Recurrence analysis on Julia sets of semigroups of complex polynomials

We introduce a recurrence function in order to analyze the dynamics of semigroups of complex polynomials. We show that under a regularity hypothesis, the recurrence function is continuous in the complex plane. This is a new notion even for the case of a semigroup with just one generator.     

Quasi-periodic bifurcations of invariant circles in low-dimensional dissipative dynamical systems

This paper first summarizes the theory of quasi-periodic bifurcations for dissipative dynamical systems. Then it presents algorithms for the computation and continuation of invariant circles and of their bifurcations. Finally several applications are given for quasiperiodic bifurcations of Hopf, saddle-node and period-doubling type.     

A Logic-Based Approach to Polymer Sequence Analysis

Polymers are compounds formed by the joining of smaller, often repeating, units linked by covalent bonds. The analysis of their sequence is a fundamental issue in many areas of chemistry, medicine and biology. Nowadays, the prevalent approach to this problem consists in using a mass spectrometry analysis that gives information about the molecular weights of the polymer and of its fragments. This information should be used in order to obtain the sequence. This is however a difficult mathematical problem, and several approaches have been proposed for it. In particular, a promising one is based on a propositional logic modeling of the problem. This paper presents conceptual improvements in this approach, principally the off-line computation of a database that substantially speeds-up the sequencing operations. This is obtained by finding a correspondence between sequences and natural numbers, so that all sequences up to a certain molecular weight can be implicitly considered in the above database, and explicitly computed only when needed. Results on real-world problems show the effectiveness of this approach.     

Quantitative evaluation of noncovalent chorismate mutase-inhibitor binding by ESI-MS

Electrospray time-of-flight mass spectrometry was used to quantitatively determine the dissociation constant of chorismate mutase and a transition state analogue inhibitor. This system presents a fairly complex stoichiometry because the native protein is a homotrimer with three equal and independent substrate binding sites. We can detect the chorismate mutase trimer as well as chorismate mutase-inhibitor complexes by choosing appropriate conditions in the ESI source. To verify that the protein-inhibitor complexes are specific, titration experiments with different enzyme variants and different inhibitors were performed. A plot of the number of bound inhibitors versus added inhibitor concentration revealed saturation behavior with 3:1 (inhibitor:functional trimer) stoichiometry for the TSA. The soft ESI conditions, the relatively high protein mass of 43.5 kDa, and the low charge state (high m/z) result in broad peaks, a typical problem in analyzing noncovalent protein complexes. Due to the low molecular weight of the TSA (226 Da) the peaks of the free protein and the protein with one, two or three inhibitors bound cannot be clearly resolved. For data analysis, relative peak areas of the deconvoluted spectra of chorismate mutase-inhibitor complexes were obtained by fitting appropriate peak shapes to the signals corresponding to the free enzyme and its complexes with one, two, or three inhibitor molecules. From the relative peak areas we were able to calculate a dissociation constant that agreed well with known solution-phase data. This method may be generally useful for interpreting mass spectra of noncovalent complexes that exhibit broad peaks in the high m/z range.     

Nanoscale chemical imaging of segregated lipid domains using tip-enhanced Raman spectroscopy

Lipid domains in supported lipid layers serve as a popular model to gain insight into the processes associated with the compartmentalization of biological membranes into so-called lipid rafts. In this paper, we present reproducible tip-enhanced Raman spectra originating from a very small number of molecules in a lipid monolayer on a gold surface, probed by the apex of a nanometer-sized silver tip. For the first time, we show large (128 × 128 pixels), high-resolution (< 50 nm) tip-enhanced Raman images of binary lipid mixtures with full spectral information at each pixel.     

Primal-dual first-order methods with $${\mathcal {O}(1/\epsilon)}$$ iteration-complexity for cone programming

In this paper we consider the general cone programming problem, and propose primal-dual convex (smooth and/or nonsmooth) minimization reformulations for it. We then discuss first-order methods suitable for solving these reformulations, namely, Nesterov’s optimal method (Nesterov in Doklady AN SSSR 269:543–547, 1983; Math Program 103:127–152, 2005), Nesterov’s smooth approximation scheme (Nesterov in Math Program 103:127–152, 2005), and Nemirovski’s prox-method (Nemirovski in SIAM J Opt 15:229–251, 2005), and propose a variant of Nesterov’s optimal method which has outperformed the latter one in our computational experiments. We also derive iteration-complexity bounds for these first-order methods applied to the proposed primal-dual reformulations of the cone programming problem. The performance of these methods is then compared using a set of randomly generated linear programming and semidefinite programming instances. We also compare the approach based on the variant of Nesterov’s optimal method with the low-rank method proposed by Burer and Monteiro (Math Program Ser B 95:329–357, 2003; Math Program 103:427–444, 2005) for solving a set of randomly generated SDP instances.     

Reaction between Dichlorophosphines and Azoalkenes: A General Synthesis of 3,4-Dihydro-2H-1,2,3-diazaphosphole Derivatives

The reaction between dichlorophosphines and azoalkenes provides a smooth method for the preparation of 3,4-dihydro-2H-1,2,3-diazaphosphole-3-oxides. When R¹ ≠ H, the cis: trans ratio is strongly dependent on the nature of the R⁴ substituent. Compounds arising from phenylazoalkenes are stable for several months, whereas compounds arising from methylazoalkenes undergo modifications during silica gel chromatography. Trichlorosilane reduction of these oxides gives the corresponding phosphines with retention of configuration. In some cases, this process competes with stereomutation of the starting phosphine oxide. Aryl-substituted phosphines are much more stable than the corresponding alkyl-substituted phosphines.