Datasets:

OleehyO
/

latex-formulas

image imagewidth (px) 15 746	latex_formula stringlengths 27 460
	\begin{align}-\div (y^{1-2m}\nabla w)=0\mbox{in}\mathbb R^n\times\mathbb R_+;w\big\|_{y=0}=\|u\|\end{align}
	\begin{align} \frac{{\rm d}\vec{X}}{{\rm d}t} = \vec{\Im} \left( \displaystyle \vec{X} \right)\end{align}
	\begin{align} L_{\vec{X}} \phi (\vec{X}) = \mbox{Tr}[J] \phi (\vec{X}) + P (\vec{V} \cdot \vec{\gamma})\end{align}
	\begin{align}z_k = (z_{1,k}, z_{2,k})\in Z, \ \ \ \ k=1,..., K,\end{align}
	\begin{align}(\nu_\mu\cdot \Phi)(A,U) = \int_P (\mu\cdot \Phi)(A\cap \varphi_p(X), U\cap \varphi_p(X)) \, dp\end{align}
	\begin{align}s\omega _{1}=\partial _{i}k^{i}, \end{align}
	\begin{align}d(\omega _kdx^k)=(\partial _i\omega _k)dx^idx^k+\omega _kd^2x^k;\end{align}
	\begin{align}K^{\epsilon '} = \tilde{M} K^{\epsilon} M^{-1} \quad .\end{align}
	\begin{align}+\delta_{\alpha\beta}[-\Delta B-{\tilde d}(\partial B)^{2}-q (\partial A\partial B) - r(\partial F\partial B)], \end{align}
	\begin{align}a^{\dagger}_m=\sqrt{\frac{2}{\zeta}}z_m\,,\quad a_m=\sqrt{\frac{2}{\zeta}}\bar{z}_m\quad\textrm{for}\quad m=1,\,3\,,\end{align}
	\begin{align}{J}:=\frac{\partial {L}} {\partial \dot q^j}\delta_v q^j+{L}\delta t,\end{align}
	\begin{align}[\tilde {J} {^{R} _3} (0,x^-) , \tilde {J} {^{R} _3} (0,y^-)] = {i\over 2 \pi} \delta'(x^- - y^-)\end{align}
	\begin{align}\Psi_{\rm{0}}(r',\theta';0)={1\over\sqrt{2\pi}\xi}\exp\left\{ikr'\cos\theta'-{1\over4\xi^2}(r'^2+r_0^2+2rr'\cos\theta')\right\}.\end{align}
	\begin{align}\left( 1+\varepsilon \delta R+O[\left( \delta \alpha \right) ^{2}]\right)\end{align}
	\begin{align}a_{I}=\bar{a}_{I}+\sum_{k=1}^{\infty}{\mu^{2k}\over (k!)^2}\partial_{t_{r}}^{2k}\bar{a}_{I},\end{align}
	\begin{align}\left(\begin{array}{cc} \mu_1 & 0 \\ 0 & \mu_2 \\ \end{array} \right)\left(\begin{array}{cc} 0 & d_1 \\ d_2 & 0 \\ \end{array} \right)\left(\begin{array}{cc}\bar{\mu_1}^{-1} & 0 \\0 & \bar{\mu_2}^{-1} \\\end{array}\right) =\left(\begin{array}{cc} 0 & 1 \\ 1 & 0 \\ \end{array} \right).\end{align}
	\begin{align}I_0 \rightarrow I_0+I_0^{\beta}\end{align}
	\begin{align}{E_i}=-F_{0i} = -\partial_0A_i + \partial_iA_0\end{align}
	\begin{align}Ev(\omega)\widetilde{{\cal D}(u)}{\cal R} = 0.\end{align}
	\begin{align}\Psi' (x) = \exp (-iD^{a}\theta_{a} )\Psi (x),\end{align}
	\begin{align}\{\Phi^A\}=\{\gamma_{\mu\nu}\, ,\, X^M\, ,\, A^a_\mu\, ,\, \xi^\mu\, ,\, c\, ,\, C^a\}.\end{align}
	\begin{align}F = \frac{1}{2\pi} \left( \delta(m) - \delta(\infty) - \pi n^++ \pi n^- - \delta(-m) + \delta(-\infty) \right),\end{align}
	\begin{align}{\cal D}_{\Gamma}(w)=\sqrt{{\rm Ber}\;\Omega_0\vert_{\Gamma}}\equiv \sqrt{{\rm Ber}\frac{\partial _r z^A}{\partial w^{\mu}}\Omega_{(0)AB}\frac{\partial _l z^B}{\partial w^{\nu}}},\end{align}
	\begin{align}\nabla^2h_{\mu\nu}=-16\pi G^D_N\left(T^{mat}_{\mu\nu}-{1\over{D-2}}\eta_{\mu\nu}T^{mat}\right)\equiv-16\pi G^D_N\bar{T}^{mat}_{\mu\nu},\end{align}
	\begin{align}Z_F(G^{})=Z_F(0)Z_{WZNW}(N)Z_{WZNW}(-N)Z_{ghost}\end{align*}
	\begin{align}Q=-\partial ^{2}_{\tau }+\mu ^{2}_{0}\left( 1-\frac{2}{\cosh ^{2}\left( \mu _{0}\tau \right) }\right)\end{align}
	\begin{align} g = kah, k \in K, a \in A^+, h \in H.\end{align}
	\begin{align}-\gamma^{cd} \partial_{c} X^{\mu}\partial_{d} X^{\nu} g_{\mu \nu} +\frac {\varepsilon^{cd}}{\sqrt{-\gamma}} F_{cd} = M\end{align}
	\begin{align}d_{i} = \sqrt{S_{i}} \ K_{i} \ exp(-S_{i}), \ \ \ i = 1, 2\end{align}
	\begin{align}M_{n+1} - M_n = C_n \exp(-4\pi M_n^2).\end{align}
	\begin{align}{\bf A}={1\over 2}\left( \begin{array}{cc} p^\dagger & q^\dagger \end{array}\right){\bf \sigma}\left(\begin{array}{c} p\\q \end{array}\right)\end{align}
	\begin{align}\alpha _{p}l^{D-2p}=\left\{\begin{array}{ll}(D-2p)^{-1}\left(\begin{array}{c}n-1 \\ p\end{array}\right) , & D=2n-1 \\ \left(\begin{array}{c}n \\ p\end{array}\right) , & D=2n.\end{array}\right. \end{align}
	\begin{align}x_H = -\,\frac{1}{2\lambda} \ln\frac{M}{2\lambda}\,,\end{align}
	\begin{align}\int_{{\cal M}_{{\rm ins}}} \left\vert \frac{1}{{\rm det}T_{0}} \right\vert,\end{align}
	\begin{align}\sum_{j \neq i} (c_j + d_j c_i) T^j =0 \, .\end{align}
