A Novel Gesture-based CAPTCHA Design for Smart Devices

CAPTCHAs have been widely used in Web applications to prevent service abuse. With the evolution of computing environment from desktop computing to ubiquitous computing, more and more users are accessing Web applications on smart devices where touch based interactions are dominant. However, the majority of CAPTCHAs are designed for use on computers and laptops which do not reflect the shift of interaction style very well. In this paper, we propose a novel CAPTCHA design to utilise the convenience of touch interface while retaining the needed security. This is achieved through using a hybrid challenge to take advantages of human’s cognitive abilities. A prototype is also developed and found to be more user friendly than conventional CAPTCHAs in the preliminary user acceptance test

The Radio Number of Grid Graphs

The radio number problem uses a graph-theoretical model to simulate optimal frequency assignments on wireless networks. A radio labeling of a connected graph $G$ is a function $f:V(G) \to \mathbb Z_{0}^+$ such that for every pair of vertices $u,v \in V(G)$, we have $\lvert f(u)-f(v)\rvert \ge \text{diam}(G) + 1 - d(u,v)$ where $\text{diam}(G)$ denotes the diameter of $G$ and $d(u,v)$ the distance between vertices $u$ and $v$. Let $\text{span}(f)$ be the difference between the greatest label and least label assigned to $V(G)$. Then, the \textit{radio number} of a graph $\text{rn}(G)$ is defined as the minimum value of $\text{span}(f)$ over all radio labelings of $G$. So far, there have been few results on the radio number of the grid graph: In 2009 Calles and Gomez gave an upper and lower bound for square grids, and in 2008 Flores and Lewis were unable to completely determine the radio number of the ladder graph (a 2 by $n$ grid). In this paper, we completely determine the radio number of the grid graph $G_{a,b}$ for $a,b>2$, characterizing three subcases of the problem and providing a closed-form solution to each. These results have implications in the optimization of radio frequency assignment in wireless networks such as cell towers and environmental sensors.Comment: 17 pages, 7 figure

Existence and convexity of local solutions to degenerate hessian equations

In this work, we prove the existence of local convex solution to the degenerate Hessian equationComment: corrections some typos in this versio