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

Physics of locked modes in ITER: Error field limits, rotation for obviation, and measurement of field errors

The existing theoretical and experimental basis for predicting the levels of resonant static error field at different components m,n that stop plasma rotation and produce a locked mode is reviewed. For ITER ohmic discharges, the slow rotation of the very large plasma is predicted to incur a locked mode (and subsequent disastrous large magnetic islands) at a simultaneous weighted error field ({Sigma}{sub 1}{sup 3}w{sub m1}B{sup 2}{sub rm1}){sup {1/2}}/B{sub T} {ge} 1.9 x 10{sup -5}. Here the weights w{sub m1} are empirically determined from measurements on DIII-D to be w{sub 11} = 0. 2, w{sub 21} = 1.0, and w{sub 31} = 0. 8 and point out the relative importance of different error field components. This could be greatly obviated by application of counter injected neutral beams (which adds fluid flow to the natural ohmic electron drift). The addition of 5 MW of 1 MeV beams at 45{degrees} injection would increase the error field limit by a factor of 5; 13 MW would produce a factor of 10 improvement. Co-injection beams would also be effective but not as much as counter-injection as the co direction opposes the intrinsic rotation while the counter direction adds to it. A means for measuring individual PF and TF coil total axisymmetric field error to less than 1 in 10,000 is described. This would allow alignment of coils to mm accuracy and with correction coils make possible the very low levels of error field needed

Practical beta limit in ITER-shaped discharges in DIII-D and its increase by higher collisionality

The maximum beta which can be sustained for a long pulse in ITER-shaped plasmas in DIII-D with q{sub 95} {approx_gt} 3, ELMs, and sawteeth is found to be limited by resistive tearing modes, particularly m/n = 3/2 and 2/1. At low collisionality comparable to that which will occur in ITER, the beta limit is a factor of two below the usually expected n = {infinity} ballooning and n = 1 kink ideal limits