On the algebra of symmetries of Laplace and Dirac operators

We consider a generalization of the classical Laplace operator, which includes the Laplace–Dunkl operator defined in terms of the differential-difference operators associated with finite reflection groups called Dunkl operators. For this Laplace-like operator, we determine a set of symmetries commuting with it, in the form of generalized angular momentum operators, and we present the algebraic relations for the symmetry algebra. In this context, the generalized Dirac operator is then defined as a square root of our Laplace-like operator. We explicitly determine a family of graded operators which commute or anticommute with our Dirac-like operator depending on their degree. The algebra generated by these symmetry operators is shown to be a generalization of the standard angular momentum algebra and the recently defined higher-rank Bannai–Ito algebra.     

Can a polynomial interpolation improve on the Kaplan-Yorke dimension?

The Kaplan-Yorke dimension can be derived using a linear interpolation between an h-dimensional Lyapunov exponent λ(h)>0 and an h+1-dimensional Lyapunov exponent λ(h+1)<0. In this Letter, we use a polynomial interpolation to obtain generalized Lyapunov dimensions and study the relationships among them for higher-dimensional systems.     

Coupled map lattices as spatio-temporal fitness functions: Landscape measures and evolutionary optimization

Coupled Map Lattices (CML) can be interpreted as spatio-temporal fitness landscapes which may pose a dynamic optimization problem. In this paper, we analyze such dynamic fitness landscapes in terms of the landscape measures modality, ruggedness, information content and epistasis. These measures account for different aspects of problem hardness. We use an evolutionary algorithm to solve the dynamic optimization problem and study the relationship between performance criteria of the algorithm and the landscape measures. In this way we relate problem hardness to expectable performance.     

The Driving Force for the Acylation of β-Lactam Antibiotics by L,D-Transpeptidase 2: Quantum Mechanics/Molecular Mechanics (QM/MM) Study

β-lactam antibiotics, which are used to treat infectious diseases, are currently the most widely used class of antibiotics. This study focused on the chemical reactivity of five- and six-membered ring systems attached to the β-lactam ring. The ring strain energy (RSE), force constant (FC) of amide (C−N), acylation transition states and second-order perturbation stabilization energies of 13 basic structural units of β-lactam derivatives were computed using the M06-2X and G3/B3LYP multistep method. In the ring strain calculations, an isodesmic reaction scheme was used to obtain the total energies. RSE is relatively greater in the five-(1a–2c) compared to the six-membered ring systems except for 4b, which gives a RSE that is comparable to five-membered ring lactams. These variations were also observed in the calculated inter-atomic amide bond distances (C−N), which is why the six-membered ring lactams C−N bond are more rigid than those with five-membered ring lactams. The calculated ΔG^# values from the acylation reaction of the lactams (involving the S−H group of the cysteine active residue from L,D transpeptidase 2) revealed a faster rate of C−N cleavage in the five-membered ring lactams especially in the 1–2 derivatives (17.58 kcal mol⁻¹). This observation is also reflected in the calculated amide bond force constant (1.26 mDyn/A) indicating a weaker bond strength, suggesting that electronic factors (electron delocalization) play more of a role on reactivity of the β-lactam ring, than ring strain.     

Cosolvent and Crowding Effects on the Temperature- and Pressure-Dependent Dissociation Process of the α/β-Tubulin Heterodimer

Tubulin is one of the main components of the cytoskeleton of eukaryotic cells. The formation of microtubules depends strongly on environmental and solution conditions, and has been found to be among the most pressure sensitive processes in vivo. We explored the effects of different types of cosolvents, such as trimethylamine-N-oxide (TMAO), sucrose and urea, and crowding agents to mimic cell-like conditions, on the temperature and pressure stability of the building block of microtubules, i. e. the α/β-tubulin heterodimer. To this end, fluorescence and FTIR spectroscopy, differential scanning and pressure perturbation calorimetry as well as fluorescence anisotropy and correlation spectroscopies were applied. The pressure and temperature of dissociation of α/β-tubulin as well as the underlying thermodynamic parameters upon dissociation, such as volume and enthalpy changes, have been determined for the different solution conditions. The temperature and pressure of dissociation of the α/β-tubulin heterodimer and hence its stability increases dramatically in the presence of TMAO and the nanocrowder sucrose. We show that by adjusting the levels of compatible cosolutes and crowders, cells are able to withstand deteriorating effects of pressure even up to the kbar-range.     

