Brunei Math Club

Continuous maps So far, we have considered multivariate functions where the domain is (a subset of) \(\mathbb{R}^n\) (\(n \in \mathbb{N}\)) and the codomain is \(\mathbb{R}\). We can generalize the codomain to \(\mathbb{R}^m\) for any \(m \in \mathbb{N}\). To do this, first consider the following \(m\) real-valued functions on \(S (\subset \mathbb{R}^n)\): \[f_1(x_1,x_2,\cdots,x_n), f_2(x_1,x_2,\cdots,x_n), \cdots, f_m(x_1,x_2,\cdots,x_n).\] Now, for each \(a = (a_1, a_2,\cdots, a_n) \in S\), we can assign a point in \(\mathbb{R}^m\) \[(f_1(a), f_2(a),\cdots,f_m(a)) = (f_1(a_1,a_2,\cdots,a_n), f_2(a_1,a_2,\cdots,a_n), \cdots, f_m(a_1,a_2,\cdots,a_n)).\] By this correspondence, we can define a map from \(S\) to \(\mathbb{R}^m\) by \[a = (a_1, a_2,\cdots, a_n) \mapsto (f_1(a), f_2(a),\cdots,f_m(a)).\] We often write \[F: S \to \mathbb{R}^m\] and \[F(a) = (f_1(a), f_2(a),\cdots,f_m(a)).\] Example . When \(n = 1\) and \(m= 2\), a map from the closed interval \([0, 1]\) to \(\mathbb{R}^...

Martingale is an abstraction of "fair play." If a stochastic process is a martingale, some problems can be greatly simplified. We continue our study of branching processes. \(X_n\) is the random variable representing the population size of the \(n\)-th generation (\(X_0 = 1\) is the initial condition). See also: Branching processes: Mean and variance Branching processes: Probability of extinction Martingales Consider the following conditional expectation:  \[\mathbb{E}(X_{n+1}|X_0,X_1,\cdots,X_n) = \sum_{x}x\Pr(X_{n+1}=x|X_0,X_1,\cdots,X_n).\] Here the sum is over all possible values of \(X_{n+1}\). Note that this conditional expectation itself is a random variable as it is conditioned on the random variables \(X_0, X_1, \cdots, X_n\) (not on their values). This expectations doesn't really depend on \(X_0,X_1,\cdots,X_{n-1}\), but it does depend on \(X_n\) (the Markov property!). Each individual out of \(X_n\) individuals produces \(\mu_1\) offsprings on average, so the ...

See also: Branching processes: Mean and variance. We continue our study of branching processes. That is, \(X_n\) is the random variable representing the population size of the \(n\)-th generation, its generating function \(G_n(s)\) satisfies the following equation: \[G_n(s) = G_{n-1}(G(s)) = G(G(\cdots(G(s))\cdots))\] where  \[G(s) = G_1(s) = \sum_{j=0}^{\infty}p_js^j.\] The initial condition \(X_0 = 1\) and the probabilities \(p_j, j = 0, 1, \cdots,\) of each individual producing \(j\) descendants are assumed to be known. \[G_1(s) = G(s) = \sum_{j=0}^{\infty}p_js^j.\] Population size of zero, \(X_n = 0\), means extinction. The probability of extinction can be computed as \(G_n(0)\) (why?). Let \(g_n = G_n(0)\). \(g_1\) is the probability of extinction in the first generation. Recall the definition \[G_1(s) = G(s) = \sum_{j=0}^{\infty}p_js^j.\] Thus, \(g_1 = G_1(0) = p_0\): the probability that \(X_1 = 0\). Assume \(0 \leq s \leq 1\). Of course, \(0 \leq p_j \leq 1\) for all \(j\)....

