## 3 Cut-Off Values for \(m\)  
As discussed in the introduction, there is a cut-off value of \(m\), say \(m_{0}\), where solutions only exist for \(m\geq m_{0}\), and do not exist for \(m<m_{0}\). Determining the cut-off values \(m_{0}\) analytically is a very interesting and important new result, which we shall obtain; the cut-off values \(m_{0}\) can be determined analytically. This was performed numerically in [6] for certain values of \(n\) over a limited range. We shall obtain numerical computations for \(m_{0}\) over a wide range of \(n\), and also derive a mathematical formula that provides the values of \(m_{0}\) (in terms of \(n\)) with relatively high accuracy. It is remarked here that the graphs in [6] may suggest that for \(n>1\) there may be a minimum value of \(|m_{0}|\) for all \(n\), with some focal point at some \(m>m_{0}\) where all graphs intersect. In fact, this can be shown to be false by contradiction. Suppose it were true, then there would be some \(m_{0}=M<0\) where solutions exist for all \(n\), but then, from (15), we would have \(h\) as an increasing function on (0,1) for large \(n\). This, in turn, implies that a solution where \(h(1)=0\) could not exist, constituting a contradiction. This shows that \(m_{0}\to 0\) as \(n\to\infty\). In fact, assuming \(m_{0}<-\frac{\kappa}{n^{\prime}}(\kappa>0)\) for large \(n\) leads to a contradiction, similar to the one above, whenever \(p<1\). On the other hand, assume \(m_{0}<-\frac{\kappa}{n}\). This would imply that the solution gradient is \(h^{\prime}(z_{0})\approx\kappa\) at some very small \(z_{0}>0\). At the same time, we would have \(h(z_{0})\approx 0\). In fact, at the point where \(z=\sqrt{-m}\), the two terms in the governing Equation (10) become comparable and almost equal for \(m\) very close to zero. Therefore, if we take \(z_{0}\) proportional to \(\sqrt{-m}\), or say \(z_{0}=(-m)^{p}\) for some \(p>1/2\) but close to \(1/2\), then the second term in (10) becomes negligible with \(m\) very close to zero and large \(n\). We can then use the estimate \(h^{\prime}(z)\approx\kappa+az^{2}/2\) on \((z_{0},1)\) leading to \(h(1)\approx\kappa+\alpha/6\). In particular, if \(\kappa>1/6\) then it must be that \(h(1)>0\), so that \(h(1)=0\) could not be satisfied and a solution would not exist. (A mathematically precise classical \(e\)-\(\delta\) argument can be made here, but we choose to omit it and focus on what is more relevant to us from the practical viewpoint). We have shown that  
Proposition 1: _A solution to (7) and (8) does not exist whenever_  
\[m<m_{0}=-\frac{1}{6n+c} \tag{17}\]  
_where \(c=c(n)\) is of an order less than one in \(n\)._  
Remark 3: _The result that we have established here for the value of \(m_{0}\) (for large \(n\)), as in (17), exhibits a significant improvement from what has been observed in [26] where it was shown that a solution does not exist for \(m<-\frac{1}{2n+1}\)._  
In fact, it was found numerically that \(c\) could be approximated to a constant where the cut-off value of \(m\) would be given as \(m_{0}\approx-\frac{1}{6n+c}\). Table 1 shows some numerical values of \(m_{0}\) and the corresponding values of \(c\) for different values of \(n\), spanning over a relatively wide range. Observe that the sequence of values of \(c\) in the table suggests a convergence to occur, for large \(n\), to a value around \(4.33\) (or some value likely greater than \(4\), or possibly greater than \(4.3\)). This is a very useful result at the practical level. At the same time, it is a very peculiar mathematical observation that requires further analytical study at the mathematical level. From a practical point of view, if one uses (17) with \(c=4.35\) as an approximation for \(m_{0}\) over the wide range \(50<n<500\), then the relative error in the corresponding estimates of \(m_{0}\) would be less than 0.002%, providing accuracy up to the fifth non-zero decimal place (or better) in most readings. Similarly, a suitable value of \(c\) can be chosen to provide the desired accuracy for different ranges of \(n\).