Williamson's many necessary existents

too many students’ congresses, are they necessary and useful?

How Many Bootstrap Replicates Are Necessary?

Phylogenetic bootstrapping (BS) is a standard technique for inferring confidence values on phylogenetic trees that is based on reconstructing many trees from minor variations of the input data, trees called replicates. BS is used with all phylogenetic reconstruction approaches, but we focus here on one of the most popular, maximum likelihood (ML). Because ML inference is so computationally dema...

Classification of emergency department visits: how many are necessary?

In a recent year, hospital emergency departments (EDs) in Rhode Island provided care to over 380,000 patients who did not require subsequent admission to an inpatient or observation bed.1 Many ED visits are for true emergencies that could not be treated in other health care settings, but there has been much discussion among providers, payers, and policy makers on whether some of these patients ...

Why many theories of shock waves are necessary: Convergence error in formally path-consistent schemes

We are interested in nonlinear hyperbolic systems in nonconservative form arising in fluid dynamics, and, for solutions containing shock waves, we investigate the convergence of finite difference schemes applied to such systems. According to Dal Maso, LeFloch, and Murat’s theory, a shock wave theory for a given nonconservative system requires prescribing a priori a family of paths in the phase ...

Necessary and Sufficient Optimality Conditions for Elliptic Control Problems with Finitely Many Pointwise State Constraints

The goal of this paper is to prove the first and second order optimality conditions for some control problems governed by semilinear elliptic equations with pointwise control constraints and finitely many equality and inequality pointwise state constraints. To carry out the analysis we formulate a regularity assumption which is equivalent to the first order optimality conditions. Though the pre...