On the maximum tolerable noise of k-input gates for reliable computation by formulas

We determine the precise threshold of component noise below which formulas composed of odd degree components can reliably compute all Boolean functions

Signal propagation and noisy circuits

The information carried by a signal decays when the signal is corrupted by random noise. This occurs when a message is transmitted over a noisy channel, as well as when a noisy component performs computation. We first study this signal decay in the context of communication and obtain a tight bound on the rate at which information decreases as a signal crosses a noisy channel. We then use this information theoretic result to obtain depth lower bounds in the noisy circuit model of computation defined by von Neumann. In this model, each component fails (produces 1 instead of 0 or vice-versa) independently with a fixed probability, and yet the output of the circuit is required to be correct with high probability. Von Neumann showed how to construct circuits in this model that reliably compute a function and are no more than a constant factor deeper than noiseless circuits for the function. We provide a lower bound on the multiplicative increase in circuit depth necessary for reliable computation, and an upper bound on the maximum level of noise at which reliable computation is possible

Information Theory and Noisy Computation

We report on two types of results. The first is a study of the rate of decay of information carried by a signal which is being propagated over a noisy channel. The second is a series of lower bounds on the depth, size, and component reliability of noisy logic circuits which are required to compute some function reliably. The arguments used for the circuit results are information-theoretic, and in particular, the signal decay result is essential to the depth lower bound. Our first result can be viewed as a quantified version of the data processing lemma, for the case of Boolean random variables

Designing Effective Questions for Classroom Response System Teaching

Classroom response systems (CRSs) can be potent tools for teaching physics. Their efficacy, however, depends strongly on the quality of the questions used. Creating effective questions is difficult, and differs from creating exam and homework problems. Every CRS question should have an explicit pedagogic purpose consisting of a content goal, a process goal, and a metacognitive goal. Questions can be engineered to fulfil their purpose through four complementary mechanisms: directing students' attention, stimulating specific cognitive processes, communicating information to instructor and students via CRS-tabulated answer counts, and facilitating the articulation and confrontation of ideas. We identify several tactics that help in the design of potent questions, and present four "makeovers" showing how these tactics can be used to convert traditional physics questions into more powerful CRS questions.Comment: 11 pages, including 6 figures and 2 tables. Submitted (and mostly approved) to the American Journal of Physics. Based on invited talk BL05 at the 2005 Winter Meeting of the American Association of Physics Teachers (Albuquerque, NM

Case Study on 3D Printing Implementation Strategies

The construction industry is one of the slowest to adapt and utilize new technologies. As a result, the construction industry has remained stagnant while almost all others are improving efficiency. It is becoming incredibly important for the construction industry to adapt to new technologies in order to counter the current worldwide shortage in skilled labor. 3D printing is one of the most advanced technologies that can help increase efficiency and reduce cost across all sectors of the construction industry. This paper will analyze implementation strategies used on three different construction projects using 3D printing and will evaluate what made them successful and what challenges were faced. When analyzing the implementation strategies, this paper will focus on what collaboration between the design and construction team was most effective, and how the use of 3D printing changed the construction process. The purpose of this paper is to provide suggestions regarding witch of the strategies are most effective for differing project types

Resolución de problemas basada en el análisis : hacer del análisis y del razonamiento el foco de la enseñanza de la física

Usando los resultados provenientes de la investigación educativa en áreas tales como concepciones alternativas, diferencias entre expertos y novatos, adquisición de esquemas, sobrecarga cognitiva y análisis jerárquico, hemos desarrollado un marco para pensar sobre la organización del conocimiento y su uso para la comunicación, y sobre el razonamiento y la resolución de problemas. Basados en este marco y en nuestras reflexiones sobre el aprendizaje y la enseñanza, hemos creado una metodología didáctica llamada resolución de problemas basada en el análisis, diseñada para promover tanto la comprensión conceptual profunda como la capacidad de resolver problemas eficientemente a través de enfocarse en el análisis y el razonamiento como un puente entre las dos. Presentamos aquí una progresión de objetivos de enseñanza que lleva a los estudiantes de ser aprendices novatos a ser «resolvedores» de problemas más eficientes y más reflexivos. Para cada objetivo, resumimos los resultados relevantes de la investigación cognitiva y describimos las recomendaciones didácticas que se pueden derivar de ella. Luego hicimos la lista de algunas estrategias didácticas útiles para promover ese objetivo en particular y, finalmente, indicamos algunas prácticas de aula que deberían ayudar a que los estudiantes progresen desde esta perspectiva.Summary. Using the results of educational research coming from such areas as alternative conceptions, expert-novice differences, schema acquisition, cognitive overload, and hierarchical analysis, we have developed a framework for thinking about knowledge organization and its use for communication, reasoning, and problem solving. Based on this framework, as well as on our own reflections on learning and teaching, we have created an instructional approach called analysis-based problem solving that is designed to promote both deep, conceptual understanding and proficient problem-solving ability, by focusing on analysis and reasoning as a bridge between them both. We report here on a progression of instructional goals that takes students from novice learners to more proficient, more thoughtful problem solvers. For each goal, we summarize the relevant cognitive research results, and describe the pedagogy that can be derived from it. Then, we list some modes of instruction useful for promoting that particular goal, and finally, indicate some classroom practices that should help students make progress within the approach

The Adversarial Noise Threshold for Distributed Protocols

We consider the problem of implementing distributed protocols, despite adversarial channel errors, on synchronous-messaging networks with arbitrary topology. In our first result we show that any $n$-party $T$-round protocol on an undirected communication network $G$ can be compiled into a robust simulation protocol on a sparse ($\mathcal{O}(n)$ edges) subnetwork so that the simulation tolerates an adversarial error rate of $\Omega\left(\frac{1}{n}\right)$; the simulation has a round complexity of $\mathcal{O}\left(\frac{m \log n}{n} T\right)$, where $m$ is the number of edges in $G$. (So the simulation is work-preserving up to a $\log$ factor.) The adversary's error rate is within a constant factor of optimal. Given the error rate, the round complexity blowup is within a factor of $\mathcal{O}(k \log n)$ of optimal, where $k$ is the edge connectivity of $G$. We also determine that the maximum tolerable error rate on directed communication networks is $\Theta(1/s)$ where $s$ is the number of edges in a minimum equivalent digraph. Next we investigate adversarial per-edge error rates, where the adversary is given an error budget on each edge of the network. We determine the exact limit for tolerable per-edge error rates on an arbitrary directed graph. However, the construction that approaches this limit has exponential round complexity, so we give another compiler, which transforms $T$-round protocols into $\mathcal{O}(mT)$-round simulations, and prove that for polynomial-query black box compilers, the per-edge error rate tolerated by this last compiler is within a constant factor of optimal.Comment: 23 pages, 2 figures. Fixes mistake in theorem 6 and various typo

Doctors at Risk: A Problem As Viewed by Decision Analysis

The authors closely analyze a case in which a Peer Review Organization cited a physician for treatment with potential for significant adverse effect. They also critique the regulatory scheme under which peer review occurs and conclude that such regulation interferes with physicians\u27 primary obligations, fails to encourage cost-effective behavior and may decrease the quality of medical care