Live mathematical reference

Mathematical compendium

A live index of the equations in published articles. Each entry leads to its article, equation guide, and the terms explained there. New or edited published articles appear automatically.

1725 equations across 1445 symbols.

d(R1,R2)=cos⁡−1 ⁣(Tr⁡(R1TR2)−12)d(R_1,R_2)= \cos^{-1}\!\left(\frac{\operatorname{Tr}(R_1^{\mathsf T}R_2)-1}{2}\right)

1 published occurrence

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This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

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D(T)=12∥∣h0⟩⟨h0∣−∣h1⟩⟨h1∣∥1D(T)=\frac12 \left\| |h_0\rangle\langle h_0|- |h_1\rangle\langle h_1| \right\|_1

1 published occurrence

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This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

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ds2=−U(Z)2 dx02+dZ2+dx2+dy2,x0:=ct,U(Z):=1+gZc2ds^2 = -U(Z)^2\,dx_0^2 + dZ^2 + dx^2 + dy^2, \qquad x_0 := ct, \qquad U(Z) := 1 + \frac{gZ}{c^2}

1 published occurrence

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This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

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h=2 σyϕ,zz=1.414×10−183.43×10−23 m−2≈203 mh = \sqrt{\frac{\sqrt2\,\sigma_y}{\phi_{,zz}}} = \sqrt{\frac{1.414\times10^{-18}}{3.43\times10^{-23}\,\mathrm{m^{-2}}}} \approx 203\,\mathrm m

1 published occurrence

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This equation gives an approximation: it relates the quantities while allowing an approximation.

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Hint(n)=ℏΩ J⋅σ(n)2j,Un=exp⁡ ⁣(−iκJ⋅σ(n)2j)H_{\mathrm{int}}^{(n)} =\hbar\Omega\, \frac{\mathbf J\cdot\boldsymbol\sigma^{(n)}}{2j}, \qquad U_n=\exp\!\left(-i\kappa \frac{\mathbf J\cdot\boldsymbol\sigma^{(n)}}{2j}\right)

1 published occurrence

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This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

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