Homotopy Groups of Spheres and Lipschitz Homotopy Groups of Heisenberg Groups

We provide a sufficient condition for the nontriviality of the Lipschitz homotopy group of the Heisenberg group, ${\pi_m^{\rm Lip}(\mathbb{H}_n)}$ , in terms of properties of the classical homotopy group of the sphere, ${\pi_m(\mathbb{S}^n)}$ . As an application we provide a new simplified proof of the fact that ${\pi_n^{\rm Lip}(\mathbb{H}_n)\neq \{0\}, n=1,2,\ldots}$ , and we prove a new result that ${\pi_{4n-1}^{\rm Lip}(\mathbb{H}_{2n})\neq \{0\}}$ for n = 1,2,… The last result is based on a new generalization of the Hopf invariant. We also prove that Lipschitz mappings are not dense in the Sobolev space ${W^{1,p}(\mathcal{M},\mathbb{H}_{2n})}$ when ${\dim \mathcal{M} \geq 4n}$ and 4n?1 ≤  p < 4n.     

Finding all steady state solutions of chemical kinetic models

Ordinary differential equations are used frequently by theoreticians to model kinetic process in chemistry and biology. These systems can have stable and unstable steady states and oscillations. This paper presents an algorithm to find all steady state solutions to a restricted class of ODE models, for which the right-hand sides are linear combinations of rational functions of variables and parameters. The algorithm converts the steady state equations into a system of polynomial equations and uses a globally convergent homotopy method to find all the roots of the system of polynomials. All steady state solutions of the original ODEs are guaranteed to be present as roots of the polynomial equations. The conversion may generate some spurious roots that do not correspond to steady state solutions. The stability properties of the steady states are not revealed. This paper explains the algorithms used and gives results for a cell cycle modeling problem.     

Quasiconvexity in the Heisenberg group

We show that if A is a closed subset of the Heisenberg group whose vertical projections are nowhere dense, then the complement of A is quasiconvex. In particular, closed sets which are null sets for the cc-Hausdorff 3-measure have quasiconvex complements. Conversely, we exhibit a compact totally disconnected set of Hausdorff dimension three whose complement is not quasiconvex.     

Factors affecting the detection limit of a flow- injection spectrophotometric procedure

Design criteria for manifolds to obtain adequated sensitivity and freedom from certain types of noise are discussed. For the determination of chloride by the mercury(II) thiocynate method, the mergin-streams manifold, needed to eliminated negative peaks at low concentrations and refractive index effects, increased the baseline noise because of inefficient mixing at the confluence point. Packed-bed reactors and single-bead-string reactors (SBSR) significantly improved the baseline noise. Refractive index effects caused by sample concentration gradients were eliminated by the injection of large volumes (>500 μl), so that the detector could view a homogeneous central part of the sample zone. This approach maximises the signal because the dispersion is governed only by the relative flow rates of the carrier and reagent streams. The detection limits for chloride were improved to 40 ng l^?1 in aqueous solutions of low ionic strength and to 1.3 mg l^?1 in 5.25% (w/v) potassium sulphate.     

Review of Solid Sample Introduction for Plasma Spectrometry and a Comparison of Results for Laser Ablation, Electrothermal Vaporization, and Slurry Nebulization

The relative merits of three solid sample introduction techniques for inductively coupled plasma atomic emission spectroscopy and inductively coupled plasma mass spectrometry have been evaluated. The techniques studied were laser ablation (LA), slurry nebulization, and slurry introduction via electrothermal vaporization. The application of each technique to the quantification of Pa and Pb in geochemical samples has been described and the results obtained for each technique have been compared. To help in assessing the potential of the systems described here a simple classification of the agreement obtained between found and known values has been proposed allowing a comparison of these results with other results published in the literature to be made. Problems encountered during the evaluation of the techniques have been described and, for LA, the associated problems have led to the development of a modified sampling system which was used in an investigation of some fundamental aspects of LA as a method of solid sample introduction. Possible routes for improvement in sampling and transport are suggested, including use of reactive gases, ablation under stopped flow conditions and removal of large particles by gravitational settling.     

Conformal dimension of the antenna set

We show that the self-similar set known as the ``antenna set' has the property that (where the infimum is over all quasiconformal mappings of the plane), but that this infimum is not attained by any quasiconformal map; indeed, is not attained for any quasisymmetric map into any metric space.

     

Peptides—XXXVI: Synthesis of the 27–37 fragment of a lysozyme analogue

Two syntheses of the fully protected 27–37 fragment have been carried out. The results indicate that coupling of the 27–32 and 33–37 fragments is superior to the alternative 27–29 plus 30–37 approach.     

Evidence for a Boat Conformation at the Transition State of GH76 α‐1,6‐Mannanases—Key Enzymes in Bacterial and Fungal Mannoprotein Metabolism

α‐Mannosidases and α‐mannanases have attracted attention for the insight they provide into nucleophilic substitution at the hindered anomeric center of α‐mannosides, and the potential of mannosidase inhibitors as cellular probes and therapeutic agents. We report the conformational itinerary of the family GH76 α‐mannanases studied through structural analysis of the Michaelis complex and synthesis and evaluation of novel aza/imino sugar inhibitors. A Michaelis complex in an ^OS₂ conformation, coupled with distortion of an azasugar in an inhibitor complex to a high energy B_2,5 conformation are rationalized through ab initio QM/MM metadynamics that show how the enzyme surface restricts the conformational landscape of the substrate, rendering the B_2,5 conformation the most energetically stable on‐enzyme. We conclude that GH76 enzymes perform catalysis using an itinerary that passes through ^OS₂ and B_2,5^≠ conformations, information that should inspire the development of new antifungal agents.     

Volatile organic compound sensing using a surface-relief D-shaped fiber Bragg grating and a polydimethylsiloxane layer

The authors use a surface-relief fiber Bragg grating with a polydimethylsiloxane (PDMS) layer as a volatile organic compound chemical sensor. A PDMS layer is used because it is compatible with the optical properties of the grating and exhibits good chemical selectivity. As the analyte is absorbed the refractive index of the PDMS changes, causing the Bragg wavelength to shift, and this shift is correlated to chemical type and concentration. The direction and amount of the Bragg wavelength shift is dependent on the absorbed chemical. The authors demonstrate chemical differentiation between dichloromethane and acetone in gaseous states.