfold, serendipity – all these elements combine to make mathematics so magical. The purpose of this note is to show how a routine algebraic identity, the binomial expansion of (x- 1)2, can be used to give a new proof of the fundamental inequality between the arithmetic and geometric means. The proof will provide further evidence that a great deal of useful mathematics can be derived from the obvious assertion that the square of a real number is never negative.

80.52 An Algebraic Identity Leading to Wilson's Theorem

1996 ◽  
Vol 80 (489) ◽  
pp. 579 ◽  
Author(s):  
Sebastian Martin Ruiz
Keyword(s):  
Algebraic Identity ◽  
Wilson’S Theorem

2350. On Note 2045: An Algebraic Identity

1953 ◽  
Vol 37 (321) ◽  
pp. 212 ◽  
Author(s):  
B. E. Lawrence
Keyword(s):  
Algebraic Identity

scholarly journals An analytic problem whose solution follows from a simple algebraic identity

1960 ◽  
Vol 10 (3) ◽  
pp. 731-742 ◽  
Author(s):  
Glen Baxter
Keyword(s):  
Algebraic Identity ◽  
Analytic Problem

scholarly journals Some Remarks on an Algebraic Identity.

1960 ◽  
Vol 8 ◽  
pp. 39 ◽  
Author(s):  
Harald Bergström
Keyword(s):  
Algebraic Identity

scholarly journals A Binomial Coefficient Identity Associated with Beukers' Conjecture on Apéry numbers

10.37236/1856 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
Wenchang Chu
Keyword(s):  
Rational Function ◽  
Binomial Coefficient ◽  
Harmonic Number ◽  
Partial Fraction ◽  
Algebraic Identity ◽  
Apéry Numbers ◽  
Partial Fraction Decomposition ◽  
Limiting Case ◽  
Binomial Coefficient Identity

By means of partial fraction decomposition, an algebraic identity on rational function is established. Its limiting case leads us to a harmonic number identity, which in turn has been shown to imply Beukers' conjecture on the congruence of Apéry numbers.

2045. An Algebraic Identity

1949 ◽  
Vol 33 (303) ◽  
pp. 37
Author(s):  
A. A. Krishnaswami Ayyangar
Keyword(s):  
Algebraic Identity
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