Helly-Type Theorems in Property Testing

Helly's theorem is a fundamental result in discrete geometry, describing the ways in which convex sets intersect with each other. If $S$ is a set of $n$ points in $R^d$, we say that $S$ is $(k,G)$-clusterable if it can be partitioned into $k$ clusters (subsets) such that each cluster can be contained in a translated copy of a geometric object $G$. In this paper, as an application of Helly's theorem, by taking a constant size sample from $S$, we present a testing algorithm for $(k,G)$-clustering, i.e., to distinguish between two cases: when $S$ is $(k,G)$-clusterable, and when it is $\epsilon$-far from being $(k,G)$-clusterable. A set $S$ is $\epsilon$-far $(0<\epsilon\leq1)$ from being $(k,G)$-clusterable if at least $\epsilon n$ points need to be removed from $S$ to make it $(k,G)$-clusterable. We solve this problem for $k=1$ and when $G$ is a symmetric convex object. For $k>1$, we solve a weaker version of this problem. Finally, as an application of our testing result, in clustering with outliers, we show that one can find the approximate clusters by querying a constant size sample, with high probability

The Separatrix Algorithm for Synthesis and Analysis of Stochastic Simulations with Applications in Disease Modeling

Decision makers in epidemiology and other disciplines are faced with the daunting challenge of designing interventions that will be successful with high probability and robust against a multitude of uncertainties. To facilitate the decision making process in the context of a goal-oriented objective (e.g., eradicate polio by ), stochastic models can be used to map the probability of achieving the goal as a function of parameters. Each run of a stochastic model can be viewed as a Bernoulli trial in which “success” is returned if and only if the goal is achieved in simulation. However, each run can take a significant amount of time to complete, and many replicates are required to characterize each point in parameter space, so specialized algorithms are required to locate desirable interventions. To address this need, we present the Separatrix Algorithm, which strategically locates parameter combinations that are expected to achieve the goal with a user-specified probability of success (e.g. 95%). Technically, the algorithm iteratively combines density-corrected binary kernel regression with a novel information-gathering experiment design to produce results that are asymptotically correct and work well in practice. The Separatrix Algorithm is demonstrated on several test problems, and on a detailed individual-based simulation of malaria

Combinatorics of linear iterated function systems with overlaps

Let $\bm p_0,...,\bm p_{m-1}$ be points in ${\mathbb R}^d$, and let $\{f_j\}_{j=0}^{m-1}$ be a one-parameter family of similitudes of ${\mathbb R}^d$: $$f_j(\bm x) = \lambda\bm x + (1-\lambda)\bm p_j, j=0,...,m-1,$$ where $\lambda\in(0,1)$ is our parameter. Then, as is well known, there exists a unique self-similar attractor $S_\lambda$ satisfying $S_\lambda=\bigcup_{j=0}^{m-1} f_j(S_\lambda)$. Each $\bm x\in S_\lambda$ has at least one address $(i_1,i_2,...)\in\prod_1^\infty\{0,1,...,m-1\}$, i.e., $\lim_n f_{i_1}f_{i_2}... f_{i_n}({\bf 0})=\bm x$. We show that for $\lambda$ sufficiently close to 1, each $\bm x\in S_\lambda\setminus\{\bm p_0,...,\bm p_{m-1}\}$ has $2^{\aleph_0}$ different addresses. If $\lambda$ is not too close to 1, then we can still have an overlap, but there exist $\bm x$'s which have a unique address. However, we prove that almost every $\bm x\in S_\lambda$ has $2^{\aleph_0}$ addresses, provided $S_\lambda$ contains no holes and at least one proper overlap. We apply these results to the case of expansions with deleted digits. Furthermore, we give sharp sufficient conditions for the Open Set Condition to fail and for the attractor to have no holes. These results are generalisations of the corresponding one-dimensional results, however most proofs are different.Comment: Accepted for publication in Nonlinearit

TB187: Forest Vegetation Monitoring in Acadia National Park

The goal of this report is to present the results of the vegetation component of the PRIMENet study at Acadia. The results include a classification of vegetation types and their locations within Cadillac Brook and Hadlock Brook watersheds; a synthesis of the primary and meta tree, sapling, and seedling data from the two study watersheds; and foliar chemical analyses using Acer rubrum and Picea rubens from Cadillac Brook and Hadlock Brook watersheds. This report provides the baseline information for long-term forest vegetation monitoring in the deciduous and coniferous forests in Cadillac Brook and Hadlock Brook watersheds. Ongoing interest and studies on the status of the natural resources within Acadia National Park makes availability of information from previous work, such as the baseline data in this report, very important.https://digitalcommons.library.umaine.edu/aes_techbulletin/1021/thumbnail.jp

