Datasets:

wulanbhai
/

latex-easy-test

image imagewidth (px) 57 742	latex_formula stringlengths 30 389
	\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}\alpha_1=\epsilon_1-\epsilon_2\,;\,\,\alpha_2=\epsilon_2-\epsilon_3\,;\,\,\alpha_3=\epsilon_1-\epsilon_3\end{align}
	\begin{align}H^2(u)=\left[\frac{e^{4u}+b^4}{e^{4u}-b^4}-\frac{1}{2u}+\frac{2K}{u \Lambda^2}\left(\frac{e^{2u}}{e^{4u}-b^4}\right)\right].\end{align}
	\begin{align}A(\lambda/e^2)\approx 100.56{\exp\left(-1.21(\lambda/e^2)^{2/3}\right)\over(\lambda/e^2)^{1/3}}\;.\end{align}
	\begin{align}\alpha=2\lambda-4x^\mu k_\mu-4\bar\theta\eta.\end{align}
	\begin{align}w^{(N)}(ab\|\alpha \beta ,\, \lambda )=e^{i\omega b}w(ab\|\alpha \beta ,\, \lambda ).\end{align}
	\begin{align}\langle \sqrt{dx' \wedge dy'}, \sqrt{dq_1 \wedge dp_2} \rangle = dq_1 \,dq_2 \,dp_2.\end{align}
	\begin{align}<G(p,\tau)> = \int_0^\infty d\tau \exp{-i \tau (m^2-p^2)} \sum_{n=0}^\infty(\frac{-e^2 A}{\epsilon})^n (\mu^2 \tau)^{n \epsilon}.\end{align}
	\begin{align}U_0(\theta)=1\;;\;\;\;\;\; U_1(\theta)=2\cos\theta=\frac{1}{\sqrt x}.\end{align}
	\begin{align}s_{1}(P,t)=-P \int_{0}^{t} \frac{X(t^{\prime})}{A(t^{\prime})} d t^{\prime},\end{align}
	\begin{align}\psi_1=f_1(x_1+\frac{2i\gamma_1\hbar}{\theta}x_2)e^{-\frac{3x_1^2}{8\gamma_1\hbar}-\frac{\gamma_1\hbar x_2^2}{2\theta^2}+\frac{ix_1x_2}{2\theta}},\end{align}
	\begin{align} N_{l}= <0,in\|a_{l}^{\dagger}(out)a_{l}(out)\|0,in>=\left\|g\left({}_{-}\|{}^{+}\right)\right\|^2.\end{align}
	\begin{align} S = \frac{1}{2\kappa^2} \int{ d^4x \sqrt{-g} \left( R + \frac{1}{6m^2} R^2 \right) }.\end{align}
	\begin{align}S\left( p\right) =\frac{1}{p\hspace{-0.2cm}/ A\left(p^{2}\right) -B\left( p^{2}\right) }\end{align}
	\begin{align}\varsigma^{} W_C= W_1(\rho)\pm W_2(\chi)\end{align*}
	\begin{align}\pi^{\mu \nu} = \frac{\partial{\cal L}}{\partial {\dot{B}}_{\mu \nu}} = \frac{1}{2}(H^{0 \mu \nu} - m \epsilon^{0 \rho \mu \nu}A_{\rho})\end{align}
	\begin{align}p(r)=\frac{1}{r^3}\int_{\infty}^{r}{\varepsilon (t)t^2dt}=\frac{2}{r^2}\int_{\infty}^{r}{p_\perp (t)tdt},\end{align}
	\begin{align}DX_{a}(t,\tau) = - \theta_{a}(t) + \tau {\dot{x}}_{a}(t) \ ,\end{align}
	\begin{align}V'(0) = 0, \; \; \; V''(0) = \rho, \; \; \; U(0) = 0, \; \; \; U''(0) = 0.\end{align}
	\begin{align}\check{t_{21}}s_{n}^{\psi}(\rho,\varphi,\phi)=e^{i\psi+\frac{nh}{2\pi}}s_{n}^{\psi}(\rho,\varphi,\phi). \end{align}
