A1  ^?? Express the differential conductivity $G = dI/dV$ as
$$G/G_{\text{t}} = 1 - \dfrac{e^2/C_{\Sigma}}{k_{\text{B}}T}\cdot g\left(\dfrac{eV}{2k_{\text{B}}T}\right),$$where $G_{\text{t}} = (2R_{\text{t}})^{-1}$ — conductivity of the system at high (room) temperature.
Obtain the explicit form of the function $g(x)$ and plot its graph.

To make differentiation easirer it's better to calculate
\[f(v) + f(-v) = \frac{e^{-v}-1+v}{4 \sinh^2 v/2}-\frac{e^{v}-1-v}{4 \sinh^2 v/2}=\frac{v- \sinh v}{2 \sinh^2 v/2}\]With staright-forward differention we have
$$g(x) = - \dfrac{2 \cosh x -2 - x \sinh x}{2(\cosh x - 1)^2}.$$

A2  ^?? Determine the depth of the differential conductivity gap$1 - G_{\text{min}}/G_{\text{t}}$ at $V=0$. Remember that $e^{x} = \sum x^n/n!$.

Expanding the expression obtained in the previous question into a Taylor series up to the fourth order, we arrive at the result
$$g(0) = \lim_{x \rightarrow 0} g(x) = \lim_{x \rightarrow 0} \dfrac{x\left(x + \frac{x^3}{4}\right) - 4\left(\frac{x}{2} + \frac{1}{4} \left(\frac{x}{2}\right)^3\right)^2}{8\left(\frac{x}{2}\right)^4} = \dfrac{1}{6}.$$

A3  ^?? Numerically determine the full width $x_{1/2}$ of $g(x)$ on the half-height:
$$g\left(\frac{1}{2}x_{1/2}\right) = \frac{1}{2} g(0).$$

After numerical solving of $g(x_{1/2}/2) = 1/12$ we have the following answer:

A4  ^?? Obtain the numerical value of the width of the conductivity gap $V_{1/2}$ at the boiling point of liquid helium $T = 4.21~\text{ K}$.

A5  ^?? The graph shows the dependence of normalized conductivity on half the voltage across the transistor ($V_{\text{BIAS}} = V/2$). For each of the four dependencies, determine the temperature at which it was studied.

For each dependence, we determine the minimum conductivity and the full width at half depth. In accordance with the formula obtained in the previous question, we calculate the temperature.
Note that we actually obtained overestimated values, since the assumption $v \ll 1$ for the proposed dependencies is poorly fulfilled.

A6  ^?? Determine the total capacitance $C_{\Sigma}$ of a single-electron transistor for which the above dependencies were obtained.

To determine the capacity, we will construct the dependence $(1 - G_{\text{min}}/G_{\text{t}})^{-1}(T)$. Using the answer for A2, we obtain
$$(1 - G_{\text{min}}/G_{\text{t}}) = \dfrac{e^2}{6k_{\text{B}}T\cdot C_{\Sigma}}.$$This means that the slope $\kappa$ of the constructed dependence determines the transistor's capacitance:
$$C_{\Sigma} = \dfrac{e^2}{6k_{\text{B}}}\cdot \kappa.$$