1,029 research outputs found
Multidisciplinary perspectives on Artificial Intelligence and the law
This open access book presents an interdisciplinary, multi-authored, edited collection of chapters on Artificial Intelligence (‘AI’) and the Law. AI technology has come to play a central role in the modern data economy. Through a combination of increased computing power, the growing availability of data and the advancement of algorithms, AI has now become an umbrella term for some of the most transformational technological breakthroughs of this age. The importance of AI stems from both the opportunities that it offers and the challenges that it entails. While AI applications hold the promise of economic growth and efficiency gains, they also create significant risks and uncertainty. The potential and perils of AI have thus come to dominate modern discussions of technology and ethics – and although AI was initially allowed to largely develop without guidelines or rules, few would deny that the law is set to play a fundamental role in shaping the future of AI. As the debate over AI is far from over, the need for rigorous analysis has never been greater. This book thus brings together contributors from different fields and backgrounds to explore how the law might provide answers to some of the most pressing questions raised by AI. An outcome of the Católica Research Centre for the Future of Law and its interdisciplinary working group on Law and Artificial Intelligence, it includes contributions by leading scholars in the fields of technology, ethics and the law.info:eu-repo/semantics/publishedVersio
Negative type and bi-lipschitz embeddings into Hilbert space
The usual theory of negative type (and -negative type) is heavily
dependent on an embedding result of Schoenberg, which states that a metric
space isometrically embeds in some Hilbert space if and only if it has
2-negative type. A generalisation of this embedding result to the setting of
bi-lipschitz embeddings was given by Linial, London and Rabinovich. In this
article we use this newer embedding result to define the concept of distorted
p-negative type and extend much of the known theory of p-negative type to the
setting of bi-lipschitz embeddings. In particular we show that a metric space
has -negative type with distortion , ) if and only if ) admits a bi-lipschitz embedding
into some Hilbert space with distortion at most . Analogues of strict
-negative type and polygonal equalities in this new setting are given and
systematically studied. Finally, we provide explicit examples of these concepts
in the bi-lipschitz setting for the bipartite graphs and the Hamming
cube
Sharp subelliptic estimates in the -Neumann problem via an uncertainty principle
A novel technique based on a "-uncertainty principle" is
introduced in the study of subellipticity of the -Neumann
problem. As an application, we determine the sharp order of subellipticity at
the origin for a class of dilation-invariant special domains in ambient
dimension .Comment: 37 page
Interpolating between Hausdorff and box dimension
Hausdorff and box dimension are two familiar notions of fractal dimension. Box dimension can be larger than Hausdorff dimension, because in the definition of box dimension, all sets in the cover have the same diameter, but for Hausdorff dimension there is no such restriction. This thesis focuses on a family of dimensions parameterised by θ ∈ (0,1), called the intermediate dimensions, which are defined by requiring that diam(U) ⩽ (diam(V))ᶿ for all sets U, V in the cover.
We begin by generalising the intermediate dimensions to allow for greater refinement in how the relative sizes of the covering sets are restricted. These new dimensions can recover the interpolation between Hausdorff and box dimension for compact sets whose intermediate dimensions do not tend to the Hausdorff dimension as θ → 0. We also use a Moran set construction to prove a necessary and sufficient condition, in terms of Dini derivatives, for a given function to be realised as the intermediate dimensions of a set.
We proceed to prove that the intermediate dimensions of limit sets of infinite conformal iterated function systems are given by the maximum of the Hausdorff dimension of the limit set and the intermediate dimensions of the set of fixed points of the contractions. This applies to sets defined using continued fraction expansions, and has applications to dimensions of projections, fractional Brownian images, and general Hölder images.
Finally, we determine a formula for the intermediate dimensions of all self-affine Bedford–McMullen carpets. The functions display features not witnessed in previous examples, such as having countably many phase transitions. We deduce that two carpets have equal intermediate dimensions if and only if the multifractal spectra of the corresponding uniform Bernoulli measures coincide. This shows that if two carpets are bi-Lipschitz equivalent then the multifractal spectra are equal."This work was supported by a Leverhulme Trust Research Project Grant (RPG-2019-034)." -- Fundin
A posteriori error control for a Discontinuous Galerkin approximation of a Keller-Segel model
We provide a posteriori error estimates for a discontinuous Galerkin scheme
for the parabolic-elliptic Keller-Segel system in 2 or 3 space dimensions. The
estimates are conditional, in the sense that an a posteriori computable
quantity needs to be small enough - which can be ensured by mesh refinement -
and optimal in the sense that the error estimator decays with the same order as
the error under mesh refinement. A specific feature of our error estimator is
that it can be used to prove existence of a weak solution up to a certain time
based on numerical results.Comment: 31 pages, 1 figure, 5 table
Quantum effects and novel physics in rotating frames
The birth of quantum physics and general relativity were two revolutions in physics. But a century later, scientists have not yet united the two theories. Attempts to combine them are mostly theoretical; controlled experiments have historically been neglected due to the comparative weakness of gravity and the corresponding precision or extreme scales assumed needed to test quantum gravity effects.
We take a new approach, inspired by Einstein’s equivalence of gravitational fields and accelerated frames. Non-inertial frames can be controlled in the lab, and allows us to experimentally test new frame-dependent effects and already-established quantum effects in new regimes. This frame-dependence is fundamentally interesting by itself, but also provides parallels to curved spacetime effects. To that effect, I have carried out experiments in rotating frames and shown new effects.
I have combined mechanical rotation with acoustics, sending sound waves through a rotating absorber. With this, I was the first to show experimental proof of the Zel’dovich effect: the amplification of waves carrying angular momentum by a rotating object. It is theorised the Zel’dovich effect should also generate electromagnetic waves out of the quantum vacuum, however the conditions are much harder to meet.
I have also done optics experiments to show how rotation can affect quantum entanglement. The Hong-Ou-Mandel effect was used as a witness for antisymmetric entanglement between photons. The symmetry of frequency entangled photon pairs can be manipulated by introducing path superpositions and controlling their phase difference. Through experiment I established that to witness antisymmetry with the Hong-Ou-Mandel effect it was much easier in the regime where the superposed paths had path length differences outwith the single-photon coherence length. Within a rotating frame, a rotation-dependent phase difference between counterpropagating beams of light appears, called the Sagnac effect. Combining Sagnac interferometers with a Hong-Ou-Mandel interferometer on a rotating platform, I have shown how rotation can control the entanglement symmetry of photon pairs.
The success of these experiments can be built on in future experiments exploring quantum effects in rotating frames and curved spacetimes. Identifying these effects has relevance in fundamental physics and to new technologies e.g. quantum communication, as it scales up to satellites in the curved spacetime around the rotating Earth
Positive-density ground states of the Gross-Pitaevskii equation
We consider the nonlinear Gross-Pitaevskii equation at positive density, that
is, for a bounded solution not tending to 0 at infinity. We focus on infinite
ground states, which are by definition minimizers of the energy under local
perturbations. When the Fourier transform of the interaction potential takes
negative values we prove the existence of a phase transition at high density,
where the constant solution ceases to be a ground state. The analysis requires
mixing techniques from elliptic PDE theory and statistical mechanics, in order
to deal with a large class of interaction potentials.Comment: Added a short proof of the uniform bounds in the simpler case of
positive interaction
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