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Welcome to a new lecture on the binomial theorem. In this lecture, we will look at certain divisibility results which can be derived based on the binomial theorem. Once again, recall the binomial theorem is given by (x+y)^n = \sum{r=0}^{n}{C_r x^{n-r} y^r}, where we use C_r to denote nC_r. To give you a flavor of the kind of results we will be looking at today, let's look at the first example which asks you to show that 3^{2n+2} - 8n - 9 is divisible by 64 when n is a natural number. It asks you to show that this term is divisible by 64 for natural numbers n. At first sight, there doesn't seem to be a binomial expansion you can apply to generate this result so we would like to create a binomial expansion to derive this result. Notice that you have an exponent here, but you also have some terms here and these terms are, in general, not divisible by 64 for n being a natural number. You can plug in certain natural numbers and you can see that this term, which is bein...

star Festival is a new CD ROM based career curriculum that enables teachers to take their students on an interactive electronic field trip to explore other cultures and their own cultural identity originally based on Research at the Massachusetts Institute of Technology the CD ROM tells the story of real life MIT Professor Shigeru miyagawa as he returns to Japan during the star festival and rediscovers his roots and Heritage finally the seventh day of the seventh month is arrived it's a beautiful clear day so the tanabata lovers will probably meet I feel like I've made a journey too from My adoptive culture back to my roots am I American or Japanese or American and Japanese maybe like the magpies of of the Tabatha my I'm a bridge between two [Music] cultures I thought I knew all about Japan and I realize now I just made a lot of [Music] assumptions the professor recorded his notes into a Hightech diary known as the PDA during the festival he loses the PDA we f...

PROFESSOR: A new way of thinking about this is based on integral equations. So it's a nice method. It's less complicated than it seems, suddenly leads to some expressions for the scattering amplitude. We'll find another formula for this quantity, f of k of theta, when we cannot calculate it with phase shifts. But phase shifts is very powerful if your problem has spherical symmetry. It's probably the first thing you try, because there's nothing to prevent you from finding a solution, in that case. OK, so integral equations, so integral scattering equation, OK, integral scattering equation. So what are we solving? We're solving minus h squared over 2m Laplacian, plus v of r of potential, that is not necessarily spherical, equals or acts on a psi of r, to give you h squared, k squared over 2m, psi of r. So it's an equation for eigenstates, the Schrodinger equation. It's good. It's nice. And this is what we've been solving. We've be...

We now introduce a new of random variable, the exponential random variable. It has a probability density function that is determined by a single parameter lambda, which is a positive number. And the form of the PDF is as shown here. Note that the PDF is equal to 0 when x is negative, which means that negative values of X will not occur. They have zero probability. And so our random variable is a non-negative random variable. The shape of the PDF is as shown in this diagram. It's 0 for negative values, and then for positive values, it starts off, it starts off at a value equal to lambda. This is because if you plug in x equal to 0 in this expression, you get lambda times e to the 0, which leaves you just with lambda. So it starts off with lambda, and then it decays at the rate of lambda. Notice that when lambda is small, the initial value of the PDF is small. But then the decay rate is also small, so that the PDF extends over a large range of x's. On the other hand...

Let's create a new abstraction called a "process" to capture the notion of a running program. A process encompasses all the resources that would be used when running a program including those of the CPU, the MMU, input and output devices, etc. Each process has a "state" that captures everything we know about its execution. The process state includes * the hardware state of the CPU, i.e., the values in the registers and program counter. * the contents of the process' virtual address space, including code, data values, the stack, and data objects dynamically allocated from the heap. Under the management of the MMU, this portion of the state can be resident in main memory or can reside in secondary storage. * the hardware state of the MMU, which, as we saw earlier, depends on the context-number and page-directory registers. Also included are the pages allocated for the hierarchical page map. * additional information about the process' input an...