6,035 research outputs found
On the total curvature and extrinsic area growth of surfaces with tamed second fundamental form
In this paper we show that a complete and non-compact surface immersed in the
Euclidean space with quadratic extrinsic area growth has finite total curvature
provided the surface has tamed second fundamental form and admits total
curvature. In such a case we obtain as well a generalized Chern-Osserman
inequality. In the particular case of a surface of nonnegative curvature, we
prove that the surface is diffeomorphic to the Euclidean plane if the surface
has tamed second fundamental form, and that the surface is isometric to the
Euclidean plane if the surface has strongly tamed second fundamental form. In
the last part of the paper we characterize the fundamental tone of any
submanifold of tamed second fundamental form immersed in an ambient space with
a pole and quadratic decay of the radial sectional curvatures.Comment: 19 pages. Title changed and several improvement of the main theorems
are done. arXiv admin note: text overlap with arXiv:0805.0323 by other
author
Probing Brownstein-Moffat Gravity via Numerical Simulations
In the standard scenario of the Newtonian gravity, a late-type galaxy (i.e.,
a spiral galaxy) is well described by a disk and a bulge embedded in a halo
mainly composed by dark matter. In Brownstein-Moffat gravity, there is a claim
that late-type galaxy systems would not need to have halos, avoiding as a
result the dark matter problem, i.e., a modified gravity (non-Newtonian) would
account for the galactic structure with no need of dark matter. In the present
paper, we probe this claim via numerical simulations. Instead of using a
"static galaxy," where the centrifugal equilibrium is usually adopted, we probe
the Brownstein-Moffat gravity dynamically via numerical -body simulations.Comment: 33 pages and 14 figures - To appear in The Astrophysical Journa
Equivalence of Statistical Mechanical Ensembles for Non-Critical Quantum Systems
We consider the problem of whether the canonical and microcanonical ensembles
are locally equivalent for short-ranged quantum Hamiltonians of spins
arranged on a -dimensional lattices. For any temperature for which the
system has a finite correlation length, we prove that the canonical and
microcanonical state are approximately equal on regions containing up to
spins. The proof rests on a variant of the Berry--Esseen
theorem for quantum lattice systems and ideas from quantum information theory
Exponential Quantum Speed-ups are Generic
A central problem in quantum computation is to understand which quantum
circuits are useful for exponential speed-ups over classical computation. We
address this question in the setting of query complexity and show that for
almost any sufficiently long quantum circuit one can construct a black-box
problem which is solved by the circuit with a constant number of quantum
queries, but which requires exponentially many classical queries, even if the
classical machine has the ability to postselect.
We prove the result in two steps. In the first, we show that almost any
element of an approximate unitary 3-design is useful to solve a certain
black-box problem efficiently. The problem is based on a recent oracle
construction of Aaronson and gives an exponential separation between quantum
and classical bounded-error with postselection query complexities.
In the second step, which may be of independent interest, we prove that
linear-sized random quantum circuits give an approximate unitary 3-design. The
key ingredient in the proof is a technique from quantum many-body theory to
lower bound the spectral gap of local quantum Hamiltonians.Comment: 24 pages. v2 minor correction
Entanglement area law from specific heat capacity
We study the scaling of entanglement in low-energy states of quantum
many-body models on lattices of arbitrary dimensions. We allow for unbounded
Hamiltonians such that systems with bosonic degrees of freedom are included. We
show that if at low enough temperatures the specific heat capacity of the model
decays exponentially with inverse temperature, the entanglement in every
low-energy state satisfies an area law (with a logarithmic correction). This
behaviour of the heat capacity is typically observed in gapped systems.
Assuming merely that the low-temperature specific heat decays polynomially with
temperature, we find a subvolume scaling of entanglement. Our results give
experimentally verifiable conditions for area laws, show that they are a
generic property of low-energy states of matter, and, to the best of our
knowledge, constitute the first proof of an area law for unbounded Hamiltonians
beyond those that are integrable.Comment: v3 now featuring bosonic system
ELT teachers' stories of resilience (summary version)
© 2020 The Author(s).This article is a summary of the research project we were funded by the British Council to conduct between 2017 and 2019. It focuses on the construction and analysis of six early career ELT teachers' stories focusing on how they fostered resilience in their workplace.Peer reviewe
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