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An important part of your image making, mostly small devices, is what you use for backgrounds, or how to view backgrounds. You know, you can make some very interesting images even more interesting if you subtlety suggest a certain kind of background that works, that's relevant. Your selection can really change your image. Sometimes you have to be kind of careful 'cause the background can overtake what you want to say and we'll see that in a moment. So take for example this image that I made of an assemblage of pieces. And I took it on a surface against a white wall, which is what a lot of you do and, the issue for me is that even though you don't think that the view sees it, the viewer does see the horizon, where the two pieces of material meet. You might say, 'Well, what what's the big deal'? Well, you should start really getting into the habit of trying to edit out anything that is ambiguous or that is not giving the viewer a very clear view ...

An important application of the central limit theorem is in the approximate calculation of the binomial probabilities. Here is what is involved. We start with random variables-- Xi-- that are independent. And they have the same distribution. They're all Bernoulli with parameter p. We add n of those random variables, and the resulting random variable, Sn, we know that it has a binomial PNF with parameters n and p. We also know its mean, and we do know its variance. What the central limit theorem tells us, in this case, since we're dealing with the sum of independent identically distributed random variables, is the following. If we take this random variable here that we have been denoting by Zn, which is a standardized version of Sn-- we subtract the mean of Sn and divide by the standard deviation-- this random variable has a CDF that approaches as n goes to infinity, the CDF of a standard normal. So let us use what we now know to calculate some probabilities. Let u...

An important reason why conditional probabilities are very useful is that they allow us to divide and conquer. They allow us to split complicated probability modes into simpler submodels that we can then analyze one at a time. Let me remind you of the Total Probability Theorem that has his particular flavor. We divide our sample space into three disjoint events-- A1, A2, and A3. And these events form a partition of the sample space, that is, they exhaust all possibilities. They correspond to three alternative scenarios, one of which is going to occur. And then we may be interested in a certain event B. That event B may occur under either scenario. And the Total Probability Theorem tells us that we can calculate the probability of event B by considering the probability that it occurs under any given scenario and weigh those probabilities according to the probabilities of the different scenarios. Now, let us bring random variables into the picture. Let us fix a particular v...

PROFESSOR: Important thing to do is to just try to understand one more thing. The creation and annihilation operators-- what do they do to those states? You see, a creation operator will I add one more a dagger, so somehow must change phi n into phi n plus 1. A destruction operator with an a will kill one of these factors, and therefore it will give you a state with lower number of phi n minus 1. And we would like to know the precise relations. So look at this. Let's do with an A on phi n. And we know it should be roughly phi n minus 1. This is one destruction operator, but we can do it. Look-- this is 1 over square root of n. A times a dagger to the n phi 0. A with a dagger to the n phi 0, we can replace by a commutator again. Commutator of a, a dagger to the n phi 0. This is 1 over square root of n factorial, and here we get a factor of n times a dagger to the n minus 1 phi 0. You know, it's all a matter of those commutators we on the left blackboard. But this s...

PROFESSOR: So the next important part of turbulent plume theory that we need is the distribution of concentration of particles or droplets, in this case, that are injected with the fluid at the source. So as we've just arrived, the concentration C in this case, we could have referred to infection quanta and infectious aerosols relative to that leaving the mouth, which is we've called C_q, scales as square root of area of the mouth divided by alpha x where alpha is the turbulent entrainment coefficient, around 0.1 or 0.15. And that leads to a jet which grows in size and grows in fluctuations as you see more and more eddies, and eventually might even bend due to flows in the room or thermal buoyancy effects. And I'd like to talk about the difference between short range transmission due to really placing yourself in this jet and breathing that air directly, which is more concentrated than the background, and then compare that with the transmission in the well-mix...

PROFESSOR: So another important source of convection in an air filled room is buoyancy due to differences in the density of the air as the temperature varies. Even relatively small variations in temperature can lead to significant flows. There's another dimensionless number, which controls the appearance and strength of such flows, which is the Rayleigh number, written Ra. And this is also a physical property of the-- it's a combination of physical properties of the fluid plus the geometry. So in this case, the relevant geometrical scale is the height, because this is a gravitational instability. So in the Rayleigh number, we have gravity. I'll just define all these-- a gravitational acceleration, which is 9.8 meters per second squared. We have-- well, if we define the change in air density relative to the initial air density that's caused by changes in temperature, if the temperature changes aren't too big, there is a linear response, which is defined...