Closed Unstretchable Knotless Ribbons and the Wunderlich Functional

In 1962, Wunderlich published the article “On a developable Möbius band,” in which he attempted to determine the equilibrium shape of a free standing Möbius band. In line with Sadowsky’s pioneering works on Möbius bands of infinitesimal width, Wunderlich used an energy minimization principle, which asserts that the equilibrium shape of the Möbius band has the lowest bending energy among all possible shapes of the band. By using the developability of the band, Wunderlich reduced the bending energy from a surface integral to a line integral without assuming that the width of the band is small. Although Wunderlich did not completely succeed in determining the equilibrium shape of the Möbius band, his dimensionally reduced energy integral is arguably one of the most important developments in the field. In this work, we provide a rigorous justification of the validity of the Wunderlich integral and fully formulate the energy minimization problem associated with finding the equilibrium shapes of closed bands, including both orientable and nonorientable bands with arbitrary number of twists. This includes characterizing the function space of the energy functional, dealing with the isometry and local injectivity constraints, and deriving the Euler–Lagrange equations. Special attention is given to connecting edge conditions, regularity properties of the deformed bands, determination of the parameter space needed to ensure that the deformation is surjective, reduction in isometry constraints, and deriving matching conditions and jump conditions associated with the Euler–Lagrange equations.

     

A conjectural extension of Hecke’s converse theorem

By using specific subsequences of two different types of generalized Stern polynomials, we obtain several related classes of finite and infinite continued fractions involving a single term \(z^{t^j}\) in their partial numerators, where z is a complex variable and t is a positive integer. This approach is extended to other, sparser, subsequences of Stern polynomials, based on certain Lucas functions; this then leads to further infinite classes of continued fractions.     

A new look at a smart polling model

We analyze a Markovian smart polling model, which is a special case of the smart polling models studied in the work of Boon et al. (Queueing Syst 66:239–274, 2010), as well as a generalization of the gated M / M / 1 queue considered in Resing and Rietman (Stat Neerlandica 58:97–110, 2004). We first derive tractable expressions for the stationary distribution (when it exists) as well as the Laplace transforms of the transition functions of this polling model—while further assuming the system is empty at time zero—and we also present simple necessary and sufficient conditions for ergodicity of the smart polling model. Finally, we conclude the paper by briefly explaining how these techniques can be used to study other interesting variants of this smart polling model.     

Cholesterol supports headgroup superlattice domain formation in fluid phospholipid/cholesterol bilayers

Fluorescence and Fourier transform infrared (FTIR) spectroscopic techniques were used to explore the effect of added cholesterol on the composition-dependent formation of putative phospholipid headgroup superlattices in fluid 1-palmitoyl-2-oleoyl-phosphatidylethanolamine/1-palmitoyl-2-oleoyl-phosphatidylcholine/cholesterol (POPE/POPC/CHOL) bilayers. Steady-state fluorescence anisotropy measurements of diphenylhexatriene (DPH) chain-labeled phosphatidylcholine (DPH-PC) revealed significant dips at several POPE-to-phospholipid mole fractions (X(PE)'s) when the cholesterol-to-lipid mole fraction (X(CHOL)) was fixed at 0.00, 0.35, 0.40, and 0.50. Most of the observed dips occur at or close to critical X(PE)'s predicted by the Headgroup Superlattice (SL) model, suggesting that phospholipid headgroups of different structures tend to adopt regular distributions even in the presence of cholesterol. Time-resolved fluorescence anisotropy measurements revealed that DPH-PC senses a disordered and highly mobile microenvironment in the POPE/POPC/CHOL bilayers at those critical X(PE)'s, indicating that this probe may partition to defect regions in the bilayers. The presence of coexisting packing defect regions and regularly distributed SL domains is a key feature predicted by the Headgroup SL model. Importantly, probe-free FTIR measurements of acyl chain C-H, interfacial carbonyl, and headgroup phosphate stretching peak frequencies revealed the presence of abrupt changes at X(PE)'s close to those observed in the fluorescence data. When X(PE) was varied from 0.60 to 0.72 and X(CHOL) from 0.34 to 0.46, a clear dip at the lipid composition coordinates (X(PE), X(CHOL)) approximately (0.68, 0.40) was observed in the three-dimensional surface plots of DPH-PC anisotropy as well as the carbonyl and phosphate stretching frequencies. The critical X(CHOL) at 0.40 agrees with the Cholesterol SL model, which assumes that cholesterol and phospholipid form SL domains at the lipid acyl chain level. In conclusion, this study provides evidence that cholesterol supports formation of phospholipid headgroup SLs in fluid state ternary lipid bilayers. The feasibility of the parallel existence of SLs at the lipid headgroup and acyl chain levels supports the relevance of the lipid SL model for the membranes of eukaryotic cells that typically contain significant amounts of cholesterol. We speculate that lipid SL formation may play a central role in the regulation of membrane lipid compositions, maintenance of organelle boundaries, and other crucial phenomena in those cells.     

