Criticality of the Exponential Rate of Decay for the Largest Nearest Neighbor Link in Random Geometric Graph

Let n points be placed independently in d-dimensional space according to the densities $f(x) = A_d e^{-\lambda \|x\|^{\alpha}}, \lambda > 0, x \in \Re^d, d \geq 2.$ Let $d_n$ be the longest edge length for the nearest neighbor graph on these points. We show that $(\log(n))^{1-1/\alpha}d_n -b_n$ converges weakly to the Gumbel distribution where $b_n \sim \log \log n.$ We also show that the strong law result, % \lim_{n \to \infty} \frac{(\lambda^{-1}\log(n))^{1-1/\alpha}d_n}{\sqrt{\log \log n}} \to \frac{d}{\alpha \lambda}, a.s. % Thus, the exponential rate of decay i.e. $\alpha = 1$ is critical, in the sense that for $\alpha > 1, d_n \to 0,$ where as $\alpha < 1, d_n \to \infty$ a.s. as $n \to \infty.$Comment: Communicated to 'Stochastic Processes and Their Applications'. Sep. 11, 2006: replaced paper uploaded on Apr. 27, 2006 by a corrected version; errors/corrections found by the authors themselve

Nonuniform random geometric graphs with location-dependent radii

We propose a distribution-free approach to the study of random geometric graphs. The distribution of vertices follows a Poisson point process with intensity function $nf(\cdot)$, where $n\in \mathbb{N}$, and $f$ is a probability density function on $\mathbb{R}^d$. A vertex located at $x$ connects via directed edges to other vertices that are within a cut-off distance $r_n(x)$. We prove strong law results for (i) the critical cut-off function so that almost surely, the graph does not contain any node with out-degree zero for sufficiently large $n$ and (ii) the maximum and minimum vertex degrees. We also provide a characterization of the cut-off function for which the number of nodes with out-degree zero converges in distribution to a Poisson random variable. We illustrate this result for a class of densities with compact support that have at most polynomial rates of decay to zero. Finally, we state a sufficient condition for an enhanced version of the above graph to be almost surely connected eventually.Comment: Published in at http://dx.doi.org/10.1214/11-AAP823 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Drugs for preventing red blood cell dehydration in people with sickle cell disease.

BACKGROUND: Sickle cell disease is an inherited disorder of hemoglobin, resulting in abnormal red blood cells. These are rigid and may block blood vessels leading to acute painful crises and other complications. Recent research has focused on therapies to rehydrate the sickled cells by reducing the loss of water and ions from them. Little is known about the effectiveness and safety of such drugs. This is an updated version of a previously published review. OBJECTIVES: To assess the relative risks and benefits of drugs to rehydrate sickled red blood cells. SEARCH METHODS: We searched the Cochrane Cystic Fibrosis and Genetic Disorders Group\u27s Haemoglobinopathies Trials Register.Last search of the Group\u27s Trials Register: 28 November 2015. SELECTION CRITERIA: Randomized or quasi-randomized controlled trials of drugs to rehydrate sickled red blood cells compared to placebo or an alternative treatment. DATA COLLECTION AND ANALYSIS: Both authors independently selected studies for inclusion, assessed study quality and extracted data. MAIN RESULTS: Of the 51 studies identified, three met the inclusion criteria. The first study tested the effectiveness of zinc sulphate to prevent sickle cell-related crises in a total of 145 participants and showed a significant reduction in painful crises over one and a half years, mean difference -2.83 (95% confidence interval -3.51 to -2.15). However, analysis was restricted due to limited statistical data. Changes to red cell parameters and blood counts were inconsistent. No serious adverse events were noted in the study.The second study was a Phase II dose-finding study of senicapoc (a Gardos channel blocker) compared to placebo. Compared to the placebo group the high dose senicapoc showed significant improvement in change in hemoglobin level, number and proportion of dense red blood cells, red blood cell count and indices and hematocrit. The results with low-dose senicapoc were similar to the high-dose senicapoc group but of lesser magnitude. There was no difference in the frequency of painful crises between the three groups. A subsequent Phase III study of senicapoc was terminated early since there was no difference observed between the treatment and control groups in the primary end point of painful crises. AUTHORS\u27 CONCLUSIONS: While the results of zinc for reducing sickle-related crises are encouraging, larger and longer-term multicenter studies are needed to evaluate the effectiveness of this therapy for people with sickle cell disease.While the Phase II and the prematurely terminated phase III studies of senicapoc showed that the drug improved red cell survival (depending on dose), this did not lead to fewer painful crises.We will continue to run searches to identify any potentially relevant trials; however, we do not plan to update other sections of the review until new trials are published

