596 research outputs found
A literature survey of low-rank tensor approximation techniques
During the last years, low-rank tensor approximation has been established as
a new tool in scientific computing to address large-scale linear and
multilinear algebra problems, which would be intractable by classical
techniques. This survey attempts to give a literature overview of current
developments in this area, with an emphasis on function-related tensors
Analysis of the mean squared derivative cost function
In this paper, we investigate the mean squared derivative cost functions that
arise in various applications such as in motor control, biometrics and optimal
transport theory. We provide qualitative properties, explicit analytical
formulas and computational algorithms for the cost functions. We also perform
numerical simulations to illustrate the analytical results. In addition, as a
by-product of our analysis, we obtain an explicit formula for the inverse of a
Wronskian matrix that is of independent interest in linear algebra and
differential equations theory.Comment: 28 page
Kramers' law: Validity, derivations and generalisations
Kramers' law describes the mean transition time of an overdamped Brownian
particle between local minima in a potential landscape. We review different
approaches that have been followed to obtain a mathematically rigorous proof of
this formula. We also discuss some generalisations, and a case in which
Kramers' law is not valid. This review is written for both mathematicians and
theoretical physicists, and endeavours to link concepts and terminology from
both fields.Comment: 26 pages, 9 figure
On the Shape of Things: From holography to elastica
We explore the question of which shape a manifold is compelled to take when
immersed in another one, provided it must be the extremum of some functional.
We consider a family of functionals which depend quadratically on the extrinsic
curvatures and on projections of the ambient curvatures. These functionals
capture a number of physical setups ranging from holography to the study of
membranes and elastica. We present a detailed derivation of the equations of
motion, known as the shape equations, placing particular emphasis on the issue
of gauge freedom in the choice of normal frame. We apply these equations to the
particular case of holographic entanglement entropy for higher curvature three
dimensional gravity and find new classes of entangling curves. In particular,
we discuss the case of New Massive Gravity where we show that non-geodesic
entangling curves have always a smaller on-shell value of the entropy
functional. Then we apply this formalism to the computation of the entanglement
entropy for dual logarithmic CFTs. Nevertheless, the correct value for the
entanglement entropy is provided by geodesics. Then, we discuss the importance
of these equations in the context of classical elastica and comment on terms
that break gauge invariance.Comment: 54 pages, 8 figures. Significantly improved version, accepted for
publication in Annals of Physics. New section on logarithmic CFTs. Detailed
derivation of the shape equations added in appendix B. Typos corrected,
clarifications adde
Exact Entanglement in the Driven Quantum Symmetric Simple Exclusion Process
Entanglement properties of driven quantum systems can potentially differ from
the equilibrium situation due to long range coherences. We confirm this
observation by studying a suitable toy model for mesoscopic transport: the open
quantum symmetric simple exclusion process (QSSEP). We prove that the average
mutual information of the open QSSEP in the steady state satisfies a volume
law, and derive exact formulae for the mutual information between different
regions of the system. Exploiting the free probability structure of QSSEP, we
obtain these results by developing a new method to determine the eigenvalue
spectrum of sub-blocks of random matrices from their so-called local free
cumulants -- a mathematical result on its own with potential applications in
the theory of random matrices. As an illustration of this method, we show how
to compute expectation values of observables in systems satisfying the
Eigenstate Thermalization Hypothesis (ETH) from the local free cumulants.Comment: 6 pages main text, 5 pages supplemental materia
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