16,669 research outputs found
High-Precision Entropy Values for Spanning Trees in Lattices
Shrock and Wu have given numerical values for the exponential growth rate of
the number of spanning trees in Euclidean lattices. We give a new technique for
numerical evaluation that gives much more precise values, together with
rigorous bounds on the accuracy. In particular, the new values resolve one of
their questions.Comment: 7 pages. Revision mentions alternative approach. Title changed
slightly. 2nd revision corrects first displayed equatio
Critical percolation of free product of groups
In this article we study percolation on the Cayley graph of a free product of
groups.
The critical probability of a free product of groups
is found as a solution of an equation involving only the expected subcritical
cluster size of factor groups . For finite groups these
equations are polynomial and can be explicitly written down. The expected
subcritical cluster size of the free product is also found in terms of the
subcritical cluster sizes of the factors. In particular, we prove that
for the Cayley graph of the modular group (with the
standard generators) is , the unique root of the polynomial
in the interval .
In the case when groups can be "well approximated" by a sequence of
quotient groups, we show that the critical probabilities of the free product of
these approximations converge to the critical probability of
and the speed of convergence is exponential. Thus for residually finite groups,
for example, one can restrict oneself to the case when each free factor is
finite.
We show that the critical point, introduced by Schonmann,
of the free product is just the minimum of for the factors
Is the Sun Lighter than the Earth? Isotopic CO in the Photosphere, Viewed through the Lens of 3D Spectrum Synthesis
We consider the formation of solar infrared (2-6 micron) rovibrational bands
of carbon monoxide (CO) in CO5BOLD 3D convection models, with the aim to refine
abundances of the heavy isotopes of carbon (13C) and oxygen (18O,17O), to
compare with direct capture measurements of solar wind light ions by the
Genesis Discovery Mission. We find that previous, mainly 1D, analyses were
systematically biased toward lower isotopic ratios (e.g., R23= 12C/13C),
suggesting an isotopically "heavy" Sun contrary to accepted fractionation
processes thought to have operated in the primitive solar nebula. The new 3D
ratios for 13C and 18O are: R23= 91.4 +/- 1.3 (Rsun= 89.2); and R68= 511 +/- 10
(Rsun= 499), where the uncertainties are 1 sigma and "optimistic." We also
obtained R67= 2738 +/- 118 (Rsun= 2632), but we caution that the observed
12C17O features are extremely weak. The new solar ratios for the oxygen
isotopes fall between the terrestrial values and those reported by Genesis
(R68= 530, R6= 2798), although including both within 2 sigma error flags, and
go in the direction favoring recent theories for the oxygen isotope composition
of Ca-Al inclusions (CAI) in primitive meteorites. While not a major focus of
this work, we derive an oxygen abundance of 603 +/- 9 ppm (relative to
hydrogen; 8.78 on the logarithmic H= 12 scale). That the Sun likely is lighter
than the Earth, isotopically speaking, removes the necessity to invoke exotic
fractionation processes during the early construction of the inner solar
system
Microwave properties of DyBa_2Cu_3O_(7-x) monodomains and related compounds in magnetic fields
We present a microwave characterization of a DyBaCuO
single domain, grown by the top-seeded melt-textured technique. We report the
(a,b) plane field-induced surface resistance, , at 48.3 GHz,
measured by means of a cylindrical metal cavity in the end-wall-replacement
configuration. Changes in the cavity quality factor Q against the applied
magnetic field yield at fixed temperatures. The temperature
range [70 K ; T_c] was explored. The magnetic field 0.8 T was
applied along the c axis. The field dependence of does not
exhibit the steep, step-like increase at low fields typical of weak-links. This
result indicates the single-domain character of the sample under investigation.
exhibits a nearly square-root dependence on H, as expected for
fluxon motion. From the analysis of the data in terms of motion of Abrikosov
vortices we estimate the temperature dependences of the London penetration
depth and the vortex viscosity , and their zero-temperature
values 165 nm and 3 10 Nsm, which are
found in excellent agreement with reported data in YBaCuO
single crystals. Comparison of microwave properties with those of related
samples indicate the need for reporting data as a function of T/T_c in order to
obtain universal laws.Comment: 6 pages, 4 figures, LaTeX, submitted to Journal of Applied Physic
G-Brownian Motion as Rough Paths and Differential Equations Driven by G-Brownian Motion
The present paper is devoted to the study of sample paths of G-Brownian
motion and stochastic differential equations (SDEs) driven by G-Brownian motion
from the view of rough path theory. As the starting point, we show that
quasi-surely, sample paths of G-Brownian motion can be enhanced to the second
level in a canonical way so that they become geometric rough paths of roughness
2 < p < 3. This result enables us to introduce the notion of rough differential
equations (RDEs) driven by G-Brownian motion in the pathwise sense under the
general framework of rough paths. Next we establish the fundamental relation
between SDEs and RDEs driven by G-Brownian motion. As an application, we
introduce the notion of SDEs on a differentiable manifold driven by GBrownian
motion and construct solutions from the RDE point of view by using pathwise
localization technique. This is the starting point of introducing G-Brownian
motion on a Riemannian manifold, based on the idea of Eells-Elworthy-Malliavin.
The last part of this paper is devoted to such construction for a wide and
interesting class of G-functions whose invariant group is the orthogonal group.
We also develop the Euler-Maruyama approximation for SDEs driven by G-Brownian
motion of independent interest
Slow movement of a random walk on the range of a random walk in the presence of an external field
In this article, a localisation result is proved for the biased random walk
on the range of a simple random walk in high dimensions (d \geq 5). This
demonstrates that, unlike in the supercritical percolation setting, a slowdown
effect occurs as soon a non-trivial bias is introduced. The proof applies a
decomposition of the underlying simple random walk path at its cut-times to
relate the associated biased random walk to a one-dimensional random walk in a
random environment in Sinai's regime
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