	\begin{align}\left\|x-a\right\|+\left\|x-b\right\|+\left\|x-c\right\|=\sqrt{\mathcal{X}}\end{align}
	\begin{align}{\cal O} = {\cal O}^{\eta } + \zeta {\cal O}^{\zeta }\,,\end{align}
	\begin{align} g = hak, h \in H, a \in A^+, k \in K,\end{align}
	\begin{align}{\bf C}P^2 = U_1 \cup \{ (0:\phi_2:\phi_3) \},\quad (\phi_2:\phi_3) \not= 0 ,\end{align}
	\begin{align}{\cal E}_0=\lim_{\delta\rightarrow 0}\frac{\pi\hbar}{\delta}{\cal E}_\delta=\int_{-\infty}^{+\infty}d\gamma\,e^{\frac{i}{\hbar}\gamma\hat{\phi}}\ \ \ .\end{align}
	\begin{align}f_M=\left(\begin{array}{c}f^1\\\Lambda f^{1}.\end{array}\right).\end{align*}
	\begin{align}f(\infty)=1,\quad \chi(\infty)=v(\infty)=u(\infty)=\phi(\infty)=\kappa(\infty)= 0,.\end{align}
	\begin{align}K_{DFF} = K_{CCM} + \frac{g}{4H_{DFF}} = w_{0,2}+{g\over4}w_{-2,0}.\end{align}
	\begin{align}m^2 = \pi^3 \frac{\Lambda^4}{g^4}\exp\{-\frac{\pi}{2}\frac{\Lambda}{g^2}\}\end{align}
	\begin{align}\delta \chi_-=-\eta_- \left( \partial_0 w +i\epsilon_{ojk}\partial^j \alpha^k -2i e \Re ( \phi^ \varphi ) \right) \,.\end{align*}
	\begin{align}Z_+(\theta)=Ai(z) \qquad Z_-(\theta) = -Ai\,^\prime(z)\end{align}
	\begin{align}s^2-\varrho^2=\left\|\frac{2\varrho}{m}\frac{(p,\xi)}{\chi}\right\|^2\end{align}
	\begin{align}K(T, {\bar T})= -\log\{( T+ {\bar T})^3 + {\cal I}_{instanton} \}\end{align}
	\begin{align}M^{w}=\left\{ t(w\bar{t}w)a\|t\in M,a=(a_{1},a_{2},...,a_{n}),a_{i}=1w(i)\neq i,a_{i}=\pm1w(i)=i\right\} .\end{align}
	\begin{align}F_{\mu\nu}(x)=\partial_\mu A_\nu(x)-\partial_\nu A_\mu(x)+i\,[A_\mu(x),A_\nu(x)].\end{align}
	\begin{align}\Lambda _{B}^{A}=R_{BCD}^{A}\nabla _{a}\Phi ^{C}\nabla ^{a}\Phi ^{D}.\end{align}
	\begin{align}w(z)={\hbar\over 2 \pi^2}\int_0^{\infty}dk_z\int_0^{\infty} dk_{\\|} {k_{\\|}\over k}\left[\left(k^2+k_{\\|}^2\right)\sin^2(k_z z) +k_z^2 \cos^2(k_z z)\right]\left[{1\over 2}+{\overline n}(k)\right].\end{align}
	\begin{align}\left. A_{\beta,1}^{(2)}\right\|_{{\cal C}_\beta}=3\left. A_{\beta,1}^{(0)}\right\|_{{\cal C}_\beta}+8(\beta-2\pi)+ 2\cdot4\pi~~~.\end{align}
	\begin{align}\zeta_{\rho}(s)=-\frac{2\, R^{2s}}{\Gamma(s+1)\,\Gamma(-s)}\sum_{l=1}^{\infty}\nu^{1-2s}\int_0^{\infty}dk\,k^{-2s}\frac{d}{dk}\ln\Delta_{\rho,l}(\nu k), \quad \nu=l+\frac{1}{2}, \;\;\rho=\pm1 \end{align}
	\begin{align}\pi \min (1,p)<\|\Im m\, \vartheta \|<\pi \frac{p+1}{2}\end{align}
	\begin{align}\frac{\partial}{\partial r}\left(e\sp{-u}\right)=\frac{\mp k}{r^{2}}\left[1-\frac{\partial H_{3}}{\partial\theta}-\cot\theta\:H_{3}-H_{2}H_{4}\right],\end{align}
	\begin{align}\partial_{\mu}j^{\mu}+\partial_{\tau}j^{5}=0\end{align}
	\begin{align}\left[ {\cal P}^2-\left( M\Omega \right) ^2-\frac q2\sigma ^{\mu \nu }F_{\mu\nu }+ibM\gamma ^0\right] \phi (x)=0\;,\;\; \end{align}
	\begin{align}\left[\begin{array}{cccc}g_{2}+w_{2}^{\ 2}h_{4}+n_{2}^{\ 2}h_{5} & w_{2}w_{3}h_{4}+n_{2}n_{3}h_{5} &w_{2}h_{4} & n_{2}h_{5} \\w_{2}w_{3}h_{4}+n_{2}n_{3}h_{5} & g_{3}+w_{3}^{\ 2}h_{4}+n_{3}^{\ 2}h_{5} &w_{3}h_{4} & n_{3}h_{5} \\w_{2}h_{4} & w_{3}h_{4} & h_{4} & 0 \\n_{2}h_{5} & n_{3}h_{5} & 0 & h_{5}\end{array}\right] , \end{align}
	\begin{align}\big(\alpha_{\kappa}\|_{M^{w}}=\chi\|_{M^{w}}\big)\Rightarrow\big(\overline{\alpha_{\kappa}}w(\alpha_{\kappa})=\bar{\chi}w(\chi)\big).\end{align}
	\begin{align}T=\left( \begin{array}{cc}A&B\\C&D \end{array}\right) .\end{align}
	\begin{align}\langle f \| g \rangle= \int \frac{dp}{(1 + \beta p^2)^{1-\alpha}}\,f^(p)\,g(p)\;,\end{align*}
	\begin{align}f= C_1 + C_2 \log \left( \frac {2-z-2\sqrt{1-z}}{z} \right),\end{align}
	\begin{align}\frac{\lambda_R}{2m^2_{f,R}}\left(\langle\Phi^2(t)\rangle-\langle\Phi^2(0)\rangle\right)\approx 1\end{align}
	\begin{align}f^{(n,i_1,i_2,\dots,i_k)}_P \mbox{tr}[(b^\dagger)^{i_1} a^\dagger\dots a^\dagger]\dots \mbox{tr}[(b^\dagger)^{i_k} a^\dagger \dots a^\dagger]\|0>,\end{align}
	\begin{align}X^I = \frac{q^I}{Z}\,, \qquad X_I = \frac{V_I}{X^J V_J}\,.\end{align}
	\begin{align}\Delta^t U(a)=U(a\cos(2\pi t)+ia\sin(2\pi t))\Delta^t\end{align}
	\begin{align}f(\rho) \sim 1, \quad L(\rho) \sim r(\rho),\end{align}
	\begin{align}C_m,C_{m}^i;\ \ \ C_{mn},C_{mn}^i;\ \ \ \ \ \ m,n=1,2,3\end{align}
