324 research outputs found
Quantitative Tverberg theorems over lattices and other discrete sets
This paper presents a new variation of Tverberg's theorem. Given a discrete
set of , we study the number of points of needed to guarantee the
existence of an -partition of the points such that the intersection of the
convex hulls of the parts contains at least points of . The proofs
of the main results require new quantitative versions of Helly's and
Carath\'eodory's theorems.Comment: 16 pages. arXiv admin note: substantial text overlap with
arXiv:1503.0611
Quantitative combinatorial geometry for continuous parameters
We prove variations of Carath\'eodory's, Helly's and Tverberg's theorems
where the sets involved are measured according to continuous functions such as
the volume or diameter. Among our results, we present continuous quantitative
versions of Lov\'asz's colorful Helly theorem, B\'ar\'any's colorful
Carath\'eodory's theorem, and the colorful Tverberg theorem.Comment: 22 pages. arXiv admin note: substantial text overlap with
arXiv:1503.0611
Quantitative Tverberg, Helly, & Carath\'eodory theorems
This paper presents sixteen quantitative versions of the classic Tverberg,
Helly, & Caratheodory theorems in combinatorial convexity. Our results include
measurable or enumerable information in the hypothesis and the conclusion.
Typical measurements include the volume, the diameter, or the number of points
in a lattice.Comment: 33 page
Helly numbers of Algebraic Subsets of
We study -convex sets, which are the geometric objects obtained as the
intersection of the usual convex sets in with a proper subset
. We contribute new results about their -Helly
numbers. We extend prior work for , , and ; we give sharp bounds on the -Helly numbers in
several new cases. We considered the situation for low-dimensional and for
sets that have some algebraic structure, in particular when is an
arbitrary subgroup of or when is the difference between a
lattice and some of its sublattices. By abstracting the ingredients of Lov\'asz
method we obtain colorful versions of many monochromatic Helly-type results,
including several colorful versions of our own results.Comment: 13 pages, 3 figures. This paper is a revised version of what was
originally the first half of arXiv:1504.00076v
Beyond Chance-Constrained Convex Mixed-Integer Optimization: A Generalized Calafiore-Campi Algorithm and the notion of -optimization
The scenario approach developed by Calafiore and Campi to attack
chance-constrained convex programs utilizes random sampling on the uncertainty
parameter to substitute the original problem with a representative continuous
convex optimization with convex constraints which is a relaxation of the
original. Calafiore and Campi provided an explicit estimate on the size of
the sampling relaxation to yield high-likelihood feasible solutions of the
chance-constrained problem. They measured the probability of the original
constraints to be violated by the random optimal solution from the relaxation
of size .
This paper has two main contributions. First, we present a generalization of
the Calafiore-Campi results to both integer and mixed-integer variables. In
fact, we demonstrate that their sampling estimates work naturally for variables
restricted to some subset of . The key elements are
generalizations of Helly's theorem where the convex sets are required to
intersect . The size of samples in both algorithms will
be directly determined by the -Helly numbers.
