Design, analysis, and implementation of a multiprecision polynomial rootfinder

We present the design, analysis, and implementation of an algorithm for the computation of any number of digits of the roots of a polynomial with complex coefficients. The real and the imaginary parts of the coefficients may be integer, rational, or floating point numbers represented with an arbitrary number of digits. The algorithm has been designed to deal also with numerically hard polynomials like those arising from the symbolic preprocessing of systems of polynomial equations, where the degree and the size of the coefficients are typically huge.The algorithm is based on an adaptive strategy which automatically exploits any specific feature of the input polynomial, like its sparsity or the conditioning of its roots, in order to speed up the computation. We introduce different concepts and tools suitably designed to arrive at an adaptive implementation, such as the concepts of root neighborhood, inclusion discs and some inclusion and conditioning theorems for their determination. The main engine for shrinking the inclusion discs is the simultaneous iteration method of Ehrlich–Aberth, complemented with a suitable technique for cluster analysis that is used for getting rid of the slow convergence in case of clustered or multiple roots.The algorithm, implemented in C, relies on the GNU multiprecision package GMP and allows many options. Counting, isolating and approximating all roots in a given set are the main goals that the algorithm provides. Automatic determination of multiplicities and the detection of real or imaginary roots can be selected as well. Polynomials having coefficients with a bounded precision may be processed too. Comparisons with the polynomial rootfinders of the packages Mathematica, Mapleand Pari, performed on a wide class of test polynomials show that our algorithm is generally much faster: in most cases the speedup factor is greater than 10 and, for certain polynomials, it is greater than 1000. The MPSolve package can be downloaded from the numeralgo library of netlib.     

Exponential Times in the One-Dimensional Gross–Pitaevskii Equation with Multiple Well Potential

We present an algorithm for constructing the Wilson operator product expansion (OPE) for perturbative interacting quantum field theory in general Lorentzian curved spacetimes, to arbitrary orders in perturbation theory. The remainder in this expansion is shown to go to zero at short distances in the sense of expectation values in arbitrary Hadamard states. We also establish a number of general properties of the OPE coefficients: (a) they only depend (locally and covariantly) upon the spacetime metric and coupling constants, (b) they satisfy an associativity property, (c) they satisfy a renormalization group equation, (d) they satisfy a certain microlocal wave front set condition, (e) they possess a “scaling expansion”. The latter means that each OPE coefficient can be written as a sum of terms, each of which is the product of a curvature polynomial at a spacetime point, times a Lorentz invariant Minkowski distribution in the tangent space of that point. The algorithm is illustrated in an example.     

On the Darboux theorem for weak symplectic manifolds

A new tool to study reducibility of a weak symplectic form to a constant one is introduced and used to prove a version of the Darboux theorem more general than previous ones. More precisely, at each point of the considered manifold a Banach space is associated to the symplectic form (dual of the phase space with respect to the symplectic form), and it is shown that the Darboux theorem holds if such a space is locally constant. The following application is given. Consider a weak symplectic manifold on which the Darboux theorem is assumed to hold (e.g. a symplectic vector space). It is proved that the Darboux theorem holds also for any finite codimension symplectic submanifolds of , and for symplectic manifolds obtained from by the Marsden-Weinstein reduction procedure.

     

An extension problem for the CR fractional Laplacian

We show that the conformally invariant fractional powers of the sub-Laplacian on the Heisenberg group are given in terms of the scattering operator for an extension problem to the Siegel upper halfspace. Remarkably, this extension problem is different from the one studied, among others, by Caffarelli and Silvestre. We also prove an energy identity that yields a sharp trace Sobolev embedding.     

Fragment-Based Stabilizers of Protein–Protein Interactions through Imine-Based Tethering

Small-molecule stabilization of protein–protein interactions (PPIs) is a promising concept in drug discovery, however the question how to identify or design chemical starting points in a “bottom-up” approach is largely unanswered. We report a novel concept for identifying initial chemical matter for PPI stabilization based on imine-forming fragments. The imine bond offers a covalent anchor for site-directed fragment targeting, whereas its transient nature enables efficient analysis of structure–activity relationships. This bond enables fragment identification and optimisation using protein crystallography. We report novel fragments that bind specifically to a lysine at the PPI interface of the p65-subunit-derived peptide of NF-κB with the adapter protein 14-3-3. Those fragments that subsequently establish contacts with the p65-derived peptide, rather than with 14-3-3, efficiently stabilize the 14-3-3/p65 complex and offer novel starting points for molecular glues.     

Suzuki–Miyaura Cross-Coupling of Bromotryptophan Derivatives at Ambient Temperature

Mild reaction conditions are highly desirable for bio-orthogonal side chain derivatizations of amino acids, peptides or proteins due to the sensitivity of these substrates. Transition metal catalysed cross-couplings such as Suzuki–Miyaura reactions are highly versatile, but usually require unfavourable reaction conditions, in particular, when applied with aryl bromides. Ligand-free solvent-stabilised Pd-nanoparticles represent an efficient and sustainable alternative to conventional phosphine-based catalysts, because the cross-coupling can be performed at considerably lower temperature. We report on the application of such a highly reactive heterogeneous catalyst for the Suzuki–Miyaura cross-coupling of brominated tryptophan derivatives. The solvent-stabilised Pd-nanoparticles are even more efficient than the literature-known ADHP-Pd precatalyst. Interestingly, the latter also leads to the formation of quasi-homogeneous Pd-nanoparticles as the catalytic species. One advantage of our approach is the compatibility with aqueous and aerobic conditions at near-ambient temperatures and short reaction times of only 2 h. The influence of different N^α-protecting groups, boronic acids as well as the impact of different amino acid side chains in bromotryptophan-containing peptides has been studied. Notably, a surprising acceleration of the catalysis was observed when palladium-coordinating side chains were present in proximal positions.     

Electron transfer between a tyrosyl radical and a cysteine residue in hemoproteins: spin trapping analysis

We investigated electron transfer between a tyrosyl radical and cysteine residue in two systems, oxyhemoglobin (oxyHb)/peroxynitrite/5,5-dimethyl-1-pyrroline N-oxide (DMPO) and myoglobin (Mb)/hydrogen peroxide/DMPO, using a combination of techniques including ESR, immuno-spin trapping (IST), and ESI/MS. These techniques show that the nitrone spin trap DMPO covalently binds to one or more amino acid radicals in the protein. Treating oxyHb with peroxynitrite and Mb with H2O2 in the presence of a low DMPO concentration yielded secondary Cys-DMPO radical adduct exclusively, whereas in the presence of high DMPO, more of the primary Tyr-DMPO radical adduct was detected. In both systems studied, we found that, at high DMPO concentrations, mainly tyrosyl radicals (Hb-Tyr42/Tyr24 and Mb-Tyr103) are trapped and the secondary electron-transfer reaction does not compete, whereas in the presence of low concentrations of DMPO, the secondary reaction predominates over tyrosyl trapping, and a thiyl radical is formed and then trapped (Hb-Cys93 or Mb-Cys110). With increasing concentrations of DMPO in the reaction medium, primary radicals have an increasing probability of being trapped. MS/MS was used to identify the specific Tyr and Cys residues forming radicals in the myoglobin system. All data obtained from this combination of approaches support the conclusion that the initial site of radical formation is a Tyr, which then abstracts an electron from a cysteine residue to produce a cysteinyl radical. This complex phenomenon of electron transfer from one radical to another has been investigated in proteins by IST, ESR, and MS.