Large Time Asymptotics of Fundamental Solution for the Diffusion Equation in Periodic Medium and Its Application to Estimates in the Theory of Averaging

The diffusion equation is considered in an infinite 1-periodic medium. For its fundamental solution we find approximations at large values of time t. Precision of approximations has pointwise and integral estimates of orders O(t(-d+j+1)/2) and O(t(-j+1)/2), j=0,1,…, respectively. Approximations are constructed based on the known fundamental solution of the averaged equation with constant coefficients, its derivatives, and solutions of a family of auxiliary problems on the periodicity cell. The family of problems on the cell is generated recurrently. These results are used for construction of approximations of the operator exponential of the diffusion equation with precision estimates in operator norms in Lp-spaces, 1≤p≤∞. For the analogous equation in an ε-periodic medium (here ε is a small parameter) we obtain approximations of the operator exponential in Lp-operator norms for a fixed time with precision of order O(εn), n=1,2,….

Majorana fermion wavefunctions in carbon nanotubes and carbynes

Electron properties of semiconducting zigzag carbon nanotubes (CNTs) can be described by two uncoupled Dirac equations of dimension (1+1) for the particle with nonzero mass. The solutions of these equations are two charge-neutral Majorana fields. An analogous equation is obtained for the carbon chains. We use the approach, wherein wavefunction of charged particle is represented as the production of a rapidly oscillating exponent and the slowly varying function amplitude depending on the longitudinal coordinate.

Reexamining the Vertical Structure of Tangential Winds in Tropical Cyclones: Observations and Theory

A few commonly held beliefs regarding the vertical structure of tropical cyclones drawn from prior studies, both observational and theoretical, are examined in this study. One of these beliefs is that the outward slope of the radius of maximum winds (RMW) is a function of the size of the RMW. Another belief is that the outward slope of the RMW is also a function of the intensity of the storm. Specifically, Shea and Gray found that the RMW becomes increasingly vertical with increasing intensity and decreasing radius. The third belief evaluated here is that the RMW is a surface of constant absolute angular momentum M. These three conventional wisdoms of vertical structure are revisited with a dataset of three-dimensional Doppler wind analyses, comprising seven hurricanes on 17 different days. Azimuthal mean tangential winds are calculated for each storm, and the slopes of the RMW and M surfaces are objectively determined. The outward slope of the RMW is shown to increase with radius, which supports prior studies. In contrast to prior results, no relationship is found between the slope of the RMW and intensity. It is shown that the RMW is indeed closely approximated by an M surface for the majority of storms. However, there is a small but systematic tendency for M to decrease upward along the RMW. Utilizing Emanuel’s analytical hurricane model, a new equation is derived for the slope of the RMW in radius–pressure space. This predicts a linear increase of slope with radius and essentially no dependence of slope on intensity. An exactly analogous equation can be derived in log-pressure height coordinates, and a numerical solution yields the same conclusions in geometric height coordinates. These conclusions are further supported by the results of simulations utilizing Emanuel’s simple, time-dependent, axisymmetric hurricane model. As both the model and the analytical theory are governed by the dual constraints of thermal wind balance and slantwise moist neutrality, it is demonstrated that it is these two assumptions that require the slope of the RMW to be a function of its size but not of the intensity of the storm. Finally, it is shown that within the context of Emanuel’s theory, the RMW must very closely approximate an M surface through most of the depth of the vortex.

Study of the ratcheting by the indentation fatigue method with a flat cylindrical indenter: Part I. Experimental study

Generally, ratcheting is studied on round specimens under tension–compression tests with a nonzero mean load. This work explored the possibility of studying ratcheting by indentation fatigue with a flat cylindrical indenter. In the experiment, emphasis was concentrated on the influence of maximum indentation load (Pmax.), indentation load variance (ΔP = Pmax − Pmin) and frequency of cycling (f) on the indentation depth–cycle curves. The experimental results showed that the indentation depth–cycle curves are analogous to the conventional strain–cycle curve of uniaxial fatigue testing, which has a primary stage of decaying indentation depth per cycle followed by a secondary stage of nearly constant rate of indentation depth per cycle. It was found that the steady-state indentation depth per cycle is an approximate linear function of maximum indentation load (Pmax) and indentation load variance (ΔP = Pmax − Pmin) in the log–log grid. This relationship can be given with a power-law expression as an analogous equation of steady-state ratcheting rate. Further study showed that the influence of frequency of cycling on the steady state indentation depth per cycle can be ignored when the frequency of cycling exceeds a certain value. Finally, comparison was made between the conventional uniaxial fatigue test and indentation fatigue test for the steady-state stage. It was shown that the conventional uniaxial fatigue parameters can be obtained by the indentation fatigue method.

Forward scattering of light, X-rays and neutrons

The central points of this paper can now be summarized. We consider here, for simplicity only, vanishing particle concentration. In equilibrium sedimentation equation (6) applies. The density increment is a measurable quantity. It can either be introduced into equation (6) to calculate M2, or it can be analysed by equations (7) and (8) to provide additional information on specific volumes and solute interactions.Light scattering is determined by the analogous equation (20). The refractive index increment is also experimentally accessible and its structure (not considered here) is similar to that of the density increment. Small angle X-ray scattering is determined by equation (31) and the electron density increment which appears in this equation cannot be directly determined by experiment. Yet it can be obtained in straightforward fashion from the mass density increment, by equation (34). Similarly, in the case of neutron scattering (equation (38)), the scattering length density increment is obtained from the mass density increment by equation (40), or it may now be directly evaluated by neutron interferometry.