# Optimized Passive Defense Measures via IR Imaging
## 5 Discussion of the Results  
In order to make an apt qualitative analysis of the results concerning (in)stability of the radiation distribution \(f(\lambda,T)\), we choose the following region  
\[\mathcal{R}:=\{(\lambda,T)|-0.1\leq\lambda,T\leq 1.0\} \tag{29}\]for observing its behavior for the subsequent discussion. Here, we have allowed small absorptions by choosing a negative \(\lambda\) or \(T\). In the subsequent figures, the wavelength \(\lambda\) is measured in \(\hat{A}\) on \(X\)-axis, and the temperature \(T\) is measured in \(K\) on \(Y\)-axis. Note that the range of the dependent variable on each plot is selected to present the best view of the chosen flow functions as in the foregoing section in the region \(\mathcal{R}\). From Figure 5, in the range \(\mathcal{R}\), we see that the radiation distribution function \(f(\lambda,T)\) has a peak in the region \(0.0<\lambda<0.3\), for a fixed \(T>0.6\). For all \(T\), we observe that there are IR singularities at \(\lambda=0\) (as it is analytically shown by Equation (4). When \(\lambda\) grows from \(0\) to \(1\) and \(T\) approaches the origin from the left side corresponding to the absorption effects, the distribution function \(f\) grows from \(-\infty\) to \(-1\). Fundamentally, the singularities of \(f\) have the quantum or statistical field theoretic type origins which lies in multi-dimension distribution theory. We leave such examinations open for a future analysis. The distribution function \(f\), when plotted in the region \(\mathcal{R}\), shows a growing nature with respect to \(T\). This can be also supported by the positive sign of the component \(f_{T}\) for \(Ab>0\). As discussed earlier, from Equation (5), the extrema of \(f\) when \(T\) is fixed, are found at the points \((\lambda,T)\) satisfying  
\[\left(5-\frac{b}{\lambda T}\right)e^{\frac{b}{\lambda T}}=5 \tag{30}\]  
in the \(\lambda T\)-plane. In the special case of \(A=1=b\), the approximate value of \(\lambda\) is given by Equation (7). In the regions \(0<\lambda<1,\ 0<T<0.5\) and \(0.6<\lambda<1,\ 0.5<T<1\), the distribution function \(f(\lambda,T)\) takes almost a constant value. In Figure 6, we depict qualitative features of the rate \(f_{T}\) of the radiation distribution function \(f(\lambda,T)\) in the region \(\mathcal{R}\). We observe that \(f_{T}\) acquires its maximum value \(110\) almost in the same region as the distribution function \(f(\lambda,T)\) itself. For the limiting wavelength \(\lambda\to-0.1\) and temperature \(T\to 1\), we notice another set of fluctuations of the amplitude \(50\) in the derivative \(f_{T}\). As in the case of the distribution function \(f(\lambda,T)\), we find that it's rate with respect to the temperature \(T\) remains nearly constant for large \(\lambda\), e.g., \(\lambda>0.7\) and any \(T\). Moreover, the same conclusion holds for \(T<0.3\) and any \(\lambda\). Figure 5: The radiation distribution function \(f(\lambda,T)\) plotted in the region \(\mathcal{R}\). In Figure 7, we provide variations of the radiation distribution function \(f(\lambda,T)\) with respect to the wavelength \(\lambda\), viz. the derivative \(f_{\lambda}\) in the region \(\mathcal{R}\). Hereby, from Figure 7, we observe that \(f_{\lambda}\) has fluctuations with their amplitude of the order 300. On the other hand, \(f_{\lambda}\) shows a dip of an approximate value of \(-80\) in the region where the associated temperature rate possesses a peak. Namely, the dip in \(f_{\lambda}\) is found for \(\lambda\approx 0.3\) and \(T\approx 1\). Notice that the peak in \(f_{\lambda}\) arises in the same region of the temperature when the wavelength \(\lambda\) lies in the interval \((0.1,0.3)\). Thus, there are fluctuations of higher amplitude in the rate \(f_{\lambda}(\lambda,T)\) as \(T\) grows, see Figure 7. Apart from the above regions, \(f_{\lambda}\) remains nearly constant almost every where, except when the \(\lambda\) or \(T\) approach the origin. Figure 8 shows that the glow capacity \(f_{TT}\) takes a positive value throughout the region \(\mathcal{R}\).