On the locus formed by the maximum heights of projectile motion with air resistance

We present an analysis on the geometrical place formed by the set of maxima of the trajectories of a projectile launched in a media with linear drag. Such a place, the locus of apexes, is written in term of the Lambert $W$ function in polar coordinates, confirming the special role played by this function in the problem. In order to characterize the locus, a study of its curvature is presented in two parameterizations, in terms of the launch angle and in the polar one. The angles of maximum curvature are compared with other important angles in the projectile problem. As an addendum, we find that the synchronous curve in this problem is a circle as in the drag-free case.Comment: 7 pages, 6 color eps figures. Synchronous curve added. Typos and style corrected

Analytical study of electro-osmosis modulated capillary peristaltic hemodynamics

A mathematical model is developed to analyse electro-kinetic effects on unsteady peristaltic transport of blood in cylindrical vessels of finite length. The Newtonian viscous model is adopted. The analysis is restricted under Debye-Hückel linearization (i.e. wall zeta potential less than or equal to 25mV is sufficiently small). The transformed, non-dimensional conservation equations are derived via lubrication theory and long wavelength and the resulting linearized boundary value problem is solved exactly. The case of a thin electric double layer (i.e. where only slip electro-osmotic velocity considered) is retrieved as a particular case of the present model. The response in pumping characteristics (axial velocity, pressure gradient or difference, volumetric flow rate, local wall shear stress) to the influence of electro-osmotic effect (inverse Debye length) and Helmholtz-Smoluchowski velocity is elaborated in detail. Visualization of trapping phenomenon is also included and the bolus dynamics evolution with electro-kinetic effects examined. A comparative study of train wave propagation and single wave propagation is presented under the effects of thickness of EDL and external electric field. The study is relevant to electrophoresis in haemotology, electrohydrodynamic therapy and biomimetic electro-osmotic pumps

Unambiguous comparison of the states of multiple quantum systems

We consider N quantum systems initially prepared in pure states and address the problem of unambiguously comparing them. One may ask whether or not all $N$ systems are in the same state. Alternatively, one may ask whether or not the states of all N systems are different. We investigate the possibility of unambiguously obtaining this kind of information. It is found that some unambiguous comparison tasks are possible only when certain linear independence conditions are satisfied. We also obtain measurement strategies for certain comparison tasks which are optimal under a broad range of circumstances, in particular when the states are completely unknown. Such strategies, which we call universal comparison strategies, are found to have intriguing connections with the problem of quantifying the distinguishability of a set of quantum states and also with unresolved conjectures in linear algebra. We finally investigate a potential generalisation of unambiguous state comparison, which we term unambiguous overlap filtering.Comment: 20 pages, no figure