Latin transversals of rectangular arrays

Let m and n be integers, $2 \leq m \leq n$. An m by n array consists of mn cells, arranged in m rows and n columns, and each cell contains exactly one symbol. A transversal of an array consists of m cells, one from each row and no two from the same column. A latin transversal is a transversal in which no symbol appears more than once. We will establish a sufficient condition that a 3 by n array has a latin transversal.Comment: Theorem 4 has been added, which provides a lower bound on L(m,n

Gresham\u27s Law: Algorithm Drives Out Thought

Gresham\u27s law in economics states, Bad money drives good money out of circulation. An application of this law in mathematical pedagogy states that Algorithm drives out thought. While universities are ideally places where classes are meant to develop students\u27 independence and critical thinking skills, often mathematics courses reflect this altered version of Gresham\u27s law. This paper demonstrates the ways traditional mathematical pedagogy has held up Gresham\u27s law and presents several suggestions for ways to change this approach to mathematical education to focus more on critical thinking without sacrificing the necessity of algorithm

Mitigating the Effect of Free-Riders in BitTorrent using Trusted Agents

Even though Peer-to-Peer (P2P) systems present a cost-effective and scalable solution to content distribution, most entertainment, media and software, content providers continue to rely on expensive, centralized solutions such as Content Delivery Networks. One of the main reasons is that the current P2P systems cannot guarantee reasonable performance as they depend on the willingness of users to contribute bandwidth. Moreover, even systems like BitTorrent, which employ a tit-for-tat protocol to encourage fair bandwidth exchange between users, are prone to free-riding (i.e. peers that do not upload). Our experiments on PlanetLab extend previous research (e.g. LargeViewExploit, BitTyrant) demonstrating that such selfish behavior can seriously degrade the performance of regular users in many more scenarios beyond simple free-riding: we observed an overhead of up to 430% for 80% of free-riding identities easily generated by a small set of selfish users. To mitigate the effects of selfish users, we propose a new P2P architecture that classifies peers with the help of a small number of {\em trusted nodes} that we call Trusted Auditors (TAs). TAs participate in P2P download like regular clients and detect free-riding identities by observing their neighbors' behavior. Using TAs, we can separate compliant users into a separate service pool resulting in better performance. Furthermore, we show that TAs are more effective ensuring the performance of the system than a mere increase in bandwidth capacity: for 80\% of free-riding identities a single-TA system has a 6\% download time overhead while without the TA and three times the bandwidth capacity we measure a 100\% overhead

Aequitas: A Trusted P2P System for Paid Content Delivery

P2P file-sharing has been recognized as a powerful and efficient distribution model due to its ability to leverage users' upload bandwidth. However, companies that sell digital content on-line are hesitant to rely on P2P models for paid content distribution due to the free file-sharing inherent in P2P models. In this paper we present Aequitas, a P2P system in which users share paid content anonymously via a layer of intermediate nodes. We argue that with the extra anonymity in Aequitas, vendors could leverage P2P bandwidth while effectively maintaining the same level of trust towards their customers as in traditional models of paid content distribution. As a result, a content provider could reduce its infrastructure costs and subsequently lower the costs for the end-users. The intermediate nodes are incentivized to contribute their bandwidth via electronic micropayments. We also introduce techniques that prevent the intermediate nodes from learning the content of the files they help transmit. In this paper we present the design of our system, an analysis of its properties and an implementation and experimental evaluation. We quantify the value of the intermediate nodes, both in terms of efficiency and their effect on anonoymity. We argue in support of the economic and technological merits of the system