Dynamics of a pair of coupled nonlinear oscillators

A pair of coupled classical oscillators with a general potential and general form of coupling is investigated. For general potentials, the single-frequency solution is shown to be stable for small excitations. For special potentials, such system remains stable for an arbitrary excitation. In both cases, the stability does not depend on the form of coupling. Transition to the instability regime follows from the way how nonlinear potential entrains the energy transfer between the oscillators. Relation between the existence of multi-frequency quasi-periodic or periodic solutions and the instability of single-frequency ones is discussed.     

A Parabolic Spline on a Nonuniform Grid

We obtain estimates for errors of interpolation on a nonuniform grid for a parabolic nonperiodic spline of defect 1.     

Growth properties for solutions of differential equations defined in the disk

Summary We investigate the growth of solutions of a class of differential equations which are defined in the unit disk and for which the coefficients have slow growth in the unit disk as measured by the Nevanlinna characteristic. These results extend earlier work of S. Bank.     

Study of the reciprocal space image of the biaxial nematic phase in the plane of the amphiphilic bilayer

High resolution synchrotron X-ray measurements have been performed on the uniaxial and biaxial phases of the lyotropic mixture potassuim laurate, decanol and water. An elegant magnetic orientational procedure allows us to obtain the cut of the reciprocal space image of the biaxial phase at the plane perpendicular to the amphiphilic bilayer. The analysis of densitometric profiles of the diffracted bands indicate an anisotropic micellar correlation in the plane of the amphiphilic bilayer. It is suggested that an anisotropic distribution of decanol and potassium laurate in this plane could be responsible for this effect.     

Inertia-preserving secant updates

Corrections to the last two equations of Ref. 1 are given.     

Glass transitions in hydrogen-bonded polymer complexes

Mutual precipitates of poly (N, N-dimethyl acrylamide) and poly (4-hydroxystyrene) were collected from dioxane, methanol, or acetone. The glass transition (T_g) temperatures of the precipitates are higher than the weight-average values. Clear films cast from dimethylformamide solutions have lower T_g values. Complexation also occurred between poly (ethyl oxazoline) and poly (4-hydroxystyrene) in dioxane and between poly (vinyl pyrrolidone) and poly (4-hydroxystyrene) in methanol. Again, the glass transition temperatures of the precipitates are higher than the values for the blend films. The ΔC_p values associated with the glass transitions of the complexes are smaller than those of the blends having the same compositions. Negative excess heat capacities of mixing have been observed for several precipitates.     

Pit your wits against young minds!

This column is a place for those bits of contagious mathematics that travel from person to person in the community, because they are so elegant, suprising, or appealing that one has an urge to pass them on. Contributions are most welcome.     

Equivalence of u–p and ζ–ψ formulations of the time-dependent Navier–Stokes equations

This paper deals with the non-stationary incompressible Navier--Stokes equations for two-dimensional flows expressed in terms of the velocity and pressure and of the vorticity and streamfunction. The equivalence of the two formulations is demonstrated, both formally and rigorously, by virtue of a condition of compatibility between the boundary and initial values of the normal component of velocity. This condition is shown to be the only compatibility condition necessary to allow for solutions of a minimal regularity, namely H¹ for the velocity, as in most current numerical schemes relying on spatial discretizations of local type.