	\begin{align}\langle T_{\mu\nu}\rangle_{ren}\stackrel{\rm def}{=}\langle T_{\mu\nu}\rangle-\langle T_{\mu\nu}\rangle^{(4)}\, .\end{align}
	\begin{align}\overline{\alpha_{\kappa}}w(\alpha_{\kappa})=\bar{\chi}w(\chi)\end{align}
	\begin{align}i_Y{\cal I}_\Omega=\Bigl[Y^\mu {\bf F}_{\mu\nu}dx^\nu\Bigr]\otimes\varepsilon^2-\Bigl[Y^\mu ({\bf F})_{\mu\nu}dx^\nu\Bigr]\otimes\varepsilon^1.\end{align*}
	\begin{align}\tilde{Q}^{(L, I)}_1 \tilde{Q}^{(L, I)}_0 \neq 0\,\end{align}
	\begin{align}H=\frac{m}{2}\sum_{a=1}^N\,\vec{v}_a^2 =\frac{1}{2m}\sum_{a=1}^N\,[\vec{p}_a-e\vec{A}(\vec{x}_a)]^2\end{align}
	\begin{align}Z_K\propto\left(\frac{8\pi G \Lambda}{9} I(\phi_{cl})\right)^{-K} K!\end{align}
	\begin{align}A\|_{I_\mu} \;=\; \left( \begin{array}{cccc} \mu & 1 & \cdots & 0 \\0 & \mu & \cdots & 0 \\ \vdots & \vdots & \ddots & 1 \\ 0 & 0 & \cdots & \mu \end{array} \right) \;.\end{align}
	\begin{align}\dot{x}^{AA'} = o^A \bar{o}^{A'}, \,\,\acute{x}^{AA'}= o^A \bar{\iota}^{A'} + \iota^A \bar{o}^{A'},\,\,(\rho^\tau o^A\bar{o}^{A'})\dot{} = {\cal F}^{AA'},\end{align}
	\begin{align}\begin{array}{c}V(z_{12}^{\prime})=U^{-1}(z_{1};g)V(z_{12})U(z_{2};g)\,,\\{}\\V(z_{21}^{\prime})=U^{-1}(z_{2};g)V(z_{21})U(z_{1};g)\,.\end{array}\end{align}
	\begin{align}{u^2 \over l^2} \sin^2 \alpha ~\rho^{12} + \left({u^2 \over l^2} \cos^2 \alpha- {u^2 \over l^2} \sin^2 \alpha -1\right) \rho^6 - {u^2 \over l^2} \cos^2 \alpha = 0\end{align}
	\begin{align}F = \frac{1}{\pi} \left( 2 \alpha ' \right)^{- d/2} \langle B, y_1 , v_1 \| D \|B, y_2 , v_2 \rangle~. \end{align}
	\begin{align}ds^2 = (dX^0)^2 - R(X^0)^2 \sum_{i=1}^{D-1}(dX^i)^2\end{align}
	\begin{align}\prod_{(i,j)\in I_{w}}\big\|\frac{t_{i}}{t_{j}}\big\|^{\kappa(i,j)+\kappa(w(i),w(j))}=\prod_{i=1}^{n}\|t_{i}\|^{\lambda_{i}+\lambda_{w(i)}}.\end{align}
	\begin{align}\tilde{\zeta}(s)=\frac{V_q}{(4\pi)^{q/2}}\frac{\Gamma(s-q/2)}{\Gamma(s)} \sum_{\bf p}(\sigma^d_{\bf p}+m^2-\mu^2)^{-s}\;.\end{align}
	\begin{align}\tan ^{-1} \left( \frac {{\sqrt {4\xi -1}}}{3} \right)=\frac {{\sqrt {4\xi -1}}}{3} -\frac {(4\xi -1)^{\frac {3}{2}}}{27} +O(4\xi -1)^{\frac {5}{2}}.\end{align}
	\begin{align}{\rm Aux}^2\;=\;\bigl\{ \sum_i\,[D,a_i][D,b_i]\;:\;\sum_i\;a_i\,[D,b_i]\;=\;0\bigr\},\end{align}
	\begin{align} T_{ab}{}^{c} \;=\; -\left[ i_{k_{(m)}}\hat{C}\right]_{ab} Q^{mn}k_{(n)}^{c} \hspace{.2cm},\hspace{.2cm} 2 K_{abc} \;=\; -T_{abc}+T_{bca}-T_{cab} \; ,\; \check{\omega} \;=\; \omega + K \; .\end{align}
	\begin{align}f\left( T(z)\right) =T'(z) \left[ f(z) -\frac{\delta w}{\epsilon w}(z-\alpha )\right] \ .\end{align}
	\begin{align}\vec{A}=\frac{\phi}{2\pi}\frac{\Theta(r-R)}{r}\vec{e}_\varphi\ , \ \ A^0=0\ ;\end{align}
	\begin{align}\Psi = x^{-\kappa} ( \Psi_{0} + \Psi_{1} + ... )\; .\end{align}
	\begin{align}\delta A_i = \theta C_{iT} \quad \qquad\delta \pi_i = \theta P_{iT}\end{align}
	\begin{align}S(g)^{-1}=1+\sum_{n=1}^\infty \frac {1}{n!} \int d^4x_{1}\ldots d^4 x_n \tilde{T}_n(x_1,\ldots x_n)g(x_1)\ldots g(x_n)\end{align}
	\begin{align}{\rm U}_0({\tau}^{(1)}) {\rm U}_0({\tau}^{(2)}) =\exp\{ 2\pi i {\omega}_{2}({\tau}^{(1)}, {\tau}^{(2)})\}{\rm U}_{0}( {\tau}^{(1)} + {\tau}^{(2)} ) ,\end{align}
	\begin{align}\gamma(s,\chi,\psi)\gamma(1-s,\overline{\chi^{-1}},\overline{\psi^{-1}})=1.\end{align}
	\begin{align}\alpha_1=\epsilon_1-\epsilon_2\,;\,\,\alpha_2=\epsilon_2-\epsilon_3\,;\,\,\alpha_3=\epsilon_1-\epsilon_3\end{align}
	\begin{align}(\sigma_i\psi)_a(x)=-\psi_a(x).\end{align}
	\begin{align}{\cal M} = - \frac{c_f g_s^2}{4} \frac{1}{k^2 - M_G^2} J_c^\mu J_{c \mu}^{\dagger}\end{align}
	\begin{align}<u^{2}><q_{u}^{2}> = 1/4 , \qquad <v^{2}><q_{v}^{2}> = 1/4 .\end{align}
	\begin{align}\tilde C_{\alpha}^{\,\alpha_{0}}(p):=\int\limits_{\alpha_{0}}^{\alpha}d\alpha'\,e^{\,-\alpha'(\,p\eta p+(\varepsilon+i)m^{2})}\quad,\qquad\varepsilon>0\ ,\ 0<\alpha_{0}\leq\alpha<\infty\quad,\end{align}
	\begin{align}g_{12}= e_1 e_2 = g_{34}= e_3 e_4 = 1, \qquad (others\quad components \quad g_{ab}) = 0 .\end{align}
	\begin{align}V_{eff}=\frac{1}{2}\mu ^{2}\varphi ^{2}+\lambda\varphi^{4}-\varphi H\, ,\end{align}