	\begin{align}p^s\approx 0,\quad p^a\approx 0,\quad p^{ij}\approx 0.\end{align}
	\begin{align}(1+\omega)/2,(1-\omega)/2,2;-x/\Sigma_0^2\end{align}
	\begin{align}F_{ab} = -i L_a A_b + iL_b A_a -\epsilon_{abc}A_c\end{align}
	\begin{align}\lim_{N\rightarrow \infty}\limsup_{M\rightarrow \infty}[ \|\|u^_t(\tau)-u^{N,M,k^{N,M}}_{j^{N,M}}\|\| +\|\|y^_t(\tau))- y^{N,M,k^{N,M}}_{j^{N,M}}\|\|] = 0. \end{align}
	\begin{align}{\rm tr}_{\rho_1 \times \rho_2} e^{iF} = \left({\rm tr}_{\rho_1 }e^{iF} \right) \left( {\rm tr}_{\rho_2} e^{iF} \right)\end{align}
	\begin{align}j_{\nu,1}\,< \sqrt{\left(\nu+\frac12\right)\left(\nu+2\sqrt{\nu +\frac 32}+\frac 52\right)}\end{align}
	\begin{align}\alpha_\phi^{ij}(x;y)=\alpha^{ij}(x)+\nabla_r\alpha^{ij}(x)\,y^r+\left(\frac12\,\nabla_r\nabla_s\alpha^{ij}(x)-\frac16\,R^{[i}_{rks}(x)\alpha^{j]k}(x)\right)\,y^ry^s+\cdots,\end{align}
	\begin{align}X(t)=\sum\limits_{k \in \mathbb{Z}}\xi_{0k}a_{0k}(t)+\sum\limits_{j=0}^{\infty}\sum\limits_{l \in \mathbb{Z}}\eta_{jl}b_{jl}(t),\end{align}
	\begin{align}\bar L = L + \partial _{\mu } F^{\mu } \,\,\, , \end{align}
	\begin{align}\sum_{\{\Gamma_{i}\}} n_{i}\left ({\cal R'} \Gamma_{i}\right).\end{align}
	\begin{align}F= \frac{1}{\tilde r} \sigma(\alpha)^{-5}\ d{\rm Vol}_{S^5} =\tilde r^4 12^{-5/2} \sigma^{-8}\sin^4\alpha\ d\alpha\wedge d\Omega_{(4)} \,.\end{align}
	\begin{align}\vec{\beta}=\vec{\beta}_{(n'_{1},n_{1})(n'_{2},n_{2})(n'_{3},n_{3})}=\sum_{a=1}^{3}(\frac{1-n'_{a}}{2}\alpha_{-}+\frac{1-n_{a}}{2}\alpha_{+})\vec{\omega}_{a}\end{align}
	\begin{align}ds^{2}=-d\tau ^{2}+\tau ^{2}dx_i dx^i \end{align}
	\begin{align}\varphi _{4}=\partial _{1}A_{1}=0. \end{align}
	\begin{align}\epsilon \{ \nabla \times \dot{\bf v} + \nabla\times (({\bf v} \cdot \nabla) {\bf v}) \} = \Delta (\nabla \times {\bf v}),\end{align}
	\begin{align}{ }^{\varepsilon}A_{\mu} \equiv A_{\mu}+\nabla_{\mu} \varepsilon ,\end{align}
	\begin{align}\frac{1}{T_f}=\frac{1}{\tilde{\omega}}\ln\left[1+\frac{1}{\bar{n}_{\rm eq}}\right]\;.\end{align}
	\begin{align}m_n \simeq \left(n + \frac{1}{4} \pm \frac{4 M_{5}^{3}}{\pi W}\right) \pi k e^{-Rk\pi }\; .\end{align}
	\begin{align}U(l)=T_l {\rm exp}\, \left(\int_0^l \eta(l')dl'\right)\,.\end{align}
	\begin{align}\ln \sqrt{b'(t)}=\sigma(b),~~~\frac{b''(t)}{2b'(t)^2}=\sigma'(b),\end{align}
	\begin{align}R^2_g\ \sim\\frac{1}{m^2_\gamma}\, L^2 =\frac{1}{m^2_\gamma} \left( \ln \frac{m_\gamma}{m_q}\right)^2\, .\end{align}
