# Optimized Passive Defense Measures via IR Imaging
## 4 Global Radiation Stability  
In this section, we provide algebraic discussion of the stability of a radiating system. Note that the radiation is well characterized via the sign of the discriminant \(D\) as in Equation (17). To simplify the subsequent presentation, let us define the flow variable \(\mu\) and discriminant prefactor \(\varepsilon\) as  
\[\mu = exp(s),\] \[\varepsilon = \frac{\lambda^{12}T^{2}}{A^{2}s}\frac{(e^{s}-1)^{6}}{e^{s}}. \tag{19}\]  
Therefore, from Equation (17), it follows that the sign of \(D\) can be described according as the roots of the cubic equation  
\[\alpha\mu^{3}+\beta\mu^{2}+\gamma\mu+\delta-\varepsilon D=0. \tag{20}\]  
To determine the radiation zones, let \(\{\mu_{1},\mu_{2},\mu_{3}\}\) be the roots of the above Equation (20). In this case, we have the following identities  
\[\mu_{1}+\mu_{2}+\mu_{3} = -\frac{\beta}{\alpha},\] \[\mu_{1}\mu_{2}+\mu_{2}\mu_{3}+\mu_{3}\mu_{1} = \frac{\gamma}{\alpha},\] \[\mu_{1}\mu_{2}\mu_{3} = \frac{\varepsilon D-\delta}{\alpha}. \tag{21}\]  
Figure 4: A schematic classification of the thermal radiation energy density profile in the \(\{\lambda,T\}\)-plane.  
Thus, from the last relation as in Equation (21), we can characterize the signature of \(D\) as per the relation  
\[D=\frac{\alpha\mu_{1}\mu_{2}\mu_{3}+\delta}{\varepsilon}. \tag{22}\]  
Hereby, when all the roots \(\{\mu_{1},\mu_{2},\mu_{3}\}\) of the Equation (20)are positive, we have a positive discriminant \(D\). This happens whenever the following conditions are satisfied  
\[\alpha\mu_{1}\mu_{2}\mu_{3}+\delta\quad\text{is positive for} \quad\varepsilon>0,\] \[\alpha\mu_{1}\mu_{2}\mu_{3}+\delta\quad\text{is negative for} \quad\varepsilon<0. \tag{23}\]  
Form the above Equation (19), it follows that the parameter \(\varepsilon\) has the same sign as the one of \(s\). Further, it is worth mentioning that we can have an optimized radiation energy whenever the Equation (23) is satisfied in the \(\lambda T\)-plane.  
On the other hand, we have a saddle point if \(D\) takes a negative value. That is, the roots \(\{\mu_{1},\mu_{2},\mu_{3}\}\) obey the contrary requirement to the Equation (23). Namely, we have a saddle point for a given value of \(s\) whenever the roots \(\{\mu_{1},\mu_{2},\mu_{3}\}\) satisfy  
\[\alpha\mu_{1}\mu_{2}\mu_{3}+\delta\quad\text{is negative for} \quad\varepsilon>0,\] \[\alpha\mu_{1}\mu_{2}\mu_{3}+\delta\quad\text{is positive for} \quad\varepsilon<0. \tag{24}\]  
Notice further that the graph of the above cubic radiation discriminant \(D(\mu)\) as in Equation (20) has \(180^{\circ}\) rotational symmetry about its points of inflection. For a given \(\mu\in\mathbb{R}\), it is well-known that the inflection point of an arbitrary cubic polynomial \(D(\mu)\) occurs [36] at \((\mu_{0},D(\mu_{0}))\) such that we have  
\[\frac{d^{2}D}{d\mu^{2}}\mid_{\mu=\mu_{0}}=0. \tag{25}\]  
In this case, from the Equation (20), it follows that the discriminant \(D(\mu)\) satisfies  
\[\frac{d^{2}D}{d\mu^{2}}=\frac{1}{\varepsilon}\bigg{(}6\alpha\mu+2\beta\bigg{)}. \tag{26}\]  
Thus, it follows that the inflection point of \(D(\mu)\) is given by  
\[(-\frac{\beta}{3\alpha},\frac{2\beta^{3}}{27\alpha^{2}\varepsilon}-\frac{ \gamma\beta}{3\alpha\varepsilon}+\frac{\delta}{\varepsilon}). \tag{27}\]  
By translating an arbitrary radiation point \(\mu\to\mu+\mu_{0}\) and using Equation (20), we observe that the translated discriminant \(D_{T}(\mu)=D(\mu+\mu_{0})-D(\mu_{0})\) reduces as per the following datum  
\[D_{T}(\mu)=\frac{1}{\varepsilon}\bigg{(}\alpha\mu^{3}+(\gamma-\frac{\beta^{2} }{3\alpha})\mu\bigg{)}. \tag{28}\]  
This shows that that radiation discriminant \(D_{T}(\mu)\) is rotationally symmetric about its point of inflection as it has only odd powers of \(\mu\), viz. given any radiation energy \(f(\lambda,T)\), we have \(D_{T}(-\mu)=-D_{T}(\mu)\) for all \(\mu\) as in Equation (19).