484 research outputs found
The Exponential Map for the Conformal Group 0(2,4)
We present a general method to obtain a closed, finite formula for the
exponential map from the Lie algebra to the Lie group, for the defining
representation of the orthogonal groups. Our method is based on the
Hamilton-Cayley theorem and some special properties of the generators of the
orthogonal group, and is also independent of the metric. We present an explicit
formula for the exponential of generators of the groups, with , in particular we are dealing with the conformal group , which
is homomorphic to the group. This result is needed in the
generalization of U(1) gauge transformations to spin gauge transformations,
where the exponential plays an essential role. We also present some new
expressions for the coefficients of the secular equation of a matrix.Comment: 16pages,plain-TeX,(corrected TeX
The exponential map for the unitary group SU(2,2)
In this article we extend our previous results for the orthogonal group,
, to its homomorphic group . Here we present a closed, finite
formula for the exponential of a traceless matrix, which can be
viewed as the generator (Lie algebra elements) of the group. We apply
this result to the group, which Lie algebra can be represented by the
Dirac matrices, and discuss how the exponential map for can be
written by means of the Dirac matrices.Comment: 10 page
Relativistic Chasles' theorem and the conjugacy classes of the inhomogeneous Lorentz group
This work is devoted to the relativistic generalization of Chasles' theorem,
namely to the proof that every proper orthochronous isometry of Minkowski
spacetime, which sends some point to its chronological future, is generated
through the frame displacement of an observer which moves with constant
acceleration and constant angular velocity. The acceleration and angular
velocity can be chosen either aligned or perpendicular, and in the latter case
the angular velocity can be chosen equal or smaller than than the acceleration.
We start reviewing the classical Euler's and Chasles' theorems both in the Lie
algebra and group versions. We recall the relativistic generalization of
Euler's theorem and observe that every (infinitesimal) transformation can be
recovered from information of algebraic and geometric type, the former being
identified with the conjugacy class and the latter with some additional
geometric ingredients (the screw axis in the usual non-relativistic version).
Then the proper orthochronous inhomogeneous Lorentz Lie group is studied in
detail. We prove its exponentiality and identify a causal semigroup and the
corresponding Lie cone. Through the identification of new Ad-invariants we
classify the conjugacy classes, and show that those which admit a causal
representative have special physical significance. These results imply a
classification of the inequivalent Killing vector fields of Minkowski spacetime
which we express through simple representatives. Finally, we arrive at the
mentioned generalization of Chasles' theorem.Comment: Latex2e, 49 pages. v2: few typos correcte
Ultra--Planck Scattering in D=3 Gravity Theories
We obtain the high energy, small angle, 2-particle gravitational scattering
amplitudes in topologically massive gravity (TMG) and its two non-dynamical
constituents, Einstein and Chern--Simons gravity. We use 't Hooft's approach,
formally equivalent to a leading order eikonal approximation: one of the
particles is taken to scatter through the classical spacetime generated by the
other, which is idealized to be lightlike. The required geometries are derived
in all three models; in particular, we thereby provide the first explicit
asymptotically flat solution generated by a localized source in TMG. In
contrast to =4, the metrics are not uniquely specified, at least by naive
asymptotic requirements -- an indeterminacy mirrored in the scattering
amplitudes. The eikonal approach does provide a unique choice, however. We also
discuss the discontinuities that arise upon taking the limits, at the level of
the solutions, from TMG to its constituents, and compare with the analogous
topologically massive vector gauge field models.Comment: 20 pages, preprint BRX TH--337, DAMTP R93/5, ADP-93-204/M1
Data-driven simulation and characterisation of gold nanoparticle melting
The simulation and analysis of the thermal stability of nanoparticles, a stepping stone towards their application in technological devices, require fast and accurate force fields, in conjunction with effective characterisation methods. In this work, we develop efficient, transferable, and interpretable machine learning force fields for gold nanoparticles based on data gathered from Density Functional Theory calculations. We use them to investigate the thermodynamic stability of gold nanoparticles of different sizes (1 to 6 nm), containing up to 6266 atoms, concerning a solid-liquid phase change through molecular dynamics simulations. We predict nanoparticle melting temperatures in good agreement with available experimental data. Furthermore, we characterize the solid-liquid phase change mechanism employing an unsupervised learning scheme to categorize local atomic environments. We thus provide a data-driven definition of liquid atomic arrangements in the inner and surface regions of a nanoparticle and employ it to show that melting initiates at the outer layers
Lightlike infinity in GCA models of Spacetime
This paper discusses a 7 dimensional conformal geometric algebra model for
spacetime based on the notion that spacelike and timelike infinities are
distinct. I show how naturally of the dimensions represents the lightlike
infinity and appears redundant in computations, yet usefull in interpretationComment: 12 page
CinÊtica da produção de gås in vitro de espÊcies comumente encontradas em pastagem nativa do bioma Pampa.
Uso de extrato aquoso e Ăłleo de eucaliptos no controle de fungos fitopatogĂŞnicos in vitro.
Resumo
DNA Electrophoretic Migration Patterns Change after Exposure of Jurkat Cells to a Single Intense Nanosecond Electric Pulse
Intense nanosecond pulsed electric fields (nsPEFs) interact with cellular membranes and intracellular structures. Investigating how cells respond to nanosecond pulses is essential for a) development of biomedical applications of nsPEFs, including cancer therapy, and b) better understanding of the mechanisms underlying such bioelectrical effects. In this work, we explored relatively mild exposure conditions to provide insight into weak, reversible effects, laying a foundation for a better understanding of the interaction mechanisms and kinetics underlying nsPEF bio-effects. In particular, we report changes in the nucleus of Jurkat cells (human lymphoblastoid T cells) exposed to single pulses of 60 ns duration and 1.0, 1.5 and 2.5 MV/m amplitudes, which do not affect cell growth and viability. A dose-dependent reduction in alkaline comet-assayed DNA migration is observed immediately after nsPEF exposure, accompanied by permeabilization of the plasma membrane (YO-PRO-1 uptake). Comet assay profiles return to normal within 60 minutes after pulse delivery at the highest pulse amplitude tested, indicating that our exposure protocol affects the nucleus, modifying DNA electrophoretic migration patterns
Revisiting special relativity: A natural algebraic alternative to Minkowski spacetime
Minkowski famously introduced the concept of a space-time continuum in 1908,
merging the three dimensions of space with an imaginary time dimension , with the unit imaginary producing the correct spacetime distance , and the results of Einstein's then recently developed theory of special
relativity, thus providing an explanation for Einstein's theory in terms of the
structure of space and time. As an alternative to a planar Minkowski space-time
of two space dimensions and one time dimension, we replace the unit imaginary , with the Clifford bivector for the plane
that also squares to minus one, but which can be included without the addition
of an extra dimension, as it is an integral part of the real Cartesian plane
with the orthonormal basis and . We find that with this model of
planar spacetime, using a two-dimensional Clifford multivector, the spacetime
metric and the Lorentz transformations follow immediately as properties of the
algebra. This also leads to momentum and energy being represented as components
of a multivector and we give a new efficient derivation of Compton's scattering
formula, and a simple formulation of Dirac's and Maxwell's equations. Based on
the mathematical structure of the multivector, we produce a semi-classical
model of massive particles, which can then be viewed as the origin of the
Minkowski spacetime structure and thus a deeper explanation for relativistic
effects. We also find a new perspective on the nature of time, which is now
given a precise mathematical definition as the bivector of the plane.Comment: 29 pages, 2 figure
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