	\begin{align}v_k^{''} + \left(k^2 - (z^{''}/z)\right)~v_k =0 \; .\end{align}
	\begin{align}\big( A_{j_1} \cdot B_{j_2} \big)_{j_3} := \left\{ \big( A_{j_1} \cdot B_{j_2} \big)_{j_3,m_3} \Big\| \; m_3=-j_3,\dots,+j_3 \right\}\end{align}
	\begin{align}\Gamma: R(\lambda, \mu) \equiv R_0(\lambda,\mu) + \sum_j R_j(\lambda,\mu) H_j = 0\end{align}
	\begin{align}{\cal P}=({\hat{\xi}}_{i}ad{\lambda}_{i})^2 \end{align}
	\begin{align} \phi\cdot f_{uu} =\frac{(f_v\cdot f_{u})_u-(f_{uv}\cdot f_u)}v =-\frac{(f_{uv}\cdot f_u)}v =-\frac{E_v}{2v}=-\frac{2E_0+v (E_0)_v}{2}. \end{align}
	\begin{align}H = \sum_{\bf n} \frac{1}{2}(1+\sigma_{3}({\bf n}))+\kappa\sum_{{\bf n},{\bf i}} ( \sigma_{+}({\bf n}) \mu_{3}({\bf n},{\bf i}) \sigma_{+}({\bf n+i})+ h.c.)\end{align}
	\begin{align}{\cal H}^0_{\alpha=\pm \infty}={ 1 \over 2} p^2_0\end{align}
	\begin{align}\sum_{k=1,\cdots,6}({\rm Tr}\gamma_{k,6})^2 = 6 \cdot 16^2.\end{align}
	\begin{align}W[{\psi},A] = \int_{- \infty}^{\infty} dt \int_{0}^{\rm L} dx\sum_{k=1}^{2} \overline{\psi}_k ({\gamma}^{\mu} i{\partial}_{\mu} - m_k) {\psi}_k + \frac{1}{2}\int_{- \infty}^{\infty} dt \int_{0}^{\rm L} dx \int_{0}^{\rm L}dy J^0(x,t) D(x,y\|{\rm L}) J^0(y,t)\end{align}
	\begin{align}\varphi(x)=\sqrt{\frac{\lambda}{2\pi}}\,e^{i\frac{N+1}{2}x},\;\;\psi(x)=\sqrt{\frac{\lambda}{2\pi}}\,e^{-i\frac{N-1}{2}x}.\end{align}
	\begin{align}\sum_{n=1/2}^{\Lambda} ( \Delta_n(\zeta) + \Delta_n(-\zeta) ) a_0 \sim (\ln \Lambda + {\rm{finite}} ) a_0\;.\end{align}
	\begin{align}{1\over 2}\int {d^4p\over (2\pi)^4}\phi(p)(p^2+m^2)\phi(-p)K^{-1}(p^2/\Lambda^2)\end{align}
	\begin{align}\phi'=\frac{b}{(r-m)^2},\hspace{5mm}r'^2=1-\frac{b^2}{(r-m)^2},\end{align*}
	\begin{align}a^{ijkl}=a^{\sigma(i)\sigma(j)\sigma(k)\sigma(l)},\end{align}
	\begin{align}N!=\sqrt{2\pi N}N^Ne^{-N}\big( 1+\frac{1}{12N}+{\cal O}\big(\frac{1}{N^2}\big) \big) \; .\end{align}
	\begin{align}c^{111122}=4c^{111122}+2c^{110022}+2c^{001122}+c^{000022},\end{align}
	\begin{align}\gamma^{ijklqrst}(\eta_{ab})+\gamma^{ijklqstr}(\eta_{ab})+\gamma^{ijklqtrs}(\eta_{ab})=0.\end{align}
	\begin{align}\mbox{\mbox{}$^\!$} F_{\mu\nu}(x) = -\mbox{\small $\frac{1}{2}$} \epsilon_{\mu\nu\rho\sigma}F^{\rho\sigma}(x),\end{align*}
	\begin{align}S(\bar{\gamma}_n)\cdot\bar{\gamma}_{n-1}/\gamma_n-\bar{\gamma}_n\cdot S(\bar{\gamma}_{n-1}/\gamma_n)=-\bar{\gamma}_n\cdot\bar{\gamma}_{n-1}/\gamma_n+\bar{\gamma}_n\cdot\bar{\gamma}_{n-1}/\gamma_n=0.\end{align}
