On Rational Sets in Euclidean Spaces and Spheres

IFor a positive rational $l$, we define the concept of an $l$-elliptic and an $l$-hyperbolic rational set in a metric space. In this article we examine the existence of (i) dense and (ii) infinite $l$-hyperbolic and $l$-ellitpic rationals subsets of the real line and unit circle. For the case of a circle, we prove that the existence of such sets depends on the positivity of ranks of certain associated elliptic curves. We also determine the closures of such sets which are maximal in case they are not dense. In higher dimensions, we show the existence of $l$-ellitpic and $l$-hyperbolic rational infinite sets in unit spheres and Euclidean spaces for certain values of $l$ which satisfy a weaker condition regarding the existence of elements of order more than two, than the positivity of the ranks of the same associated elliptic curves. We also determine their closures. A subset $T$ of the $k$-dimensional unit sphere $S^k$ has an antipodal pair if both $x,-x\in T$ for some $x\in S^k$. In this article, we prove that there does not exist a dense rational set $T\subset S^2$ which has an antipodal pair by assuming Bombieri-Lang Conjecture for surfaces of general type. We actually show that the existence of such a dense rational set in $S^k$ is equivalent to the existence of a dense $2$-hyperbolic rational set in $S^k$ which is further equivalent to the existence of a dense 1-elliptic rational set in the Euclidean space $\mathbb{R}^k$.Comment: 20 page

The Contribution of Hot Electron Spin Polarization to the Magnetotransport in a Spin-Valve Transistor at Finite Temperatures

The effect of spin mixing due to thermal spin waves and temperature dependence of hot electron spin polarization to the collector current in a spin-valve transistor has been theoretically explored. We calculate the collector current as well as the temperature dependence of magnetocurrent at finite temperatures to investigate the relative importance of spin mixing and hot electron spin polarization. In this study the inelastic scattering events in ferromagnetic layers have been taken into account to explore our interests. The theoretical calculations suggest that the temperature dependence of hot electron spin polarization has substantial contribution to the magnetotransport in the spin-valve transistor.Comment: 8 pages and 6 figure