The movement towards a more experimental approach to problem solving in mathematics using coding

Motivated by a problem proposed in a coding competition for secondary students, I will show on this paper how coding substantially changed the problem-solving process towards a more experimental approach.     

Invariant convex sets in polar representations

We study a compact invariant convex set E in a polar representation of a compact Lie group. Polar representations are given by the adjoint action of K on p, where K is a maximal compact subgroup of a real semisimple Lie group G with Lie algebra g = k ⊕ p. If a ? p is a maximal abelian subalgebra, then P = E ∩ a is a convex set in a. We prove that up to conjugacy the face structure of E is completely determined by that of P and that a face of E is exposed if and only if the corresponding face of P is exposed. We apply these results to the convex hull of the image of a restricted1 momentum map.     

Accelerating molecular modeling applications with graphics processors

Molecular mechanics simulations offer a computational approach to study the behavior of biomolecules at atomic detail, but such simulations are limited in size and timescale by the available computing resources. State-of-the-art graphics processing units (GPUs) can perform over 500 billion arithmetic operations per second, a tremendous computational resource that can now be utilized for general purpose computing as a result of recent advances in GPU hardware and software architecture. In this article, an overview of recent advances in programmable GPUs is presented, with an emphasis on their application to molecular mechanics simulations and the programming techniques required to obtain optimal performance in these cases. We demonstrate the use of GPUs for the calculation of long-range electrostatics and nonbonded forces for molecular dynamics simulations, where GPU-based calculations are typically 10-100 times faster than heavily optimized CPU-based implementations. The application of GPU acceleration to biomolecular simulation is also demonstrated through the use of GPU-accelerated Coulomb-based ion placement and calculation of time-averaged potentials from molecular dynamics trajectories. A novel approximation to Coulomb potential calculation, the multilevel summation method, is introduced and compared with direct Coulomb summation. In light of the performance obtained for this set of calculations, future applications of graphics processors to molecular dynamics simulations are discussed.     

Electrospray ionization mass spectrometric characterization of key Te(IV) cationic intermediates for the addition of TeCl4 to alkynes

Tellurium tetrachloride adds to alkynes via two pathways: a concerted syn-addition that yields Z-tri- and -tetrasubstituted alkenes or an anti-addition that yields E-alkenes. The mechanistic aspects of these divergent pathways for TeCl4 addition to alkynes have been investigated by on-line electrospray ionization (tandem) mass spectrometry (ESI-MS(/MS)). Via ESI-MS(/MS), we have been able to intercept and characterize the active electrophile TeCl3+ in tetrahydrofuran (THF) solutions of TeCl4, as well as its THF complex and several TeClx(OH)y+ derivatives. For the first time, also, key Te(IV) cationic intermediates of the electrophilic addition of TeCl4 to alkynes were captured for gas-phase MS investigation. The detailed structural data of cyclic tellurane intermediates intercepted herein seems to provide insights into the coordinative behavior of the Te(IV) atom and its mode of action towards biological targets.     

DNA three-way junction with a dinuclear iron(II) supramolecular helicate at the center: a NMR structural study

A tetracationic supramolecular helicate, [Fe2L3]4+ (L = C25H20N4), with a triple-helical architecture is found to induce the formation of a three-way junction (3WJ) of deoxyribonucleotides with the helicate located in the center of the junction. NMR spectroscopic studies of the interaction between the M enantiomer of the helicate and two different oligonucleotides, [5'-d(TATGGTACCATA)]2 and [5'-d(CGTACG)]2, show that, in each case, the 2-fold symmetry of the helicate is lifted, while the 3-fold symmetry around the helicate axis is retained. The 1:3 helicate/DNA stoichiometry estimated from 1D NMR spectra supports a molecular model of a three-way junction composed of three strands. Three separate double-helical arms of the three-way junction are chemically identical giving rise to one set of proton resonances. The NOE contacts between the helicate and DNA unambiguously show that the helicate is fitted into the center of the three-way junction experiencing a hydrophobic 3-fold symmetric environment. Close stacking interactions between the ligand phenyl groups and the nucleotide bases are demonstrated through unusually large downfield shifts (1-2 ppm) of the phenyl protons. The unprecedented 3WJ arrangement observed in solution has also been found to exist in the crystal structure of the helicate adduct of [d(CGTACG)2] (Angew. Chem., Int. Ed. 2006, 45, 1227).     

Nuclear electric quadrupole moment of gold

The nuclear quadrupole moment for (197)Au has been determined on the base of the state-of-art relativistic molecular calculations. The experimental shifts in the nuclear coupling constants in the series of molecules AuF, XeAuF, KrAuF, ArAuF, (OC)AuF, and AuH have been combined with highly accurate determinations of the electric field gradient (EFG) at the gold nucleus, obtained by molecular relativistic Dirac-Coulomb-Gaunt Hartree-Fock calculations. The electronic correlation contribution to the EFG is included with the CCSD(T) and CCSD-T approaches, also in the four-component framework, using a finite-difference method. In order to estimate the accuracy of their approach the authors have thoroughly investigated the convergence of the results with respect to the basis set employed and the size of the correlated orbital space. The effect of the full Breit electron-electron interaction on the nuclear quadrupole moment of gold has also been considered explicitly for the AuF molecule. They obtain for (197)Au a nuclear quadrupole moment of 510+/-15 mb, which deviates by about 7% from the currently accepted muonic value.     

Nanoscale conformational ordering in polyanilines investigated by SAXS and AFM

Understanding the adsorption mechanisms in nanostructured polymer films has become crucial for their use in technological applications, since film properties vary considerably with the experimental conditions utilized for film fabrication. In this paper, we employ small-angle X-ray scattering (SAXS) to investigate solutions of polyanilines and correlate the chain conformations with morphological features of the nanostructured films obtained with atomic force microscopy (AFM). It is shown that aggregates formed already in solution affect the film morphology; in particular, at early stages of adsorption film morphology appears entirely governed by the chain conformation in solution and adsorption of aggregates. We also use SAXS data for modeling poly(o-ethoxyaniline) (POEA) particle shape through an ab initio procedure based on simulated annealing using the dummy atom model (DAM), which is then compared to the morphological features of POEA films fabricated with distinct pHs and doping acids. Interestingly, when the derivative POEA is doped with p-toluene sulfonic acid (TSA), the resulting films exhibit a fibrillar morphology-seen with atomic force microscopy and transmission electron microscopy-that is consistent with the cylindrical shape inferred from the SAXS data. This is in contrast with the globular morphology observed for POEA films doped with other acids.     

Non-ionic contributions to the electric-field gradient at 111Cd/181Ta impurities in bixbyites

The electric-field-gradient (EFG) tensor at both cation sites of the bixbyite structure in ¹⁸¹Hf-implanted Lu- and Sm-sesquioxides was determined by the PAC technique. The cumulated EFG data at Ta-impurity sites in binary oxides enable us to discuss the “universal” character of the empirical correlation between local and ionic contributions to the EFG in these systems. An EFG factorization in terms of the electronic characteristics of the probe and the geometry of the cation coordination is proposed, which explains the experimental EFG results at Ta/Cd impurity sites in bixbyites and agrees with a simplified decomposition of the EFG valence contribution coming from ab-initio calculations. This revised version was published online in August 2006 with corrections to the Cover Date.