Miscellaneous Thoughts

David K. Zhang
Last Modified 2023-12-31

If a first-order ODE can be written in the form

y'(t) = \frac{p(t)}{q(y(t))}

for some functions $p$ and $q$ , then it is separable. Such an ODE admits a one-parameter family of solutions

y(t) = Q^{-1}(P(t) + C)

where $P$ is an antiderivative of $p$ , $Q$ is an antiderivative of $q$ , and $C$ is an arbitrary constant.

If a first-order ODE can be written in the form

y'(t) = a(t) y(t) + b(t)

for some functions $a$ and $b$ , then it is linear. Such an ODE admits a one-parameter family of solutions

y(t) = e^{A(t)} [ Q(t) + C ]

where $A$ is an antiderivative of $a$ , $Q(t)$ is an antiderivative of $e^{-A(t)} b(t)$ with respect to $t$ , and $C$ is an arbitrary constant. If $b$ is the zero function, then we say that the equation is homogeneous, and its corresponding family of solutions simplifies to

y(t) = C e^{A(t)}.

If a first-order ODE can be written in the form

y'(t) = -\frac{p(t, y(t))}{q(t, y(t))}

where $p$ and $q$ are functions satisfying ${\partial p} / {\partial y} = {\partial q} / {\partial t}$ , then it is exact. Such an ODE admits a one-parameter family of solutions obtained by solving the equation

U(t, y(t)) = C

for $y(t)$ , where $C$ is an arbitrary constant and $U = U(t, y)$ is a function that satisfies $p = \partial U \! / \partial t$ and $q = \partial U \! / \partial y$ .

The method of variation of parameters is used to construct a particular solution to an inhomogeneous linear ODE

y^{(n)}(t) = a_{n-1}(t) y^{(n-1)}(t) + \dots + a_{1}(t) y'(t) + a_{0}(t) y(t) + f(t)

when we already know a fundamental set of complementary solutions $y_1, \dots, y_n$ to the corresponding homogeneous ODE

y^{(n)}(t) = a_{n-1}(t) y^{(n-1)}(t) + \dots + a_{1}(t) y'(t) + a_{0}(t) y(t).

In particular, variation of parameters tells us that

y(t) = u_1(t) y_1(t) + \dots + u_n(t) y_n(t)

is a solution to the inhomogeneous ODE, where the functions $u_1, \dots, u_n$ are the antiderivatives of the functions $u_1', \dots, u_n'$ determined by the following linear system:

\begin{bmatrix} y_1(t) & y_2(t) & \cdots & y_n(t) \\ y_1'(t) & y_2'(t) & \cdots & y_n'(t) \\ \vdots & \vdots & \ddots & \vdots \\ y_1^{(n-1)}(t) & y_2^{(n-1)}(t) & \cdots & y_n^{(n-1)}(t) \end{bmatrix} \begin{bmatrix} u_1'(t) \\ u_2'(t) \\ \vdots \\ u_n'(t) \end{bmatrix} = \begin{bmatrix} 0 \\ \vdots \\ 0 \\ f(t) \end{bmatrix}

Definition: Relation, Binary Relation, $a \mathrel{R} b$

Let $A$ and $B$ be sets. A relation between $A$ and $B$ , also known as a binary relation, is a subset of the Cartesian product $A \times B$ . Given a relation $R \subseteq A \times B$ and two elements $a \in A$ and $b \in B$ , we write $a \mathrel{R} b$ to denote that $(a, b) \in R$ .

Definition: Reflexive, Symmetric, Antisymmetric, Transitive

Let $A$ be a set.

A binary relation $R \subseteq A \times A$ is reflexive if $a \mathrel{R} a$ for all $a \in A$ .
A binary relation $R \subseteq A \times A$ is symmetric if $a \mathrel{R} b$ implies $b \mathrel{R} a$ for all $a, b \in A$ .
A binary relation $R \subseteq A \times A$ is antisymmetric if $a \mathrel{R} b$ and $b \mathrel{R} a$ implies $a = b$ for all $a, b \in A$ .
A binary relation $R \subseteq A \times A$ is transitive if $a \mathrel{R} b$ and $b \mathrel{R} c$ implies $a \mathrel{R} c$ for all $a, b, c \in A$ .

