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$\mathrm{A\,B\,C\,D\,E\,F\,G\,H\,I\,J\,K\,L\,M\,N\,O\,P\,Q\,R\,S\,T\,U\,V\,W\,X\,Y\,Z}$

$A\,B\,C\,D\,E\,F\,G\,H\,I\,J\,K\,L\,M\,N\,O\,P\,Q\,R\,S\,T\,U\,V\,W\,X\,Y\,Z$

$\mathrm{a\,b\,c\,d\,e\,f\,g\,h\,i\,j\,k\,l\,m\,n\,o\,p\,q\,r\,s\,t\,u\,v\,w\,x\,y\,z}$

$a\,b\,c\,d\,e\,f\,g\,h\,i\,j\,k\,l\,m\,n\,o\,p\,q\,r\,s\,t\,u\,v\,w\,x\,y\,z$

$0\;1\;2\;3\;4\;5\;6\;7\;8\;9$

$\Gamma\;\Delta\;\Theta\;\Lambda\;\Xi\;\Pi\;\Sigma\;\Upsilon\;\Phi\;\Psi\;\Omega$

$\alpha\;\beta\;\gamma\;\delta\;\epsilon\;\zeta\;\eta\;\theta\;\iota\;\kappa\;\lambda\;\mu\;\nu\;\xi\;\pi\;\rho\;\sigma\;\tau\;\upsilon\;\phi\;\chi\;\psi\;\omega$

$\varepsilon\;\vartheta\;\varkappa\;\varpi\;\varrho\;\varsigma\;\varphi$

$\mathbb{A\,B\,C\,D\,E\,F\,G\,H\,I\,J\,K\,L\,M\,N\,O\,P\,Q\,R\,S\,T\,U\,V\,W\,X\,Y\,Z}$

$\mathcal{A\,B\,C\,D\,E\,F\,G\,H\,I\,J\,K\,L\,M\,N\,O\,P\,Q\,R\,S\,T\,U\,V\,W\,X\,Y\,Z}$

$\mathfrak{A\,B\,C\,D\,E\,F\,G\,H\,I\,J\,K\,L\,M\,N\,O\,P\,Q\,R\,S\,T\,U\,V\,W\,X\,Y\,Z}$

$+ \;-\; \times\; \div\; \pm\; \mp\; \cdot\; \circ\; \star\; \ast\; \bullet\; \oplus\; \otimes\; \odot$

$= \;\neq\; <\; >\; \leq\; \geq\; \ll\; \gg\; \approx\; \sim\; \simeq\; \cong\; \equiv\; \propto$

$\in\; \notin\; \ni\; \subset\; \supset\; \subseteq\; \supseteq\; \cap\; \cup\; \setminus\; \emptyset$

$\forall\; \exists\; \nexists\; \neg\; \land\; \lor\; \Rightarrow\; \Leftrightarrow\; \vdash\; \models\; \top\; \bot$

$\infty\; \partial\; \nabla\; \triangle\; \square\; \hbar\; \ell\; \wp\; \Re\; \Im\; \aleph$

Let $V$ be a finite-dimensional vector space over $\F$ with basis $\{e_1, \ldots, e_n\}$. A linear map $T : V \to W$ is represented by a matrix $\mathbf{A} \in \F^{m \times n}$ whose entries are $a_{ij} = \langle w_i^*, T(e_j) \rangle$.

The eigenvalue problem seeks $\lambda \in \C$ and $\mathbf{v} \neq \mathbf{0}$ such that $\mathbf{A}\mathbf{v} = \lambda\mathbf{v}$, equivalently $\det(\mathbf{A} - \lambda \mathbf{I}) = 0$. The characteristic polynomial is

where $e_k(\mathbf{A})$ is the $k$-th elementary symmetric polynomial of the eigenvalues. By the Cayley–Hamilton theorem, $p(\mathbf{A}) = \mathbf{0}$.

