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

Effect of "dipolar-biasing" on the tunability of tunneling magnetoresistance in transition metal oxide systems

We observe an unusual tunneling magnetoresistance (TMR) phenomenon in a composite of La$_{2/3}$Sr$_{1/3}$MnO$_{3}$ with CoFe$_{2}$O$_{4}$ where the TMR versus applied magnetic field loop suggests a "negative coercive field". Tracing its origin back to a "dipolar-biasing" of La$_{2/3}$Sr$_{1/3}$MnO$_{3}$ by CoFe$_{2}$O$_{4}$, we show that the TMR of even a single composite can be tuned continuously so that the resistance peak or the highest sensitivity of the TMR can be positioned anywhere on the magnetic field axis with a suitable magnetic history of the sample. This phenomenon of an unprecedented tunability of the TMR should be present in general in all such composites.Comment: Copyright (2012) American Institute of Physics. This article may be downloaded for personal use only. Any other use requires prior permission of the author and the American Institute of Physic

Temperature Dependence of Magneto Current in Spin Valve Transistor: A phenomenological Study

The temperature dependence of magneto current in the spin spin valve transistor system is theoretically explored based on phenomenological model. We find that the collector current strongly depends on the relative orientation of magnetic moment of ferromagnetic metals due to spin mixing effect. For example, the collector current is decreasing in the parallel case with increasing temperature, and it is increasing in anti-parallel configuration. We then obtain decreasing magneto current with increasing temperature. The result accords with the experimental data in qualitative manner. This phenomenological model calculations suggest that spin mixing effect may play an important role in the spin valve transistor system at finite temperature.Comment: 8 pages and 4 figure

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