233 research outputs found
An extension of Fourier analysis for the n-torus in the magnetic field and its application to spectral analysis of the magnetic Laplacian
We solved the Schr{\"o}dinger equation for a particle in a uniform magnetic
field in the n-dimensional torus. We obtained a complete set of solutions for a
broad class of problems; the torus T^n = R^n / {\Lambda} is defined as a
quotient of the Euclidean space R^n by an arbitrary n-dimensional lattice
{\Lambda}. The lattice is not necessary either cubic or rectangular. The
magnetic field is also arbitrary. However, we restrict ourselves within
potential-free problems; the Schr{\"o}dinger operator is assumed to be the
Laplace operator defined with the covariant derivative. We defined an algebra
that characterizes the symmetry of the Laplacian and named it the magnetic
algebra. We proved that the space of functions on which the Laplacian acts is
an irreducible representation space of the magnetic algebra. In this sense the
magnetic algebra completely characterizes the quantum mechanics in the magnetic
torus. We developed a new method for Fourier analysis for the magnetic torus
and used it to solve the eigenvalue problem of the Laplacian. All the
eigenfunctions are given in explicit forms.Comment: 32 pages, LaTeX, minor corrections are mad
Magnetic translation groups in an n-dimensional torus
A charged particle in a uniform magnetic field in a two-dimensional torus has
a discrete noncommutative translation symmetry instead of a continuous
commutative translation symmetry. We study topology and symmetry of a particle
in a magnetic field in a torus of arbitrary dimensions. The magnetic
translation group (MTG) is defined as a group of translations that leave the
gauge field invariant. We show that the MTG on an n-dimensional torus is
isomorphic to a central extension of a cyclic group Z_{nu_1} x ... x
Z_{nu_{2l}} x T^m by U(1) with 2l+m=n. We construct and classify irreducible
unitary representations of the MTG on a three-torus and apply the
representation theory to three examples. We shortly describe a representation
theory for a general n-torus. The MTG on an n-torus can be regarded as a
generalization of the so-called noncommutative torus.Comment: 29 pages, LaTeX2e, title changed, re-organized, to be published in
Journal of Mathematical Physic
Neel probability and spin correlations in some nonmagnetic and nondegenerate states of hexanuclear antiferromagnetic ring Fe6: Application of algebraic combinatorics to finite Heisenberg spin systems
The spin correlations \omega^z_r, r=1,2,3, and the probability p_N$ of
finding a system in the Neel state for the antiferromagnetic ring Fe(III)6 (the
so-called `small ferric wheel') are calculated. States with magnetization M=0,
total spin 0<=S<=15 and labeled by two (out of four) one-dimensional
irreducible representations (irreps) of the point symmetry group D_6 are taken
into account. This choice follows from importance of these irreps in analyzing
low-lying states in each S-multiplet. Taking into account the Clebsch--Gordan
coefficients for coupling total spins of sublattices (SA=SB=15/2) the global
Neel probability p*_N can be determined. Dependencies of these quantities on
state energy (per bond and in the units of exchange integral J) and the total
spin S are analyzed. Providing we have determined p_N(S) etc. for other
antiferromagnetic rings (Fe10, for instance) we could try to approximate
results for the largest synthesized ferric wheel Fe18. Since thermodynamic
properties of Fe6 have been investigated recently, in the present
considerations they are not discussed, but only used to verify obtained values
of eigenenergies. Numerical results re calculated with high precision using two
main tools: (i) thorough analysis of symmetry properties including methods of
algebraic combinatorics and (ii) multiple precision arithmetic library GMP. The
system considered yields more than 45 thousands basic states (the so-called
Ising configurations), but application of the method proposed reduces this
problem to 20-dimensional eigenproblem for the ground state (S=0). The largest
eigenproblem has to be solved for S=4; its dimension is 60. These two facts
(high precision and small resultant eigenproblems) confirm efficiency and
usefulness of such an approach, so it is briefly discussed here.Comment: 13 pages, 7 figs, 5 tabs, revtex
Geometric entropy, area, and strong subadditivity
The trace over the degrees of freedom located in a subset of the space
transforms the vacuum state into a density matrix with non zero entropy. This
geometric entropy is believed to be deeply related to the entropy of black
holes. Indeed, previous calculations in the context of quantum field theory,
where the result is actually ultraviolet divergent, have shown that the
geometric entropy is proportional to the area for a very special type of
subsets. In this work we show that the area law follows in general from simple
considerations based on quantum mechanics and relativity. An essential
ingredient of our approach is the strong subadditive property of the quantum
mechanical entropy.Comment: Published versio
Analysis of Agglomerative Clustering
The diameter -clustering problem is the problem of partitioning a finite
subset of into subsets called clusters such that the maximum
diameter of the clusters is minimized. One early clustering algorithm that
computes a hierarchy of approximate solutions to this problem (for all values
of ) is the agglomerative clustering algorithm with the complete linkage
strategy. For decades, this algorithm has been widely used by practitioners.