The singular value decomposition factors any $\mathbf{A} \in \R^{m \times n}$ as

where $\mathbf{U} \in \R^{m \times m}$ and $\mathbf{V} \in \R^{n \times n}$ are orthogonal, $\sigma_1 \geq \cdots \geq \sigma_r > 0$, and $r = \rank(\mathbf{A})$. The Neumann series $(I - \lambda \mathbf{A})^{-1} = \sum_{k=0}^{\infty} \lambda^k \mathbf{A}^k$ converges when $|\lambda| \cdot \|\mathbf{A}\| < 1$.

Let $\Omega \subset \R^n$ be a bounded domain with smooth boundary $\partial\Omega$. Consider the Dirichlet problem for the Laplacian:

where $\Delta u = \sum_{i=1}^{n} \dfrac{\partial^2 u}{\partial x_i^2}$ is the Laplacian. A weak solution $u \in H^1(\Omega)$ satisfies

By the Lax–Milgram theorem, existence and uniqueness follow from the coercivity $\|\nabla u\|_{L^2}^2 \geq C \|u\|_{H^1}^2$ (Poincaré inequality).

The heat equation $\partial_t u - \Delta u = 0$ on $\R^n \times (0,\infty)$ has fundamental solution

The Fourier transform $\hat{f}(\xi) = \int_{\R^n} f(x)\, e^{-2\pi i \langle x, \xi\rangle}\, dx$ satisfies Parseval's identity $\|\hat{f}\|_{L^2} = \|f\|_{L^2}$ and the convolution theorem $\widehat{f * g} = \hat{f} \cdot \hat{g}$.

Let $(X, \tau)$ be a topological space. A sequence of covers $\mathcal{U}_1 \succ \mathcal{U}_2 \succ \cdots$ refining each other gives rise to the Čech cohomology groups

where $\mathcal{F}$ is a sheaf on $X$. For a compact oriented $n$-manifold $M$ without boundary, Poincaré duality gives an isomorphism

and the Euler characteristic satisfies $\chi(M) = \sum_{k=0}^{n} (-1)^k \, b_k$ where $b_k = \rank H_k(M; \, \Z)$ are the Betti numbers.

The Gauss–Bonnet theorem for a compact surface $\Sigma$ states

where $K$ is the Gaussian curvature and $\kappa_g$ is the geodesic curvature of the boundary. For a closed surface, this reduces to $\int_\Sigma K \, dA = 2\pi\,\chi(\Sigma)$.

The long exact sequence of a fibration $F \hookrightarrow E \twoheadrightarrow B$ is

The number of ways to partition a set of $n$ elements into $k$ non-empty subsets is the Stirling number of the second kind, given by

The exponential generating function is $\sum_{n \geq k} S(n,k)\, \frac{x^n}{n!} = \frac{(e^x - 1)^k}{k!}$.

The Catalan numbers $C_n = \frac{1}{n+1}\binom{2n}{n}$ count the number of Dyck paths from $(0,0)$ to $(2n,0)$. Their generating function satisfies

By the matrix-tree theorem, the number of spanning trees of a graph $G$ on $n$ vertices equals any cofactor of the Laplacian $\mathbf{L} = \mathbf{D} - \mathbf{A}$:

where $0 = \lambda_0 \leq \lambda_1 \leq \cdots \leq \lambda_{n-1}$ are the eigenvalues of $\mathbf{L}$. The chromatic polynomial $\chi_G(k)$ counts proper $k$-colorings and satisfies the deletion–contraction recurrence $\chi_G(k) = \chi_{G-e}(k) - \chi_{G/e}(k)$.

Let $X_1, X_2, \ldots$ be i.i.d. random variables with $\E[X_i] = \mu$ and $\operatorname{Var}(X_i) = \sigma^2 < \infty$. The central limit theorem states

where $\bar{X}_n = \frac{1}{n}\sum_{i=1}^n X_i$. The density of $\mathcal{N}(\mu, \sigma^2)$ is

For a Markov chain with transition matrix $\mathbf{P} = (p_{ij})$, the stationary distribution $\boldsymbol{\pi}$ satisfies $\boldsymbol{\pi}^\top \mathbf{P} = \boldsymbol{\pi}^\top$ and $\sum_i \pi_i = 1$. The ergodic theorem gives

Bayes' theorem: $\P(A \mid B) = \dfrac{\P(B \mid A)\,\P(A)}{\P(B)}$, or in density form, $f_{\Theta|X}(\theta \mid x) \propto f_{X|\Theta}(x \mid \theta) \, f_\Theta(\theta)$.

