We identify a collection of primitive elements generating the Hopf algebra NCSym of symmetric functions in noncommuting variables and give a combinatorial formula for the antipode.

We outline the description of Quantum Mechanics with noncommuting coordinates within the framework of star operation. We discuss simple cases of integrability.

The coefficients of K. Goldberg in the infinite series for \o$(exey) for noncommuting x and y are computed as far as words of length twenty.

We study non-linear surjective mappings on subsets of Mn(F), which preserve the zeros of some fixed polynomials in noncommuting variables. Mathematics subject classification (2000): 15A99, 16W99.

It is shown that a nite system T of matrices whose real linear combinations have real spectrum satisses a bound of the form ke ihT;;i k C(1 + jj) s e rj=j. The proof appeals to the monogenic functional calculus.

Using precision electroweak data, we put limits on non-commuting extended technicolor models. We conclude that these models are viable only if the ETCinteractions are strong. Interestingly, these models predict a pattern of deviations from the standard model which can fit the data significantly better than the standard model does, even after taking into account the extra parameters involved. ∗e...