Miscellaneous Thoughts

David K. Zhang
Last Modified 2022-03-20


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

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

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

y(t)=Q1(P(t)+C) y(t) = Q^{-1}(P(t) + C)

where PP is an antiderivative of pp, QQ is an antiderivative of qq, and CC is an arbitrary constant.


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

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

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

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

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

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

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

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

where pp and qq are functions satisfying p/y=q/t{\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 U(t, y(t)) = C

for y(t)y(t), where CC is an arbitrary constant and U=U(t,y)U = U(t, y) is a function that satisfies p=U ⁣/tp = \partial U \! / \partial t and q=U ⁣/yq = \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)=an1(t)y(n1)(t)++a1(t)y(t)+a0(t)y(t)+f(t) 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 y1,,yny_1, \dots, y_n to the corresponding homogeneous ODE

y(n)(t)=an1(t)y(n1)(t)++a1(t)y(t)+a0(t)y(t). 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)=u1(t)y1(t)++un(t)yn(t) 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 u1,,unu_1, \dots, u_n are the antiderivatives of the functions u1,,unu_1', \dots, u_n' determined by the following linear system:

[y1(t)y2(t)yn(t)y1(t)y2(t)yn(t)y1(n1)(t)y2(n1)(t)yn(n1)(t)][u1(t)u2(t)un(t)]=[00f(t)] \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, aRba \mathrel{R} b

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

Definition: Reflexive, Symmetric, Antisymmetric, Transitive

Let AA be a set.

  • A binary relation RA×AR \subseteq A \times A is reflexive if aRaa \mathrel{R} a for all aAa \in A.
  • A binary relation RA×AR \subseteq A \times A is symmetric if aRba \mathrel{R} b implies bRab \mathrel{R} a for all a,bAa, b \in A.
  • A binary relation RA×AR \subseteq A \times A is antisymmetric if aRba \mathrel{R} b and bRab \mathrel{R} a implies a=ba = b for all a,bAa, b \in A.
  • A binary relation RA×AR \subseteq A \times A is transitive if aRba \mathrel{R} b and bRcb \mathrel{R} c implies aRca \mathrel{R} c for all a,b,cAa, b, c \in A.

Definition: Equivalence Relation

An equivalence relation on a set AA is a binary relation RA×AR \subseteq A \times A that is reflexive, symmetric, and transitive.

Definition: Converse, Inverse

Let AA and BB be sets. The converse or inverse of a relation RA×BR \subseteq A \times B is the relation R1B×AR^{-1} \subseteq B \times A defined by

R1{(b,a)B×A:(a,b)R}. R^{-1} \coloneqq \{ (b, a) \in B \times A : (a, b) \in R \}.

In other words, bR1ab \mathrel{R^{-1}} a if and only if aRba \mathrel{R} b.

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

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

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

Let AA and BB be sets. A function f:ABf: A \to B is injective if a1a2a_1 \ne a_2 implies f(a1)f(a2)f(a_1) \ne f(a_2) for all a1,a2Aa_1, a_2 \in A. We call an injective function an injection, and we write f:ABf: A \injto B to denote that ff is an injective function from AA to BB.

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 ⁣ ⁣ ⁣ ⁣ ⁣Bf: A \surjto B

Let AA and BB be sets. A function f:ABf: A \to B is surjective if, for every bBb \in B, there exists aAa \in A such that f(a)=bf(a) = b. We call a surjective function a surjection, and we write f:A ⁣ ⁣ ⁣ ⁣ ⁣Bf: A \surjto B to denote that ff is a surjective function from AA to BB.

Definition: Bijective, Bijection, f:A ⁣ ⁣ ⁣ ⁣ ⁣Bf: A \bijto B

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

Only Bijections have Inverses

Theorem: Let AA and BB be sets, and let f:ABf: A \to B be a function. The converse relation f1f^{-1} is a function if and only if ff is a bijection.


Proof: Suppose f1f^{-1} is a function. This means that, for every bBb \in B, there exists a unique aAa \in A such that bf1ab \mathrel{f^{-1}} a, or equivalently, afba \mathrel{f} b. The existence of aa implies that ff is surjective, and the uniqueness of aa implies that ff is injective.

Conversely, suppose ff is a bijection. For every bBb \in B, the surjectivity of ff implies that there exists an aAa \in A such that afba \mathrel{f} b, and the injectivity of ff implies that this aa is unique. Hence, f1f^{-1} is a function.

Definition: Composition, gfg \circ f

Let AA, BB, and CC be sets. The composition of two functions f:ABf: A \to B and g:BCg: B \to C is the function gf:ACg \circ f: A \to C defined by

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

Composition Preserves Injectivity and Surjectivity

Theorem: Let AA, BB, and CC be sets. Let f:ABf: A \to B and g:BCg: B \to C.

  • If ff and gg are both injective, then gfg \circ f is injective.
  • If ff and gg are both surjective, then gfg \circ f is surjective.

Proof:

  • Let a1,a2Aa_1, a_2 \in A be given. If a1a2a_1 \ne a_2, then by the injectivity of ff, we have f(a1)f(a2)f(a_1) \ne f(a_2). It follows by the injectivity of gg that g(f(a1))g(f(a2))g(f(a_1)) \ne g(f(a_2)), or equivalently, (gf)(a1)(gf)(a2)(g \circ f)(a_1) \ne (g \circ f)(a_2).
  • Let cCc \in C be given. By the surjectivity of gg, there exists bBb \in B such that g(b)=cg(b) = c. Now, by the surjectivity of ff, there exists aAa \in A such that f(a)=bf(a) = b. Hence, we have (gf)(a)=g(f(a))=g(b)=c(g \circ f)(a) = g(f(a)) = g(b) = c.


Definition: Hausdorff Space

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

Definition: Basis

Let (X,T)(X, T) be a topological space. A basis of (X,T)(X, T) is a collection of open sets BTB \subseteq T that has the following property: for every open set UTU \in T, there exists a sub-collection CBC \subseteq B such that U=CU = \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 nNn \in \N, also called an n\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 nNn \in \N. The factorial of nn, denoted by n!n!, is the product of the numbers {1,2,,n}\{1, 2, \dots, n\}. In other words,

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

We define 0!10! \coloneqq 1 in accordance with the convention that the product of the empty set \varnothing is the multiplicative identity 11.

Definition: Gauss’s Pi Function

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

Π(z)0tzetdt. \Pi(z) \coloneqq \int_0^\infty t^z e^{-t} \,\mathrm{d}t.

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

Π(z)=0tzetlogtdt, \Pi'(z) = \int_0^\infty t^z e^{-t} \log t \,\mathrm{d}t,

converges absolutely.

Π(0)=0etdt=(ee0)=1 \Pi(0) = \int_0^\infty e^{-t} \,\mathrm{d}t = -(e^{-\infty} - e^0) = 1

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

Π(z+1)=0tz+1etdt=(z+1)0tzetdt=(z+1)Π(z) \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,)(0,)f: (0, \infty) \to (0, \infty) that has the following properties:

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


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

(nk){n \choose k} is the number of kk-element subsets of [n][n].

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

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

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

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

xnx^{\underline{n}}

xnx^{\overline{n}}