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𝑩𝑰𝑮 𝑵𝑬𝑾𝑺‼️

📮 [2025-08] We have updated to a larger dataset, which contains nearly 80 million samples, with significant improvements in both data quality and diversity. [link]

📮 [2024-02] We trained a formula recognition model, 𝐓𝐞𝐱𝐓𝐞𝐥𝐥𝐞𝐫, using the latex-formulas dataset. It can convert LaTeX formulas into images and boasts high accuracy and strong generalization capabilities, covering most formula recognition scenarios.

For more details, please refer to the 𝐓𝐞𝐱𝐓𝐞𝐥𝐥𝐞𝐫 GitHub repository.

Dataset Description

中文版本

There are two datasets: raw_formulas and cleaned_formulas(This dataset has 550K formula-image pairs).

We scraped approximately 1 million LaTeX formula image-text pairs from arxiv that were uncleaned and without text segmentation to create the raw_formulas dataset. After cleaning the raw_formulas dataset and integrating it with the im2latex-100K dataset, we obtained the cleaned_formulas dataset, which has 550K formula-image pairs.

To render the images corresponding to the formulas, the following external packages are needed:

amsmath
amsfonts
amssymb
mathtools

Usage

for raw_formulas dataset:

from datasets import load_dataset
data = load_dataset("OleehyO/latex-formulas", "raw_formulas")

for cleaned_formulas dataset:

from datasets import load_dataset
data = load_dataset("OleehyO/latex-formulas", "cleaned_formulas")

Details About the raw_formulas Dataset

We scraped LaTeX formulas containing the following environments:

equation
align
align*
gather
gather*

The formulas do not include the following content:

\label
%
\quad
\qquad
\vspace
\hspace
\resizebox
\scalebox
\rotatebox
\parbox
\fbox
\makebox
\raisebox
\addvspace
\hfill
\vfill
\textwidth
\textheight
\rule

Preprocessing Details of the cleaned_formulas Dataset

Cleaning

We removed some useless junk data from both raw_formulas and im2latex-100K.
We deleted overly complex formulas from both raw_formulas and im2latex-100K:
- Formulas were deleted if the aspect ratio of the corresponding rendered image was greater than 0.8.
- Formulas with a character length greater than 200 were deleted.
In the formulas from both raw_formulas and im2latex-100K, the following content was removed:
- \tag
- \text
- \begin{split}
- \end{split}
- \nonumber
- \notag
The equation, equation*, align, \[...\] environments in raw_formulas were all replaced with the align* environment.
We deleted formulas from raw_formulas that contained custom macros.

Downloads last month: 549

Size of downloaded dataset files:

2.87 GB

Size of the auto-converted Parquet files:

2.87 GB

Number of rows:

