The Spend-It-All Region and Small Time Results for the Continuous Bomber Problem

A problem of optimally allocating partially effective ammunition $x$ to be used on randomly arriving enemies in order to maximize an aircraft's probability of surviving for time~$t$, known as the Bomber Problem, was first posed by \citet{Klinger68}. They conjectured a set of apparently obvious monotonicity properties of the optimal allocation function $K(x,t)$. Although some of these conjectures, and versions thereof, have been proved or disproved by other authors since then, the remaining central question, that $K(x,t)$ is nondecreasing in~$x$, remains unsettled. After reviewing the problem and summarizing the state of these conjectures, in the setting where $x$ is continuous we prove the existence of a ``spend-it-all'' region in which $K(x,t)=x$ and find its boundary, inside of which the long-standing, unproven conjecture of monotonicity of~$K(\cdot,t)$ holds. A new approach is then taken of directly estimating~$K(x,t)$ for small~$t$, providing a complete small-$t$ asymptotic description of~$K(x,t)$ and the optimal probability of survival

The technique of time in fiction

Thesis (M.A.)--Boston University, 1937. This item was digitized by the Internet Archive

On Optimal Allocation of a Continuous Resource Using an Iterative Approach and Total Positivity

We study a class of optimal allocation problems, including the well-known Bomber Problem, with the following common probabilistic structure. An aircraft equipped with an amount~$x$ of ammunition is intercepted by enemy airplanes arriving according to a homogenous Poisson process over a fixed time duration~$t$. Upon encountering an enemy, the aircraft has the choice of spending any amount~$0\le y\le x$ of its ammunition, resulting in the aircraft's survival with probability equal to some known increasing function of $y$. Two different goals have been considered in the literature concerning the optimal amount~$K(x,t)$ of ammunition spent: (i)~Maximizing the probability of surviving for time~$t$, which is the so-called Bomber Problem, and (ii) maximizing the number of enemy airplanes shot down during time~$t$, which we call the Fighter Problem. Several authors have attempted to settle the following conjectures about the monotonicity of $K(x,t)$: [A] $K(x,t)$ is decreasing in $t$, [B] $K(x,t)$ is increasing in $x$, and [C] the amount~$x-K(x,t)$ held back is increasing in $x$. [A] and [C] have been shown for the Bomber Problem with discrete ammunition, while [B] is still an open question. In this paper we consider both time and ammunition continuous, and for the Bomber Problem prove [A] and [C], while for the Fighter we prove [A] and [C] for one special case and [B] and [C] for another. These proofs involve showing that the optimal survival probability and optimal number shot down are totally positive of order 2 (\mbox{TP}_2) in the Bomber and Fighter Problems, respectively. The \mbox{TP}_2 property is shown by constructing convergent sequences of approximating functions through an iterative operation which preserves \mbox{TP}_2 and other properties.Comment: 2 figure

Identifying the Harm in Racial Gerrymandering Claims

This Article proceeds along two lines. First, it reviews the theories of harm set forth in the Justices\u27 various opinions, i.e., the articulated risks to individual rights that may or may not be presented by racial gerrymandering. What is learned from this survey is that Shaw and its progeny serve different purposes for different members of the Court. Four members of the Shaw, Miller v. Johnson, and United States v. Hays majorities-Chief Justice Rehnquist, along with Justices Scalia, Kennedy, and Thomas- are far more concerned with race than gerrymandering. In particular, they consider all race-based government classifications to be inherently injurious, and they appear to view the racial gerrymandering cases as a vehicle for moving the Court\u27s interpretation of the Fourteenth Amendment closer to the ideal of colorblindness.

An ectobiont-bearing foraminiferan, Bolivina pacifica, that inhabits microxic pore waters : cell-biological and paleoceanographic insights

Author Posting. © The Author(s), 2009. This is the author's version of the work. It is posted here by permission of John Wiley & Sons for personal use, not for redistribution. The definitive version was published in Environmental Microbiology 12 (2010): 2107-2119, doi:10.1111/j.1462-2920.2009.02073.x.The presence of tests (shells) in foraminifera could be taken as an indicator that this protist taxon is unlikely to possess ectosymbionts. Here, however, we describe an association between Bolivina pacifica, a foraminiferan with a calcareous test, and a rod-shaped microbe (bacterium or archaeon) that is directly associated with the pores of the foraminiferan’s test. In addition to these putative ectosymbionts, B. pacifica has previously undescribed cytoplasmic plasma membrane invaginations (PMIs). These adaptations (i.e., PMIs, ectobionts), along with the clustering of mitochondria under the pores and at the cell periphery, suggest active exchange between the host and ectobiont. The B. pacifica specimens examined were collected from sediments overlain by oxygen-depleted bottom waters (0.7 μM) of the Santa Barbara Basin (SBB; California, USA). An ultrastructural comparison between B. pacifica from the SBB and a congener (Bolivina cf. B. lanceolata) collected from well-oxygenated sediments (Florida Keys) suggests that PMIs, ectobionts, and peripherally distributed mitochondria are all factors that promote inhabitation of microxic environments by B. pacifica. The calcitic δ13C signatures of B. pacifica and a of co-occurring congener (B. argentea) that lacks ectobionts differ by >1.5‰, raising the possibility that the presence of ectobionts can affect incorporation of paleoceanographic proxies.This work was supported by a W. Storrs Cole Memorial Research Award through the Geological Society of America (to JMB), as well as by NASA Exobiology NRA-01-01-EXB-057 (to JMB), NSF MCB-0702491 (to JMB), and NSF DEB0445181 (to SSB and STG)