Let $G$ be a finite group acting on a set $S$. By Burnside's lemma, the number of distinct orbits is

where $S^g = \{s \in S : g \cdot s = s\}$ is the fixed-point set of $g$.

For a Galois extension $L/K$ with $\Gal(L/K) \cong G$, the fundamental theorem of Galois theory establishes a bijection

given by $E \mapsto \Gal(L/E)$ and $H \mapsto L^H$, where $[E : K] = [G : H]$.

In a principal ideal domain $R$, every finitely generated module $M$ decomposes as

where $r = \rank(M)$ and $p_1^{a_1}, \ldots, p_s^{a_s}$ are the elementary divisors. For $R = \Z$, this gives the classification of finitely generated abelian groups.

Maxwell's equations in differential form (Gaussian units):

The Schrödinger equation for a particle of mass $m$ in potential $V$ is

and the time-independent version $\hat{H}\psi = E\psi$ with Hamiltonian $\hat{H} = -\frac{\hbar^2}{2m}\nabla^2 + V$ has eigenvalues $E_n$ forming the energy spectrum.

Einstein's field equations of general relativity:

where $R_{\mu\nu}$ is the Ricci tensor, $R = g^{\mu\nu}R_{\mu\nu}$ is the scalar curvature, $g_{\mu\nu}$ is the metric tensor, $\Lambda$ is the cosmological constant, and $T_{\mu\nu}$ is the stress–energy tensor.

$\hat{a}\;\hat{f}\;\hat{x}\;\hat{y}\;\hat{A}\;\hat{F}$   $\tilde{a}\;\tilde{f}\;\tilde{x}\;\tilde{y}\;\tilde{A}\;\tilde{F}$   $\bar{a}\;\bar{f}\;\bar{x}\;\bar{y}\;\bar{A}\;\bar{F}$

$\dot{a}\;\dot{f}\;\dot{x}\;\dot{y}\;\dot{A}\;\dot{F}$   $\ddot{a}\;\ddot{f}\;\ddot{x}\;\ddot{y}\;\ddot{A}\;\ddot{F}$   $\vec{a}\;\vec{f}\;\vec{x}\;\vec{y}\;\vec{A}\;\vec{F}$

$\check{a}\;\check{f}\;\check{x}\;\check{y}\;\check{A}\;\check{F}$   $\breve{a}\;\breve{f}\;\breve{x}\;\breve{y}\;\breve{A}\;\breve{F}$   $\acute{a}\;\acute{f}\;\acute{x}\;\acute{y}\;\acute{A}\;\acute{F}$

$\grave{a}\;\grave{f}\;\grave{x}\;\grave{y}\;\grave{A}\;\grave{F}$

$\widehat{x}\quad \widehat{xy}\quad \widehat{xyz}\quad \widehat{xyzw}\quad \widehat{xyzwv}$

$\widetilde{x}\quad \widetilde{xy}\quad \widetilde{xyz}\quad \widetilde{xyzw}\quad \widetilde{xyzwv}$

$\overline{x}\quad \overline{xy}\quad \overline{xyz}\quad \overline{xyzw}\quad \overline{x+y+z+w+v}$

$\hat{\alpha}\;\hat{\beta}\;\hat{\lambda}$   $\tilde{\sigma}\;\tilde{\omega}$   $\bar{\mu}\;\dot{\theta}\;\vec{\rho}$