1,558,585

Models trained or fine-tuned on OleehyO/latex-formulas

image imagewidth (px) 15 746	latex_formula stringlengths 27 460
	\begin{align}-\div (y^{1-2m}\nabla w)=0\mbox{in}\mathbb R^n\times\mathbb R_+;w\big\|_{y=0}=\|u\|\end{align}
	\begin{align} \frac{{\rm d}\vec{X}}{{\rm d}t} = \vec{\Im} \left( \displaystyle \vec{X} \right)\end{align}
	\begin{align} L_{\vec{X}} \phi (\vec{X}) = \mbox{Tr}[J] \phi (\vec{X}) + P (\vec{V} \cdot \vec{\gamma})\end{align}
	\begin{align}z_k = (z_{1,k}, z_{2,k})\in Z, \ \ \ \ k=1,..., K,\end{align}
	\begin{align}(\nu_\mu\cdot \Phi)(A,U) = \int_P (\mu\cdot \Phi)(A\cap \varphi_p(X), U\cap \varphi_p(X)) \, dp\end{align}
	\begin{align}s\omega _{1}=\partial _{i}k^{i}, \end{align}
	\begin{align}d(\omega _kdx^k)=(\partial _i\omega _k)dx^idx^k+\omega _kd^2x^k;\end{align}
	\begin{align}K^{\epsilon '} = \tilde{M} K^{\epsilon} M^{-1} \quad .\end{align}
	\begin{align}+\delta_{\alpha\beta}[-\Delta B-{\tilde d}(\partial B)^{2}-q (\partial A\partial B) - r(\partial F\partial B)], \end{align}
	\begin{align}a^{\dagger}_m=\sqrt{\frac{2}{\zeta}}z_m\,,\quad a_m=\sqrt{\frac{2}{\zeta}}\bar{z}_m\quad\textrm{for}\quad m=1,\,3\,,\end{align}
	\begin{align}{J}:=\frac{\partial {L}} {\partial \dot q^j}\delta_v q^j+{L}\delta t,\end{align}
	\begin{align}[\tilde {J} {^{R} _3} (0,x^-) , \tilde {J} {^{R} _3} (0,y^-)] = {i\over 2 \pi} \delta'(x^- - y^-)\end{align}
	\begin{align}\Psi_{\rm{0}}(r',\theta';0)={1\over\sqrt{2\pi}\xi}\exp\left\{ikr'\cos\theta'-{1\over4\xi^2}(r'^2+r_0^2+2rr'\cos\theta')\right\}.\end{align}
	\begin{align}\left( 1+\varepsilon \delta R+O[\left( \delta \alpha \right) ^{2}]\right)\end{align}
	\begin{align}a_{I}=\bar{a}_{I}+\sum_{k=1}^{\infty}{\mu^{2k}\over (k!)^2}\partial_{t_{r}}^{2k}\bar{a}_{I},\end{align}
	\begin{align}\left(\begin{array}{cc} \mu_1 & 0 \\ 0 & \mu_2 \\ \end{array} \right)\left(\begin{array}{cc} 0 & d_1 \\ d_2 & 0 \\ \end{array} \right)\left(\begin{array}{cc}\bar{\mu_1}^{-1} & 0 \\0 & \bar{\mu_2}^{-1} \\\end{array}\right) =\left(\begin{array}{cc} 0 & 1 \\ 1 & 0 \\ \end{array} \right).\end{align}
	\begin{align}I_0 \rightarrow I_0+I_0^{\beta}\end{align}
	\begin{align}{E_i}=-F_{0i} = -\partial_0A_i + \partial_iA_0\end{align}
	\begin{align}Ev(\omega)\widetilde{{\cal D}(u)}{\cal R} = 0.\end{align}
	\begin{align}\Psi' (x) = \exp (-iD^{a}\theta_{a} )\Psi (x),\end{align}
	\begin{align}\{\Phi^A\}=\{\gamma_{\mu\nu}\, ,\, X^M\, ,\, A^a_\mu\, ,\, \xi^\mu\, ,\, c\, ,\, C^a\}.\end{align}
	\begin{align}F = \frac{1}{2\pi} \left( \delta(m) - \delta(\infty) - \pi n^++ \pi n^- - \delta(-m) + \delta(-\infty) \right),\end{align}
	\begin{align}{\cal D}_{\Gamma}(w)=\sqrt{{\rm Ber}\;\Omega_0\vert_{\Gamma}}\equiv \sqrt{{\rm Ber}\frac{\partial _r z^A}{\partial w^{\mu}}\Omega_{(0)AB}\frac{\partial _l z^B}{\partial w^{\nu}}},\end{align}
	\begin{align}\nabla^2h_{\mu\nu}=-16\pi G^D_N\left(T^{mat}_{\mu\nu}-{1\over{D-2}}\eta_{\mu\nu}T^{mat}\right)\equiv-16\pi G^D_N\bar{T}^{mat}_{\mu\nu},\end{align}
	\begin{align}Z_F(G^{})=Z_F(0)Z_{WZNW}(N)Z_{WZNW}(-N)Z_{ghost}\end{align*}
