10,254 research outputs found
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 , and
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
- …