Lower Bounds for Sparse Recovery

We consider the following k-sparse recovery problem: design an m x n matrix A, such that for any signal x, given Ax we can efficiently recover x' satisfying ||x-x'||_1 <= C min_{k-sparse} x"} ||x-x"||_1. It is known that there exist matrices A with this property that have only O(k log (n/k)) rows. In this paper we show that this bound is tight. Our bound holds even for the more general /randomized/ version of the problem, where A is a random variable and the recovery algorithm is required to work for any fixed x with constant probability (over A).Comment: 11 pages. Appeared at SODA 201

Sums and differences of correlated random sets

Many fundamental questions in additive number theory (such as Goldbach's conjecture, Fermat's last theorem, and the Twin Primes conjecture) can be expressed in the language of sum and difference sets. As a typical pair of elements contributes one sum and two differences, we expect that $|A-A| > |A+A|$ for a finite set $A$. However, in 2006 Martin and O'Bryant showed that a positive proportion of subsets of $\{0, \dots, n\}$ are sum-dominant, and Zhao later showed that this proportion converges to a positive limit as $n \to \infty$. Related problems, such as constructing explicit families of sum-dominant sets, computing the value of the limiting proportion, and investigating the behavior as the probability of including a given element in $A$ to go to zero, have been analyzed extensively. We consider many of these problems in a more general setting. Instead of just one set $A$, we study sums and differences of pairs of \emph{correlated} sets $(A,B)$. Specifically, we place each element $a \in \{0,\dots, n\}$ in $A$ with probability $p$, while $a$ goes in $B$ with probability $\rho_1$ if $a \in A$ and probability $\rho_2$ if $a \not \in A$. If $|A+B| > |(A-B) \cup (B-A)|$, we call the pair $(A,B)$ a \emph{sum-dominant $(p,\rho_1, \rho_2)$-pair}. We prove that for any fixed $\vec{\rho}=(p, \rho_1, \rho_2)$ in $(0,1)^3$, $(A,B)$ is a sum-dominant $(p,\rho_1, \rho_2)$-pair with positive probability, and show that this probability approaches a limit $P(\vec{\rho})$. Furthermore, we show that the limit function $P(\vec{\rho})$ is continuous. We also investigate what happens as $p$ decays with $n$, generalizing results of Hegarty-Miller on phase transitions. Finally, we find the smallest sizes of MSTD pairs.Comment: Version 1.0, 19 pages. Keywords: More Sum Than Difference sets, correlated random variables, phase transitio

Pediatric Resuscitation: Evaluation of a Clinical Curriculum

Objective: To assess the impact of a 6-hour pediatric resuscitation curriculum on the comfort levels of resident physicians’ evaluation and treatment of critically ill pediatric patients. Methods: An evaluation instrument assessed resident comfort levels, measured on a seven digit Likert scale ranging from significantly uncomfortable to significantly comfortable, in 13 areas of pediatric resuscitation. To complete the curriculum, residents had to demonstrate proficiency in knowledge and procedural skills during mock resuscitation scenarios and on both written and oral examinations. Results: Thirty-one residents participated in the study: 51.6% were pediatric, 12.9% were medicine/pediatric and 35.5% were emergency medicine residents. Participants in the curriculum had little previous experience with pediatric resuscitation (83% had been involved in five or fewer pediatric resuscitations). In all 13 areas of pediatric resuscitation tested, residents reported improvement in comfort levels following the course (p<0.002; Wilcoxon Signed Rank Tests). The most significant changes were observed for the following items: resuscitation of pulseless arrest, performance of cardioversion and defibrillation, performance of intraosseous needle insertion, and drug selection and dosing for rapid sequence intubation. Fewer than 48% of learners rated themselves as comfortable in these areas prior to training, but after completion, more than 80% rated themselves in the comfortable range. All residents but one received passing scores on their written examinations (97%). During the mock resuscitation scenarios and oral examination, 100% of the residents were assessed to have ‘completely’ met the learning objectives and critical actions Conclusion: Implementation of a pediatric resuscitation curriculum improves pediatric and emergency medicine residents’ comfort with the evaluation and treatment of critically ill pediatric patients. This curriculum can be used in residency training to document the acquisition of core competencies, knowledge and procedural skills needed for the evaluation and treatment of the critically ill child. The results reported in this study support using this model of instructional design to implement educational strategies, which will meet the requirements of graduate education

The Ottoman empire in early modern Austrian history : assessment and perspectives

This paper discusses the over-focalisation on the Mediterranean area of historians of the early modern Euro-Ottoman relationship and it offers a critical assessment of the numerous studies conducted by historians of the Habsburg Monarchy over thirty years. It shows that the histories of the Austrian monarchy and of the Ottoman Empire were interdependent and that war is a marginal element in their relationship. This paper emphasises the political use of the Ottoman history by Austrian scholars from Hammer-Purgstall’s essential enterprise to the violent contestation of Samuel Huntington and his civilizational pattern. Cultural history, trade and diplomacy appear as the three ways of the Austrian historiographical shift, which nevertheless calls nowadays for a more pragmatic approach