Photoelastic stress separation along lines of symmetry

By utilizing the compatibility and equilibrium field equations a differential relationship for the variation of the separated stress components along a line of symmetry is derived. This is in terms of only normal-incidence isochromatic data and does not use any isoclinic data. Its application to two problems using experimental data is demonstrated.     

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.     

General Relativistic Shock Waves that Extend the Oppenheimer-Snyder Model

In earlier work we constructed a class of spherically symmetric, fluid dynamical shock waves that satisfy the Einstein equations of general relativity. These shock waves extend the celebrated Oppenheimer-Snyder result to the case of non-zero pressure. Our shock waves are determined by a system of ordinary differential equations that describe the matching of a Friedmann-Robertson-Walker metric (a cosmological model for the expanding universe) to an Oppenheimer-Tolman metric (a model for the interior of a star) across a shock interface. In this paper we derive an alternate version of these ordinary differential equations, which are used to demonstrate that our theory generates a large class of physically meaningful (Lax-admissible) outgoing shock waves that model blast waves in a general relativistic setting. We also obtain formulas for the shock speed and other important quantities that evolve according to the equations. The resulting formulas are important for the numerical simulation of these solutions. (Accepted January 19, 1996)     

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.