	\begin{align}J^{V_L}=\lambda^{(d-1)} \left(-\frac{16\pi^2 \lambda^{\varepsilon}}{15g} \right)^{\frac{d+1}{2}} [1+{\cal O}(\varepsilon,g)] \, .\end{align}
	\begin{align}\dot{C}_{1(2)}({\vec k},t) = \frac{\dot{\nu}_k}{{\nu}_k}{\cal N}^t({\vec k},t) \mp 2{\nu}_k C_{1(2)}({\vec k},t) .\end{align}
	\begin{align}\delta_F ( \left[ v \theta^6 \right] )+ \delta_B ( \left[ v^2 \theta^4 \right] )+ v^i \frac{\partial}{\partial \phi^i } \left( \epsilon L^{(4)} \theta \right) = 0 ~ ,\end{align}
	\begin{align}S_{gauge} = -\frac{1}{2\kappa^2}\int d^{10} x e^{-2\Phi}\frac{\alpha^\prime}{8} tr F^2,\end{align}
	\begin{align}X^{\alpha\beta\mu\lambda}{}_{,\lambda \alpha} = 0\end{align}
	\begin{align}\Delta_l^{TE}(i\hat \omega, a\rightarrow \infty)=\Delta_l^{TM}(i\hat \omega, a\rightarrow\infty)=\frac{1}{2}(\sqrt{\epsilon_1\mu_2}+\sqrt{\epsilon_2\mu_1}).\end{align}
	\begin{align}{\cal L}=\|\partial_{\mu}\vec{\varphi}\|^2 +\frac{i}{2} \bar{\vec{\psi}} \sigma^{\mu}\partial_{\mu} \vec{\psi} +\frac{i}{2} \vec{\psi} \bar{\sigma}^{\mu}\partial_{\mu} \bar{\vec{\psi}}.\end{align}
	\begin{align}\left\langle f(a^a_\mu) g(A^m_\nu) \right\rangle = \left\langle f(a^a_\mu) \right\rangle \left\langle g(A^m_\mu) \right\rangle\end{align}
	\begin{align}\Psi[\sigma\ ;A]=\psi[\sigma]e^{-iA{\cal E}}\end{align}
	\begin{align}G(z)=\dfrac{1}{2\pi i }\sum_{j=1}^n\int_{\gamma_j}\dfrac{Q(s)}{(s-a)^M(p(s)-z)}ds,\end{align}
	\begin{align}\chi_{\{h\}}(A)=\det_{_{\hskip -2pt (k,l)}} \bigl(P_{h_k+1-l}(\theta)\bigr)\end{align}
	\begin{align}S_{\rm CS}(A) = {k\over 4\pi} \int_{\cal M}{\rm Tr}\left( A\wedge dA + {2\over 3} A\wedge A\wedge A\right).\end{align}
	\begin{align}f[\phi]=i\int d^4x\Biggl[{\partial\cdot A^\alpha\over\lambda}(\partial\cdot A^\alpha-\eta\cdot A^\alpha)+{\bar c}(\partial\cdot{\rm D}-\eta\cdot{\rm D})c^\alpha\Biggr].\end{align}
	\begin{align}D_{i\omega }(k\xi _{1},k\xi _{2})=0. \end{align}
	\begin{align}\psi _{\pm }^{(3)}=C_1^{\prime \prime }e^{ip_{\bot }r}+C_2^{\prime\prime }e^{-ip_{\bot }r}.\end{align}
	\begin{align}V_g(\Phi) = g_n\, { \|\Phi\|^{2m}\, \Phi^{4-2m+n}\over M_{\rm P}^n} + h.c.\ ,\end{align}
	\begin{align}{ \bf (3,1)_{-x} \otimes (1,2)_{-y} \rightarrow (3,2)_{-(x+y)} } \qquad\quad { \bf (\bar{3},1)_x \otimes (1,2)_y \rightarrow (\bar{3},2)_{x+y} }\end{align}
