343 research outputs found
Semilinear mixed problems on Hilbert complexes and their numerical approximation
Arnold, Falk, and Winther recently showed [Bull. Amer. Math. Soc. 47 (2010),
281-354] that linear, mixed variational problems, and their numerical
approximation by mixed finite element methods, can be studied using the
powerful, abstract language of Hilbert complexes. In another recent article
[arXiv:1005.4455], we extended the Arnold-Falk-Winther framework by analyzing
variational crimes (a la Strang) on Hilbert complexes. In particular, this gave
a treatment of finite element exterior calculus on manifolds, generalizing
techniques from surface finite element methods and recovering earlier a priori
estimates for the Laplace-Beltrami operator on 2- and 3-surfaces, due to Dziuk
[Lecture Notes in Math., vol. 1357 (1988), 142-155] and later Demlow [SIAM J.
Numer. Anal., 47 (2009), 805-827], as special cases. In the present article, we
extend the Hilbert complex framework in a second distinct direction: to the
study of semilinear mixed problems. We do this, first, by introducing an
operator-theoretic reformulation of the linear mixed problem, so that the
semilinear problem can be expressed as an abstract Hammerstein equation. This
allows us to obtain, for semilinear problems, a priori solution estimates and
error estimates that reduce to the Arnold-Falk-Winther results in the linear
case. We also consider the impact of variational crimes, extending the results
of our previous article to these semilinear problems. As an immediate
application, this new framework allows for mixed finite element methods to be
applied to semilinear problems on surfaces.Comment: 22 pages; v2: major revision, particularly sharpening of error
estimates in Section
Gibbs' paradox and black-hole entropy
In statistical mechanics Gibbs' paradox is avoided if the particles of a gas
are assumed to be indistinguishable. The resulting entropy then agrees with the
empirically tested thermodynamic entropy up to a term proportional to the
logarithm of the particle number. We discuss here how analogous situations
arise in the statistical foundation of black-hole entropy. Depending on the
underlying approach to quantum gravity, the fundamental objects to be counted
have to be assumed indistinguishable or not in order to arrive at the
Bekenstein--Hawking entropy. We also show that the logarithmic corrections to
this entropy, including their signs, can be understood along the lines of
standard statistical mechanics. We illustrate the general concepts within the
area quantization model of Bekenstein and Mukhanov.Comment: Contribution to Mashhoon festschrift, 13 pages, 4 figure
Observational Constraints of Modified Chaplygin Gas in Loop Quantum Cosmology
We have considered the FRW universe in loop quantum cosmology (LQC) model
filled with the dark matter (perfect fluid with negligible pressure) and the
modified Chaplygin gas (MCG) type dark energy. We present the Hubble parameter
in terms of the observable parameters , and
with the redshift and the other parameters like , , and .
From Stern data set (12 points), we have obtained the bounds of the arbitrary
parameters by minimizing the test. The best-fit values of the
parameters are obtained by 66%, 90% and 99% confidence levels. Next due to
joint analysis with BAO and CMB observations, we have also obtained the bounds
of the parameters () by fixing some other parameters and .
From the best fit of distance modulus for our theoretical MCG model in
LQC, we concluded that our model is in agreement with the union2 sample data.Comment: 14 pages, 10 figures, Accepted in EPJC. arXiv admin note: text
overlap with arXiv:astro-ph/0311622 by other author
Running coupling: Does the coupling between dark energy and dark matter change sign during the cosmological evolution?
In this paper we put forward a running coupling scenario for describing the
interaction between dark energy and dark matter. The dark sector interaction in
our scenario is free of the assumption that the interaction term is
proportional to the Hubble expansion rate and the energy densities of dark
sectors. We only use a time-variable coupling (with the scale factor
of the universe) to characterize the interaction . We propose a
parametrization form for the running coupling in which the
early-time coupling is given by a constant , while today the coupling is
given by another constant, . For investigating the feature of the running
coupling, we employ three dark energy models, namely, the cosmological constant
model (), the constant model (), and the time-dependent
model (). We constrain the models with the current
observational data, including the type Ia supernova, the baryon acoustic
oscillation, the cosmic microwave background, the Hubble expansion rate, and
the X-ray gas mass fraction data. The fitting results indicate that a
time-varying vacuum scenario is favored, in which the coupling crosses
the noninteracting line () during the cosmological evolution and the sign
changes from negative to positive. The crossing of the noninteracting line
happens at around , and the crossing behavior is favored at about
1 confidence level. Our work implies that we should pay more attention
to the time-varying vacuum model and seriously consider the phenomenological
construction of a sign-changeable or oscillatory interaction between dark
sectors.Comment: 8 pages, 5 figures; refs added; to appear in EPJ
Physics, Topology, Logic and Computation: A Rosetta Stone
In physics, Feynman diagrams are used to reason about quantum processes. In
the 1980s, it became clear that underlying these diagrams is a powerful analogy
between quantum physics and topology: namely, a linear operator behaves very
much like a "cobordism". Similar diagrams can be used to reason about logic,
where they represent proofs, and computation, where they represent programs.
With the rise of interest in quantum cryptography and quantum computation, it
became clear that there is extensive network of analogies between physics,
topology, logic and computation. In this expository paper, we make some of
these analogies precise using the concept of "closed symmetric monoidal
category". We assume no prior knowledge of category theory, proof theory or
computer science.Comment: 73 pages, 8 encapsulated postscript figure
Optical Properties of III-Mn-V Ferromagnetic Semiconductors
We review the first decade of extensive optical studies of ferromagnetic,
III-Mn-V diluted magnetic semiconductors. Mn introduces holes and local moments
to the III-V host, which can result in carrier mediated ferromagnetism in these
disordered semiconductors. Spectroscopic experiments provide direct access to
the strength and nature of the exchange between holes and local moments; the
degree of itineracy of the carriers; and the evolution of the states at the
Fermi energy with doping. Taken together, diversity of optical methods reveal
that Mn is an unconventional dopant, in that the metal to insulator transition
is governed by the strength of the hybridization between Mn and its p-nictogen
neighbor. The interplay between the optical, electronic and magnetic properties
of III-Mn-V magnetic semiconductors is of fundamental interest and may enable
future spin-optoelectronic devices.Comment: Topical Revie
An Integrated TCGA Pan-Cancer Clinical Data Resource to Drive High-Quality Survival Outcome Analytics
For a decade, The Cancer Genome Atlas (TCGA) program collected clinicopathologic annotation data along with multi-platform molecular profiles of more than 11,000 human tumors across 33 different cancer types. TCGA clinical data contain key features representing the democratized nature of the data collection process. To ensure proper use of this large clinical dataset associated with genomic features, we developed a standardized dataset named the TCGA Pan-Cancer Clinical Data Resource (TCGA-CDR), which includes four major clinical outcome endpoints. In addition to detailing major challenges and statistical limitations encountered during the effort of integrating the acquired clinical data, we present a summary that includes endpoint usage recommendations for each cancer type. These TCGA-CDR findings appear to be consistent with cancer genomics studies independent of the TCGA effort and provide opportunities for investigating cancer biology using clinical correlates at an unprecedented scale. Analysis of clinicopathologic annotations for over 11,000 cancer patients in the TCGA program leads to the generation of TCGA Clinical Data Resource, which provides recommendations of clinical outcome endpoint usage for 33 cancer types
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