2,463 research outputs found
On Longevity of I-ball/Oscillon
We study I-balls/oscillons, which are long-lived, quasi-periodic, and
spatially localized solutions in real scalar field theories. Contrary to the
case of Q-balls, there is no evident conserved charge that stabilizes the
localized configuration. Nevertheless, in many classical numerical simulations,
it has been shown that they are extremely long-lived. In this paper, we clarify
the reason for the longevity, and show how the exponential separation of time
scales emerges dynamically. Those solutions are time-periodic with a typical
frequency of a mass scale of a scalar field. This observation implies that they
can be understood by the effective theory after integrating out relativistic
modes. We find that the resulting effective theory has an approximate global
U(1) symmetry reflecting an approximate number conservation in the
non-relativistic regime. As a result, the profile of those solutions is
obtained via the bounce method, just like Q-balls, as long as the breaking of
the U(1) symmetry is small enough. We then discuss the decay processes of the
I-ball/oscillon by the breaking of the U(1) symmetry, namely the production of
relativistic modes via number violating processes. We show that the imaginary
part is exponentially suppressed, which explains the extraordinary longevity of
I-ball/oscillon. In addition, we find that there are some attractor behaviors
during the evolution of I-ball/oscillon that further enhance the lifetime. The
validity of our effective theory is confirmed by classical numerical
simulations. Our formalism may also be useful to study condensates of ultra
light bosonic dark matter, such as fuzzy dark matter, and axion stars, for
instance.Comment: 31 pages, 8 figures; v2: typos fixed, published version; v3: typos in
the figures fixe
The dynamics of digits: Calculating pi with Galperin's billiards
In Galperin billiards, two balls colliding with a hard wall form an analog
calculator for the digits of the number . This classical, one-dimensional
three-body system (counting the hard wall) calculates the digits of in a
base determined by the ratio of the masses of the two particles. This base can
be any integer, but it can also be an irrational number, or even the base can
be itself. This article reviews previous results for Galperin billiards
and then pushes these results farther. We provide a complete explicit solution
for the balls' positions and velocities as a function of the collision number
and time. We demonstrate that Galperin billiard can be mapped onto a
two-particle Calogero-type model. We identify a second dynamical invariant for
any mass ratio that provides integrability for the system, and for a sequence
of specific mass ratios we identify a third dynamical invariant that
establishes superintegrability. Integrability allows us to derive some new
exact results for trajectories, and we apply these solutions to analyze the
systematic errors that occur in calculating the digits of with Galperin
billiards, including curious cases with irrational number bases.Comment: 30 pages, 13 figure
A geometrical calibration method for the PIXSCAN micro-CT scanner
Reconstruction in Cone-Beam Tomography can suffer from artifacts due to geometrical misalignments of the source-detector system. They can be avoided by a complete and precise description of the system. We present a high precision method for the geometric calibration for the PIXSCAN, a small animal X-ray CT scanner demonstrator based on hybrid pixel detectors (XPAD2). The specificities of the XPAD2 detectors (dead pixels, tilts and gaps between modules...) made the calibration of the PIXSCAN quite difficult. The method uses a calibration object consisting of a hollow cylinder of polycarbonate on which we positioned four metallic balls. It requires 360 X-ray images (1° increments). An analytic expression of the 3 image ellipses has been derived. It is used for a least square regression of the 13 alignment parameters after a correction of the internal XPAD2 geometry. Our method is fast and completely automated, achieving a precision of about 30 μm
Approximating the Maximum Overlap of Polygons under Translation
Let and be two simple polygons in the plane of total complexity ,
each of which can be decomposed into at most convex parts. We present an
-approximation algorithm, for finding the translation of ,
which maximizes its area of overlap with . Our algorithm runs in
time, where is a constant that depends only on and .
This suggest that for polygons that are "close" to being convex, the problem
can be solved (approximately), in near linear time
Learning Generative Models with Sinkhorn Divergences
The ability to compare two degenerate probability distributions (i.e. two
probability distributions supported on two distinct low-dimensional manifolds
living in a much higher-dimensional space) is a crucial problem arising in the
estimation of generative models for high-dimensional observations such as those
arising in computer vision or natural language. It is known that optimal
transport metrics can represent a cure for this problem, since they were
specifically designed as an alternative to information divergences to handle
such problematic scenarios. Unfortunately, training generative machines using
OT raises formidable computational and statistical challenges, because of (i)
the computational burden of evaluating OT losses, (ii) the instability and lack
of smoothness of these losses, (iii) the difficulty to estimate robustly these
losses and their gradients in high dimension. This paper presents the first
tractable computational method to train large scale generative models using an
optimal transport loss, and tackles these three issues by relying on two key
ideas: (a) entropic smoothing, which turns the original OT loss into one that
can be computed using Sinkhorn fixed point iterations; (b) algorithmic
(automatic) differentiation of these iterations. These two approximations
result in a robust and differentiable approximation of the OT loss with
streamlined GPU execution. Entropic smoothing generates a family of losses
interpolating between Wasserstein (OT) and Maximum Mean Discrepancy (MMD), thus
allowing to find a sweet spot leveraging the geometry of OT and the favorable
high-dimensional sample complexity of MMD which comes with unbiased gradient
estimates. The resulting computational architecture complements nicely standard
deep network generative models by a stack of extra layers implementing the loss
function
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