How does negative power lead to amplification? I am currently investigating semiconductor superlattices and I am analyzing the negative differential velocity (NDV) after a certain limit. I understand how NDV leads to negative power, but I am struggling to see how this leads to signal amplification? 
Any help or 'points in the right direction' would be greatly appreciated!
Thanks
 A: Too many  words.
Signal amplification is a fact. Negative power is an attempt to clarify a concept of stray reactances leading to a mathematical solution known as a "power factor" -  not a fact.
"P:ower" refers to the ability to perform work when force(pressure/voltage) is combined with the capacity to deliver "energy" (electrical current. or current of water or flow of air). In electrical terms Power is defined as 
volts  x  amps.
When dealng with alternating-currents the definitions must be modified
instantaneous power  = instantaneoas voltage  x  instantaneous current 
but the presence of (stray) reactances causes the waveforms of voltage and current to lose synchrony (they develop a difference in relative phase).     Now multiply values of instantaneous voltag and current, plot them as a graph and you have a sinewave
BUT
note that this is a graphical representation of a mathematical function and has no entity as a waveFORM.
The relative phase-shift is referred to as a "power factor" which expresses the ratio between "DC-power" and "AC-power" and it pleases some to regard the lower half-cycles of that graph as representing "negative  power"..
In truth the concept can be useful especially when makimg measurements with an ac-bridge.  Be careful not to fall into the easy trap of forgetting that you are desling with "short speak".
To cvcalculate he gain provided by an amplifier requires a venture into Algebra and the technique varies depending on thr amplifier concerned is a voltage-Amp, a current-amp or a power-amp.
Follow through the algebraic procedure and find  that you are confronted with parts of the circuit where increasing (say) the voltage causes a decrease in current ... a negative value for the circuit resistance. But that is only in the world of mathematics.
