$\require{cancel}$ A list of mathematical terms I have come across and needed to look up.
Table of Contents
- Spaces
- Topology-esque
- Differential Geometry
- Group Theory
- Abstract Algebra
- Control Theory
- Still Fuzzy On
Spaces
Topological Space
Informally, a set of points, along with a set of neighbourhoods for each point satisfying a set of axioms relating points and neighbourhoods. Formally, let $X$ be a set, and let $\mathscr{U}$ be a collection of subsets of $X$. $\mathscr{U}$ is called a topology on X if
- $X,\emptyset\in\mathscr{U}$;
- $\{U_{\alpha}\}$, an arbitrary collection of elements of $\mathscr{U}$, implies that $\cup_{\alpha}U_\alpha\in\mathscr{U}$;
- $\{U_j \}^n_{j=1}$, a finite collection of elements of $\mathscr{U}$, implies that $\cap_{j=1}^n U_j\in\mathscr{U}$
The pair $(X,\mathscr{U})$ is called a topological space, and the elements of $\mathscr{U}$ are called open sets.
Metric Space
Informally, a metric space is a space imbued with a “sense of distance”. Formally, let $X$ be a set. A function $d:X\times X\to\mathbb{R}$ is said to be a matric on $X$ if
- $d(u,v)\ge0$ for all $u,v\in X$, and $d(u,v)=0$ if and only if $u=v$;
- $d(u,v)=d(v,u)$ for all $u,v\in X$; and
- $d(u,v)\le d(u,w)+d(w,v)$ for all $u,v,w\in X$.
The pair $(X,d)$ is called a metric space.
Dual Space
Any vector space $V$ has a corresponding dual vector space, $V^*$, consisting of all linear functionals on V. Formally, given any vector space $V$ over a field $\mathbb{F}$, the (algebraic) dual space $V^*$ is the set of all linear maps $\varphi:V\to\mathbb{F}$.
Topology-esque
Morphism
A morphism is a structure-preserving map between mathematical structures of the same type.
Endomorphism
A morphism from an object to itself. An endomorphism that is also an isomorphism is an automorphism.
Example
The orthogonal projection onto a line is an endomorphism on the plane which is not an automorphism.
Isomorphism
A bijective morphism.
Homomorphism
A mapping between two algebraic objects, which preserves operation in those objects.
Homeomorphism
A continuous topological isomorphism. In other words, it is a continuous function between topological spaces with continuous inverse.
Diffeomorphism
A differentiable homeomorphism, i.e. a continuously differentiable mapping between topological spaces with continuously differentiable inverse.
Examples
Let $(G, * )$ and $(H,\cdot)$ be two groups. A group homomorphism is a map $\rho:G\to H$ that satisfies $\rho (a*b)=\rho(a)\cdot \rho(b)$. If $\rho$ is bijective, then it is a group isomorphism. In this case, $G$ and $H$ are isomorphic, written as $G\cong H$.
Differential Geometry
Manifold
Informally, a manifold is a “surface”, i.e. a space that is locally Euclidean. Formally, a (topological) manifold is a second countable Hausdorff space that is locally homeomorphic to Euclidean space.
Foliage
An equivalence relation on an $n$-manifold, the equivalence classes being connected, injectively immersed submanifolds, all of the same dimension $p$, modeled on the decomposition of the real coordinate space $\mathbb{R}^n$ into the cosets $x+\mathbb{R}^p$ of the standardly embedded subspace $\mathbb{R}^p$.
Fiber Bundle
Informally, a fiber bundle is locally a product space, but globally may have a different topological structure. If the fiber bundle is just a product space, it is known as the trivial bundle.
Sections of Fiber Bundles
Let $B$ be the base space, $F$ be the fiber, and $E$ be the total space of the fiber bundle. A section of the fiber bundle $E$ is given by a continuous map $\sigma:B\to E$ such that $\pi(\sigma(x))=x$ for all $x\in B$. A section of the tangent bundle $TM$ is a vector field on $M$.
Examples
Consider the base space $B=\mathbb{S}^1$, the Fiber $F=\mathbb{R}$, and the trivial bundle $E=B\times F$, which manifests itself as a cylinder. An example of a nontrivial fiber bundle with the same form would be the M"obius strip. If instead $F=\mathbb{S}^1$, the trivial bundle would be the torus and the klein bottle would be an example of a nontrivial bundle.
Vector Bundle
A fiber bundle whose fibers are vector spaces.
Tangent Bundle
The tangent bundle to a manifold $M$ is a manifold $TM$ that is the disjoint union of the tangent spaces of $M$. The tangent bundle is the prototypical example of a vector bundle.
Cotangent Bundle
The dual bundle of the Tangent Bundle, conistent of all of the cotangent spaces at every point in the manifold.
Lie Derivative
The Lie derivative evaluates the change of a tensor field along the flow defined by another vector field.
Example
$\dot{V}=\frac{dV}{dx}\dot{x}$ represents the Lie derivative of the scalar field $V$ (Lyapunov function) along the flow defined by the vector field given by the dynamics of a system.
Distributions
A distribution is a collection of subspaces of the tangent bundle assigned to every point on a manifold. That is, for every point $x\in M$, we assign an $n$-dimensional subspace $\Delta_x\subset T_xM$, which depends smoothly on $x$ and have constant dimension for all $x$.
