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Birth process

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  Defining the birth process  Consider a colony of bacteria that never dies. We study the following process known as  the birth process , also known as  the Yule process . The colony starts with \(n_0\) cells at time \(t = 0\). Assume that the probability that any individual cell divides in the time interval \((t, t + \delta t)\) is proportional to \(\delta t\) for small \(\delta t\). Further assume that each cell division is independent of others. Let \(\lambda\) be the  birth rate.  The probability of a cell division for a population of \(n\) cells during \(\delta t\) is \(\lambda n \delta t\). We assume that the probability that two or more births take place in the time interval \(\delta t\) is \(o(\delta t)\). That is, it can be ignored. Consequently, the probability that no cell divides during \(\delta t\) is \(1 - \lambda n \delta t - o(\delta t)\). Note that this process is an example of the Markov chain with states \({n_0}, {n_0 + 1}, {n_0 + 2}...
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Informal introduction to formal logic

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 In mathematics, we must prove (almost) everything and the proofs must be done logically and rigorously. Therefore, we need some understanding of basic logic. Here, I will informally explain some rudimentary formal logic. Definitions (Proposition): A proposition is a statement that is either true or false. "True" and "false" are called the truth values, and are often denoted \(\top\) and \(\bot\). Here is an example. "Dr. Akira teaches at UBD." is a statement that is either true or false (we understand the existence of Dr. Akira and UBD), hence a proposition. The following statement is also a proposition, although we don't know if it's true or false (yet): Any even number greater than or equal to 4 is equal to a sum of two primes. See also: Goldbach's conjecture   Next, we define several operations on propositions. Note that propositions combined with these operations are again propositions.   (Conjunction, logical "and"):  Let \(P\)...
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Introduction to Sets

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Defining sets Set theory is the foundation of modern mathematics. Every mathematical notion is built on some set. What is a set? This is a very deep question beyond the scope of this lecture. Instead, we give the following very rough, informal definition. Definition (very informal) A set is a collection of distinct objects. Objects of a set are called elements of the set. To denote a set, we can enumerate its elements, enclosed by curly brackets. For example,  \[\{1, 2, 3\}\] denotes a set consisting of three elements that are 1, 2, and 3. We may give a set a name as in \[S = \{1, 2, 3\},\] and say, for example, "the set \(S\) contains the elements 1, 2, and 3. Suppose \(S\) is any set. To denote that an object \(x\) is an element of \(S\), we write \[x \in S\] and say "\(x\) is in \(S\)" and so on. To denote that \(x\) is not in \(S\), we write \[x \not\in S.\] It is important that the elements of a set are distinct. For example, \[\{a, a, b, c\}\] is not a set beca...
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