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

Exact on-event expressions for discrete potential systems

The properties of systems composed of atoms interacting though discrete potentials are dictated by a series of events which occur between pairs of atoms. There are only four basic event types for pairwise discrete potentials and the square-well/shoulder systems studied here exhibit them all. Closed analytical expressions are derived for the on-event kinetic energy distribution functions for an atom, which are distinct from the Maxwell-Boltzmann distribution function. Exact expressions are derived that directly relate the pressure and temperature of equilibrium discrete potential systems to the rates of each type of event. The pressure can be determined from knowledge of only the rate of core and bounce events. The temperature is given by the ratio of the number of bounce events to the number of disassociation/association events. All these expressions are validated with event-driven molecular dynamics simulations and agree with the data within the statistical precision of the simulations

A variational approach to moment-closure approximations for the kinetics of biomolecular reaction networks

Approximate solutions of the chemical master equation and the chemical Fokker-Planck equation are an important tool in the analysis of biomolecular reaction networks. Previous studies have highlighted a number of problems with the moment-closure approach used to obtain such approximations, calling it an ad-hoc method. In this article, we give a new variational derivation of moment-closure equations which provides us with an intuitive understanding of their properties and failure modes and allows us to correct some of these problems. We use mixtures of product-Poisson distributions to obtain a flexible parametric family which solves the commonly observed problem of divergences at low system sizes. We also extend the recently introduced entropic matching approach to arbitrary ansatz distributions and Markov processes, demonstrating that it is a special case of variational moment closure. This provides us with a particularly principled approximation method. Finally, we extend the above approaches to cover the approximation of multi-time joint distributions, resulting in a viable alternative to process-level approximations which are often intractable.Comment: Minor changes and clarifications; corrected some typo

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

HI Imaging of LGS 3 and an Apparently Interacting High-Velocity Cloud

We present a 93' by 93' map of the area near the Local Group dwarf galaxy LGS 3, centered on an HI cloud 30' away from the galaxy. Previous authors associated this cloud with LGS 3 but relied on observations made with a 36' beam. Our high-resolution (3.4'), wide-field Arecibo observations of the region reveal that the HI cloud is distinct from the galaxy and suggest an interaction between the two. We point out faint emission features in the map that may be gas that has been tidally removed from the HI cloud by LGS 3. We also derive the rotation curve of the cloud and find that it is in solid-body rotation out to a radius of 10', beyond which the rotation velocity begins to decline. Assuming a spherical geometry for the cloud, the implied mass is 2.8 x 10^7 (d/Mpc) M_{Sun}, where d is the distance in Mpc. The observed HI mass is 5.5 x 10^6 (d/Mpc)^2 M_{Sun}, implying that the cloud is dark-matter dominated unless its distance is at least 1.9 Mpc. We propose that the cloud is a high-velocity cloud that is undergoing a tidal interaction with LGS 3 and therefore is located roughly 700 kpc away from the Milky Way. The cloud then contains a total mass of ~2.0 x 10^7 M_{Sun}, 82% of which consists of dark matter.Comment: 5 pages, 2 color figures. Accepted for publication in ApJ Letter