English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Open

A problem of optimally allocating partially effective ammunition $x$ to beused on randomly arriving enemies in order to maximize an aircraft'sprobability of surviving for time~$t$, known as the Bomber Problem, was firstposed by \citet{Klinger68}. They conjectured a set of apparently obviousmonotonicity properties of the optimal allocation function $K(x,t)$. Althoughsome of these conjectures, and versions thereof, have been proved or disprovedby other authors since then, the remaining central question, that $K(x,t)$ isnondecreasing in~$x$, remains unsettled. After reviewing the problem andsummarizing the state of these conjectures, in the setting where $x$ iscontinuous 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, unprovenconjecture of monotonicity of~$K(\cdot,t)$ holds. A new approach is then takenof 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 Spend-It-All Region and Small Time Results for the Continuous Bomber Problem