Definition: Equivalence Relation

An equivalence relation on a set $A$ is a binary relation $R \subseteq A \times A$ that is reflexive, symmetric, and transitive.

Definition: Converse, Inverse

Let $A$ and $B$ be sets. The converse or inverse of a relation $R \subseteq A \times B$ is the relation $R^{-1} \subseteq B \times A$ defined by

R^{-1} \coloneqq \{ (b, a) \in B \times A : (a, b) \in R \}.

In other words, $b \mathrel{R^{-1}} a$ if and only if $a \mathrel{R} b$ .

Definition: Function, Map, Domain, Codomain, $f: A \to B$ , $f(a)$ , Maps

Let $A$ and $B$ be sets. A function from $A$ to $B$ , also known as a map, is a relation $f \subseteq A \times B$ that has the following property: for every $a \in A$ , there exists a unique $b \in B$ such that $a \mathrel{f} b$ . We call the set $A$ the domain of $f$ , we call the set $B$ the codomain of $f$ , and we write $f: A \to B$ to denote that $f$ is a function from $A$ to $B$ . Given $a \in A$ , we write $f(a)$ to denote the unique element of $B$ that $a$ is related to, and we say that $f$ maps $a$ to $f(a)$ .

Definition: Injective, Injection, $f: A \injto B$

Let $A$ and $B$ be sets. A function $f: A \to B$ is injective if $a_1 \ne a_2$ implies $f(a_1) \ne f(a_2)$ for all $a_1, a_2 \in A$ . We call an injective function an injection, and we write $f: A \injto B$ to denote that $f$ is an injective function from $A$ to $B$ .

In other words, a function is injective if it maps distinct elements of its domain to distinct elements of its codomain.

Definition: Surjective, Surjection, $f: A \surjto B$

Let $A$ and $B$ be sets. A function $f: A \to B$ is surjective if, for every $b \in B$ , there exists $a \in A$ such that $f(a) = b$ . We call a surjective function a surjection, and we write $f: A \surjto B$ to denote that $f$ is a surjective function from $A$ to $B$ .

Definition: Bijective, Bijection, $f: A \bijto B$

Let $A$ and $B$ be sets. A function $f: A \to B$ is bijective if it is both injective and surjective. We call a bijective function a bijection, and we write $f: A \bijto B$ to denote that $f$ is a bijective function from $A$ to $B$ .

Only Bijections have Inverses

Theorem: Let $A$ and $B$ be sets, and let $f: A \to B$ be a function. The converse relation $f^{-1}$ is a function if and only if $f$ is a bijection.

Proof: Suppose $f^{-1}$ is a function. This means that, for every $b \in B$ , there exists a unique $a \in A$ such that $b \mathrel{f^{-1}} a$ , or equivalently, $a \mathrel{f} b$ . The existence of $a$ implies that $f$ is surjective, and the uniqueness of $a$ implies that $f$ is injective.

Conversely, suppose $f$ is a bijection. For every $b \in B$ , the surjectivity of $f$ implies that there exists an $a \in A$ such that $a \mathrel{f} b$ , and the injectivity of $f$ implies that this $a$ is unique. Hence, $f^{-1}$ is a function. ∎

Definition: Composition, $g \circ f$

Let $A$ , $B$ , and $C$ be sets. The composition of two functions $f: A \to B$ and $g: B \to C$ is the function $g \circ f: A \to C$ defined by

(g \circ f)(a) \coloneqq g(f(a)) \text{ for all } a \in A.

Composition Preserves Injectivity and Surjectivity

Theorem: Let $A$ , $B$ , and $C$ be sets. Let $f: A \to B$ and $g: B \to C$ .

If $f$ and $g$ are both injective, then $g \circ f$ is injective.
If $f$ and $g$ are both surjective, then $g \circ f$ is surjective.

