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

In the addition of HX to an unsymmetrically substituted alkene, the hydrogen atom adds to the carbon atom of the double bond that already has the greater number of hydrogen atoms and that the X group adds to the carbon having more alkyl substituents. Markovnikov’s rule can be restated by saying that, in the addition of HX to an alkene, the more stable carbocation intermediate is formed

Functional groups The structural features that make it possible to classify compounds by reactivity are called functional groups. A functional group is a part of a larger molecule and is composed of an atom or a group of atoms that have a characteristic chemical behavior. The Chemistry of every organic molecules

Quinones are an interesting and valuable class of compounds because of their oxidation-reduction, or redox, properties. The redox properties of quinones are important to the functioning of living cells, where compounds called ubiquinones(泛醌) act as biochemical oxidizing agents to mediate the electro-transfer processes involved in energy production

17.1 The Acidity of the α Hydrogens of Carbonyl Compounds: Enolate Anions 17.2 Keto and Enol Tautomers 17.3 Reactions via Enols and Enolate Anions

A UV-Vis spectrophotometer measures the amount of light absorbed at each wavelength of the UV and visible regions of the electromagnetic spectrum. λmax— the wavelength of maximum absorption

10.3 The Reactions of Alkanes with Halogens 10.3 A Multiple Substitution Reactions Versus Selectivity 10.4 Chlorination of Methane:Mechanism of Reaction

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

Stability Condition in Terms of the Pole Locations A causal LTI digital filter is BIBO stable if and only if its impulse response h[n] is absolutely summable, i.e., We now develop a stability condition in terms of the pole locations of the transfer function H(z)

The dtft provides a frequency-domain representation of discrete-time signals and lti discrete-time systems Because of the convergence condition, in many cases. the dtft of a sequence may not exist as a result. it is not possible to make use of such frequency-domain characterization in these cases Copyright C 2001, S K. Mitra