Minimum product set sizes in nonabelian groups of order pq

Let G be a nonabelian group of order pq, where p and q are distinct odd primes. We analyze the minimum product set cardinality μG(r,s)=min|AB|μG(r,s)=min|AB|, where A and B range over all subsets of G of cardinalities r and s , respectively. In this paper, we completely determine μG(r,s)μG(r,s) in the case where G has order 3p and conjecture that this result can be extended to all nonabelian groups of order pq. We also prove that for every nonabelian group of order pq there exist 1⩽r,s⩽pq1⩽r,s⩽pq such that μG(r,s)>μZ/pqZ(r,s)μG(r,s)>μ[subscript Z over pqZ(r,s)].National Science Foundation (U.S.) (Grant DMS-0447070-001)United States. National Security Agency (Grant H98230-06-1-0013

Strong Duality for a Multiple-Good Monopolist

We characterize optimal mechanisms for the multiple‐good monopoly problem and provide a framework to find them. We show that a mechanism is optimal if and only if a measure µ derived from the buyer's type distribution satisfies certain stochastic dominance conditions. This measure expresses the marginal change in the seller's revenue under marginal changes in the rent paid to subsets of buyer types. As a corollary, we characterize the optimality of grand‐bundling mechanisms, strengthening several results in the literature, where only sufficient optimality conditions have been derived. As an application, we show that the optimal mechanism for n independent uniform items each supported on [c,c,+1] is a grand‐bundling mechanism, as long as c is sufficiently large, extending Pavlov's result for two items Pavlov, 2011. At the same time, our characterization also implies that, for all c and for all sufficiently large n, the optimal mechanism for n independent uniform items supported on [c,c+1] is not a grand‐bundling mechanism

Near-optimal no-regret algorithms for zero-sum

We propose a new no-regret learning algorithm. When used against an adversary, our algorithm achieves average regret that scales as O (1/√T) with the number T of rounds. This regret bound is optimal but not rare, as there are a multitude of learning algorithms with this regret guarantee. However, when our algorithm is used by both players of a zero-sum game, their average regret scales as O (ln T/T), guaranteeing a near-linear rate of convergence to the value of the game. This represents an almost-quadratic improvement on the rate of convergence to the value of a game known to be achieved by any no-regret learning algorithm, and is essentially optimal as we show a lower bound of Ω (1/T). Moreover, the dynamics produced by our algorithm in the game setting are strongly-uncoupled in that each player is oblivious to the payoff matrix of the game and the number of strategies of the other player, has limited private storage, and is not allowed funny bit arithmetic that can trivialize the problem; instead he only observes the performance of his strategies against the actions of the other player and can use private storage to remember past played strategies and observed payoffs, or cumulative information thereof. Here, too, our rate of convergence is nearly-optimal and represents an almost-quadratic improvement over the best previously known strongly-uncoupled dynamics.National Science Foundation (U.S.) (CAREER Award CCF-0953960

Global economic burden of unmet surgical need for appendicitis

Background There is a substantial gap in provision of adequate surgical care in many low- and middle-income countries. This study aimed to identify the economic burden of unmet surgical need for the common condition of appendicitis. Methods Data on the incidence of appendicitis from 170 countries and two different approaches were used to estimate numbers of patients who do not receive surgery: as a fixed proportion of the total unmet surgical need per country (approach 1); and based on country income status (approach 2). Indirect costs with current levels of access and local quality, and those if quality were at the standards of high-income countries, were estimated. A human capital approach was applied, focusing on the economic burden resulting from premature death and absenteeism. Results Excess mortality was 4185 per 100 000 cases of appendicitis using approach 1 and 3448 per 100 000 using approach 2. The economic burden of continuing current levels of access and local quality was US $92 492 million using approach 1 and$73 141 million using approach 2. The economic burden of not providing surgical care to the standards of high-income countries was $95 004 million using approach 1 and$75 666 million using approach 2. The largest share of these costs resulted from premature death (97.7 per cent) and lack of access (97.0 per cent) in contrast to lack of quality. Conclusion For a comparatively non-complex emergency condition such as appendicitis, increasing access to care should be prioritized. Although improving quality of care should not be neglected, increasing provision of care at current standards could reduce societal costs substantially