Achieving Non-Zero Information Velocity in Wireless Networks

In wireless networks, where each node transmits independently of other nodes in the network (the ALOHA protocol), the expected delay experienced by a packet until it is successfully received at any other node is known to be infinite for signal-to-interference-plus-noise-ratio (SINR) model with node locations distributed according to a Poisson point process. Consequently, the information velocity, defined as the limit of the ratio of the distance to the destination and the time taken for a packet to successfully reach the destination over multiple hops, is zero, as the distance tends to infinity. A nearest neighbor distance based power control policy is proposed to show that the expected delay required for a packet to be successfully received at the nearest neighbor can be made finite. Moreover, the information velocity is also shown to be non-zero with the proposed power control policy. The condition under which these results hold does not depend on the intensity of the underlying Poisson point process.Comment: to appear in Annals of Applied Probabilit

Drugs for preventing red blood cell dehydration in people with sickle cell disease.

BACKGROUND: Sickle cell disease is an inherited disorder of hemoglobin, resulting in abnormal red blood cells. These are rigid and may block blood vessels leading to acute painful crises and other complications. Recent research has focused on therapies to rehydrate the sickled cells by reducing the loss of water and ions from them. Little is known about the effectiveness and safety of such drugs. This is an updated version of a previously published review. OBJECTIVES: To assess the relative risks and benefits of drugs to rehydrate sickled red blood cells. SEARCH METHODS: We searched the Cochrane Cystic Fibrosis and Genetic Disorders Group\u27s Haemoglobinopathies Trials Register. We also searched online trials registries for any ongoing trials (01 July 2018).Last search of the Group\u27s Haemoglobinopathies Trials Register: 08 October 2018. SELECTION CRITERIA: Randomized or quasi-randomized controlled trials of drugs to rehydrate sickled red blood cells compared to placebo or an alternative treatment. DATA COLLECTION AND ANALYSIS: Both authors independently selected studies for inclusion, assessed study quality and extracted data. MAIN RESULTS: Of the 51 studies identified, three met the inclusion criteria, including 524 people with sickle cell disease aged between 12 and 65 years of age. One study tested the effectiveness of zinc sulphate as compared to placebo and the remaining two assessed senicapoc versus placebo. No deaths were seen in any of the studies (low-quality evidence). The zinc sulphate study showed a significant reduction in painful crises (in a total of 145 participants) over one and a half years, mean difference -2.83 (95% confidence interval -3.51 to -2.15) (moderate-quality evidence). However, analysis was restricted due to limited statistical data. Changes to red blood cell parameters and blood counts were inconsistent (very low-quality evidence). No serious adverse events were noted in the study. The Phase II dose-finding study of senicapoc (a Gardos channel blocker) compared to placebo showed that the high dose senicapoc showed significant improvement in change in hemoglobin level, the number and proportion of dense red blood cells, red blood cell count and indices and hematocrit value (very low-quality evidence). The results with low-dose senicapoc were similar to the high-dose senicapoc group but of lesser magnitude. There was no difference in the frequency of painful crises between the three groups (low-quality evidence). A subsequent Phase III study of senicapoc was terminated early since there was no difference observed between the treatment and control groups in the primary end point of painful crises. AUTHORS\u27 CONCLUSIONS: While the results of zinc for reducing sickle-related crises are encouraging, larger and longer-term multicenter studies are needed to evaluate the effectiveness of this therapy for people with sickle cell disease.While the Phase II and the prematurely terminated phase III studies of senicapoc showed that the drug improved red blood cell survival (depending on dose), this did not lead to fewer painful crises.Given this is no longer an active area of research, this review will no longer be regularly updated

Limit laws for k-coverage of paths by a Markov-Poisson-Boolean model

Let P := {X_i,i >= 1} be a stationary Poisson point process in R^d, {C_i,i >= 1} be a sequence of i.i.d. random sets in R^d, and {Y_i^t; t \geq 0, i >= 1} be i.i.d. {0,1}-valued continuous time stationary Markov chains. We define the Markov-Poisson-Boolean model C_t := {Y_i^t(X_i + C_i), i >= 1}. C_t represents the coverage process at time t. We first obtain limit laws for k-coverage of an area at an arbitrary instant. We then obtain the limit laws for the k-coverage seen by a particle as it moves along a one-dimensional path.Comment: 1 figure. 24 Pages. Accepted at Stochastic Models. Theorems 6 and 7 corrected. Theorem 9 and Appendix adde