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The effect of more than one pair of degenerate vibrational modes on the energy levels and spectrum of a doubly degenerate electronic state is discussed. The dynamic Jahn-Teller effect for this case is treated by exact numerical methods and by approximation schemes. The e~ect .of quadrat!c term~ in the vibrational coordinates is also treated by these methods. The use of adiabatic surfaces m treatmg these problems is critically examined
Time-evolution of the Rule 150 cellular automaton activity from a Fibonacci iteration
The total activity of the single-seeded cellular rule 150 automaton does not
follow a one-step iteration like other elementary cellular automata, but can be
solved as a two-step vectorial, or string, iteration, which can be viewed as a
generalization of Fibonacci iteration generating the time series from a
sequence of vectors of increasing length. This allows to compute the total
activity time series more efficiently than by simulating the whole
spatio-temporal process, or even by using the closed expression.Comment: 4 pages (3 figs included
Masses, luminosities, and orbital coplanarities of the µ Orionis quadruple-star system from phases differential astrometry
μ Orionis was identified by spectroscopic studies as a quadruple-star system. Seventeen high-precision differential astrometry measurements of μ Ori have been collected by the Palomar High-precision Astrometric Search for Exoplanet Systems (PHASES). These show both the motion of the long-period binary orbit and short-period perturbations superimposed on that caused by each of the components in the long-period system being themselves binaries. The new measurements enable the orientations of the long-period binary and short-period subsystems to be determined. Recent theoretical work predicts the distribution of relative inclinations between inner and outer orbits of hierarchical systems to peak near 40 and 140 degrees. The degree of coplanarity of this complex system is determined, and the angle between the planes of the A–B and Aa–Ab orbits is found to be 136.7 ± 8.3 degrees, near the predicted distribution peak at 140 degrees; this result is discussed in the context of the handful of systems with established mutual inclinations. The system distance and masses for each component are obtained from a combined fit of the PHASES astrometry and archival radial velocity observations. The component masses have relative precisions of 5% (component Aa), 15% (Ab), and 1.4% (each of Ba and Bb). The median size of the minor axes of the uncertainty ellipses for the new measurements is 20 micro-arcseconds (μas). Updated orbits for δ Equulei, κ Pegasi, and V819 Herculis are also presented
A Fascinating Polynomial Sequence arising from an Electrostatics Problem on the Sphere
A positive unit point charge approaching from infinity a perfectly spherical
isolated conductor carrying a total charge of +1 will eventually cause a
negatively charged spherical cap to appear. The determination of the smallest
distance ( is the dimension of the unit sphere) from the point
charge to the sphere where still all of the sphere is positively charged is
known as Gonchar's problem. Using classical potential theory for the harmonic
case, we show that is equal to the largest positive zero of a
certain sequence of monic polynomials of degree with integer
coefficients which we call Gonchar polynomials. Rather surprisingly,
is the Golden ratio and the lesser known Plastic number. But Gonchar
polynomials have other interesting properties. We discuss their factorizations,
investigate their zeros and present some challenging conjectures.Comment: 12 pages, 6 figures, 1 tabl
Multiqubit symmetric states with high geometric entanglement
We propose a detailed study of the geometric entanglement properties of pure
symmetric N-qubit states, focusing more particularly on the identification of
symmetric states with a high geometric entanglement and how their entanglement
behaves asymptotically for large N. We show that much higher geometric
entanglement with improved asymptotical behavior can be obtained in comparison
with the highly entangled balanced Dicke states studied previously. We also
derive an upper bound for the geometric measure of entanglement of symmetric
states. The connection with the quantumness of a state is discussed
Random perfect lattices and the sphere packing problem
Motivated by the search for best lattice sphere packings in Euclidean spaces
of large dimensions we study randomly generated perfect lattices in moderately
large dimensions (up to d=19 included). Perfect lattices are relevant in the
solution of the problem of lattice sphere packing, because the best lattice
packing is a perfect lattice and because they can be generated easily by an
algorithm. Their number however grows super-exponentially with the dimension so
