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

A topological central point theorem

In this paper a generalized topological central point theorem is proved for maps of a simplex to finite-dimensional metric spaces. Similar generalizations of the Tverberg theorem are considered.Comment: In this version some typos were corrected after the official publicatio

Analogues of the central point theorem for families with dd-intersection property in Rd\mathbb R^d

In this paper we consider families of compact convex sets in $\mathbb R^d$ such that any subfamily of size at most $d$ has a nonempty intersection. We prove some analogues of the central point theorem and Tverberg's theorem for such families

Bounding Helly numbers via Betti numbers

We show that very weak topological assumptions are enough to ensure the existence of a Helly-type theorem. More precisely, we show that for any non-negative integers $b$ and $d$ there exists an integer $h(b,d)$ such that the following holds. If $\mathcal F$ is a finite family of subsets of $\mathbb R^d$ such that $\tilde\beta_i\left(\bigcap\mathcal G\right) \le b$ for any $\mathcal G \subsetneq \mathcal F$ and every $0 \le i \le \lceil d/2 \rceil-1$ then $\mathcal F$ has Helly number at most $h(b,d)$. Here $\tilde\beta_i$ denotes the reduced $\mathbb Z_2$-Betti numbers (with singular homology). These topological conditions are sharp: not controlling any of these $\lceil d/2 \rceil$ first Betti numbers allow for families with unbounded Helly number. Our proofs combine homological non-embeddability results with a Ramsey-based approach to build, given an arbitrary simplicial complex $K$, some well-behaved chain map $C_*(K) \to C_*(\mathbb R^d)$.Comment: 29 pages, 8 figure

Small contribution of gold mines to the ongoing tuberculosis epidemic in South Africa: a modeling-based study.

BACKGROUND: Gold mines represent a potential hotspot for Mycobacterium tuberculosis (Mtb) transmission and may be exacerbating the tuberculosis (TB) epidemic in South Africa. However, the presence of multiple factors complicates estimation of the mining contribution to the TB burden in South Africa. METHODS: We developed two models of TB in South Africa, a static risk model and an individual-based model that accounts for longer-term trends. Both models account for four populations - mine workers, peri-mining residents, labor-sending residents, and other residents of South Africa - including the size and prevalence of latent TB infection, active TB, and HIV of each population and mixing between populations. We calibrated to mine- and country-level data and used the static model to estimate force of infection (FOI) and new infections attributable to local residents in each community compared to other residents. Using the individual-based model, we simulated a counterfactual scenario to estimate the fraction of overall TB incidence in South Africa attributable to recent transmission in mines. RESULTS: We estimated that the majority of FOI in each community is attributable to local residents: 93.9% (95% confidence interval 92.4-95.1%), 91.5% (91.4-91.5%), and 94.7% (94.7-94.7%) in gold mining, peri-mining, and labor-sending communities, respectively. Assuming a higher rate of Mtb transmission in mines, 4.1% (2.6-5.8%), 5.0% (4.5-5.5%), and 9.0% (8.8-9.1%) of new infections in South Africa are attributable to gold mine workers, peri-mining residents, and labor-sending residents, respectively. Therefore, mine workers with TB disease, who constitute ~ 2.5% of the prevalent TB cases in South Africa, contribute 1.62 (1.04-2.30) times as many new infections as TB cases in South Africa on average. By modeling TB on a longer time scale, we estimate 63.0% (58.5-67.7%) of incident TB disease in gold mining communities to be attributable to recent transmission, of which 92.5% (92.1-92.9%) is attributable to local transmission. CONCLUSIONS: Gold mine workers are estimated to contribute a disproportionately large number of Mtb infections in South Africa on a per-capita basis. However, mine workers contribute only a small fraction of overall Mtb infections in South Africa. Our results suggest that curtailing transmission in mines may have limited impact at the country level, despite potentially significant impact at the mining level

Capillary electrophoresis of ultrasmall carboxylate functionalized silicon nanoparticles

Capillary electrophoresis is used to separate ultrasmall ( approximately 1 nm) carboxylate functionalized Si nanoparticles (Si-np-COO(-)) prepared via hydrosilylation with an omega-ester 1-alkene. The electropherograms show a monodisperse Si core size with one or two carboxylate groups added to the surface. On-column detection of their laser-induced fluorescence demonstrates that the individual Si-np-COO(-) have narrow emissions (full width at half maximum = 30-40 nm) with a nearly symmetric lineshape. Preparative scale electrophoresis should be a viable route for purification of the Si-np-COO(-) for further study and future applications

Advantages and Limitations of Commercially Available Electrocuting Grids for Studying Mosquito Behaviour.

