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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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Properties of differentiation

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 We show some basic properties of differentiation, such as linearity, the product rule, the quotient rule, the chain rule, etc. We also introduce higher-order derivatives and differentiability classes. Theorem (Properties of differentiation) Let \(f(x)\) and \(g(x)\) be differentiable functions on an open interval \(I\). (Linearity) \[\frac{d}{dx}[kf(x) + lg(x)] = kf'(x) + lg'(x)\] where \(k, l\) are constants.  (Product rule) \[\frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x).\]  (Quotient rule) \(\frac{f(x)}{g(x)}\) is differentiable on \(\{x \mid g(x) \neq 0, x\in I\}\) and \[\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}.\] Proof .  Exercise. Since \(f(x)\) is differentiable at \(x=a\), it is continuous at \(x=a\) so \(f(x) \to f(a)\) as \(x \to a\). For any \(a\in I\), \[\begin{eqnarray*} \frac{f(x)g(x) - f(a)g(a)}{x - a} &=& \frac{f(x)g(x) - f(x)g(a) + f(x)g(a) - f(a)g(a)}{x - a}\\ &=& f(x)\cdot\frac{g(x) ...
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