A Synthetic Null Energy Condition McCann, Robert J.
Communications in mathematical physics,
02/2024, Letnik:
405, Številka:
2
Journal Article
Recenzirano
Odprti dostop
We give a simpler approach to Kunzinger and Sämann’s theory of Lorentzian length spaces in the globally hyperbolic case; these provide a nonsmooth framework for general relativity. We close a gap in ...the regularly localizable setting, by showing consistency of two potentially different notions of timelike geodesic segments used in the literature. In the smooth pseudo-Riemannian setting, we show Penrose’ null energy condition is equivalent to a variable lower bound on the timelike Ricci curvature. This allows us to give a nonsmooth reformulation of the null energy condition using the timelike curvature-dimension conditions of Cavalletti and Mondino (and Braun). Although this definition is consistent with the smooth setting, it proves unstable relative to the notion of pointed measured convergence for which timelike curvature-dimensions conditions are known to be stable. We illustrate this instability using a sequence of smooth weighted Lorentzian manifolds-with-boundary that satisfy it, yet converge to a disconnected pair of timelike related points that violate it in the limit.
Consider a collection of particles interacting through an attractive-repulsive potential given as a difference of power laws and normalized so that its unique minimum occurs at unit separation. For a ...range of exponents corresponding to mild repulsion and strong attraction, we show that the minimum energy configuration is uniquely attained—apart from translations and rotations—by equidistributing the particles over the vertices of a regular top-dimensional simplex (i.e. an equilateral triangle in two dimensions and regular tetrahedron in three). If the attraction is not assumed to be strong, we show that these configurations are at least local energy minimizers in the relevant
d
∞
metric from optimal transportation, as are all of the other uncountably many unbalanced configurations with the same support. We infer the existence of phase transitions. The proof is based in part on a simple isodiametric variance bound which characterizes regular simplices; it shows that among probability measures on
R
n
whose supports have at most unit diameter, the variance around the mean is maximized precisely by those measures which assign mass
1
/
(
n
+
1
)
to each vertex of a (unit-diameter) regular simplex.
On a smooth bounded Euclidean domain, Sobolev-subcritical fast diffusion with vanishing boundary trace is known to lead to finite-time extinction, with a vanishing profile selected by the initial ...datum. In rescaled variables, we quantify the rate of convergence to this profile uniformly in relative error, showing the rate is either exponentially fast (with a rate constant predicted by the spectral gap), or algebraically slow (which is only possible in the presence of non-integrable zero modes). In the first case, the nonlinear dynamics are well-approximated by exponentially decaying eigenmodes up to at least twice the gap; this refines and confirms a 1980 conjecture of Berryman and Holland. We also improve on a result of Bonforte and Figalli by providing a new and simpler approach which is able to accommodate the presence of zero modes, such as those that occur when the vanishing profile fails to be isolated (and possibly belongs to a continuum of such profiles).
Among probability measures on
d
-dimensional real projective space, one which maximizes the expected angle
arccos
(
x
|
x
|
·
y
|
y
|
)
between independently drawn projective points
x
and
y
was ...conjectured to equidistribute its mass over the standard Euclidean basis
{
e
0
,
e
1
,
…
,
e
d
}
by Fejes Tóth (Acta Math Acad Sci Hung 10:13–19, 1959.
https://doi.org/10.1007/BF02063286
). If true, this conjecture evidently implies the same measure maximizes the expectation of
arccos
α
(
x
|
x
|
·
y
|
y
|
)
for any exponent
α
>
1
. The kernel
arccos
α
(
x
|
x
|
·
y
|
y
|
)
represents the objective of an infinite-dimensional quadratic program. We verify discrete and continuous versions of this milder conjecture in a non-empty range
α
>
α
Δ
d
≥
1
, and establish uniqueness of the resulting maximizer
μ
^
up to rotation. We show
μ
^
no longer maximizes when
α
<
α
Δ
d
. At the endpoint
α
=
α
Δ
d
of this range, we show another maximizer
μ
must also exist which is not a rotation of
μ
^
. For the continuous version of the conjecture, an “Appendix A” provided by Bilyk et al in response to an earlier draft of this work combines with the present improvements to yield
α
Δ
d
<
2
. The original conjecture
α
Δ
d
=
1
remains open (unless
d
=
1
). However, in the maximum possible range
α
>
1
, we show
μ
^
and its rotations maximize the aforementioned expectation uniquely on a sufficiently small ball in the
L
∞
-Kantorovich–Rubinstein–Wasserstein metric
d
∞
from optimal transportation; the same is true for any measure
μ
which is mutually absolutely continuous with respect to
μ
^
, but the size of the ball depends on
α
,
d
, and
‖
d
μ
^
d
μ
‖
∞
.
In the context of malaria elimination, interventions will need to target high burden areas to further reduce transmission. Current tools to monitor and report disease burden lack the capacity to ...continuously detect fine-scale spatial and temporal variations of disease distribution exhibited by malaria. These tools use random sampling techniques that are inefficient for capturing underlying heterogeneity while health facility data in resource-limited settings are inaccurate. Continuous community surveys of malaria burden provide real-time results of local spatio-temporal variation. Adaptive geostatistical design (AGD) improves prediction of outcome of interest compared to current random sampling techniques. We present findings of continuous malaria prevalence surveys using an adaptive sampling design.
