In this example the predictors are the measurements at the individual frequencies. Because the frequencies lie in a systematic order (850–1,050 nm), the predictors have a high degree of correlation. Hence, the data lie in a smaller dimension than the total number of predictors (100). Use PCA to determine the effective dimension of these data. What is the effective dimension?

Materials will have different energies associated with the electrons inside them. this is related to how strongly bound they are by their work function. if we sandwich a material with a very low energy between two materials with a high energy, we create a trap for an electron. if this trap is very narrow in one direction in comparison to the others, this is approximately a 1-dimensional box that confines the electron. what is the lowest energy of an electron in the box if the thinnest direction across the materials is 4.81 nm?