C-bifurcations and period-adding in one-dimensional piecewise-smooth maps

This paper examines the behaviour of piecewise-smooth, continuous, one-dimensional maps that have been derived in the literature as normal forms for grazing and sliding bifurcations. These maps are linear for negative values of the parameter and non-linear for positive values of the parameter. Both C¹ and C² maps of this form are considered. These maps display both period-adding and period-doubling behaviour. For maps with a squared or 3/2 term the stability and existence conditions of fixed points and period-2 orbits in the vicinity of the border-collision are found analytically. These agree with the Feigin classification proposed by di Bernardo et al. [Chaos Solitons and Fractals 10 (1999) 1881]. The period-adding behaviour is examined in these maps, where analytical solutions for the boundaries of periodic solutions are found. Implicit equations for the boundaries of periodic windows for varying power term are also found and plotted. Thus, it is proved that period-adding scenarios are generic in maps of this form.     

Poly(dimethylsiloxane) cross-linked carbon paste electrodes for microfluidic electrochemical sensing

Recently, the development of electrochemical biosensors as part of microfluidic devices has garnered a great deal of attention because of the small instrument size and portability afforded by the integration of electrochemistry in microfluidic systems. Electrode fabrication, however, has proven to be a major obstacle in the field. Here, an alternative method to create integrated, low cost, robust, patternable carbon paste electrodes (CPEs) for microfluidic devices is presented. The new CPEs are composed of graphite powder and a binder consisting of a mixture of poly(dimethylsiloxane) (PDMS) and mineral oil. The electrodes are made by filling channels molded in previously cross-linked PDMS using a method analogous to screen printing. The optimal binder composition was investigated to obtain electrodes that were physically robust and performed well electrochemically. After studying the basic electrochemistry, the PDMS-oil CPEs were modified with multi-walled carbon nanotubes (MWCNT) and cobalt phthalocyanine (CoPC) for the detection of catecholamines and thiols, respectively, to demonstrate the ease of electrode chemical modification. Significant improvement of analyte signal detection was observed from both types of modified CPEs. A nearly 2-fold improvement in the electrochemical signal for 100 μM dithiothreitol (DTT) was observed when using a CoPC modified electrode (4.0 ± 0.2 nA (n = 3) versus 2.5 ± 0.2 nA (n = 3)). The improvement in signal was even more pronounced when looking at catecholamines, namely dopamine, using MWCNT modified CPEs. In this case, an order of magnitude improvement in limit of detection was observed for dopamine when using the MWCNT modified CPEs (50 nM versus 500 nM). CoPC modified CPEs were successfully used to detect thiols in red blood cell lysate while MWCNT modified CPEs were used to monitor temporal changes in catecholamine release from PC12 cells following stimulation with potassium.     

Coordinating Numeric and Linear Units: Elementary Students’ Strategies for Locating Whole Numbers on the Number Line

Two investigations of fifth graders’ strategies for locating whole numbers on number lines revealed patterns in students’ coordination of numeric and linear units. In Study 1, we investigated the effects of context on students’ placements of three numbers on an open number line. For one group (n?=?24), the line was presented in a thematic context as a “race course,” and, for a second group (n?=?24), the line was presented as a conventional number line. Most students in both groups placed consecutive whole numbers at appropriate linear distances, but the thematic context group was more likely to place nonconsecutive whole numbers at appropriate linear distances. In Study 2 (n?=?24), students placed numbers on lines marked with two numbers. Most students placed a third number appropriately when the marked numbers were consecutive whole numbers, but not when the labeled numbers were nonconsecutive whole numbers. The findings reveal fifth graders’ conceptual difficulties in coordinating numeric and linear units on the number line and a thematic context that can support this coordination.     

Solution-phase library synthesis of furanoses

The solution-phase synthesis of amido-, urea-, and aminofuranoses was achieved. Alkylated furanose aldehydes were treated with primary amines in the presence of sodium triacetoxyborohydride to give secondary amines. Subsequent acylation with acid chlorides and isocyanates afforded amidofuranoses and ureafuranoses, respectively. Second, reductive amination of furanose aldehydes with secondary amines yielded tertiary amines. The resulting acetonides were treated with alcohols in the presence of acid to yield mixed acetals. In the library syntheses, functionalized scavenger resins were used in the purification of intermediates and products.     