However, it is not well studied theoretically. In this paper, we analyze the
agglomerative complete linkage clustering algorithm. Assuming that the
dimension is a constant, we show that for any the solution computed by
this algorithm is an -approximation to the diameter -clustering
problem. Our analysis does not only hold for the Euclidean distance but for any
metric that is based on a norm. Furthermore, we analyze the closely related
-center and discrete -center problem. For the corresponding agglomerative
algorithms, we deduce an approximation factor of as well.Comment: A preliminary version of this article appeared in Proceedings of the
28th International Symposium on Theoretical Aspects of Computer Science
(STACS '11), March 2011, pp. 308-319. This article also appeared in
Algorithmica. The final publication is available at
http://link.springer.com/article/10.1007/s00453-012-9717-
On morphological hierarchical representations for image processing and spatial data clustering
Hierarchical data representations in the context of classi cation and data
clustering were put forward during the fties. Recently, hierarchical image
representations have gained renewed interest for segmentation purposes. In this
paper, we briefly survey fundamental results on hierarchical clustering and
then detail recent paradigms developed for the hierarchical representation of
images in the framework of mathematical morphology: constrained connectivity
and ultrametric watersheds. Constrained connectivity can be viewed as a way to
constrain an initial hierarchy in such a way that a set of desired constraints
are satis ed. The framework of ultrametric watersheds provides a generic scheme
for computing any hierarchical connected clustering, in particular when such a
hierarchy is constrained. The suitability of this framework for solving
practical problems is illustrated with applications in remote sensing
The WSO, a world-class observatory for the ultraviolet
The World Space Observatory is an unconventional space project proceeding via distributed studies. The present design, verified for feasibilty, consists of a 1.7-meter telescope operating at the second Largangian point of the Earth-Sun system. The focal plane instruments consist of three UV spectrometers covering the spectral band from Lyman alpha to the atmospheric cutoff with R~55,000 and offering long-slit capability over the same band with R~1,000. In addition, a number of UV and optical imagers view adjacent fields to that sampled by the spectrometers. Their performance compares well with that of HST/ACS and the spectral capabilities of WSO rival those of HST/COS
Custom Made Candy Plug for Distal False Lumen Occlusion in Aortic Dissection: International Experience
Objective: To evaluate early and midterm outcomes of the Candy Plug (CP) technique for distal false lumen (FL) occlusion in thoracic endovascular aortic repair for aortic dissection (AD) in a more real world cohort of patients from an international multicentre registry. Methods: A multicentre retrospective study was conducted of all consecutive patients from the contributing centres with subacute and chronic AD treated with the CP technique from October 2013 to April 2020 at 18 centres. Results: A custom made CP was used in 155 patients (92 males, mean age 62 ± 11 years). Fourteen (9%) presented with ruptured false lumen aneurysms. Technical success was achieved in all patients (100%). Clinical success was achieved in 138 patients (89%). The median hospital stay was 7 days (1 â 77). The 30 day mortality rate was 3% (n = 5). Stroke occurred in four patients (3%). Spinal cord ischaemia occurred in three patients (2%). The 30 day computed tomography angiogram (CTA) confirmed successful CP placement at the intended level in all patients. Early complete FL occlusion was achieved in 120 patients (77%). Early (30 day) CP related re-intervention was required in four patients (3%). The early (30 day) stent graft related re-intervention rate was 8% (n = 12). Follow up CTA was available in 142 patients (92%), with a median follow up of 23 months (6 â 87). Aneurysmal regression was achieved in 68 of 142 patients (47%); the aneurysm diameter remained stable in 69 of 142 patients (49%) and increased in five of 142 patients (4%). A higher rate of early FL occlusion was detected in the largest volume centre patients (50 [88%] vs. 70 [71%] from other centres; p = .019). No other differences in outcome were identified regarding volume of cases or learning curve. Conclusion: This international CP technique experience confirmed its feasibility and low mortality and morbidity rates. Aortic remodelling and false lumen thrombosis rates were high and support the concept of distal FL occlusion in AD using the CP technique
Implementation of U.K. Earth system models for CMIP6
We describe the scientific and technical implementation of two models for a core set of
experiments contributing to the sixth phase of the Coupled Model Intercomparison Project (CMIP6).
The models used are the physical atmosphere-land-ocean-sea ice model HadGEM3-GC3.1 and the
Earth system model UKESM1 which adds a carbon-nitrogen cycle and atmospheric chemistry to
HadGEM3-GC3.1. The model results are constrained by the external boundary conditions (forcing data)
and initial conditions.We outline the scientific rationale and assumptions made in specifying these.
Notable details of the implementation include an ozone redistribution scheme for prescribed ozone
simulations (HadGEM3-GC3.1) to avoid inconsistencies with the model's thermal tropopause, and land use
change in dynamic vegetation simulations (UKESM1) whose influence will be subject to potential biases in
the simulation of background natural vegetation.We discuss the implications of these decisions for
interpretation of the simulation results. These simulations are expensive in terms of human and CPU
resources and will underpin many further experiments; we describe some of the technical steps taken to
ensure their scientific robustness and reproducibility
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