Solving simple quaternionic differential equations

The renewed interest in investigating quaternionic quantum mechanics, in particular tunneling effects, and the recent results on quaternionic differential operators motivate the study of resolution methods for quaternionic differential equations. In this paper, by using the real matrix representation of left/right acting quaternionic operators, we prove existence and uniqueness for quaternionic initial value problems, discuss the reduction of order for quaternionic homogeneous differential equations and extend to the non-commutative case the method of variation of parameters. We also show that the standard Wronskian cannot uniquely be extended to the quaternionic case. Nevertheless, the absolute value of the complex Wronskian admits a non-commutative extension for quaternionic functions of one real variable. Linear dependence and independence of solutions of homogeneous (right) H-linear differential equations is then related to this new functional. Our discussion is, for simplicity, presented for quaternionic second order differential equations. This involves no loss of generality. Definitions and results can be readily extended to the n-order case.Comment: 9 pages, AMS-Te

A Semiotic Analysis on the Perceived Meanings of Coca Cola “Anthem” Video Commercial

This study observes the perceived meanings produced by young adult (18 to 30 years old) and older (50 to 65 years old) respondents as respondents from different age group can produce different perceived meanings from each other. The writer's finding is that in perceiving, young adult respondents tend to emphasize on Coca Cola's emotional roles. On the other hand, the older respondents emphasize on Coca Cola's physical roles

Right eigenvalue equation in quaternionic quantum mechanics

We study the right eigenvalue equation for quaternionic and complex linear matrix operators defined in n-dimensional quaternionic vector spaces. For quaternionic linear operators the eigenvalue spectrum consists of n complex values. For these operators we give a necessary and sufficient condition for the diagonalization of their quaternionic matrix representations. Our discussion is also extended to complex linear operators, whose spectrum is characterized by 2n complex eigenvalues. We show that a consistent analysis of the eigenvalue problem for complex linear operators requires the choice of a complex geometry in defining inner products. Finally, we introduce some examples of the left eigenvalue equations and highlight the main difficulties in their solution.Comment: 24 pages, AMS-Te

Graphene tests of Klein phenomena

Graphene is characterized by chiral electronic excitations. As such it provides a perfect testing ground for the production of Klein pairs (electron/holes). If confirmed, the standard results for barrier phenomena must be reconsidered with, as a byproduct, the accumulation within the barrier of holes.Comment: 8 page

Quaternionic eigenvalue problem

We discuss the (right) eigenvalue equation for $\mathbb{H}$, $\mathbb{C}$ and $\mathbb{R}$ linear quaternionic operators. The possibility to introduce an isomorphism between these operators and real/complex matrices allows to translate the quaternionic problem into an {\em equivalent} real or complex counterpart. Interesting applications are found in solving differential equations within quaternionic formulations of quantum mechanics.Comment: 13 pages, AMS-Te

Dense Gas in the Milky Way

We present a study of dense gas emission in the Milky Way in order to serve as a basis for comparison with extragalactic results. This study combines new observations of HCN, CS, and CO in individual GMCs and in the Milky Way plane with published studies of emission from these molecules in the inner 500 pc of the Milky Way. We find a strong trend in the fraction of emission from dense gas tracers as a function of location in the Milky Way: in the bulge, I_{HCN}/I_{CO} = 0.081 \pm 0.004, in the plane, I_{HCN}/I_{CO} = 0.026 \pm 0.008 on average, and over the full extent of nearby GMCs, I_{HCN}/I_{CO} = 0.014 \pm 0.020. Similar trends are seen in I_{CS}/I_{CO}. The low intensities of the HCN and CS emission in the plane suggests that these lines are produced by gas at moderate densities; they are thus not like the emission produced by the dense, pc-scale star forming cores in nearby GMCs. The contrast between the bulge and disk ratios in the Milky Way is likely to be caused by a combination of higher kinetic temperatures as well as a higher dense gas fraction in the bulge of the Milky Way.Comment: 34 pages LaTeX, AASTEX macros, includes 11 postscript figures. To appear in ApJ 478, March 199

A New Phase Time Formula for Opaque Barrier Tunneling

After a brief review of the derivation of the standard phase time formula, based on the use of the stationary phase method, we propose, in the opaque limit, an alternative method to calculate the phase time. The new formula for the phase time is in excellent agreement with the numerical simulations and shows that for wave packets whose upper limit of the momentum distribution is very close to the barrier height, the transit time is proportional to the barrier width.Comment: 9 pages, 2 figure

Dirac Equation Studies in the Tunnelling Energy Zone

We investigate the tunnelling zone V0 < E < V0+m for a one-dimensional potential within the Dirac equation. We find the appearance of superluminal transit times akin to the Hartman effect.Comment: 12 pages, 4 figure