Sums, Approximations, and Asymptotics II Block Stacking How far can a stack of identical blocks overhang the end of a table without toppling over? Can a block be suspended entirely beyond the table's edge? Table Physics imposes some constraints on the arrangement of the blocks. In particular, the stack falls off the desk if its center of mass lies beyond the desk's edge. Moreover, the center of mass of the top k blocks must lie above the(k+1)-st block;

1 The Number-Picking Game Here is a game that you and I could play that reveals a strange property of expectation. 3, First, you think of a probability density function on the natural numbers. Your distri- bution can be absolutely anything you like. For example, you might choose a uniform distribution on 1, 2, ... 6, like the outcome of a fair die roll. Or you might choose a bi- probability, provided that,...,n. You can even give every natural number a non-zero nomial distribution on 0, 1 he sum of all probabilities is 1

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

Key to practice exercise I. GRAMMAR 1.C 2.A在 hope,Ibet结构后的that-分句中可使用一般现在时表示将来的动作。如: bet it rains tomorrow. 3.B.主句中若有 require, demand, suggest insist order, prefer, propose recommend, request等动词,其后的that分句中应使用be型虚拟式,即动次原形或 should+动词原形

地球内部过程 Tsunamis海啸 and Coastal Flooding Coastal areas, especially around the Pacific Ocean basin where so many large earthquakes occur, may also be vulnerable易于to tsunamis

Microelectronics Processing Technology Spring Term, 200. Quiz 1 90 min/90 points total March 19 2003 Note: Be brief and relevant in your answers, and use sketches. Show your work. k=1.3810-23J/K=8.6210-eV/K 40 1. Film deposition, CVD )Discuss briefly what factors affect the microstructure a film grown on a substrate

Microelectronics Processing technology Fall Term 2003 Instructions for the mems Lab report Your lab report should follow the format of the IEEE Electron Device Letters. The format from the IC Lab report will be used Contents: Your Letter should include the following sections

Covers materials in lectures from 10/15 through 11/26 Does not include Lab Lectures A formula sheet will be provided (if needed) Lecture on Monday, Dec. 8th Lab tour of Analog Devices MEMS Facility We will leave from the classroom at 2: 35PM SHARP

We saw how dopants were introduced into a wafer by using diffusion (predepositionanddrive-in') This process is limited cannot exceed solid solubility of dopant -difficult to achieve light dopin Ion implantation is preferred because controlled, low or high dose can be

Why spend a whole lecture on oxidation of Si? Ge has high ue, Wh, Ge stable but no oxide GaAs has high ue and direct band Why sio? Sio, is stable down to 10 9 Torr. T>900C Sio can be etched with hf which leaves si unaffected