Doodles and commutator identities

Identities for the Ternary Commutator

If n is any positive integer, then an n-algebra is a vector space A over a field F together with a linear map v : Amn a A, where Amn denotes the n-fold tensor power of A. In the case n s 3 we say that A is a ternary Ž . algebra or triple system . To simplify notation, we write a a ??? a 1 2 n Ž . Ž . instead of v a m a m ??? m a for a g A 1 F i F n . 1 2 n i Given the n-algebra A, we can define...

More on Five Commutator Identities

We prove that five well-known identities universally satisfied by commutators in a group generate all universal commutator identities for commutators of weight 4.

Higher Identities for the Ternary Commutator

We use computer algebra to study polynomial identities for the trilinear operation [a, b, c] = abc − acb − bac + bca + cab − cba in the free associative algebra. It is known that [a, b, c] satisfies the alternating property in degree 3, no new identities in degree 5, a multilinear identity in degree 7 which alternates in 6 arguments, and no new identities in degree 9. We use the representation ...

Computer Doodles Using Simulated Gravitational Fields

Doodles for Authentication: Recognition and User Study Results

Traditional means of computer based authentication based on username and password combinations become unwieldy as the number of password accounts one manages increases. The average computer user needs to remember a large number of text username and password combinations for different applications, which places a large cognitive load on the user. While biometric login based systems can free the ...