Normal form for generalized Hopf bifurcation with non-semisimple 1 ∶ 1 resonance

The primary result of this research is the derivation of an explicit formula for the Poincaré-Birkhoff normal form of the generalized Hopf bifurcation with non-semisimple 1:1 resonance. The classical nonuniqueness of the normal form is resolved by the choice of complementary space which yields a unique equivariant normal form. The 4 leading complex constants in the normal form are calculated in terms of the original coefficients of both the quadratic and cubic nonlinearities by two different algorithms. In addition, the universal unfolding of the degenerate linear operator is explicitly determined. The dominant normal forms are then obtained by rescaling the variables. Finally, the methods of averaging and normal forms are compared. It is shown that the dominant terms of the equivariant normal form are, indeed, the same as those of the averaged equations with a particular choice for the constant of integration.Partially supported by NSF through grant MSS 90-57437, AFOSR through grant 91-0041 and NSERC of Canada.     

Characteristic solutions of scalar newton equations in one space dimension

Newton equations are dynamical systems on the space of fields. The solutions of a given equation which are curves of characteristic fields for its force are planar and have constant angular momentum. Separable solutions are characteristic with angular momentum equal to zero. A Newton equation is separable if and only if its characteristic equation is homogeneous. Separable equations correspond to invariants of homogeneous ordinary differential equations, and those associated with a given homogenous equation correspond to its generalized dilation symmetries. A Newton equation is compatible with the characteristic condition if and only if its characteristic equation is linear. Such equations correspond to invariants of linear ordinary differential equations. Those associated with a given linear equation correspond to the central force problems on its solution space. Regardless of compatibility, any Newton equation with a plane of characteristic fields has non-separable characteristic solutions.     

Dialkyl Ether Formation by Nickel‐Catalyzed Cross‐Coupling of Acetals and Aryl Iodides

A new substrate class for nickel‐catalyzed C(sp³) cross‐coupling reactions is reported. α‐Oxy radicals generated from benzylic acetals, TMSCl, and a mild reductant can participate in chemoselective cross‐coupling with aryl iodides using a 2,6‐bis(N‐pyrazolyl)pyridine (bpp)/Ni catalyst. The mild, base‐free conditions are tolerant of a variety of functional groups on both partners, thus representing an attractive C? C bond‐forming approach to dialkyl ether synthesis. Characterization of a [(bpp)NiCl] complex relevant to the proposed catalytic cycle is also described.     

Sensitive and Multiplexed On‐chip microRNA Profiling in Oil‐Isolated Hydrogel Chambers

Although microRNAs (miRNAs) have been shown to be excellent indicators of disease state, current profiling platforms are insufficient for clinical translation. Here, we demonstrate a versatile hydrogel‐based microfluidic approach and novel amplification scheme for entirely on‐chip, sensitive, and highly specific miRNA detection without the risk of sequence bias. A simulation‐driven approach is used to engineer the hydrogel geometry and the gel‐reaction environment is chemically optimized for robust detection performance. The assay provides 22.6 fM sensitivity over a three log range, demonstrates multiplexing across at least four targets, and requires just 10.3 ng of total RNA input in a 2 hour and 15 minutes assay.     

Divergent pathways of β,γ-unsaturated α-diazocarbonyl compounds catalyzed by dirhodium and Lewis acids catalysts separately or in combination

β,γ-Unsaturated a-diazocarbonyl compounds possess two reactive sites for electrophilic addition-one at the diazo carbon and the other at the vinylogous γ-position.Controlled by catalyst,divergent transformations are achieved starting from the same starting materials,either by Lewis acid-catalyzed addition or by dirhodium-catalyzed metal carbene reactions.In select cases two catalysts working in combination or in sequence provide a relay for cascade transformations.In this review,we summarize advances in catalyst-dependent divergent transformations of β,γ-unsaturated α-diazocarbonyl compounds and highlight the potential of this exciting research area and the many challenges that remain.     

