Continuity and differentiability of Nemytskii operators on the Hardy space\mathcal{H}^{1,1} (T^1 )

Let denote the Hardy space of real-valued functions on the unit circle with weak derivatives in the usual real Hardy space . It is shown that when the weak derivative of a locally Lipschitz continuous functionf has bounded variation on compact sets the Nemytskii operatorF, defined byF(u)=f·u, maps continuously into itself. A further condition sufficient for the continuous Fréchet differentiability ofF is then added.     

Chirped area coupled resonator optical waveguide gyroscope

We study the transmission of an optical field through a rotating coupled resonator optical waveguide (CROW) in which the size of the ring resonators changes from one ring to the next. We focus on symmetric integer wavelength chirps of the circumference of the rings relative to the central ring in the array. The transfer matrix method is used to obtain the transmission as a function of the inertial rotation rate Ω resulting from the Sagnac effect. Chirping increases the slope of the oscillations in the transmission as a function of Ω, which can be exploited to further enhance the rotation sensitivity beyond that of a CROW with uniform resonators.     

Standing Waves on an Infinitely Deep Perfect Fluid Under Gravity

The existence of two-dimensional standing waves on the surface of an infinitely deep perfect fluid under gravity is established. When formulated as a second-order equation for a real-valued function w on the 2-torus and a positive parameter μ, the problem is fully nonlinear (the highest order x-derivative appears in the nonlinear term but not in the linearization at 0) and completely resonant (there are infinitely many linearly independent eigenmodes of the linearization at 0 for all rational values of the parameter μ). Moreover, for any prescribed order of accuracy there exists an explicit approximate solution of the nonlinear problem in the form of a trigonometric polynomial. Using a Nash-Moser method to seek solutions of the nonlinear problem as perturbations of the approximate solutions, the existence question can be reduced to one of estimating the inverses of linearized operators at non-zero points. After changing coordinates these operators become first order and non-local in space and second order in time. After further changes of variables the main parts become diagonal with constant coefficients and the remainder is regularizing, or quasi-one-dimensional in the sense of [22]. The operator can then be inverted for two reasons. First, the explicit formula for the approximate solution means that, restricted to the infinite-dimensional kernel of the linearization at zero, the inverse exists and can be estimated. Second, the small-divisor problems that arise on the complement of this kernel can be overcome by considering only particular parameter values selected according to their Diophantine properties. A parameter-dependent version of the Nash-Moser implicit function theorem now yields the existence of a set of unimodal standing waves on flows of infinite depth, corresponding to a set of values of the parameter μ>1 which is dense at 1. Unimodal means that the term of smallest order in the amplitude is cos x cos t, which is one of many eigenfunctions of the completely resonant linearized problem.     

Experimental study of multiple interior impacts

In this paper, experiments involving interior impacts on soft, lead targets are described. The term “interior impact” refers to the impact of a projectile at the bottom of a predrilled hole in the target. It is known that impact of low-velocity projectiles into such soft targets results in plastic flow and cavities similar to those obtained in high-velocity impact of hard targets. Therefore, comparatively simple rifle-propelled projectile impacts on soft targets may yield useful information for high-speed hard-target impacts. Double impacts with a short-time interval, on the order of microseconds, between projectile arrivals were conducted. The leading projectile creates a plastic flow in the target which partially seals the entrance hole. The trailing projectile must first force open the passage hole before achieving additional penetration. The total penetration was measured for different time intervals between the projectiles. A method was developed to obtain double impacts at the same point. The method utilizes a duplex round, which is made up of two separate projectiles fired from the same cartridge. With this round, a series of impact experiments was conducted. It was learned that occlusion, or the close up of the hole, is negligible for nearly simultaneous projectiles (less than 10 μs between impacts), and greatest with about 80 to 100 μs between impacts. The projectile velocity was measured by the use of properly placed photodiodes.     

Solitary waves with surface tension I: Trajectories homoclinic to periodic orbits in four dimensions

Communicated by the Editor     

Nonlinear elliptic eigenvalue problems on an infinite strip — Global theory of bifurcation and asymptotic bifurcation

Mathematische Annalen -     

Evaluation of screen-printed gold on low-temperature co-fired ceramic as a substrate for the immobilization of electrochemical immunoassays

Screen-printed gold (SPG, Dupont gold conductor 5734) on low-temperature co-fired ceramic (LTCC) materials (Dupont dielectric tape 951, mostly composed of alumina and silica) has been demonstrated to be a substrate for electrochemical enzyme-linked immunosorbant assays. The effect of two different cleaning treatments and the extent of nonspecific adsorption on the SPG/LTCC and plain LTCC surfaces were also evaluated. LTCC materials hold promise for constructing a new generation of devices for microelectrochemical sensing and assays. Facile fabrication in three dimensions with integrated conducting elements makes them attractive. A standard sandwich immunoassay for a model analyte, mouse IgG, was used to evaluate the LTCC materials. After the assembly of components onto chips of SPG/LTCC and plain LTCC, p-aminophenol that was generated enzymatically by the enzyme label was detected electrochemically with a separate glassy carbon electrode. Cleaning SPG/LTCC with a piranha solution (7:1 vol/vol of concentrated H2SO4/30% H2O2), traditionally used for other gold surfaces prior to SAM assembly, resulted in a notable decrease in assay signal and an increase in nonspecific adsorption when compared to cleaning with water alone. Assay components assemble specifically on plain LTCC, with only a small percent attributed to NSA. Environmental scanning electron microscopy, energy-dispersive X-ray spectroscopy, and X-ray photoelectron spectroscopy reveal the tremendous chemical heterogeneity and complexity of both SPG/LTCC and plain LTCC surfaces and aid in the explanation of assay results. A 10% acetate Tween bovine serum albumin solution and an ethanolic solution of 4 mM 1-butanol eliminate assay signals originating from plain LTCC. The outcomes of these studies can now be used to achieve miniaturized electrochemical immunoassays on LTCC materials where both plain and SPG surfaces are present.     

The Regularity and Local Bifurcation of¶Steady Periodic Water Waves

Steady periodic water waves on infinite depth, satisfying exactly the kinematic and dynamic boundary conditions on the free surface, with or without surface tension, are given by solutions of a rather tidy nonlinear pseudo-differential operator equation for a 2π-periodic function of a real variable. Being an Euler-Lagrange equation, this formulation has the advantage of gradient structure, but is complicated by the fact that it involves a non-local operator, namely the Hilbert transform, and is quasi-linear. This paper is a mathematical study of the equation in question. First it is shown that its W ^1,2 solutions are real analytic. Then bifurcation theory for gradient operators is used to prove the existence of (non-zero) small amplitude waves near every eigenvalue (irrespective of multiplicity) of the linearised problem. Finally it is shown that when surface tension is absent there are no sub-harmonic bifurcations or turning points at the outset of the branches of Stokes waves which bifurcate from the trivial solution. Accepted: December 6, 1999