Quantitative Tverberg theorems over lattices and other discrete sets

This paper presents a new variation of Tverberg's theorem. Given a discrete set $S$ of $R^d$, we study the number of points of $S$ needed to guarantee the existence of an $m$-partition of the points such that the intersection of the $m$ convex hulls of the parts contains at least $k$ points of $S$. 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 Rd\mathbb R^d

We study $S$-convex sets, which are the geometric objects obtained as the intersection of the usual convex sets in $\mathbb R^d$ with a proper subset $S\subset \mathbb R^d$. We contribute new results about their $S$-Helly numbers. We extend prior work for $S=\mathbb R^d$, $\mathbb Z^d$, and $\mathbb Z^{d-k}\times\mathbb R^k$; we give sharp bounds on the $S$-Helly numbers in several new cases. We considered the situation for low-dimensional $S$ and for sets $S$ that have some algebraic structure, in particular when $S$ is an arbitrary subgroup of $\mathbb R^d$ or when $S$ 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 SS-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 $N$ convex constraints which is a relaxation of the original. Calafiore and Campi provided an explicit estimate on the size $N$ 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 $N$. 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 $S$ of $\mathbb R^d$. The key elements are generalizations of Helly's theorem where the convex sets are required to intersect $S \subset \mathbb R^d$. The size of samples in both algorithms will be directly determined by the $S$-Helly numbers. Motivated by the first half of the paper, for any subset $S \subset \mathbb R^d$, we introduce the notion of an $S$-optimization problem, where the variables take on values over $S$. It generalizes continuous, integer, and mixed-integer optimization. We illustrate with examples the expressive power of $S$-optimization to capture sophisticated combinatorial optimization problems with difficult modular constraints. We reinforce the evidence that $S$-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 $S$.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

Whole of Systems Trial of Prevention Strategies for Childhood Obesity: WHO STOPS Childhood Obesity

Background: Community-based initiatives show promise for preventing childhood obesity. They are characterized by community leaders and members working together to address complex local drivers of energy balance. Objectives: To present a protocol for a stepped wedge cluster randomized trial in ten communities in the Great South Coast Region of Victoria, Australia to test whether it is possible to: (1) strengthen community action for childhood obesity prevention, and (2) measure the impact of increased action on risk factors for childhood obesity. Methods: The WHO STOPS intervention involves a facilitated community engagement process that: creates an agreed systems map of childhood obesity causes for a community; identifies intervention opportunities through leveraging the dynamic aspects of the system; and, converts these understandings into community-built, systems-oriented action plans. Ten communities will be randomized (1:1) to intervention or control in year one and all communities will be included by year three. The primary outcome is childhood obesity prevalence among grade two (ages 7–8 y), grade four (9–10 y) and grade six (11–12 y) students measured using our established community-led monitoring system (69% school and 93% student participation rate in government and independent schools). An additional group of 13 external communities from other regions of Victoria with no specific interventions will provide an external comparison. These communities will also allow us to assess diffusion of the intervention to control communities during the first three years of the trial. Conclusion: This trial will test effectiveness, over a five-year period, of community-owned, -supported and -led strategies designed to address complex and dynamic causes of childhood obesity

Composition of Titan's ionosphere

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/94758/1/grl21212.pd

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

Titan's ionosphere: Model comparisons with Cassini Ta data

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/94829/1/grl20028.pd