	\begin{align}\{f\circ g\}=\sum_k{\partial f\over\partial a_k}{\partial g\over\partial a^_k}={\partial f\over\partial{\vec a}}\cdot{\partial g\over\partial{\vec a}^}.\end{align}
	\begin{align}g^{\mu,\l}_{hom,2n}= k^{n-n^2}{\prod_{i=1}^{2n-1}(i!)^{-1}}\times\det\left\{\frac{1}{\sqrt{k}}h_{\mu,\l}^{(2j+2k-3)}(0)\right\},\end{align}
	\begin{align}e^{ \varphi(z,\bar {z})}={\|{J_H^{-1}(z)}'\|^2\over({\rm Im}\,J_H^{-1}(z))^2},\end{align}
	\begin{align} \tilde{M}_{(D-1)a}:= \tilde{R}^{bc} \wedge \tilde{E}^{\wedge (D-3)}_{abc} + \ldots = J_{(D-1)a} \; ,\end{align}
	\begin{align}R^i_aZ^a_{a_1}=2y_jf^{ji}_{a_1}\ , \end{align}
	\begin{align}H^{0} (dP_9, {\cal O}_{dP_9}(9 \sigma\|_{dP_{9}} -F))=0.\end{align}
	\begin{align}\lbrack M_{1}\sigma,M_{2}\sigma]=0\,\,. \end{align}
	\begin{align}X^+(\sigma,\tau) = x^{+} + \alpha' P^+ \tau\end{align}
	\begin{align}L = {1\over 2} g_{ij}(z) \left[\dot z^i \dot z^j +i \bar \psi^i \gamma^0 D_t \psi^j \right] + {1 \over 6} R_{ijkl} (\bar \psi^i \psi^j)(\bar \psi^k \psi^l)\end{align}
	\begin{align}(E\times H)^k = (D\times B)^k\ ,\end{align}
	\begin{align}\frac{1}{2\pi i} \oint du j_u \ +\ 3. \end{align}
	\begin{align}\left\| d_{a}-d_{b}\right\| <<<1,\quad \forall a,b=1,...,n,\end{align}
	\begin{align}D^{j}_{mn}(u)=(e_{m}, T^{j}e_{n})\end{align}
	\begin{align}e^{\alpha} _{\underline {\alpha}}(x, \phi, \eta)= (f (\phi,\eta_i))^\alpha_\beta\left [ 1 -{1\over 2} x^m \Gamma_m \Gamma_r (1- \tilde \Gamma)\right]^\beta_{\underline {\alpha}} \ .\end{align}
	\begin{align}{\tilde M}_n(u) = M^{1/n}(u),\end{align}
	\begin{align}a_{ {\bf{k}} }({\bf{q}}) = \frac{1}{\sqrt{n_{ {\bf{k}} - {\bf{q}}/2 }}}c^{\dagger}_{ {\bf{k}} - {\bf{q}}/2 }M({\bf{k}}, {\bf{q}})c_{ {\bf{k}} + {\bf{q}}/2 }\end{align}
	\begin{align}f\;f^{\dagger }\rightarrow (\frac{ {\cal D} }{ {\cal D} z}\frac z{w-z})\;\;\theta(\|w\|-\|z\|).\end{align}
	\begin{align}\det(Q)^2= \lim_{n \rightarrow \infty} [ \det(P_n(Q^2))]^{-1} \end{align}
	\begin{align}c_{\epsilon}= \frac{3 - 2 \epsilon}{(4\pi)^{1-\epsilon}\Gamma(2-\epsilon)},\quad (c_0=\frac{3}{4\pi}).\end{align}
	\begin{align}M_i^{(s)}(u-\lambda)L_i^{(x)}(u)=L_i^{(y)}(u)M_i^{(t)}(u-\lambda)~.\end{align}
	\begin{align}K=-\ln\{i(S - {\bar S}) -2iG^{(1)} \} + G^{(o)},\end{align}
	\begin{align}e^{2\Phi}d(e^{-2\Phi}\phi^{\pm}) = \mp H .\end{align*}
	\begin{align}{\cal L} = R + \gamma \,( R_{A B C D} \, R^{A B C D} - 4 \, R_{A B} \, R^{A B} + R^2 ) \, , \end{align}