Thermodynamics of forming water clusters at various temperatures and pressures by Gaussian-2, Gaussian-3, complete basis set-QB3, and complete basis set-APNO model chemistries; implications for atmospheric chemistry

The Gaussian-2, Gaussian-3, complete basis set- (CBS-) QB3, and CBS-APNO methods have been used to calculate Delta H degrees and Delta G degrees values for neutral clusters of water, (H(2)O)(n), where n = 2-6. The structures are similar to those determined from experiment and from previous high-level calculations. The thermodynamic calculations by the G2, G3, and CBS-APNO methods compare well against the estimated MP2(CBS) limit. The cyclic pentamer and hexamer structures release the most heat per hydrogen bond formed of any of the clusters. While the cage and prism forms of the hexamer are the lowest energy structures at very low temperatures, as temperature is increased the cyclic structure is favored. The free energies of cluster formation at different temperatures reveal interesting insights, the most striking being that the cyclic trimer, cyclic tetramer, and cyclic pentamer, like the dimer, should be detectable in the lower troposphere. We predict water dimer concentrations of 9 x 10(14) molecules/cm(3), water trimer concentrations of 2.6 x 10(12) molecules/cm(3), tetramer concentrations of approximately 5.8 x 10(11) molecules/cm(3), and pentamer concentrations of approximately 3.5 x 10(10) molecules/cm(3) in saturated air at 298 K. These results have important implications for understanding the gas-phase chemistry of the lower troposphere.     

Chemistry of the t-butoxyl radical: evidence that most hydrogen abstractions from carbon are entropy-controlled

Absolute rate constants and Arrhenius parameters for hydrogen abstractions (from carbon) by the t-butoxyl radical ((t) BuO.) are reported for several hydrocarbons and tertiary amines in solution. Combined with data already in the literature, an analysis of all the available data reveals that most hydrogen abstractions (from carbon) by (t) BuO. are entropy controlled (i.e., TdeltaS > deltaH, in solution at room temperature). For substrates with C-H bond dissociation energies (BDEs) > 92 kcal/mol, the activation energy for hydrogen abstraction decreases with decreasing BDE in accord with the Evans-Polanyi equation, with alpha approximately 0.3. For substrates with C-H BDEs in the range from 79 to 92 kcal/mol, the activation energy does not vary significantly with C-H BDE. The implications of these results in the context of the use of (t) BuO. as a chemical model for reactive oxygen-centered radicals is discussed.     

Butyl acrylate porous polymer monoliths in fused-silica capillaries for use in capillary electrochromatography

Capillary electrochromatography incorporates features of both capillary electrophoresis and liquid chromatography. Butyl acrylate polymers, cast in-situ with heat initiated polymerization and no retaining frits have been made. Van Deemter plots of chrysene have been examined at a variety of operating temperatures to examine column behavior. Hmin moves to faster flow-rates and increases slightly in magnitude as temperature is increased. The longevity and reproducibility of the columns have been examined with a homologous series. Performance is very reproducible between two different columns of different diameters, operated on different systems and prepared from the same polymeriation batch. The relative standard deviation of retention factors is a maximum of 3.1% with most values calculated at less than 1%. The uniformity of the polymers as a function of length has also been studied with a series of polycyclic aromatic hydrocarbons, and the columns have proved to be very uniform across their length as measured by the consistency of retention factors with a maximum relative standard deviation of 3.4% and most values calculated between 1 and 2%. Plate numbers of between 65000 and 80000 plates/m have been attained for compounds with retention factors of 3 to 12. These columns have proved easy to make, are quite reproducible, and long lived.     

Binary and gray-scale patterning of chemical functionality on polymer films

We present a facile technique for the gray-scale chemical functionalization of polymer surfaces with high dynamic range. We demonstrate the use of this technique to create amine-functionalized substrates that are used for the patterned binding of fluorophores and the patterned synthesis of peptides. Studies of the behavior of the model organism Dictyostelium discoideum indicate the biocompatibility of the functionalized substrates.     

Positioning lipid membrane domains in giant vesicles by micro-organization of aqueous cytoplasm mimic

We report localization of lipid membrane microdomains to specific "poles" of asymmetric giant vesicles (GVs) in response to local internal composition. Interior aqueous microdomains were generated in a simple model cytoplasm composed of a poly(ethyleneglycol) (PEG)/dextran aqueous two-phase system (ATPS) encapsulated in the vesicles. The GV membrane composition used here was a modification of a DOPC/DPPC/cholesterol mixture known to form micrometer-scale liquid ordered and liquid disordered domains; we added lipids with PEG 2000 Da-modified headgroups. Osmotically induced budding of the ATPS-containing GVs led to structures where the PEG-rich and dextran-rich interior aqueous phases were in contact with different regions of the vesicle membrane. Liquid ordered (L o) membrane domains rich in PEG-terminated lipids preferentially coated the PEG-rich aqueous phase vesicle "body", while coexisting liquid disordered (L d) membrane domains coated the dextran-rich aqueous phase "bud". Membrane domain positioning resulted from interactions between lipid headgroups and the interior aqueous polymer solutions, e.g., PEGylated headgroups with PEG and dextran polymers. Heating resulted first in patchy membranes where L o and L d domains no longer showed any preference for coating the PEG-rich vs dextran-rich interior aqueous volumes, and eventually complete lipid mixing. Upon cooling lipid domains again coated their preferred interior aqueous microvolume. This work shows that nonspecific interactions between interior aqueous contents and the membrane that encapsulates them can drive local chemical heterogeneity, and offers a primitive experimental model for membrane and cytoplasmic polarity in biological cells.