to get an idea of their properties we propose to study a randomized version of
the algorithm and to define a random ensemble with an effective temperature in
a way reminiscent of a Monte-Carlo simulation. We therefore study the
distribution of packing fractions and kissing numbers of these ensembles and
show how as the temperature is decreased the best know packers are easily
recovered. We find that, even at infinite temperature, the typical perfect
lattices are considerably denser than known families (like A_d and D_d) and we
propose two hypotheses between which we cannot distinguish in this paper: one
in which they improve Minkowsky's bound phi\sim 2^{-(0.84+-0.06) d}, and a
competitor, in which their packing fraction decreases super-exponentially,
namely phi\sim d^{-a d} but with a very small coefficient a=0.06+-0.04. We also
find properties of the random walk which are suggestive of a glassy system
already for moderately small dimensions. We also analyze local structure of
network of perfect lattices conjecturing that this is a scale-free network in
all dimensions with constant scaling exponent 2.6+-0.1.Comment: 19 pages, 22 figure
Masses, Luminosities, and Orbital Coplanarities of the mu Orionis Quadruple Star System from PHASES Differential Astrometry
mu Orionis was identified by spectroscopic studies as a quadruple star
system. Seventeen high precision differential astrometry measurements of mu Ori
have been collected by the Palomar High-precision Astrometric Search for
Exoplanet Systems (PHASES). These show both the motion of the long period
binary orbit and short period perturbations superimposed on that caused by each
of the components in the long period system being themselves binaries. The new
measurements enable the orientations of the long period binary and short period
subsystems to be determined. Recent theoretical work predicts the distribution
of relative inclinations between inner and outer orbits of hierarchical systems
to peak near 40 and 140 degrees. The degree of coplanarity of this complex
system is determined, and the angle between the planes of the A-B and Aa-Ab
orbits is found to be 136.7 +/- 8.3 degrees, near the predicted distribution
peak at 140 degrees; this result is discussed in the context of the handful of
systems with established mutual inclinations. The system distance and masses
for each component are obtained from a combined fit of the PHASES astrometry
and archival radial velocity observations. The component masses have relative
precisions of 5% (component Aa), 15% (Ab), and 1.4% (each of Ba and Bb). The
median size of the minor axes of the uncertainty ellipses for the new
measurements is 20 micro-arcseconds. Updated orbits for delta Equulei, kappa
Pegasi, and V819 Herculis are also presented.Comment: 12 Pages, Accepted for publication in A
Immunodetectable cyclin D 1 is associated with oestrogen receptor but not Ki67 in normal, cancerous and precancerous breast lesions
Cyclin D1 is associated with cell cycle regulation and has more recently been shown to stimulate the transcriptional functions of the oestrogen receptor (ER). Furthermore, in normal breast there is a negative association between expression of ER and the proliferation marker Ki67 indicating that either ER positive cells are non-dividing or that the receptor is down-regulated as cells enter cycle. This important relationship breaks down in many ER-positive cancers and precancerous breast lesions where the receptor is often detected on proliferating cells. The aims of the present study were to determine the interplay between ER, Ki67 and cyclin D 1 in individual cells within the spectrum of human breast lesions ranging from normal to invasive carcinoma by using dual staining immunofluorescence. We found that in normal breast there was a strong positive association between ER and cyclin D 1 expression. In contrast there was a strong negative association between cyclin D 1 and Ki67 expression. Similar findings were seen for the other precancerous and cancerous breast lesions. Thus immunodetectable cyclin D 1 within individual cells does not appear to be associated with cell cycle progression in the benign or malignant breast but instead may have important interactions with ER. © 2001 Cancer Research Campaign http://www.bjcancer.co
Entropy calculation for a toy black hole
In this note we carry out the counting of states for a black hole in loop
quantum gravity, however assuming an equidistant area spectrum. We find that
this toy-model is exactly solvable, and we show that its behavior is very
similar to that of the correct model. Thus this toy-model can be used as a nice
and simplifying `laboratory' for questions about the full theory.Comment: 18 pages, 4 figures. v2: Corrected mistake in bibliography, added
appendix with further result
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