39,108 research outputs found
The mathematical research of William Parry FRS
In this article we survey the mathematical research of the late William (Bill) Parry, FRS
Holomorphic vector fields and quadratic differentials on planar triangular meshes
Given a triangulated region in the complex plane, a discrete vector field
assigns a vector to every vertex. We call such a vector
field holomorphic if it defines an infinitesimal deformation of the
triangulation that preserves length cross ratios. We show that each holomorphic
vector field can be constructed based on a discrete harmonic function in the
sense of the cotan Laplacian. Moreover, to each holomorphic vector field we
associate in a M\"obius invariant fashion a certain holomorphic quadratic
differential. Here a quadratic differential is defined as an object that
assigns a purely imaginary number to each interior edge. Then we derive a
Weierstrass representation formula, which shows how a holomorphic quadratic
differential can be used to construct a discrete minimal surface with
prescribed Gau{\ss} map and prescribed Hopf differential.Comment: 17 pages; final version, to appear in "Advances in Discrete
Differential Geometry", ed. A. I. Bobenko, Springer, 2016; references adde
Superior memory efficiency of quantum devices for the simulation of continuous-time stochastic processes
Continuous-time stochastic processes pervade everyday experience, and the
simulation of models of these processes is of great utility. Classical models
of systems operating in continuous-time must typically track an unbounded
amount of information about past behaviour, even for relatively simple models,
enforcing limits on precision due to the finite memory of the machine. However,
quantum machines can require less information about the past than even their
optimal classical counterparts to simulate the future of discrete-time
processes, and we demonstrate that this advantage extends to the
continuous-time regime. Moreover, we show that this reduction in the memory
requirement can be unboundedly large, allowing for arbitrary precision even
with a finite quantum memory. We provide a systematic method for finding
superior quantum constructions, and a protocol for analogue simulation of
continuous-time renewal processes with a quantum machine.Comment: 13 pages, 8 figures, title changed from original versio
A Discrete Theory of Connections on Principal Bundles
Connections on principal bundles play a fundamental role in expressing the
equations of motion for mechanical systems with symmetry in an intrinsic
fashion. A discrete theory of connections on principal bundles is constructed
by introducing the discrete analogue of the Atiyah sequence, with a connection
corresponding to the choice of a splitting of the short exact sequence.
Equivalent representations of a discrete connection are considered, and an
extension of the pair groupoid composition, that takes into account the
principal bundle structure, is introduced. Computational issues, such as the
order of approximation, are also addressed. Discrete connections provide an
intrinsic method for introducing coordinates on the reduced space for discrete
mechanics, and provide the necessary discrete geometry to introduce more
general discrete symmetry reduction. In addition, discrete analogues of the
Levi-Civita connection, and its curvature, are introduced by using the
machinery of discrete exterior calculus, and discrete connections.Comment: 38 pages, 11 figures. Fixed labels in figure
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