	\begin{align}Q=-\partial ^{2}_{\tau }+\mu ^{2}_{0}\left( 1-\frac{2}{\cosh ^{2}\left( \mu _{0}\tau \right) }\right)\end{align}
	\begin{align} g = kah, k \in K, a \in A^+, h \in H.\end{align}
	\begin{align}-\gamma^{cd} \partial_{c} X^{\mu}\partial_{d} X^{\nu} g_{\mu \nu} +\frac {\varepsilon^{cd}}{\sqrt{-\gamma}} F_{cd} = M\end{align}
	\begin{align}d_{i} = \sqrt{S_{i}} \ K_{i} \ exp(-S_{i}), \ \ \ i = 1, 2\end{align}
	\begin{align}M_{n+1} - M_n = C_n \exp(-4\pi M_n^2).\end{align}
	\begin{align}{\bf A}={1\over 2}\left( \begin{array}{cc} p^\dagger & q^\dagger \end{array}\right){\bf \sigma}\left(\begin{array}{c} p\\q \end{array}\right)\end{align}
	\begin{align}\alpha _{p}l^{D-2p}=\left\{\begin{array}{ll}(D-2p)^{-1}\left(\begin{array}{c}n-1 \\ p\end{array}\right) , & D=2n-1 \\ \left(\begin{array}{c}n \\ p\end{array}\right) , & D=2n.\end{array}\right. \end{align}
	\begin{align}x_H = -\,\frac{1}{2\lambda} \ln\frac{M}{2\lambda}\,,\end{align}
	\begin{align}\int_{{\cal M}_{{\rm ins}}} \left\vert \frac{1}{{\rm det}T_{0}} \right\vert,\end{align}
	\begin{align}\sum_{j \neq i} (c_j + d_j c_i) T^j =0 \, .\end{align}
	\begin{align}\left\|x-a\right\|+\left\|x-b\right\|+\left\|x-c\right\|=\sqrt{\mathcal{X}}\end{align}
	\begin{align}{\cal O} = {\cal O}^{\eta } + \zeta {\cal O}^{\zeta }\,,\end{align}
	\begin{align} g = hak, h \in H, a \in A^+, k \in K,\end{align}
	\begin{align}{\bf C}P^2 = U_1 \cup \{ (0:\phi_2:\phi_3) \},\quad (\phi_2:\phi_3) \not= 0 ,\end{align}
	\begin{align}{\cal E}_0=\lim_{\delta\rightarrow 0}\frac{\pi\hbar}{\delta}{\cal E}_\delta=\int_{-\infty}^{+\infty}d\gamma\,e^{\frac{i}{\hbar}\gamma\hat{\phi}}\ \ \ .\end{align}
	\begin{align}f_M=\left(\begin{array}{c}f^1\\\Lambda f^{1}.\end{array}\right).\end{align*}
	\begin{align}f(\infty)=1,\quad \chi(\infty)=v(\infty)=u(\infty)=\phi(\infty)=\kappa(\infty)= 0,.\end{align}
	\begin{align}K_{DFF} = K_{CCM} + \frac{g}{4H_{DFF}} = w_{0,2}+{g\over4}w_{-2,0}.\end{align}
	\begin{align}m^2 = \pi^3 \frac{\Lambda^4}{g^4}\exp\{-\frac{\pi}{2}\frac{\Lambda}{g^2}\}\end{align}
	\begin{align}\delta \chi_-=-\eta_- \left( \partial_0 w +i\epsilon_{ojk}\partial^j \alpha^k -2i e \Re ( \phi^ \varphi ) \right) \,.\end{align*}
	\begin{align}Z_+(\theta)=Ai(z) \qquad Z_-(\theta) = -Ai\,^\prime(z)\end{align}
	\begin{align}s^2-\varrho^2=\left\|\frac{2\varrho}{m}\frac{(p,\xi)}{\chi}\right\|^2\end{align}
	\begin{align}K(T, {\bar T})= -\log\{( T+ {\bar T})^3 + {\cal I}_{instanton} \}\end{align}
	\begin{align}M^{w}=\left\{ t(w\bar{t}w)a\|t\in M,a=(a_{1},a_{2},...,a_{n}),a_{i}=1w(i)\neq i,a_{i}=\pm1w(i)=i\right\} .\end{align}
	\begin{align}F_{\mu\nu}(x)=\partial_\mu A_\nu(x)-\partial_\nu A_\mu(x)+i\,[A_\mu(x),A_\nu(x)].\end{align}
	\begin{align}\Lambda _{B}^{A}=R_{BCD}^{A}\nabla _{a}\Phi ^{C}\nabla ^{a}\Phi ^{D}.\end{align}
	\begin{align}w(z)={\hbar\over 2 \pi^2}\int_0^{\infty}dk_z\int_0^{\infty} dk_{\\|} {k_{\\|}\over k}\left[\left(k^2+k_{\\|}^2\right)\sin^2(k_z z) +k_z^2 \cos^2(k_z z)\right]\left[{1\over 2}+{\overline n}(k)\right].\end{align}
	\begin{align}\left. A_{\beta,1}^{(2)}\right\|_{{\cal C}_\beta}=3\left. A_{\beta,1}^{(0)}\right\|_{{\cal C}_\beta}+8(\beta-2\pi)+ 2\cdot4\pi~~~.\end{align}
