Number Theory I Number theory is the study of the integers. Number theory is right at the core of math ematics; even Ug the Caveman surely had some grasp of the integers- at least the posi tive ones. In fact, the integers are so elementary that one might ask, What's to study?

It's really sort of amazing that people manage to communicate in the English language Here are some typical sentences: 1. You may have cake or you may have ice cream 2. If pigs can fly, then you can understand the Chernoff bound 3. If you can solve any problem we come up with then you get an a for the course. 4. Every American has a dream What precisely do these sentences mean? Can you have both cake and ice cream or must you choose just one desert? If the second sentence is true, then is the Chernoff bound incomprehensible? If you can solve some problems we come up with but not all, then do you get an a for the course? And can you still get an a even if you cant solve any of the problems? Does the last sentence imply that all Americans have the same dream or might

Random walks 1 Random walks a drunkard stumbles out of a bar staggers one step to the right, with a canal lies y steps to his right. Thi I equal p second, he either staggers one step to the left or probability. His home lies r steps to his left, and everal natural questions, including 1. What is the probability that the drunkard arrives safely at home instead of falling into the canal? 2. What is the expected duration of his journey however it ends? The drunkard's meandering path is called a random walk. Random walks are an im- portant subject, because they can model such a wide array of phenomena. For example

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

3-1一台四极直流电机,元件数S=120,每元件电阻为0.2,当转速为1000r/min时,每元件的平 均电动势为10V。求当电枢绕组分别为单叠和单波时,正负电刷间的电压U和电枢绕组电阻R各 为多少?

一、动量 1、质点的动量:P=mv大小为mv方向由确定; 单位为:kg.m/s 2、质点系的动量:P=∑mv 质点系总质量为:M=m

Electrical Potential Ener and Electric Potentia 17.1 Electrical potential energy and electric potential 1. Electric force is conservative force Coulomb's force:

Solubility limit for P at 1273 K is Co 2 1 x 10 cm\. Intrinsic carrier concentration is 10 cm\. Thus, before including higher order terms, Do=13x10-4 cm2/s. But with first order and second order correction terms

Gas flux across concentration gradient: J=h(Cg-Cs ), Cs=concentration at surface Flux for chemical reaction: J(#/vol s)=k Cs, where k=koexp[-AG/(kBT)I Film growth velos/ hk ii where Nr is number of deposited species/cm Diffusio

Ron Suiter, BSEE Lehigh, MBA USC Ron began his career as an aircraft system designer in Douglas Aircraft Company's Hydro-Mechanical system design group. He became Unit Chief of the braking system design