$\mathrm{A\;B\;C\;D\;E\;F\;G\;H\;I\;J\;K\;L\;M\;N\;O\;P\;Q\;R\;S\;T\;U\;V\;W\;X\;Y\;Z}$

$\mathrm{a\;b\;c\;d\;e\;f\;g\;h\;i\;j\;k\;l\;m\;n\;o\;p\;q\;r\;s\;t\;u\;v\;w\;x\;y\;z}$

$\mathrm{0\;1\;2\;3\;4\;5\;6\;7\;8\;9}$

$A\;B\;C\;D\;E\;F\;G\;H\;I\;J\;K\;L\;M\;N\;O\;P\;Q\;R\;S\;T\;U\;V\;W\;X\;Y\;Z$

$a\;b\;c\;d\;e\;f\;g\;h\;i\;j\;k\;l\;m\;n\;o\;p\;q\;r\;s\;t\;u\;v\;w\;x\;y\;z$

$0\;1\;2\;3\;4\;5\;6\;7\;8\;9$

$\mathbf{A\;B\;C\;D\;E\;F\;G\;H\;I\;J\;K\;L\;M\;N\;O\;P\;Q\;R\;S\;T\;U\;V\;W\;X\;Y\;Z}$

$\mathbf{a\;b\;c\;d\;e\;f\;g\;h\;i\;j\;k\;l\;m\;n\;o\;p\;q\;r\;s\;t\;u\;v\;w\;x\;y\;z}$

$\mathbf{0\;1\;2\;3\;4\;5\;6\;7\;8\;9}$

$\boldsymbol{A\;B\;C\;D\;E\;F\;G\;H\;I\;J\;K\;L\;M\;N\;O\;P\;Q\;R\;S\;T\;U\;V\;W\;X\;Y\;Z}$

$\boldsymbol{a\;b\;c\;d\;e\;f\;g\;h\;i\;j\;k\;l\;m\;n\;o\;p\;q\;r\;s\;t\;u\;v\;w\;x\;y\;z}$

$\boldsymbol{0\;1\;2\;3\;4\;5\;6\;7\;8\;9}$

$\Gamma\;\Delta\;\Theta\;\Lambda\;\Xi\;\Pi\;\Sigma\;\Upsilon\;\Phi\;\Psi\;\Omega$

$\boldsymbol{\Gamma}\;\boldsymbol{\Delta}\;\boldsymbol{\Theta}\;\boldsymbol{\Lambda}\;\boldsymbol{\Xi}\;\boldsymbol{\Pi}\;\boldsymbol{\Sigma}\;\boldsymbol{\Upsilon}\;\boldsymbol{\Phi}\;\boldsymbol{\Psi}\;\boldsymbol{\Omega}$

$\alpha\;\beta\;\gamma\;\delta\;\epsilon\;\zeta\;\eta\;\theta\;\iota\;\kappa\;\lambda\;\mu\;\nu\;\xi\;\pi\;\rho\;\sigma\;\tau\;\upsilon\;\phi\;\chi\;\psi\;\omega$

$\boldsymbol{\alpha}\;\boldsymbol{\beta}\;\boldsymbol{\gamma}\;\boldsymbol{\delta}\;\boldsymbol{\epsilon}\;\boldsymbol{\zeta}\;\boldsymbol{\eta}\;\boldsymbol{\theta}\;\boldsymbol{\iota}\;\boldsymbol{\kappa}\;\boldsymbol{\lambda}\;\boldsymbol{\mu}\;\boldsymbol{\nu}\;\boldsymbol{\xi}\;\boldsymbol{\pi}\;\boldsymbol{\rho}\;\boldsymbol{\sigma}\;\boldsymbol{\tau}\;\boldsymbol{\upsilon}\;\boldsymbol{\phi}\;\boldsymbol{\chi}\;\boldsymbol{\psi}\;\boldsymbol{\omega}$

$\varepsilon\;\vartheta\;\varkappa\;\varpi\;\varrho\;\varsigma\;\varphi$

$\boldsymbol{\varepsilon}\;\boldsymbol{\vartheta}\;\boldsymbol{\varkappa}\;\boldsymbol{\varpi}\;\boldsymbol{\varrho}\;\boldsymbol{\varsigma}\;\boldsymbol{\varphi}$

