Effect of zero-till, snow trapping, and fertilization on spring wheat and winter wheat (1983-84)

Non-Peer Reviewe

Effect of stubble height and source, rate, time, and method of application of N on yield of spring and winter wheat grown under zero-till

Non-Peer Reviewe

Interaction of snow management × N & P fertilizer and their effect on spring wheat yields

Non-Peer Reviewe

Levinson's Theorem for Dirac Particles

Levinson's theorem for Dirac particles constraints the sum of the phase shifts at threshold by the total number of bound states of the Dirac equation. Recently, a stronger version of Levinson's theorem has been proven in which the value of the positive- and negative-energy phase shifts are separately constrained by the number of bound states of an appropriate set of Schr\"odinger-like equations. In this work we elaborate on these ideas and show that the stronger form of Levinson's theorem relates the individual phase shifts directly to the number of bound states of the Dirac equation having an even or odd number of nodes. We use a mean-field approximation to Walecka's scalar-vector model to illustrate this stronger form of Levinson's theorem. We show that the assignment of bound states to a particular phase shift should be done, not on the basis of the sign of the bound-state energy, but rather, in terms of the nodal structure (even/odd number of nodes) of the bound state.Comment: Latex with Revtex, 7 postscript figures (available from the author), SCRI-06109

First normal stress difference and crystallization in a dense sheared granular fluid

The first normal stress difference (${\mathcal N}_1$) and the microstructure in a dense sheared granular fluid of smooth inelastic hard-disks are probed using event-driven simulations. While the anisotropy in the second moment of fluctuation velocity, which is a Burnett-order effect, is known to be the progenitor of normal stress differences in {\it dilute} granular fluids, we show here that the collisional anisotropies are responsible for the normal stress behaviour in the {\it dense} limit. As in the elastic hard-sphere fluids, ${\mathcal N}_1$ remains {\it positive} (if the stress is defined in the {\it compressive} sense) for dilute and moderately dense flows, but becomes {\it negative} above a critical density, depending on the restitution coefficient. This sign-reversal of ${\mathcal N}_1$ occurs due to the {\it microstructural} reorganization of the particles, which can be correlated with a preferred value of the {\it average} collision angle $\theta_{av}=\pi/4 \pm \pi/2$ in the direction opposing the shear. We also report on the shear-induced {\it crystal}-formation, signalling the onset of fluid-solid coexistence in dense granular fluids. Different approaches to take into account the normal stress differences are discussed in the framework of the relaxation-type rheological models.Comment: 21 pages, 13 figure