Novel Copolymers as Dispersants/Intercalants/Exfoliants for Polypropylene-Clay Nanocomposites

The use of various copolymers as dispersants/intercalants/exfoliants in polypropylene (PP)-clay nanocomposites based on unmodified montmorillonite clays (NaMMT) has been explored. The primary objective of this research has been to find dispersants that allow PP nanocomposites to be formed by direct melt mixing, that are effective with unmodified clays and that comprise only a minor component of the overall composition both with respect to both clay and PP. Two classes of dispersants were investigated: PEO-based nonionic surfactants and amphiphilic copolymers based on a long chain (meth)acrylate (e.g. octadecyl acrylate) and a more polar comonomer (e.g. maleic anhydride, N-vinylpyrrolidone, methyl methacrylate). The state of dispersion achieved and the properties of the derived nanocomposites were found to depend strongly on both on the level of dispersant and its overall composition but interestingly properties are not particularly dependent on the dispersant architecture (i.e. whether statistical, gradient or block copolymer). The nanocomposites possess a tensile modulus up to 40% greater that the precursor PP while elongation at break and impact strength are significantly improved over “clay alone” composites and reference organoclay-based nanocomposites. Also notable are significantly better thermal and thermo-oxidative stability as compared to both PP and “clay alone” composites. For optimal properties, it is both necessary and desirable that the surfactant should only be a minor constituent (20–50 wt-%) of the composition with respect to clay.     

FTIR spectra and normal-mode analysis of a tetranuclear manganese adamantane-like complex in two electrochemically prepared oxidation states: relevance to the oxygen-evolving complex of photosystem II

The IR spectra and normal-mode analysis of the adamantane-like compound [Mn(4)O(6)(bpea)(4)](n+) (bpea = N,N-bis(2-pyridylmethyl)ethylamine) in two oxidation states, Mn(IV)(4) and Mn(III)Mn(IV)(3), that are relevant to the oxygen-evolving complex of photosystem II are presented. Mn-O vibrational modes are identified with isotopic exchange, (16)O-->(18)O, of the mono-micro-oxo bridging atoms in the complex. IR spectra of the Mn(III)Mn(IV)(3) species are obtained by electrochemical reduction of the Mn(IV)(4) species using a spectroelectrochemical cell, based on attenuated total reflection [Visser, H.; et al. Anal. Chem. 2001, 73, 4374-4378]. A novel method of subtraction is used to reduce background contributions from solvent and ligand modes, and the difference and double-difference spectra are used in identifying Mn-O bridging modes that are sensitive to oxidation state change. Two strong IR bands are observed for the Mn(IV)(4) species at 745 and 707 cm(-1), and a weaker band is observed at 510 cm(-1). Upon reduction, the Mn(III)Mn(IV)(3) species exhibits two strong IR bands at 745 and 680 cm(-1), and several weaker bands are observed in the 510-425 cm(-1) range. A normal-mode analysis is performed to assign all the relevant bridging modes in the oxidized Mn(IV)(4) and reduced Mn(III)Mn(IV)(3) species. The calculated force constants for the Mn(IV)(4) species are f(r)(IV)= 3.15 mdyn/A, f(rOr) = 0.55 mdyn/A, and f(rMnr) = 0.20 mdyn/A. The force constants for the Mn(III)Mn(IV)(3) species are f(r)(IV)= 3.10 mdyn/A, f(r)(III)= 2.45 mdyn/A, f(rOr) = 0.40 mdyn/A, and f(rMnr) = 0.15 mdyn/A. This study provides insights for the identification of Mn-O modes in the IR spectra of the photosynthetic oxygen-evolving complex during its catalytic cycle.     

Euclidean number fields 2

The Mathematical Intelligencer -