We conducted repeated cross sectional surveys guided by an adaptive sampling design to monitor the prevalence of malaria parasitaemia and anaemia in children below five years old in the communities living around Majete Wildlife Reserve in Chikwawa district, Southern Malawi. AGD sampling uses previously collected data to sample new locations of high prediction variance or, where prediction exceeds a set threshold. We fitted a geostatistical model to predict malaria prevalence in the area.
We conducted five rounds of sampling, and tested 876 children aged 6-59 months from 1377 households over a 12-month period. Malaria prevalence prediction maps showed spatial heterogeneity and presence of hotspots-where predicted malaria prevalence was above 30%; predictors of malaria included age, socio-economic status and ownership of insecticide-treated mosquito nets.
Continuous malaria prevalence surveys using adaptive sampling increased malaria prevalence prediction accuracy. Results from the surveys were readily available after data collection. The tool can assist local managers to target malaria control interventions in areas with the greatest health impact and is ready for assessment in other diseases.
Optimal transportation with capacity constraints KORMAN, JONATHAN; MCCANN, ROBERT J.
Transactions of the American Mathematical Society,
March 1, 2015, 20150301, 2015-3-00, Letnik:
367, Številka:
3
Journal Article
Recenzirano
Odprti dostop
The classical problem of optimal transportation can be formulated as a linear optimization problem on a convex domain: among all joint measures with fixed marginals find the optimal one, where ...optimality is measured against a cost function. Here we consider a natural but largely unexplored variant of this problem by imposing a pointwise constraint on the joint (absolutely continuous) measures: among all joint densities with fixed marginals and which are dominated by a given density, find the optimal one. For this variant, we show that local non-degeneracy of the cost function implies every minimizer is extremal in the convex set of competitors, hence unique. An appendix develops rudiments of a duality theory for this problem, which allows us to compute several suggestive examples.
Remotely sensed data can serve as an independent source of information about the location of residential structures in areas under demographic and health surveillance. We report on results obtained ...combining satellite imagery, imported from Bing, with location data routinely collected using the built-in GPS sensors of tablet computers, to assess completeness of population coverage in a Health and Demographic Surveillance System in Malawi. The Majete Malaria Project Health and Demographic Surveillance System, in Malawi, started in 2014 to support a project with the aim of studying the reduction of malaria using an integrated control approach by rolling out insecticide treated nets and improved case management supplemented with house improvement and larval source management. In order to support the monitoring of the trial a Health and Demographic Surveillance System was established in the area that surrounds the Majete Wildlife Reserve (1600 km2), using the OpenHDS data system. We compared house locations obtained using GPS recordings on mobile devices during the demographic surveillance census round with those acquired from satellite imagery. Volunteers were recruited through the crowdcrafting.org platform to identify building structures on the images, which enabled the compilation of a database with coordinates of potential residences. For every building identified on these satellite images by the volunteers (11,046 buildings identified of which 3424 (ca. 30%) were part of the censused area), we calculated the distance to the nearest house enumerated on the ground by fieldworkers during the census round of the HDSS. A random sample of buildings (85 structures) identified on satellite images without a nearby location enrolled in the census were visited by a fieldworker to determine how many were missed during the baseline census survey, if any were missed. The findings from this ground-truthing effort suggest that a high population coverage was achieved in the census survey, however the crowd-sourcing did not locate many of the inhabited structures (52.3% of the 6543 recorded during the census round). We conclude that using auxiliary data can play a useful role in quality assurance in population based health surveillance, but improved algorithms would be needed if crowd-sourced house locations are to be used as the basis of population databases.
Densities of particles on
R
n
which interact pairwise through an attractive-repulsive power-law potential
W
α
,
β
(
x
)
=
|
x
|
α
/
α
-
|
x
|
β
/
β
have often been used to explain patterns produced ...by biological and physical systems. In the mildly repulsive regime
α
>
β
≥
2
with
n
≥
2
, we show there exists a decreasing homeomorphism
α
Δ
n
from 2, 4 to itself such that: distributing the particles uniformly over the vertices of a regular unit diameter
n
-simplex minimizes the potential energy if and only if
α
≥
α
Δ
n
(
β
)
. Moreover this minimum is uniquely attained up to rigid motions when
α
>
α
Δ
n
(
β
)
. We estimate
α
Δ
n
(
β
)
above and below, and identify its limit as the dimension grows large. These results are derived from a new northeast comparison principle in the space of exponents. At the endpoint
(
α
,
β
)
=
(
4
,
2
)
of this transition curve, we characterize all minimizers by showing they lie on a sphere and share all first and second moments with the spherical shell. Suitably modified versions of these statements are also established (i) for
W
α
,
β
and corresponding energies in the case where
n
=
1
, and (ii) for the attractive-repulsive potentials
D
α
(
x
)
=
|
x
|
α
(
α
log
|
x
|
-
1
)
that arise in the limit
β
↗
α
.
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