	\begin{align}\zeta_{\rho}(s)=-\frac{2\, R^{2s}}{\Gamma(s+1)\,\Gamma(-s)}\sum_{l=1}^{\infty}\nu^{1-2s}\int_0^{\infty}dk\,k^{-2s}\frac{d}{dk}\ln\Delta_{\rho,l}(\nu k), \quad \nu=l+\frac{1}{2}, \;\;\rho=\pm1 \end{align}
	\begin{align}\pi \min (1,p)<\|\Im m\, \vartheta \|<\pi \frac{p+1}{2}\end{align}
	\begin{align}\frac{\partial}{\partial r}\left(e\sp{-u}\right)=\frac{\mp k}{r^{2}}\left[1-\frac{\partial H_{3}}{\partial\theta}-\cot\theta\:H_{3}-H_{2}H_{4}\right],\end{align}
	\begin{align}\partial_{\mu}j^{\mu}+\partial_{\tau}j^{5}=0\end{align}
	\begin{align}\left[ {\cal P}^2-\left( M\Omega \right) ^2-\frac q2\sigma ^{\mu \nu }F_{\mu\nu }+ibM\gamma ^0\right] \phi (x)=0\;,\;\; \end{align}
	\begin{align}\left[\begin{array}{cccc}g_{2}+w_{2}^{\ 2}h_{4}+n_{2}^{\ 2}h_{5} & w_{2}w_{3}h_{4}+n_{2}n_{3}h_{5} &w_{2}h_{4} & n_{2}h_{5} \\w_{2}w_{3}h_{4}+n_{2}n_{3}h_{5} & g_{3}+w_{3}^{\ 2}h_{4}+n_{3}^{\ 2}h_{5} &w_{3}h_{4} & n_{3}h_{5} \\w_{2}h_{4} & w_{3}h_{4} & h_{4} & 0 \\n_{2}h_{5} & n_{3}h_{5} & 0 & h_{5}\end{array}\right] , \end{align}
	\begin{align}\big(\alpha_{\kappa}\|_{M^{w}}=\chi\|_{M^{w}}\big)\Rightarrow\big(\overline{\alpha_{\kappa}}w(\alpha_{\kappa})=\bar{\chi}w(\chi)\big).\end{align}
	\begin{align}T=\left( \begin{array}{cc}A&B\\C&D \end{array}\right) .\end{align}
	\begin{align}\langle f \| g \rangle= \int \frac{dp}{(1 + \beta p^2)^{1-\alpha}}\,f^(p)\,g(p)\;,\end{align*}
	\begin{align}f= C_1 + C_2 \log \left( \frac {2-z-2\sqrt{1-z}}{z} \right),\end{align}
	\begin{align}\frac{\lambda_R}{2m^2_{f,R}}\left(\langle\Phi^2(t)\rangle-\langle\Phi^2(0)\rangle\right)\approx 1\end{align}
	\begin{align}f^{(n,i_1,i_2,\dots,i_k)}_P \mbox{tr}[(b^\dagger)^{i_1} a^\dagger\dots a^\dagger]\dots \mbox{tr}[(b^\dagger)^{i_k} a^\dagger \dots a^\dagger]\|0>,\end{align}
	\begin{align}X^I = \frac{q^I}{Z}\,, \qquad X_I = \frac{V_I}{X^J V_J}\,.\end{align}
	\begin{align}\Delta^t U(a)=U(a\cos(2\pi t)+ia\sin(2\pi t))\Delta^t\end{align}
	\begin{align}f(\rho) \sim 1, \quad L(\rho) \sim r(\rho),\end{align}
	\begin{align}C_m,C_{m}^i;\ \ \ C_{mn},C_{mn}^i;\ \ \ \ \ \ m,n=1,2,3\end{align}
	\begin{align}\langle T_{\mu\nu}\rangle_{ren}\stackrel{\rm def}{=}\langle T_{\mu\nu}\rangle-\langle T_{\mu\nu}\rangle^{(4)}\, .\end{align}
	\begin{align}\overline{\alpha_{\kappa}}w(\alpha_{\kappa})=\bar{\chi}w(\chi)\end{align}
	\begin{align}i_Y{\cal I}_\Omega=\Bigl[Y^\mu {\bf F}_{\mu\nu}dx^\nu\Bigr]\otimes\varepsilon^2-\Bigl[Y^\mu ({\bf F})_{\mu\nu}dx^\nu\Bigr]\otimes\varepsilon^1.\end{align*}
	\begin{align}\tilde{Q}^{(L, I)}_1 \tilde{Q}^{(L, I)}_0 \neq 0\,\end{align}
	\begin{align}H=\frac{m}{2}\sum_{a=1}^N\,\vec{v}_a^2 =\frac{1}{2m}\sum_{a=1}^N\,[\vec{p}_a-e\vec{A}(\vec{x}_a)]^2\end{align}
	\begin{align}Z_K\propto\left(\frac{8\pi G \Lambda}{9} I(\phi_{cl})\right)^{-K} K!\end{align}
	\begin{align}A\|_{I_\mu} \;=\; \left( \begin{array}{cccc} \mu & 1 & \cdots & 0 \\0 & \mu & \cdots & 0 \\ \vdots & \vdots & \ddots & 1 \\ 0 & 0 & \cdots & \mu \end{array} \right) \;.\end{align}
	\begin{align}\dot{x}^{AA'} = o^A \bar{o}^{A'}, \,\,\acute{x}^{AA'}= o^A \bar{\iota}^{A'} + \iota^A \bar{o}^{A'},\,\,(\rho^\tau o^A\bar{o}^{A'})\dot{} = {\cal F}^{AA'},\end{align}