Motivated by the first half of the paper, for any subset , we introduce the notion of an -optimization problem, where the
variables take on values over . It generalizes continuous, integer, and
mixed-integer optimization. We illustrate with examples the expressive power of
-optimization to capture sophisticated combinatorial optimization problems
with difficult modular constraints. We reinforce the evidence that
-optimization is "the right concept" by showing that the well-known
randomized sampling algorithm of K. Clarkson for low-dimensional convex
optimization problems can be extended to work with variables taking values over
.Comment: 16 pages, 0 figures. This paper has been revised and split into two
parts. This version is the second part of the original paper. The first part
of the original paper is arXiv:1508.02380 (the original article contained 24
pages, 3 figures
MHD stability and disruptions in the SPARC tokamak
SPARC is being designed to operate with a normalized beta of beta(N) = 1.0, a normalized density of n(G) = 0.37 and a safety factor of q(95) approximate to 3.4, providing a comfortable margin to their respective disruption limits. Further, a low beta poloidal beta(p) = 0.19 at the safety factor q = 2 surface reduces the drive for neoclassical tearing modes, which together with a frozen-in classically stable current profile might allow access to a robustly tearing-free operating space. Although the inherent stability is expected to reduce the frequency of disruptions, the disruption loading is comparable to and in some cases higher than that of ITER. The machine is being designed to withstand the predicted unmitigated axisymmetric halo current forces up to 50 MN and similarly large loads from eddy currents forced to flow poloidally in the vacuum vessel. Runaway electron (RE) simulations using GO+CODE show high flattop-to-RE current conversions in the absence of seed losses, although NIMROD modelling predicts losses of similar to 80 %; self-consistent modelling is ongoing. A passive RE mitigation coil designed to drive stochastic RE losses is being considered and COMSOL modelling predicts peak normalized fields at the plasma of order 10(-2) that rises linearly with a change in the plasma current. Massive material injection is planned to reduce the disruption loading. A data-driven approach to predict an oncoming disruption and trigger mitigation is discussed
A whole systems approach to integrating physical activity to aid mental health recovery – Translating theory into practice
Improving health outcomes for people with severe mental illness (SMI) through increased physical activity (PA) on a large scale remains an elusive goal. There is promising evidence that increasing levels of PA in people with SMI can improve psychological and physical health outcomes. However, SMI is associated with reduced levels of physical activity and more sedentary behaviour than is usual in people without SMI.
Increasing PA and reducing sedentary behaviour among people with SMI is a complex process, as there are drivers of these behaviours at the individual, household, community and policy levels. Examples of these include the symptoms associated with SMI, poverty, unemployment, social isolation and stigma. Such drivers affect opportunities to take part in PA and individuals’ abilities to do so, creating negative reinforcing loops of behaviours and health outcomes.
Most previous approaches to PA for this population have focused largely on individual behaviour change, with limited success. To increase levels of PA effectively for people with SMI at scale also requires consideration of the wider determinants and complex dynamic drivers of PA behaviour in this population.
This position paper sets out a rationale and recommendations for the utilisation of whole systems approaches to PA in people with SMI and the improvement of physical and psychological outcomes. Such approaches should be delivered in conjunction with bespoke, individual-level interventions which address the unique needs of those with SMI
Composition of Titan's ionosphere
Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/94758/1/grl21212.pd
An upgrade of the magnetic diagnostic system of the DIII-D tokamak for non-axisymmetric measurements
The DIII-D tokamak magnetic diagnostic system [E. J. Strait, Rev. Sci. Instrum. 77, 023502 (2006)] has been upgraded to significantly expand the measurement of the plasma response to intrinsic and applied non-axisymmetric "3D" fields. The placement and design of 101 additional sensors allow resolution of toroidal mode numbers 1 ≤ n ≤ 3, and poloidal wavelengths smaller than MARS-F, IPEC, and VMEC magnetohydrodynamic model predictions. Small 3D perturbations, relative to the equilibrium field (10(-5) < δB/B0 < 10(-4)), require sub-millimeter fabrication and installation tolerances. This high precision is achieved using electrical discharge machined components, and alignment techniques employing rotary laser levels and a coordinate measurement machine. A 16-bit data acquisition system is used in conjunction with analog signal-processing to recover non-axisymmetric perturbations. Co-located radial and poloidal field measurements allow up to 14.2 cm spatial resolution of poloidal structures (plasma poloidal circumference is ~500 cm). The function of the new system is verified by comparing the rotating tearing mode structure, measured by 14 BP fluctuation sensors, with that measured by the upgraded B(R) saddle loop sensors after the mode locks to the vessel wall. The result is a nearly identical 2/1 helical eigenstructure in both cases.S. R.
Haskey wishes to thank AINSE Ltd. for providing financial
assistance
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