Involutivity
A distribution $\Delta$ is involutive if for any two vector fields $X,Y\in \Delta$, $[X,Y]\in \Delta$ Examples: Take $M=\mathbb{R}^3$ and coordinates $(x,y,z)$. Consider the distribution $\Delta=\left\{\frac{\partial}{\partial x}, \frac{\partial}{\partial y}\right\}$. Because $[X,X]=0$ for any vector field $X$, we simply need to notice that $\left[\frac{\partial}{\partial x}, \frac{\partial}{\partial y}\right]=0\in\Delta$ to conclude that $\Delta$ is involutive. Instead, consider $\Delta=\left\{\frac{\partial}{\partial x}, x\frac{\partial}{\partial y}+\frac{\partial}{\partial z}\right\}$. Now, \[ \begin{align} \left[\frac{\partial}{\partial x}, x\frac{\partial}{\partial y}+\frac{\partial}{\partial z}\right] &= \left[\frac{\partial}{\partial x}, x\frac{\partial}{\partial y}\right]+\cancelto{0}{\left[\frac{\partial}{\partial x}, \frac{\partial}{\partial z}\right]} \notag \\ &=\frac{\partial}{\partial x}\left(x\frac{\partial}{\partial y}\right) - \cancelto{0}{x\frac{\partial}{\partial y}\left(\frac{\partial}{\partial x}\right)} \notag \\ &= \frac{\partial}{\partial y} \notag \end{align} \] We require this to be a linear combination of the elements of $\Delta$; however, note that \[ \frac{\partial}{\partial y} = a\frac{\partial}{\partial x}+b\left(x\frac{\partial}{\partial y}+\frac{\partial}{\partial z}\right) \] does not hold for any functions $a,b$ and therefore $\Delta$ is not involutive.
Integrability
A submanifold $N$ of $M$ is said to be an integrable manifold of $\Delta$ if $T_xN=\Delta$ for any $x\in N$. $\Delta$ is said to be completely integrable if there exists an integral manifold of $\Delta$ through every point $x\in M$.
Lie Bracket
The Lie derivative of a vector field with respect to another vector field. The lie bracket assigns any two vector fields $X4 and $Y$ on a smooth manifold $M$ a third vector field, denoted $[X,Y]$.
Group Theory
Magma/Groupoid
A set equipped with a binary operation which sends any two elements to another element, and satisfies closure, i.e. for all $x,y\in S$, $x\cdot y\in S$.
Semigroup
A magma with associativity.
Quasigroup
A magma with divisibility.
Monoid
A semigroup with the identity.
Loop
A quasigroup with the identity.
Group
A group is a set, $G$ equipped with a binary operation $\cdot$ which satisfies the group axioms, given by
- Closure: $\forall\ a,b\in G,\ a\cdot b\in G$
- Associativity: $\forall\ a,b,c\in G, (a\cdot b)\cdot c = a\cdot (b\cdot c)$
- Identity Element: $\exists\ e\in G\ s.t.\ \forall\ a\in G, e\cdot a=a\cdot e = a$
- Inverse Element: $\forall\ a\in G,\ \exists\ b\in G s.t. a\cdot b = b\cdot a = e$
In the above terms, it is a magma which satisfies the group axioms.
Abelian ($\equiv$ Commutative) Group
A group for which the group operation is commutative.
Group Action
If $G$ is a group and $X$ is a set, then a (left) group action $\phi$ of G on X is a function \[ \phi:G\times X\to X, (g,x)\to\phi(g,x) \] that satisfies identity and compatability.
Lie Group
A group whose elements are organized continuously and smoothly. A real Lie group is a group that is a finite-dimensional real smooth manifold, in which the group operations of multiplication and inversion are smooth maps.
Example
Consider the group $S$ of continuous rotations about the origin, and the group $D_4$ of 90 degree rotations (i.e. the respective sets of rotations with the binary operation of composition). Now consider two objects, the square and the circle. $D_4$ constitutes a group action on the square, and $S$ constitutes a group action on the circle, because they preserve the symmetry of the underlying sets. Because the elements of $S$ are continuous, it is a Lie Group.
Lie Groupoid
Informally, a Lie groupois is a “many-object generalization” of a Lie group. A Lie groupoid is a groupoid where the set of objects $Ob$ and the set of morphisms $Mor$ are both manifolds, the source and target operations $s,t:Mor\to Ob$ are submersions, and the category operations are smooth.
Abstract Algebra
Ring
A ring is a set equipped with two binary operations that generalize the arithmetic operations of addition and multiplication. More rigorously, a ring is a set $\mathbf{R}$ equipped with two binary operations $+$ and $\cdot$ satisfying the ring axioms, given by
- $\mathbf{R}$ is an abelian group under addition
- $\mathbf{R}$ is a monoid under multiplication
- Multiplication is distributive with respect to addition
Module
A module over a ring is a generalization of the notion of a vector space over a fieds, wherein the corresponding scalars are the elemnts of a given ring, and multiplication is defined between elements of the ring and elements of the module.
Algebra over a Field (Algebra)
A vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure, which consists of a set, together with operations of multiplication, addition, and scalar multiplication by elements of the underlying field, and satisfies the axioms implied by “vector space” and “bilinear”.
Lie Algebra
A Lie Algebra is a vector space $\mathfrak{g}$ with a Lie bracket. Any Lie group gives ruse to a Lie algebra, which is its tangent space at the identity.
Lie Algebroid
Lie algebroids serce the same role for Lie groupoids that Lie algebras serve for Lie groups.
Control Theory
Zero Dynamics
Informally, zero dynamics are the residual dynamics after the outputs of a system have been zeroed.