Comparing hierarchies of total functionals

In this paper we consider two hierarchies of hereditarily total and continuous functionals over the reals based on one extensional and one intensional representation of real numbers, and we discuss under which asumptions these hierarchies coincide. This coincidense problem is equivalent to a statement about the topology of the Kleene-Kreisel continuous functionals. As a tool of independent interest, we show that the Kleene-Kreisel functionals may be embedded into both these hierarchies.Comment: 28 page

Relative energetics of acetyl-histidine protomers with and without Zn²⁺ and a benchmark of energy methods

We studied acetylhistidine (AcH), bare or microsolvated with a zinc cation by simulations in isolation. First, a global search for minima of the potential energy surface combining both, empirical and first-principles methods, is performed individually for either one of five possible protonation states. Comparing the most stable structures between tautomeric forms of negatively charged AcH shows a clear preference for conformers with the neutral imidazole ring protonated at the N-epsilon-2 atom. When adding a zinc cation to the system, the situation is reversed and N-delta-1-protonated structures are energetically more favorable. Obtained minima structures then served as basis for a benchmark study to examine the goodness of commonly applied levels of theory, i.e. force fields, semi-empirical methods, density-functional approximations (DFA), and wavefunction-based methods with respect to high-level coupled-cluster calculations, i.e. the DLPNO-CCSD(T) method. All tested force fields and semi-empirical methods show a poor performance in reproducing the energy hierarchies of conformers, in particular of systems involving the zinc cation. Meta-GGA, hybrid, double hybrid DFAs, and the MP2 method are able to describe the energetics of the reference method within chemical accuracy, i.e. with a mean absolute error of less than 1kcal/mol. Best performance is found for the double hybrid DFA B3LYP+XYG3 with a mean absolute error of 0.7 kcal/mol and a maximum error of 1.8 kcal/mol. While MP2 performs similarly as B3LYP+XYG3, computational costs, i.e. timings, are increased by a factor of 4 in comparison due to the large basis sets required for accurate results

Integrable systems and holomorphic curves

In this paper we attempt a self-contained approach to infinite dimensional Hamiltonian systems appearing from holomorphic curve counting in Gromov-Witten theory. It consists of two parts. The first one is basically a survey of Dubrovin's approach to bihamiltonian tau-symmetric systems and their relation with Frobenius manifolds. We will mainly focus on the dispersionless case, with just some hints on Dubrovin's reconstruction of the dispersive tail. The second part deals with the relation of such systems to rational Gromov-Witten and Symplectic Field Theory. We will use Symplectic Field theory of $S^1\times M$ as a language for the Gromov-Witten theory of a closed symplectic manifold $M$. Such language is more natural from the integrable systems viewpoint. We will show how the integrable system arising from Symplectic Field Theory of $S^1\times M$ coincides with the one associated to the Frobenius structure of the quantum cohomology of $M$.Comment: Partly material from a working group on integrable systems organized by O. Fabert, D. Zvonkine and the author at the MSRI - Berkeley in the Fall semester 2009. Corrected some mistake