Divergence of sample quantiles

We show that the left (right) sample quantile tends to the left (right) distribution quantile at p in [0,1], if the left and right quantiles are identical at p. We show that the sample quantiles diverge almost surely otherwise. The latter can be considered as a generalization of the well-known result that the sum of a random sample of a fair coin with 1 denoting heads and -1 denoting tails is 0 infinitely often. In the case that the sample quantiles do not converge we show that the limsup is the right quantile and the liminf is the left quantile

Algorithm and Related Application for Smart Wearable Devices to Reduce the Risk of Death and Brain Damage in Diabetic Coma

Diabetes is an epidemic disease of the 21st century and is growing globally. Although, final diabetes treatments and cure are still on research phase, related complications of diabetes endanger life of diabetic patients. Diabetic coma which happens with extreme high or low blood glucose is one of the risk factor for diabetic patients and if it remains unattended will lead to patient death or permanent brain damage. To reduce the risk of such deaths or damages, a novel algorithm for wearable devices application, especially for smart watches are proposed. Such application can inform the patients relatives or emergency centers, if the person falls in coma or irresponsive condition based on readouts from smart watches sensors including mobility, heart rate and skin moisture. However; such an application is not a final solution to detect all types of coma, but it potentially could save lives of many patients, if widely used among the diabetic patients around the world.Comment: 4 pages, 1 figur

The Pure derived category of quasi-coherent sheaves

Let X be a quasi-compact and quasi-separated (not necessarily semiseparated) scheme. The category QcoX of all quasi-coherent sheaves of OX-modules has several diferent pure derived categories. Recently, categorical pure derived categories of X have been studied in more details. In this work, we focus on the geometrical purity and find replacements for geometrical pure derived categories of X.Comment: 0

A Survey of Bandwidth and Latency Enhancement Approaches for Mobile Cloud Game Multicasting

Among mobile cloud applications, mobile cloud gaming has gained a significant popularity in the recent years. In mobile cloud games, textures, game objects, and game events are typically streamed from a server to the mobile client. One of the challenges in cloud mobile gaming is how to efficiently multicast gaming contents and updates in Massively Multi-player Online Games (MMOGs). This report surveys the state of art techniques introduced for game synchronization and multicasting mechanisms to decrease latency and bandwidth consumption, and discuss several schemes that have been proposed in this area that can be applied to any networked gaming context. From our point of view, gaming applications demand high interactivity. Therefore, concentrating on gaming applications will eventually cover a wide range of applications without violating the limited scope of this survey.Comment: 4 pages, Technical Report, School of Computing Science, Simon Fraser University, 201

Two Metropolis-Hastings algorithms for posterior measures with non-Gaussian priors in infinite dimensions

We introduce two classes of Metropolis-Hastings algorithms for sampling target measures that are absolutely continuous with respect to non-Gaussian prior measures on infinite-dimensional Hilbert spaces. In particular, we focus on certain classes of prior measures for which prior-reversible proposal kernels of the autoregressive type can be designed. We then use these proposal kernels to design algorithms that satisfy detailed balance with respect to the target measures. Afterwards, we introduce a new class of prior measures, called the Bessel-K priors, as a generalization of the gamma distribution to measures in infinite dimensions. The Bessel-K priors interpolate between well-known priors such as the gamma distribution and Besov priors and can model sparse or compressible parameters. We present concrete instances of our algorithms for the Bessel-K priors in the context of numerical examples in density estimation, finite-dimensional denoising and deconvolution on the circle.Comment: Minor revision

Quantiles symmetry

This paper finds a symmetry relation (between quantiles of a random variable and its negative) that is intuitively appealing. We show this symmetry is quite useful in finding new relations for quantiles, in particular an equivariance property for quantiles under continuous decreasing transformations

Quantiles Equivariance

It is widely claimed that the quantile function is equivariant under increasing transformations. We show by a counterexample that this is not true (even for strictly increasing transformations). However, we show that the quantile function is equivariant under left continuous increasing transformations. We also provide an equivariance relation for continuous decreasing transformations. In the case that the transformation is not continuous, we show that while the transformed quantile at p can be arbitrarily far from the quantile of the transformed at p (in terms of absolute difference), the probability mass between the two is zero. We also show by an example that weighted definition of the median is not equivariant under even strictly increasing continuous transformations

A Sharp Inequality for Conditional Distribution of the First Exit Time of Brownian Motion

Let $U$ be a domain, convex in $x$ and symmetric about the y-axis, which is contained in a centered and oriented rectangle $R$. \linebreak If $\tau_A$ is the first exit time of Brownian motion from $A$ and $A^+=A\cap \{(x,y):x>0\}$, it is proved that $P^z(\tau_{U^+}>s\mid \tau_{R^+}>t)\leq P^z(\tau_{U}>s\mid \tau_{R}>t)$ for every $s,t>0$ and every $z\in U^+$.Comment: 12 page

Utilizing wind in spatial covariance

This work develops a covariance function which allows for a stronger spatial correlation for pairs of points in the direction of a vector such as wind and weaker for pairs which are perpendicular to it. It derives a simple covariance function by stretching the space along the wind axes (upwind and across wind axes). It is shown that this covariance function is anisotropy in the original space and the functions is explicitly calculated

Flat quasi-coherent sheaves of finite cotorsion dimension

Let X be e quasi-compact and semi-separated scheme. If every at quasi- coherent sheaf has finite cotorsion dimension, we prove that X is n-perfect for some n > 0. If X is coherent and n-perfect(not necessarily of finite krull dimension), we prove that every at quasi-coherent sheaf has finite pure injective dimension. Also, we show that there is an equivalence K(PinfX)---> D(FlatX) of homotopy categories, whenever K(PinfX) is the homotopy category of pure injective at quasi-coherent sheaves and D(FlatX) is the pure derived category of at quasi-coherent sheaves