Mosquito feeding behaviour plays a major role in determining malaria transmission intensity and the impact of specific prevention measures. Human Landing Catch (HLC) is currently the only method that can directly and consistently measure the biting rates of anthropophagic mosquitoes, both indoors and outdoors. However, this method exposes the participant to mosquito-borne pathogens, therefore new exposure-free methods are needed to replace it. Commercially available electrocuting grids (EGs) were evaluated as an alternative to HLC using a Latin Square experimental design in Dar es Salaam, Tanzania. Both HLC and EGs were used to estimate the proportion of human exposure to mosquitoes occurring indoors (πi), as well as its two underlying parameters: the proportion of mosquitoes caught indoors (Pi) and the proportion of mosquitoes caught between the first and last hour when most people are indoors (Pfl). HLC and EGs methods accounted for 69% and 31% of the total number of female mosquitoes caught respectively and both methods caught more mosquitoes outdoors than indoors. Results from the gold standard HLC suggest that An. gambiae s.s. in Dar es Salaam is neither exophagic nor endophagic (Pi ≈ 0.5), whereas An. arabiensis is exophagic (Pi < < 0.5). Both species prefer to feed after 10 pm when most people are indoors (Pfl > >0.5). EGs yielded estimates of Pi for An. gambiae s.s., An. arabiensis and An. coustani, that were approximately equivalent to those with HLC but significantly underestimated Pfl for An. gambiae s.s. and An. coustani. The relative sampling sensitivity of EGs declined over the course of the night (p ≤ 0.001) for all mosquito taxa except An. arabiensis. Commercial EGs sample human-seeking mosquitoes with high sensitivity both indoors and outdoors and accurately measure the propensity of Anopheles malaria vectors to bite indoors rather than outdoors. However, further modifications are needed to stabilize sampling sensitivity over a full nocturnal cycle so that they can be used to survey patterns of human exposure to mosquitoes

Astrocytes Do Not Forfeit Their Neuroprotective Roles After Surviving Intense Oxidative Stress

In order to fulfill their evolutionary role as support cells, astrocytes have to tolerate intense oxidative stress under conditions of brain injury and disease. It is well known that astrocytes exposed to mild oxidative stress are preconditioned against subsequent stress exposure in dual hit models. However, it is unclear whether severe oxidative stress leads to stress tolerance, stress exacerbation, or no change in stress resistance in astrocytes. Furthermore, it is not known whether reactive astrocytes surviving intense oxidative stress can still support nearby neurons. The data in this Brief Report suggest that primary cortical astrocytes surviving high concentrations of the oxidative toxicant paraquat are completely resistant against subsequent oxidative challenges of the same intensity. Inhibitors of multiple endogenous defenses (e.g., glutathione, heme oxygenase 1, ERK1/2, Akt) failed to abolish or even reduce their stress resistance. Stress-reactive cortical astrocytes surviving intense oxidative stress still managed to protect primary cortical neurons against subsequent oxidative injuries in neuron/astrocyte co-cultures, even at concentrations of paraquat that otherwise led to more than 80% neuron loss. Although our previous work demonstrated a lack of stress tolerance in primary neurons exposed to dual paraquat hits, here we show that intensely stressed primary neurons can resist a second hit of hydrogen peroxide. These collective findings suggest that stress-reactive astroglia are not necessarily neurotoxic, and that severe oxidative stress does not invariably lead to stress exacerbation in either glia or neurons. Therefore, interference with the natural functions of stress-reactive astrocytes might have the unintended consequence of accelerating neurodegeneration

Steam exploded pine wood burning properties with particle size dependence

Power generation using waste material from the processing of agricultural crops can be a viable biomass energy source. However, there is scant data on their burning properties and this work presents measurements of the minimum explosion concentration (MEC), flame speed, deflagration index (Kst), and peak pressure for pulverised pine wood and steam exploded pine wood (SEPW). The ISO 1 m3 dust explosion vessel was used, modified to operate on relatively coarse particles, using a hemispherical dust disperser on the floor of the vessel and an external blast of 20 bar compressed air. The pulverized material was sieved into the size fractions <500 μm, <63 μm, 63–150 μm, 150–300 μm, 300–500 μm to study the coarse particles used in biomass power generation. The MEC (Ø) was measured to be leaner for finer size fraction with greater sensitivity of explosion. The measured peak Kst was 43–122 bar m/s and the maximum turbulent flame speeds ∼1.4–5.4 m/s depending on the size distribution of the fraction. These results show that the steam exploded pine biomass was more reactive than the raw pine, due to the finer particle size for the steam exploded biomass

Angle-resolved photoemission study and first principles calculation of the electronic structure of GaTe

The electronic band structure of GaTe has been calculated by numerical atomic orbitals density-functional theory, in the local density approximation. In addition, the valence-band dispersion along various directions of the GaTe Brillouin zone has been determined experimentally by angle-resolved photoelectron spectroscopy. Along these directions, the calculated valence-band structure is in good concordance with the valence-band dispersion obtained by these measurements. It has been established that GaTe is a direct-gap semiconductor with the band gap located at the Z point, that is, at Brillouin zone border in the direction perpendicular to the layers. The valence-band maximum shows a marked \textit{p}-like behavior, with a pronounced anion contribution. The conduction band minimum arises from states with a comparable \textit{s}- \textit{p}-cation and \textit{p}-anion orbital contribution. Spin-orbit interaction appears to specially alter dispersion and binding energy of states of the topmost valence bands lying at $\Gamma$. By spin-orbit, it is favored hybridization of the topmost \textit{p}$_z$-valence band with deeper and flatter \textit{p$_x$}-\textit{p$_y$} bands and the valence-band minimum at $\Gamma$ is raised towards the Fermi level since it appears to be determined by the shifted up \textit{p$_x$}-\textit{p$_y$} bands.Comment: 7 text pages, 6 eps figures, submitted to PR