Proof:

Let $a_1, a_2 \in A$ be given. If $a_1 \ne a_2$ , then by the injectivity of $f$ , we have $f(a_1) \ne f(a_2)$ . It follows by the injectivity of $g$ that $g(f(a_1)) \ne g(f(a_2))$ , or equivalently, $(g \circ f)(a_1) \ne (g \circ f)(a_2)$ .
Let $c \in C$ be given. By the surjectivity of $g$ , there exists $b \in B$ such that $g(b) = c$ . Now, by the surjectivity of $f$ , there exists $a \in A$ such that $f(a) = b$ . Hence, we have $(g \circ f)(a) = g(f(a)) = g(b) = c$ . ∎

Definition: Hausdorff Space

A Hausdorff space is a topological space $(X, T)$ that has the following property: for all $x, y \in X$ , if $x \ne y$ , then there exist $U, V \in T$ such that $x \in U$ , $y \in V$ , and $U \cap V = \varnothing$ .

Definition: Basis

Let $(X, T)$ be a topological space. A basis of $(X, T)$ is a collection of open sets $B \subseteq T$ that has the following property: for every open set $U \in T$ , there exists a sub-collection $C \subseteq B$ such that $U = \bigcup C$ .

In other words, a basis is a collection of open sets that allows every open set to be written as a union of basis elements.

Definition: Second-Countable

A topological space is second-countable if it admits a countable basis.

Definition: Manifold, Dimension

A manifold of dimension $n \in \N$ , also called an $\boldsymbol{n}$ -manifold, is a second-countable locally Euclidean Hausdorff space.

If manifolds were not required to be Hausdorff, then the line with two origins would be a one-dimensional manifold. Similarly, if manifolds were not required to be second-countable, then the long line would be a one-dimensional manifold.

Definition: Factorial

Let $n \in \N$ . The factorial of $n$ , denoted by $n!$ , is the product of the numbers $\{1, 2, \dots, n\}$ . In other words,

n! \coloneqq 1 \times 2 \times \dots \times n.

We define $0! \coloneqq 1$ in accordance with the convention that the product of the empty set $\varnothing$ is the multiplicative identity $1$ .

Definition: Gauss’s Pi Function

Gauss’s pi function is the function $\Pi: \{ z \in \C : \Re(z) > -1 \} \to \C$ defined on the half-plane $\Re(z) > -1$ by the absolutely convergent improper integral

\Pi(z) \coloneqq \int_0^\infty t^z e^{-t} \,\mathrm{d}t.

Note that $\Pi(z)$ is holomorphic on the half-plane $\Re(z) > -1$ because the integral representation of its derivative, i.e.,

\Pi'(z) = \int_0^\infty t^z e^{-t} \log t \,\mathrm{d}t,

converges absolutely.

\Pi(0) = \int_0^\infty e^{-t} \,\mathrm{d}t = -(e^{-\infty} - e^0) = 1

For $\Re(z) > -1$ , integration by parts gives

\Pi(z + 1) = \int_0^\infty t^{z + 1} e^{-t} \,\mathrm{d}t = (z + 1) \int_0^\infty t^z e^{-t} \,\mathrm{d}t = (z + 1) \Pi(z)

By putting these results together, we conclude that Gauss’s pi function interpolates the factorial function.

Bohr–Mollerup Theorem

Theorem: The gamma function is the unique function $f: (0, \infty) \to (0, \infty)$ that has the following properties:

$f(1) = 1$ .
$f(x + 1) = xf(x)$ for all $x > 0$ .
$\log f$ is convex.

\begin{CD} A @>>> B \end{CD}

${n \choose k}$ is the number of $k$ -element subsets of $[n]$ .

${n \brack k}$ is the number of permutations of $[n]$ that have $k$ disjoint cycles.

${n \brace k}$ is the number of partitions of $[n]$ into $k$ non-empty subsets.

{n \choose k} \coloneqq \frac{n!}{k!(n-k)!}

{n \brace k} \coloneqq \frac{1}{k!} \sum_{i=0}^k (-1)^i {k \choose i} (k-i)^n

$k! {n \brace k}$ is the number of surjective functions $[n] \to [k]$ .

$x^{\underline{n}}$

$x^{\overline{n}}$