Two transparent plastic sheets say of red and blue color overlap as shown in figure . An observer looks at a clear sky through the sheets. He will observe light coming through sections as, SECTION 1: Red color, less bright than usual SECTION 2: Blue color, less bright than usual SECTION 3:Dark,a...

What's the best way to determine this up $x^3$ terms? I thought it would be to take the series for $\ln(1+x)$ and the series for $e^x$ up to $x^3$ and sub the second series into the first. $$\ln(1+x) = x - \frac {x^2}{2} +\frac {x^3}{3} + \cdots$$ $$e^x = 1+ x + \frac {x^2}{2} +\frac {x^3}{6} +...

Almost anyone that has taken a Solid State Physics course will know, that any 3D crystal must be in one of the 14 Bravais lattices due to symmetry operations and space filling. In 2015, a paper was published in Scientific Reports1 (from Nature publishing group) that reports the findings of a crys...