$\mathbb{A\;B\;C\;D\;E\;F\;G\;H\;I\;J\;K\;L\;M\;N\;O\;P\;Q\;R\;S\;T\;U\;V\;W\;X\;Y\;Z}$

$\mathbb{a\;b\;c\;d\;e\;f\;g\;h\;i\;j\;k\;l\;m\;n\;o\;p\;q\;r\;s\;t\;u\;v\;w\;x\;y\;z}$

$\mathcal{A\;B\;C\;D\;E\;F\;G\;H\;I\;J\;K\;L\;M\;N\;O\;P\;Q\;R\;S\;T\;U\;V\;W\;X\;Y\;Z}$

$\mathfrak{A\;B\;C\;D\;E\;F\;G\;H\;I\;J\;K\;L\;M\;N\;O\;P\;Q\;R\;S\;T\;U\;V\;W\;X\;Y\;Z}$

$\mathfrak{a\;b\;c\;d\;e\;f\;g\;h\;i\;j\;k\;l\;m\;n\;o\;p\;q\;r\;s\;t\;u\;v\;w\;x\;y\;z}$

$\mathscr{A\;B\;C\;D\;E\;F\;G\;H\;I\;J\;K\;L\;M\;N\;O\;P\;Q\;R\;S\;T\;U\;V\;W\;X\;Y\;Z}$

$+\;\; -\;\; \times\;\; \div\;\; \pm\;\; \mp\;\; \cdot\;\; \circ\;\; \star\;\; \ast\;\; \bullet$

$\oplus\;\; \otimes\;\; \odot\;\; \ominus\;\; \oslash\;\; \wedge\;\; \vee\;\; \cap\;\; \cup$

$=\;\; \neq\;\; <\;\; >\;\; \leq\;\; \geq\;\; \ll\;\; \gg\;\; \approx\;\; \sim\;\; \simeq\;\; \cong\;\; \equiv\;\; \propto$

$\prec\;\; \succ\;\; \preceq\;\; \succeq$

$\in\;\; \notin\;\; \ni\;\; \subset\;\; \supset\;\; \subseteq\;\; \supseteq$

$\sqsubset\;\; \sqsupset\;\; \sqsubseteq\;\; \sqsupseteq$

$\rightarrow\;\; \leftarrow\;\; \Rightarrow\;\; \Leftarrow\;\; \leftrightarrow\;\; \Leftrightarrow\;\; \mapsto$

$\hookrightarrow\;\; \hookleftarrow\;\; \uparrow\;\; \downarrow\;\; \updownarrow$

$\nearrow\;\; \searrow\;\; \nwarrow\;\; \swarrow$

$\longrightarrow\;\; \longleftarrow\;\; \Longrightarrow\;\; \Longleftarrow$

$\forall\;\; \exists\;\; \nexists\;\; \neg\;\; \land\;\; \lor\;\; \vdash\;\; \models\;\; \top\;\; \bot\;\; \vDash\;\; \Vdash$

$\infty\;\; \partial\;\; \nabla\;\; \triangle\;\; \square\;\; \diamond\;\; \hbar\;\; \ell\;\; \wp\;\; \Re\;\; \Im\;\; \aleph\;\; \beth$

$a_1, a_2, \ldots, a_n \qquad a_1 + a_2 + \cdots + a_n$

1x: $( ) \quad [ ] \quad \{ \} \quad \langle \rangle \quad | \quad \|$

Display style: $\displaystyle \sum_{k=0}^{\infty} \frac{x^k}{k!} = e^x$

Text style: $\textstyle \sum_{k=0}^{\infty} \frac{x^k}{k!} = e^x$

Script style: $\scriptstyle \sum_{k=0}^{\infty} \frac{x^k}{k!} = e^x$

Scriptscript style: $\scriptscriptstyle \sum_{k=0}^{\infty} \frac{x^k}{k!} = e^x$