1 Induction Recall the principle of induction: Principle of Induction. Let P(n) be a predicate. If ·P(0) is true,an for all nE N, P(n) implies P(n+1), then P(n) is true for all nE N As an example let's try to find a simple expression equal to the following sum and then use induction to prove our guess correct 1·2+2·3+3:4+…+n·(mn+1) To help find an equivalent expression, we could try evaluating the sum for some small n and(with the help of a computer) some larger n sum
1 Logic A proposition is a statement that is either true or false. Propositions can be joined by \and\, \or\, \not\, \implies\, or \if and only if\. For each of these connective, the defini- tion and notational shorthand are given in the table below. Here A and B denote arbitrary propositions
1 Case analysis The proof of a statement can sometimes be broken down into can be tackled individually 1.1 The method In order to prove a proposition P using case analysis Write, We use case analysis Identify a sequence of conditions, at least one of which must hold. (If this is not obvious, you must prove it
Diffusion: Read Plummer Chap. 7, sections 7.1-7.4,, 7.53, 7.5.8 Show that c(zt) Wroexpfzla)I,with a=2VDf, is a solution to Fick's second law of diffusion. ac(z, t) dc(zt) 2. a) What is the intrinsic carrier concentration in Si at 1100 C? b) Calculate the effective diffusivity (including first-order, charged-vacancy corrections)for As impurities in Si at 1100 C for two cases: 1)CAs=10 cm and ii)
I. Plot resolution and depth of field as a function of exposure wavelength for a projection aligner with 100nm< A <500nm. Assume NA=0.26. Recalculate on the same plot for NA=0.41. Discuss the implication of these plots for the technologist that must manufacture transistors with 0.5 um
