2,019 research outputs found
Representations of the discrete inhomogeneous Lorentz group and Dirac wave equation on the lattice
We propose the fundamental and two dimensional representation of the Lorentz
groups on a (3+1)-dimensional hypercubic lattice, from which representations of
higher dimensions can be constructed. For the unitary representation of the
discrete translation group we use the kernel of the Fourier transform. From the
Dirac representation of the Lorentz group (including reflections) we derive in
a natural way the wave equation on the lattice for spin 1/2 particles. Finally
the induced representation of the discrete inhomogeneous Lorentz group is
constructed by standard methods and its connection with the continuous case is
discussed.Comment: LaTeX, 20 pages, 1 eps figure, uses iopconf.sty (late submission
Surface embedding, topology and dualization for spin networks
Spin networks are graphs derived from 3nj symbols of angular momentum. The
surface embedding, the topology and dualization of these networks are
considered. Embeddings into compact surfaces include the orientable sphere S^2
and the torus T, and the not orientable projective space P^2 and Klein's bottle
K. Two families of 3nj graphs admit embeddings of minimal genus into S^2 and
P^2. Their dual 2-skeletons are shown to be triangulations of these surfaces.Comment: LaTeX 17 pages, 6 eps figures (late submission to arxiv.org
Finite-Dimensional Calculus
We discuss topics related to finite-dimensional calculus in the context of
finite-dimensional quantum mechanics. The truncated Heisenberg-Weyl algebra is
called a TAA algebra after Tekin, Aydin, and Arik who formulated it in terms of
orthofermions. It is shown how to use a matrix approach to implement analytic
representations of the Heisenberg-Weyl algebra in univariate and multivariate
settings. We provide examples for the univariate case. Krawtchouk polynomials
are presented in detail, including a review of Krawtchouk polynomials that
illustrates some curious properties of the Heisenberg-Weyl algebra, as well as
presenting an approach to computing Krawtchouk expansions. From a mathematical
perspective, we are providing indications as to how to implement in finite
terms Rota's "finite operator calculus".Comment: 26 pages. Added material on Krawtchouk polynomials. Additional
references include
Energetics and stability of dangling-bond silicon wires on H passivated Si(100)
We evaluate the electronic, geometric and energetic properties of quasi 1-D
wires formed by dangling bonds on Si(100)-H (2 x 1). The calculations are
performed with density functional theory (DFT). Infinite wires are found to be
insulating and Peierls distorted, however finite wires develop localized
electronic states that can be of great use for molecular-based devices. The
ground state solution of finite wires does not correspond to a geometrical
distortion but rather to an antiferromagnetic ordering. For the stability of
wires, the presence of abundant H atoms in nearby Si atoms can be a problem. We
have evaluated the energy barriers for intradimer and intrarow diffusion
finding all of them about 1 eV or larger, even in the case where a H impurity
is already sitting on the wire. These results are encouraging for using
dangling-bond wires in future devices.Comment: 8 pages, 6 figure
The discretised harmonic oscillator: Mathieu functions and a new class of generalised Hermite polynomials
We present a general, asymptotical solution for the discretised harmonic
oscillator. The corresponding Schr\"odinger equation is canonically conjugate
to the Mathieu differential equation, the Schr\"odinger equation of the quantum
pendulum. Thus, in addition to giving an explicit solution for the Hamiltonian
of an isolated Josephon junction or a superconducting single-electron
transistor (SSET), we obtain an asymptotical representation of Mathieu
functions. We solve the discretised harmonic oscillator by transforming the
infinite-dimensional matrix-eigenvalue problem into an infinite set of
algebraic equations which are later shown to be satisfied by the obtained
solution. The proposed ansatz defines a new class of generalised Hermite
polynomials which are explicit functions of the coupling parameter and tend to
ordinary Hermite polynomials in the limit of vanishing coupling constant. The
polynomials become orthogonal as parts of the eigenvectors of a Hermitian
matrix and, consequently, the exponential part of the solution can not be
excluded. We have conjectured the general structure of the solution, both with
respect to the quantum number and the order of the expansion. An explicit proof
is given for the three leading orders of the asymptotical solution and we
sketch a proof for the asymptotical convergence of eigenvectors with respect to
norm. From a more practical point of view, we can estimate the required effort
for improving the known solution and the accuracy of the eigenvectors. The
applied method can be generalised in order to accommodate several variables.Comment: 18 pages, ReVTeX, the final version with rather general expression
Inelastic current-voltage characteristics of atomic and molecular junctions
We report first-principles calculations of the inelastic current-voltage
(I-V) characteristics of a gold point contact and a molecular junction in the
nonresonant regime. Discontinuities in the I-V curves appear in correspondence
to the normal modes of the structures. Due to the quasi-one-dimensional nature
of these systems, specific modes with large longitudinal component dominate the
inelastic I-V curves. In the case of the gold point contact, our results are in
good agreement with recent experimental data. For the molecular junction, we
find that the inelastic I-V curves are quite sensitive to the structure of the
contact between the molecule and the electrodes thus providing a powerful tool
to extract the bonding geometry in molecular wires.Comment: 4 pages, 3 figure
Automated Plankton Classification With a Dynamic Optimization and Adaptation Cycle
With recent advances in Machine Learning techniques based on Deep Neural Networks (DNNs), automated plankton image classification is becoming increasingly popular within the marine ecological sciences. Yet, while the most advanced methods can achieve human-level performance on the classification of everyday images, plankton image data possess properties that frequently require a final manual validation step. On the one hand, this is due to morphological properties manifesting in high intra-class and low inter-class variability, and, on the other hand is due to spatial-temporal changes in the composition and structure of the plankton community. Composition changes enforce a frequent updating of the classifier model via training with new user-generated training datasets. Here, we present a Dynamic Optimization Cycle (DOC), a processing pipeline that systematizes and streamlines the model adaptation process via an automatic updating of the training dataset based on manual-validation results. We find that frequent adaptation using the DOC pipeline yields strong maintenance of performance with respect to precision, recall and prediction of community composition, compared to more limited adaptation schemes. The DOC is therefore particularly useful when analyzing plankton at novel locations or time periods, where community differences are likely to occur. In order to enable an easy implementation of the DOC pipeline, we provide an end-to-end application with graphical user interface, as well as an initial dataset of training images. The DOC pipeline thus allows for high-throughput plankton classification and quick and systematized model adaptation, thus providing the means for highly-accelerated plankton analysis
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