823 research outputs found
Calculus on surfaces with general closest point functions
The Closest Point Method for solving partial differential equations (PDEs) posed on surfaces was recently introduced by Ruuth and Merriman [J. Comput. Phys. 2008] and successfully applied to a variety of surface PDEs. In this paper we study the theoretical foundations of this method. The main idea is that surface differentials of a surface function can be replaced with Cartesian differentials of its closest point extension, i.e., its composition with a closest point function. We introduce a general class of these closest point functions (a subset of differentiable retractions), show that these are exactly the functions necessary to satisfy the above idea, and give a geometric characterization this class. Finally, we construct some closest point functions and demonstrate their effectiveness numerically on surface PDEs
Evolving neural networks with genetic algorithms to study the String Landscape
We study possible applications of artificial neural networks to examine the
string landscape. Since the field of application is rather versatile, we
propose to dynamically evolve these networks via genetic algorithms. This means
that we start from basic building blocks and combine them such that the neural
network performs best for the application we are interested in. We study three
areas in which neural networks can be applied: to classify models according to
a fixed set of (physically) appealing features, to find a concrete realization
for a computation for which the precise algorithm is known in principle but
very tedious to actually implement, and to predict or approximate the outcome
of some involved mathematical computation which performs too inefficient to
apply it, e.g. in model scans within the string landscape. We present simple
examples that arise in string phenomenology for all three types of problems and
discuss how they can be addressed by evolving neural networks from genetic
algorithms.Comment: 17 pages, 7 figures, references added, typos corrected, extended
introductory sectio
Ancient solutions of geometric flows with curvature pinching
We prove rigidity theorems for ancient solutions of geometric flows of
immersed submanifolds. Specifically, we find pinching conditions on the second
fundamental form that characterize the shrinking sphere among compact ancient
solutions for the mean curvature flow in codimension greater than one, and for
some nonlinear curvature flows of hypersurfaces.Comment: Minor changes, bibliography updated. To appear on The Journal of
Geometric Analysis. The final version is available online at
https://doi.org/10.1007/s12220-018-0036-
Integration over curves and surfaces defined by the closest point mapping
We propose a new formulation for integrating over smooth curves and surfaces
that are described by their closest point mappings. Our method is designed for
curves and surfaces that are not defined by any explicit parameterization and
is intended to be used in combination with level set techniques. However,
contrary to the common practice with level set methods, the volume integrals
derived from our formulation coincide exactly with the surface or line
integrals that one wish to compute. We study various aspects of this
formulation and provide a geometric interpretation of this formulation in terms
of the singular values of the Jacobian matrix of the closest point mapping.
Additionally, we extend the formulation - initially derived to integrate over
manifolds of codimension one - to include integration along curves in three
dimensions. Some numerical examples using very simple discretizations are
presented to demonstrate the efficacy of the formulation.Comment: Revised the pape
Can We Make a Bohmian Electron Reach the Speed of Light, at Least for One Instant?
In Bohmian mechanics, a version of quantum mechanics that ascribes world
lines to electrons, we can meaningfully ask about an electron's instantaneous
speed relative to a given inertial frame. Interestingly, according to the
relativistic version of Bohmian mechanics using the Dirac equation, a massive
particle's speed is less than or equal to the speed of light, but not
necessarily less. That is, there are situations in which the particle actually
reaches the speed of light---a very non-classical behavior. That leads us to
the question of whether such situations can be arranged experimentally. We
prove a theorem, Theorem 5, implying that for generic initial wave functions
the probability that the particle ever reaches the speed of light, even if at
only one point in time, is zero. We conclude that the answer to the question is
no. Since a trajectory reaches the speed of light whenever the quantum
probability current psi-bar gamma^mu psi is a lightlike 4-vector, our analysis
concerns the current vector field of a generic wave function and may thus be of
interest also independently of Bohmian mechanics. The fact that the current is
never spacelike has been used to argue against the possibility of
faster-than-light tunnelling through a barrier, a somewhat similar question.
Theorem 5, as well as a more general version provided by Theorem 6, are also
interesting in their own right. They concern a certain property of a function
psi: R^4 --> C^4 that is crucial to the question of reaching the speed of
light, namely being transverse to a certain submanifold of C^4 along a given
compact subset of space-time. While it follows from the known transversality
theorem of differential topology that this property is generic among smooth
functions psi: R^4 --> C^4, Theorem 5 asserts that it is also generic among
smooth solutions of the Dirac equation.Comment: 17 pages LaTeX, no figures; v4 minor additions and improvement
Dirichlet sigma models and mean curvature flow
The mean curvature flow describes the parabolic deformation of embedded
branes in Riemannian geometry driven by their extrinsic mean curvature vector,
which is typically associated to surface tension forces. It is the gradient
flow of the area functional, and, as such, it is naturally identified with the
boundary renormalization group equation of Dirichlet sigma models away from
conformality, to lowest order in perturbation theory. D-branes appear as fixed
points of this flow having conformally invariant boundary conditions. Simple
running solutions include the paper-clip and the hair-pin (or grim-reaper)
models on the plane, as well as scaling solutions associated to rational (p, q)
closed curves and the decay of two intersecting lines. Stability analysis is
performed in several cases while searching for transitions among different
brane configurations. The combination of Ricci with the mean curvature flow is
examined in detail together with several explicit examples of deforming curves
on curved backgrounds. Some general aspects of the mean curvature flow in
higher dimensional ambient spaces are also discussed and obtain consistent
truncations to lower dimensional systems. Selected physical applications are
mentioned in the text, including tachyon condensation in open string theory and
the resistive diffusion of force-free fields in magneto-hydrodynamics.Comment: 77 pages, 21 figure
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