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Spot size $b \approx \text{geometric spread} + \text{diffraction spread} = a + \frac{L\lambda}{a}$.
To minimize $b$, differentiate with respect to $a$ and set to zero:
$\frac{db}{da} = 1 - \frac{L\lambda}{a^2} = 0 \implies a = \sqrt{\lambda L}$.
Substituting back: $b_{min} = \sqrt{\lambda L} + \frac{L\lambda}{\sqrt{\lambda L}} = 2\sqrt{\lambda L} = \sqrt{4\lambda L}$.
Let $S$ be the set of distinct values of $b$ for which the linear system $x+y+z=1$, $x+ay+z=1$, $ax+by+z=0$ has no solution. Describe the set $S$.
Subtract equation 1 from equation 2: $(a-1)y = 0$.
Case $a\neq1$: $y=0$. Then from eq.1 and eq.3: $x+z=1$ and $ax+z=0$, giving $(a-1)x=-1$, so $x=\frac{-1}{a-1}$. This always has a solution for $a\neq1$.
Case $a=1$: System becomes $x+y+z=1$, $x+y+z=1$, $x+by+z=0$. The third equation forces $-1=by-... $ i.e. subtracting eq.1: $(b-1)y=-1$. For no solution we need $b=1$.
So $S=\{1\}$ — a single value. Hence $S$ is a singleton.
The text's emphasis on 'most of these elements will affect you at some point in your career' suggests HRM knowledge is:
HRM knowledge is universally applicable - whether you work in HRM or not, whether you work with organizations or have people working for you, you will be dealing with people and affected by HR practices. It's essential knowledge for all professionals.
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