A classical result in the theory of Hopf algebras concerns the uniqueness and existence of inte-grals: for an arbitrary Hopf algebra, the integral space has dimension ≤ 1, and for a finite dimensional Hopf algebra, this dimension is exaclty one. We generalize these results to quasi-Hopf algebras and dual quasi-Hopf algebras. In particular, it will follow that the bijectivity of the antipode fol...

We study the local convergence of classical quasi-Newton methods for nonlinear optimization. Although it was well established a long time ago that asymptotically these converge superlinearly, corresponding rates still remain unknown. In this paper, we address problem. obtain first explicit non-asymptotic superlinear standard methods, which are based on updating formulas from convex Broyden clas...

A frequently encountered source of systematic error in quantum computations is imperfections the control pulses which are classical fields that qubit gate operations. From an analysis mechanical time-evolution operator spin wavefunction, it has been demonstrated composite can mitigate certain errors and appealing geometric interpretation was developed for design error-suppressing pulses. Here w...

We rigorously define the Liouville action functional for the finitely generated, purely loxodromic quasi-Fuchsian group using homology and cohomology double complexes naturally associated with the group action. We prove that classical action – the critical value of the Liouville action functional, considered as a function on the quasiFuchsian deformation space, is an antiderivative of a 1-form ...