	\begin{align}\begin{array}{c}V(z_{12}^{\prime})=U^{-1}(z_{1};g)V(z_{12})U(z_{2};g)\,,\\{}\\V(z_{21}^{\prime})=U^{-1}(z_{2};g)V(z_{21})U(z_{1};g)\,.\end{array}\end{align}
	\begin{align}{u^2 \over l^2} \sin^2 \alpha ~\rho^{12} + \left({u^2 \over l^2} \cos^2 \alpha- {u^2 \over l^2} \sin^2 \alpha -1\right) \rho^6 - {u^2 \over l^2} \cos^2 \alpha = 0\end{align}
	\begin{align}F = \frac{1}{\pi} \left( 2 \alpha ' \right)^{- d/2} \langle B, y_1 , v_1 \| D \|B, y_2 , v_2 \rangle~. \end{align}
	\begin{align}ds^2 = (dX^0)^2 - R(X^0)^2 \sum_{i=1}^{D-1}(dX^i)^2\end{align}
	\begin{align}\prod_{(i,j)\in I_{w}}\big\|\frac{t_{i}}{t_{j}}\big\|^{\kappa(i,j)+\kappa(w(i),w(j))}=\prod_{i=1}^{n}\|t_{i}\|^{\lambda_{i}+\lambda_{w(i)}}.\end{align}
	\begin{align}\tilde{\zeta}(s)=\frac{V_q}{(4\pi)^{q/2}}\frac{\Gamma(s-q/2)}{\Gamma(s)} \sum_{\bf p}(\sigma^d_{\bf p}+m^2-\mu^2)^{-s}\;.\end{align}
	\begin{align}\tan ^{-1} \left( \frac {{\sqrt {4\xi -1}}}{3} \right)=\frac {{\sqrt {4\xi -1}}}{3} -\frac {(4\xi -1)^{\frac {3}{2}}}{27} +O(4\xi -1)^{\frac {5}{2}}.\end{align}
	\begin{align}{\rm Aux}^2\;=\;\bigl\{ \sum_i\,[D,a_i][D,b_i]\;:\;\sum_i\;a_i\,[D,b_i]\;=\;0\bigr\},\end{align}
	\begin{align} T_{ab}{}^{c} \;=\; -\left[ i_{k_{(m)}}\hat{C}\right]_{ab} Q^{mn}k_{(n)}^{c} \hspace{.2cm},\hspace{.2cm} 2 K_{abc} \;=\; -T_{abc}+T_{bca}-T_{cab} \; ,\; \check{\omega} \;=\; \omega + K \; .\end{align}
	\begin{align}f\left( T(z)\right) =T'(z) \left[ f(z) -\frac{\delta w}{\epsilon w}(z-\alpha )\right] \ .\end{align}
	\begin{align}\vec{A}=\frac{\phi}{2\pi}\frac{\Theta(r-R)}{r}\vec{e}_\varphi\ , \ \ A^0=0\ ;\end{align}
	\begin{align}\Psi = x^{-\kappa} ( \Psi_{0} + \Psi_{1} + ... )\; .\end{align}
	\begin{align}\delta A_i = \theta C_{iT} \quad \qquad\delta \pi_i = \theta P_{iT}\end{align}
	\begin{align}S(g)^{-1}=1+\sum_{n=1}^\infty \frac {1}{n!} \int d^4x_{1}\ldots d^4 x_n \tilde{T}_n(x_1,\ldots x_n)g(x_1)\ldots g(x_n)\end{align}
	\begin{align}{\rm U}_0({\tau}^{(1)}) {\rm U}_0({\tau}^{(2)}) =\exp\{ 2\pi i {\omega}_{2}({\tau}^{(1)}, {\tau}^{(2)})\}{\rm U}_{0}( {\tau}^{(1)} + {\tau}^{(2)} ) ,\end{align}
	\begin{align}\gamma(s,\chi,\psi)\gamma(1-s,\overline{\chi^{-1}},\overline{\psi^{-1}})=1.\end{align}
	\begin{align}\alpha_1=\epsilon_1-\epsilon_2\,;\,\,\alpha_2=\epsilon_2-\epsilon_3\,;\,\,\alpha_3=\epsilon_1-\epsilon_3\end{align}
	\begin{align}(\sigma_i\psi)_a(x)=-\psi_a(x).\end{align}
	\begin{align}{\cal M} = - \frac{c_f g_s^2}{4} \frac{1}{k^2 - M_G^2} J_c^\mu J_{c \mu}^{\dagger}\end{align}
	\begin{align}<u^{2}><q_{u}^{2}> = 1/4 , \qquad <v^{2}><q_{v}^{2}> = 1/4 .\end{align}
	\begin{align}\tilde C_{\alpha}^{\,\alpha_{0}}(p):=\int\limits_{\alpha_{0}}^{\alpha}d\alpha'\,e^{\,-\alpha'(\,p\eta p+(\varepsilon+i)m^{2})}\quad,\qquad\varepsilon>0\ ,\ 0<\alpha_{0}\leq\alpha<\infty\quad,\end{align}
	\begin{align}g_{12}= e_1 e_2 = g_{34}= e_3 e_4 = 1, \qquad (others\quad components \quad g_{ab}) = 0 .\end{align}
	\begin{align}V_{eff}=\frac{1}{2}\mu ^{2}\varphi ^{2}+\lambda\varphi^{4}-\varphi H\, ,\end{align}