47,956 research outputs found
Strong laws of large numbers for sub-linear expectations
We investigate three kinds of strong laws of large numbers for capacities
with a new notion of independently and identically distributed (IID) random
variables for sub-linear expectations initiated by Peng. It turns out that
these theorems are natural and fairly neat extensions of the classical
Kolmogorov's strong law of large numbers to the case where probability measures
are no longer additive. An important feature of these strong laws of large
numbers is to provide a frequentist perspective on capacities.Comment: 10 page
Wormhole Effect in a Strong Topological Insulator
An infinitely thin solenoid carrying magnetic flux Phi (a `Dirac string')
inserted into an ordinary band insulator has no significant effect on the
spectrum of electrons. In a strong topological insulator, remarkably, such a
solenoid carries protected gapless one-dimensional fermionic modes when
Phi=hc/2e. These modes are spin-filtered and represent a distinct bulk
manifestation of the topologically non-trivial insulator. We establish this
`wormhole' effect by both general qualitative considerations and by numerical
calculations within a minimal lattice model. We also discuss the possibility of
experimental observation of a closely related effect in artificially engineered
nanostructures.Comment: 4 pages, 3 figures. For related work and info visit
http://www.physics.ubc.ca/~fran
Stability of Majorana Fermions in Proximity-Coupled Topological Insulator Nanowires
It has been shown previously that a finite-length topological insulator
nanowire, proximity-coupled to an ordinary bulk s-wave superconductor and
subject to a longitudinal applied magnetic field, realizes a one-dimensional
topological superconductor with an unpaired Majorana fermion (MF) localized at
each end of the nanowire. Here, we study the stability of these MFs with
respect to various perturbations that are likely to occur in a physical
realization of the proposed device. We show that the unpaired Majorana fermions
persist in this system for any value of the chemical potential inside the bulk
band gap of order 300 meV in BiSe by computing the Majorana number.
From this calculation, we also show that the unpaired Majorana fermions persist
when the magnetic flux through the nanowire cross-section deviates
significantly from half flux quantum. Lastly, we demonstrate that the unpaired
Majorana fermions persist in strongly disordered wires with fluctuations in the
on-site potential ranging in magnitude up to several times the size of the bulk
band gap. These results suggest this solid-state system should exhibit unpaired
Majorana fermions under accessible conditions likely important for experimental
study or future applications.Comment: 17 pages, 13 figure
An Invariance Principle of G-Brownian Motion for the Law of the Iterated Logarithm under G-expectation
The classical law of the iterated logarithm (LIL for short)as fundamental
limit theorems in probability theory play an important role in the development
of probability theory and its applications. Strassen (1964) extended LIL to
large classes of functional random variables, it is well known as the
invariance principle for LIL which provide an extremely powerful tool in
probability and statistical inference. But recently many phenomena show that
the linearity of probability is a limit for applications, for example in
finance, statistics. As while a nonlinear expectation--- G-expectation has
attracted extensive attentions of mathematicians and economists, more and more
people began to study the nature of the G-expectation space. A natural question
is: Can the classical invariance principle for LIL be generalized under
G-expectation space? This paper gives a positive answer. We present the
invariance principle of G-Brownian motion for the law of the iterated logarithm
under G-expectation
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