Multi sensor transducer and weight factor

A multi-sensor transducer and processing method allow insitu monitoring of the senor accuracy and transducer `health`. In one embodiment, the transducer has multiple sensors to provide corresponding output signals in response to a stimulus, such as pressure. A processor applies individual weight factors to reach of the output signals and provide a single transducer output that reduces the contribution from inaccurate sensors. The weight factors can be updated and stored. The processor can use the weight factors to provide a `health` of the transducer based upon the number of accurate versus in-accurate sensors in the transducer

Long paths in first passage percolation on the complete graph I. Local PWIT dynamics

We study the random geometry of first passage percolation on the complete graph equipped with independent and identically distributed edge weights. We find classes with different behaviour depending on a sequence of parameters (sn)n≥1 that quantifies the extreme-value behavior of small weights. We consider both n-independent as well as n-dependent edge weights and illustrate our results in many examples. In particular, we investigate the case where sn → ∞, and focus on the exploration process that grows the smallest-weight tree from a vertex. We establish that the smallest-weight tree process locally converges to the invasion percolation cluster on the Poisson-weighted infinite tree, and we identify the scaling limit of the weight of the smallest-weight path between two uniform vertices. In addition, we show that over a long time interval, the growth of the smallest-weight tree maintains the same volume-height scaling exponent – volume proportional to the square of the height – found in critical Galton–Watson branching trees and critical Erdős-Rényi random graphs

Long paths in first passage percolation on the complete graph II. Global branching dynamics

We study the random geometry of first passage percolation on the complete graph equipped with independent and identically distributed positive edge weights. We consider the case where the lower extreme values of the edge weights are highly separated. This model exhibits strong disorder and a crossover between local and global scales. Local neighborhoods are related to invasion percolation that display self-organised criticality. Globally, the edges with relevant edge weights form a barely supercritical Erdős–Rényi random graph that can be described by branching processes. This near-critical behaviour gives rise to optimal paths that are considerably longer than logarithmic in the number of vertices, interpolating between random graph and minimal spanning tree path lengths. Crucial to our approach is the quantification of the extreme-value behavior of small edge weights in terms of a sequence of parameters (sn)n≥1 that characterises the different universality classes for first passage percolation on the complete graph. We investigate the case where sn→ ∞ with sn= o(n1 / 3) , which corresponds to the barely supercritical setting. We identify the scaling limit of the weight of the optimal path between two vertices, and we prove that the number of edges in this path obeys a central limit theorem with mean approximately snlog(n/sn3) and variance sn2log(n/sn3). Remarkably, our proof also applies to n-dependent edge weights of the form Esn, where E is an exponential random variable with mean 1, thus settling a conjecture of Bhamidi et al. (Weak disorder asymptotics in the stochastic meanfield model of distance. Ann Appl Probab 22(1):29–69, 2012). The proof relies on a decomposition of the smallest-weight tree into an initial part following invasion percolation dynamics, and a main part following branching process dynamics. The initial part has been studied in Eckhoff et al. (Long paths in first passage percolation on the complete graph I. Local PWIT dynamics. Electron. J. Probab. 25:1–45, 2020. https://doi.org/10.1214/20-EJP484); the current paper focuses on the global branching dynamics

Lines pinning lines

A line g is a transversal to a family F of convex polytopes in 3-dimensional space if it intersects every member of F. If, in addition, g is an isolated point of the space of line transversals to F, we say that F is a pinning of g. We show that any minimal pinning of a line by convex polytopes such that no face of a polytope is coplanar with the line has size at most eight. If, in addition, the polytopes are disjoint, then it has size at most six. We completely characterize configurations of disjoint polytopes that form minimal pinnings of a line.Comment: 27 pages, 10 figure

Thermal ageing phenomena and strategies towards reactivation of NO x - storage catalysts

The thermal ageing and reactivation of Ba/CeO2 and Ba/Al2O3 based NO x -storage/ reduction (NSR) catalysts was studied on model catalysts and catalyst systems at the engine. The mixed oxides BaAl2O4 and BaCeO3, which lower the storage activity, are formed during ageing above 850°C and 900°C, respectively. Interestingly, the decomposition of BaCeO3 in an atmosphere containing H2O/NO2 leads again to NO x -storage active species, as evidenced by comparison of fresh, aged and reactivated Pt-Ba/CeO2 based model catalysts. This can be technically exploited, particularly for the Ba/CeO2 catalysts, as reactivation studies on thermally aged Ba/CeO2 and Ba/Al2O3 based NSR catalysts on an engine bench showed. An on-board reactivation procedure is presented, that improved the performance of a thermally aged catalyst significantl