Why is Rose Bengal More Phototoxic to Fibroblasts In Vitro Than In Vivo?

Photosensitized protein cross‐linking has been recently developed to seal wounds and strengthen tissue. Although the photosensitizing dye, Rose Bengal (RB), is phototoxic to cultured cells, cytotoxicity does not accompany RB‐photosensitized tissue repair in vivo. We investigated whether the environment surrounding cells in tissue or the high irradiances used for photo–cross‐linking inhibited RB phototoxicity. Fibroblasts (FB) grown within collagen gels to mimic a tissue environment and monolayer cultured FB were treated with RB (0.01–1 mm ) and the high 532 nm laser irradiances used in vivo for tissue repair (0.10–0.50 W cm^?2). Monolayer FB were substantially more sensitive to RB photosensitization: the LD₅₀ was >200‐fold lower than that in collagen gels. Collagen gel protection was associated with increased Akt phosphorylation, a prosurvival pathway. RB phototoxicity in collagen gels was 25‐fold greater at low (0.030 W cm^?2) that at high (0.50 W cm^?2) irradiances. Oxygen depletion at high irradiance only partially accounted for the irradiance dependence of phototoxicity as replacing air with nitrogen only increased the LD₅₀ by four‐fold in monolayers. These results indicate that the lack of RB phototoxicity during in vivo tissue repair results from upregulation of prosurvival pathways in tissue cells, oxygen depletion and irradiance‐dependent RB photochemistry.     

Asymmeric formal [3 + 3]-cycloaddition reactions of nitrones with electrophilic vinylcarbene intermediates

With metal carbene access to dipolar intermediates, 3,6-dihydro-1,2-oxazines are produced in high yields by dirhodium(II) carboxylate catalyzed reactions between nitrones and a β-TBSO-substituted vinyldiazoacetate. High enantiocontrol occurs with catalysis by N-phthaloyl-(S)-(amino acid)-ligated dirhodium carboxylates for [3 + 3]-cycloaddition reactions with both acyclic and cyclic nitrones.     

Thiocyanate and nitrite analysis using miniaturised isotachophoresis on a planar polymer chip

A new method has been developed to improve the determination of thiocyanate using isotachophoresis. This method uses complexation with copper(II) as a mechanism for improving the separation of thiocyanate from chlorate and perchlorate. By using a pH of 3.25 the method can also be used to analyse nitrite. Separations were carried out using a miniaturised poly(methyl methacrylate) (PMMA) separation device. Linearity was observed from 1.25 to 75 mg dm(-3) with a correlation coefficient of 0.998 for both thiocyanate and nitrite. Limits of detection for these two species were calculated to be 0.8 mg dm(-3) and 0.9 mg dm(-3) respectively. The method was successfully applied to the analysis of these anions in a range of samples including explosive residues.     

Synthesis, cytotoxicity and cellular uptake studies of N3 functionalized Re(CO)3 thymidine complexes

Nucleoside-derived drugs play an important role in the treatment of cancer. Here, we present the synthesis and characterization of an intriguing series of N3 conjugated Re(CO)(3) thymidine complexes. The complexes were characterized by NMR spectroscopy and mass spectrometry and their cytotoxicity was assessed against A549 cells. A similar dependence on the spacer length and the toxicity has been found for N3 functionalized complexes as recently reported for their C5' counterparts. A remarkable cytotoxic complex 22, carrying a dodecylene spacer at position N3 with a bis-quinoline metal chelate moiety, with an IC(50) value of 3.4 ± 1.6 μM, has been identified. Addition of a 100-fold excess of thymidine did not statistically reduce the observed cytotoxicity of all complexes. Cellular uptake studies of complex 22 have been performed by fluorescent microscopy, showing that compound 22 was clearly internalized into A549 cells. Temperature dependent uptake studies, blocking experiments with thymidine, and endosomal co-localization suggest that uptake of 22 occurs via passive diffusion and endocytosis.