Cobalt ultrathin film catalyzed ethanol chemical vapor deposition of single-walled carbon nanotubes

We report a simple and efficient chemical vapor deposition (CVD) process that can grow oriented and long single-walled carbon nanotubes (SWNTs) using a cobalt ultrathin film ( approximately 1 nm) as the catalyst and ethanol as carbon feedstock. In the process, millimeter- to centimeter-long, oriented and high-quality SWNTs can grow horizontally on various flat substrate surfaces, traverse slits as large as hundreds of micrometers wide, or grow over vertical barriers as high as 20 microm. Such observations demonstrate that the carbon nanotubes are suspended in the gas flow during the growth. The trace amount of self-contained water (0.2-5 wt %) in ethanol may act as a mild oxidizer to clean the nanotubes and to elongate the lifetime of the catalysts, but no yield improvement was observed at the CVD temperature of 850 degrees C. We found that tilting the substrates supporting the Co ultrathin film catalysts can grow more, longer carbon nanotubes. A mechanism is discussed for the growth of long SWNTs.     

Determination of the interfacial characteristics of a series of bolaamphiphilic poly(fluorooxetane) surfactants through molecular dynamics simulation

Constant surface tension (NgammaT) and constant volume (NVT) molecular dynamics simulations have been conducted on a series of bolaamphiphilic alpha,varpi-(diammonium disulfato)poly(fluorooxetane)s and on a typical "long-chain" anionic fluorosurfactant used to improve the flow-and-leveling characteristics of aqueous coatings, to compare their behavior at a water/air interface. Recent research has shown that the poly(fluorooxetane) surfactants considered in this paper could serve as an effective substitute for traditional fluorosurfactants used in flow-and-leveling applications.(1) From molecular dynamics simulation, we have determined the saturated interfacial area per surfactant, interfacial area per surfactant as a function of surface tension, density profiles, the degree of hydration for various atoms in each surfactant, the degree of counterion binding, and order parameters. Our results for saturated interfacial area per surfactant molecule are greater than what has been obtained by other researchers through parametric fitting of interfacial area from experimental surface tension data using the Davies isotherm. Possible explanations for this difference are discussed. The low interfacial areas occupied by each poly(fluorooxetane) at the water/air interface are the result of their ability to adopt a "looped" conformation, in which the carbon and oxygen backbone of each surfactant and the attached perfluoroalkyl chains are forced into the air phase. A geometrically defined penetration parameter was calculated from the density profiles, which reveals that each poly(fluorooxetane) surfactant is more effective at separating the air and water phases than the "long-chain" anionic fluorosurfactant. The degree of hydration measured for different atoms in poly(fluorooxetane) during simulation confirms that a "looped" conformation is adopted in which the surfactant backbone and the perfluoroalkyl chains are lifted away from the water surface. Calculation of order parameters revealed a much lower degree of ordering for the perfluoroalkyl side chains in each bolaamphiphile than in the "long chain" anionic fluorosurfactant. When viewed in the context of the penetration parameter analysis, the density profiles and hydration data suggest why each poly(fluorooxetane) is capable of significantly reducing surface tension when other fluorosurfactants with similarly short perfluoroalkyl moieties provide inadequate surface tension reduction for practical flow-and-leveling applications.     

Theoretical investigation of the temperature dependence of the fifth-order Raman response function of fluid and liquid xenon

The temperature dependence of the fifth-order Raman response function, R(5)(t1,t2), is calculated for fluid xenon by employing a recently developed time-correlation function (TCF) theory. The TCF theory expresses the two-dimensional (2D) Raman quantum response function in terms of a two-time, computationally tractable, classical TCF. The theory was shown to be in excellent agreement with existing exact classical MD calculations for liquid xenon as well as reproducing line shape characteristics predicted by earlier theoretical work. It is applied here to investigate the temperature dependence of the fifth-order Raman response function in fluid xenon. In general, the characteristic line shapes are preserved over the temperature range investigated (for the reduced temperature points T* = 0.5, 1.0, and 2.0); differences in the signal decay times and a large decline in intensity with decreasing temperature (and associated anharmonicity) are observed. In addition, there are some signature features that were not observed in earlier results for T* = 1. The most dramatic difference in line shape is observed for the polarization condition, xxzzxx, that shows a vibrational echo peak. In contrast, the fully polarized signal changes mainly in magnitude.     

A versatile approach toward the ansamycin antibiotics

[STRUCTURE: SEE TEXT] The ansamycin antibiotics contain metacyclophanic macrolactams, many of which possess potent antitumor activity. Only a few total syntheses of this family of natural products have been reported, and modifications to increase potency have not been described. Therefore, a method was developed to prepare the trienomycin A core via resin-bound triphenylphosphonium salts, which serve as both a reagent and a traceless linker to afford olefinic products that undergo ring-closing metathesis (RCM) to give macrocyclic scaffolds of varying ring sizes.     

Advances in Lewis acid controlled carbon-carbon bond-forming reactions enable a concise and convergent total synthesis of bullatacin

Lewis acids control regioselectivity in the alkylation of epi-chlorohydrin and the stereochemistry of an alkyne addition to set the C24/C23 anti relationship. These advances facilitate an efficient total synthesis of bullatacin in 13.3% overall yield from commercial starting materials.     

Template-directed synthesis of mechanically interlocked molecular bundles using dynamic covalent chemistry

Mixing the dipyrido[24]crown-8 derivatives carrying one or two formyl group(s) on the 4 position(s) of their pyridine ring(s) with a 3-fold symmetrical trisammonium ion template in a 3:1 ratio in CD3NO2 results in the formation of thermodynamically stable [4]pseudorotaxanes which, upon addition of a 1,3,5-trisaminobenzene cap, form mechanically interlocked molecular bundles with one and two caps, respectively, by virtue of dynamic imine bond formation.