We now consider the functions of the form \[y = f(x_1, x_2, \cdots, x_n),\] that is, functions multiple independent variables \(x_1, x_2, \cdots, x_n\): \[f: \mathbb{R}^n \longrightarrow \mathbb{R}.\] An \(n\)-variable function like \(f(x_1, x_2, \cdots, x_n)\) has a subset of \(\mathbb{R}^n =(\mathbb{R}\times\mathbb{R}\times\cdots\times\mathbb{R})\) as its domain. In particular, we study the meaning of limits and continuity in high-dimensional spaces. Remark . More generally, we may consider a function with its codomain in \(\mathbb{R}^m\). □ In the following, we mainly work with functions on \(\mathbb{R}^2\) rather than more general \(\mathbb{R}^n\). There is a huge gap between functions on \(\mathbb{R}\) and those on \(\mathbb{R}^2\). Compared to this gap, the conceptual difference between functions on \(\mathbb{R}^2\) and those on \(\mathbb{R}^n, n > 2\) is relatively small. Therefore, for most cases, understanding functions on \(\mathbb{R}^2\) is (mostly) sufficient. The a...

Generational growth Consider the following scenario (see the figure below): A single individual (cell, organism, etc.) produces \(j (= 0, 1, 2, \cdots)\) descendants with probability \(p_j\), independently of other individuals. The probability of this reproduction, \(\{p_j\}\), is known.  That individual produces no further descendants after the first (if any) reproduction. These descendants each produce further descendants at the next subsequent time with the same probabilities.  This process carries on, creating successive generations. Figure 1. An example of the branching process. Let \(X_n\) be the random variable representing the population size (number of individuals) of generation \(n\). In the above figure, we have \(X_0 = 1\), \(X_1=4\), \(X_2 = 7\), \(X_3=12\), \(X_4 = 9.\) We shall assume \(X_0 = 1\) as the initial condition. Ideally, our goal would be to find how the population size grows through generations, that is, to find the probability \(\Pr(X_n = k)\) for e...

Open sets In \(\mathbb{R}\), we have the notion of an open interval such as \((a, b) = \{x \in \mathbb{R} | a < x < b\}\). We want to extend this idea to apply to \(\mathbb{R}^n\). We also introduce the notions of bounded sets and closed sets in \(\mathbb{R}^n\). Recall that the \(\varepsilon\)-neighbor of a point \(x\in\mathbb{R}^n\) is defined as \(N_{\varepsilon}(x) = \{y \in \mathbb{R}^n | d(x, y) < \varepsilon \}\) where \(d(x,y)\) is the distance between \(x\) and \(y\). Definition (Open set) A subset \(U\) of \(\mathbb{R}^n\) is said to be an  open set  if the following holds: \[\forall x \in U ~ \exists \delta > 0 ~ (N_{\delta}(x) \subset U).\tag{Eq:OpenSet}\] That is, for every point in an open set \(U\), we can always find an open ball centered at that point, that is included in \(U\). See the following figure. Perhaps, it is instructive to see what is  not  an open set. Negating (Eq:OpenSet), we have \[\exists x \in U ~ \forall \delta > 0 ~ ...

We would like to study multivariate functions (i.e., functions of many variables), continuous multivariate functions in particular. To define continuity, we need a measure of "closeness" between points. One measure of closeness is the Euclidean distance. The set \(\mathbb{R}^n\) (with \(n \in \mathbb{N}\)) with the Euclidean distance function is called a Euclidean space. This is the space where our functions of interest live. The real line is a geometric representation of \(\mathbb{R}\), the set of all real numbers. That is, each \(a \in \mathbb{R}\) is represented as the point \(a\) on the real line. The coordinate plane , or the \(x\)-\(y\) plane , is a geometric representation of \(\mathbb{R}^2\), the set of all pairs of real numbers. Each pair of real numbers \((a, b)\) is visualized as the point \((a, b)\) in the plane. Remark . Recall that \(\mathbb{R}^2 = \mathbb{R}\times\mathbb{R} = \{(x, y) | x, y \in \mathbb{R}\}\) is the Cartesian product of \(\mathbb{R}\) with i...