1 Strong Induction Recall the principle of strong induction: Principle of Strong Induction. Let(n) be a predicate. If ·P() is true,and for all n, P(O)A P(1)...A P(n) implies P(n+1), then P() is true for all n E N. As an example, let's derive the fundamental theorem of arithmetic

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

The expectation or expected value of a random variable is a single number that tells you a lot about the behavior of the variable. Roughly, the expectation is the average value, where each value is weighted according to the probability that it comes up. Formally, the expected value of a random variable r defined on a sample space s is: (B)=∑R()Pr(o) To appreciate its signficance, suppose S is the set of students in a class, and we select a student uniformly at random. Let r be the selected student's exam score. Then

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

AARON COHEN Professor Emeritus of Mechanical Engineering Texas A&M University Professor Aaron Cohen received a B.S. degree in Mechanical Engineering from Texas A&M University in 1952 and an M.S. degree in Applied Mathematics from Stevens

Radiation from a Uniformly Accelerated Charge and the Equivalence Principle Stephen Parrott Department of Mathematics University of Massachusetts at Boston

1 Physics as Natural Philosophy Physics is one of the liberal arts, combining elements of reason, philosophy, mathematics, language, and rhetoric. The term natural philosophy reflects the creative and dynamic interplay that exists in physics among experiment, theory, logic, insight, inspiration, symmetry, beauty, and language

3-1 Measures of Random Syntactic Information Shannon Theory of Information Key point: 1. Information is something that can be used to remove uncertainty. 2. The amount of information can then be measured by the amount of uncertainty it removed. 3. In the cases of communications, only waveform is concerned while meaning and value are ignored. 4. Uncertainty and thus information are statistic in nature and statistical mathematics is enough

Annals of Mathematics, 141(1995),443-552 Modular elliptic curves and Fermat’ s Last theoren By ANDREW JOHN WILES* For Nada. Claire. Kate and Olivia Pierre de fermat Andrew ohn wiles Cubun autem in duos cubos, aut quadratoquadratum in duos quadra toquadratos, et generaliter mullam in infinitum ultra quadratum potestatum in duos ejusdem nominis fas est divider: cujes rei demonstrationem mirabile sane deter. Hanc marginis eaiguitas non capered

A function is a term used in mathematics and logic to denote a relationship between input and output variables. Each variable is restricted to binary (0, 1) values The relationship is the complex of three primitive functions (And \\Not\\Or)