Network-based Metric for Measuring Combat Effectiveness

A conceptual definition of combat effectiveness is the overall capability of a force to produce a desiredoutcome from combat against an enemy force. An ability to measure combat effectiveness is critical to strategic andtactical decision making; however, it is a challenging task to develop an operational metric for combat effectivenessdue to the large complexity presented by the rich context of a combat environment. The present paper contendsthat, under a direct fire engagement, combat effectiveness can be reasonably assessed by the prevalence of attack opportunities a given force creates in a combat environment. The paper proposes a method to quantitatively measurecombat effectiveness of a military force in a direct fire engagement environment. The proposed metric is basedon a meta-network representation that captures various aspects of a combat environment. Using a meta-networkrepresentation, two types of basic unit structures of attack opportunity – isolated and networked – are identified,which are then used as a basic element for measuring combat effectiveness. Prevalence of network motifs in anetworked combat environment and availability of attack opportunities are computed as a measure of a militaryforce’s combat effectiveness.Defence Science Journal, 2014, 64(2), pp. 115-122. DOI: http://dx.doi.org/10.14429/dsj.64.553

On local well-posedness of nonlinear dispersive equations with partially regular data

We revisit the local well-posedness theory of nonlinear Schr\"odinger and wave equations in Sobolev spaces $H^s$ and $\dot{H}^s$, $0< s\leq 1$. The theory has been well established over the past few decades under Sobolev initial data regular with respect to all spatial variables. But here, we reveal that the initial data do not need to have complete regularity like Sobolev spaces, but only partially regularity with respect to some variables is sufficient. To develop such a new theory, we suggest a refined Strichartz estimate which has a different norm for each spatial variable. This makes it possible to extract a different integrability/regularity of the data from each variable.Comment: To appear in J. Differential Equations, 15 page

Pointwise convergence of sequential Schr\"odinger means

We study pointwise convergence of the fractional Schr\"odinger means along sequences $t_n$ which converge to zero. Our main result is that bounds on the maximal function $\sup_{n} |e^{it_n(-\Delta)^{\alpha/2}} f|$ can be deduced from those on $\sup_{0<t\le 1} |e^{it(-\Delta)^{\alpha/2}} f|$ when $\{t_n\}$ is contained in the Lorentz space $\ell^{r,\infty}$. Consequently, our results provide seemingly optimal results in higher dimensions, which extend the recent work of Dimou-Seeger, and Li-Wang-Yan to higher dimensions. Our approach based on a localization argument also works for other dispersive equations and provides alternative proofs of previous results on sequential convergence