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LaTeX Easy Test Dataset

Dataset ini adalah subset "easy" dari OleehyO/latex-formulas - formula LaTeX yang tidak termasuk kategori "hard" (single-line atau formula sederhana).

Dataset Details

Source Dataset: OleehyO/latex-formulas (config: cleaned_formulas)
Total Source Samples: 552,340
Hard 2-Line Samples: 99,305
Easy Samples Available: 453,035
Easy Test Set Size: 10,000
Seed: 42

Splits

Split	Samples
Test	10,000

Definition

"Easy" subset includes all formulas that are NOT in the "hard_2line" category:

Single-line formulas
Simple expressions without complex multi-line structures
Formulas without \\ line breaks or complex environments

Usage

from datasets import load_dataset

# Load easy test set
easy_test = load_dataset("wulanbhai/latex-easy-test", split="test")

# Example: Iterate through samples
for sample in easy_test:
    print(sample['latex_formula'])

Use Cases

Baseline comparison for model evaluation
Testing model performance on simpler formulas
Sanity checks for LaTeX rendering systems
Training data for early-stage models

Related Datasets

Hard 2-Line Dataset - Complex multi-line formulas (train/val/test splits)

Generation Details

Generated: 2025-10-28T04:42:57+00:00
Selection Method: Random sampling from non-hard_2line indices
Seed: 42 (reproducible)

License

Indices only - no images or original LaTeX content redistributed. Refer to source dataset for licensing.

Downloads last month: 28

Size of downloaded dataset files:

42.6 MB

Size of the auto-converted Parquet files:

42.6 MB

Number of rows:

10,000

image imagewidth (px) 57 742	latex_formula stringlengths 30 389
	\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}\alpha_1=\epsilon_1-\epsilon_2\,;\,\,\alpha_2=\epsilon_2-\epsilon_3\,;\,\,\alpha_3=\epsilon_1-\epsilon_3\end{align}
	\begin{align}H^2(u)=\left[\frac{e^{4u}+b^4}{e^{4u}-b^4}-\frac{1}{2u}+\frac{2K}{u \Lambda^2}\left(\frac{e^{2u}}{e^{4u}-b^4}\right)\right].\end{align}
	\begin{align}A(\lambda/e^2)\approx 100.56{\exp\left(-1.21(\lambda/e^2)^{2/3}\right)\over(\lambda/e^2)^{1/3}}\;.\end{align}
	\begin{align}\alpha=2\lambda-4x^\mu k_\mu-4\bar\theta\eta.\end{align}
	\begin{align}w^{(N)}(ab\|\alpha \beta ,\, \lambda )=e^{i\omega b}w(ab\|\alpha \beta ,\, \lambda ).\end{align}
	\begin{align}\langle \sqrt{dx' \wedge dy'}, \sqrt{dq_1 \wedge dp_2} \rangle = dq_1 \,dq_2 \,dp_2.\end{align}
	\begin{align}<G(p,\tau)> = \int_0^\infty d\tau \exp{-i \tau (m^2-p^2)} \sum_{n=0}^\infty(\frac{-e^2 A}{\epsilon})^n (\mu^2 \tau)^{n \epsilon}.\end{align}
	\begin{align}U_0(\theta)=1\;;\;\;\;\;\; U_1(\theta)=2\cos\theta=\frac{1}{\sqrt x}.\end{align}
	\begin{align}s_{1}(P,t)=-P \int_{0}^{t} \frac{X(t^{\prime})}{A(t^{\prime})} d t^{\prime},\end{align}
	\begin{align}\psi_1=f_1(x_1+\frac{2i\gamma_1\hbar}{\theta}x_2)e^{-\frac{3x_1^2}{8\gamma_1\hbar}-\frac{\gamma_1\hbar x_2^2}{2\theta^2}+\frac{ix_1x_2}{2\theta}},\end{align}
	\begin{align} N_{l}= <0,in\|a_{l}^{\dagger}(out)a_{l}(out)\|0,in>=\left\|g\left({}_{-}\|{}^{+}\right)\right\|^2.\end{align}
	\begin{align} S = \frac{1}{2\kappa^2} \int{ d^4x \sqrt{-g} \left( R + \frac{1}{6m^2} R^2 \right) }.\end{align}
	\begin{align}S\left( p\right) =\frac{1}{p\hspace{-0.2cm}/ A\left(p^{2}\right) -B\left( p^{2}\right) }\end{align}
	\begin{align}\varsigma^{} W_C= W_1(\rho)\pm W_2(\chi)\end{align*}
	\begin{align}\pi^{\mu \nu} = \frac{\partial{\cal L}}{\partial {\dot{B}}_{\mu \nu}} = \frac{1}{2}(H^{0 \mu \nu} - m \epsilon^{0 \rho \mu \nu}A_{\rho})\end{align}
	\begin{align}p(r)=\frac{1}{r^3}\int_{\infty}^{r}{\varepsilon (t)t^2dt}=\frac{2}{r^2}\int_{\infty}^{r}{p_\perp (t)tdt},\end{align}
	\begin{align}DX_{a}(t,\tau) = - \theta_{a}(t) + \tau {\dot{x}}_{a}(t) \ ,\end{align}
	\begin{align}V'(0) = 0, \; \; \; V''(0) = \rho, \; \; \; U(0) = 0, \; \; \; U''(0) = 0.\end{align}
	\begin{align}\check{t_{21}}s_{n}^{\psi}(\rho,\varphi,\phi)=e^{i\psi+\frac{nh}{2\pi}}s_{n}^{\psi}(\rho,\varphi,\phi). \end{align}
	\begin{align}p^s\approx 0,\quad p^a\approx 0,\quad p^{ij}\approx 0.\end{align}
	\begin{align}(1+\omega)/2,(1-\omega)/2,2;-x/\Sigma_0^2\end{align}
	\begin{align}F_{ab} = -i L_a A_b + iL_b A_a -\epsilon_{abc}A_c\end{align}
	\begin{align}\lim_{N\rightarrow \infty}\limsup_{M\rightarrow \infty}[ \|\|u^_t(\tau)-u^{N,M,k^{N,M}}_{j^{N,M}}\|\| +\|\|y^_t(\tau))- y^{N,M,k^{N,M}}_{j^{N,M}}\|\|] = 0. \end{align}
	\begin{align}{\rm tr}_{\rho_1 \times \rho_2} e^{iF} = \left({\rm tr}_{\rho_1 }e^{iF} \right) \left( {\rm tr}_{\rho_2} e^{iF} \right)\end{align}
	\begin{align}j_{\nu,1}\,< \sqrt{\left(\nu+\frac12\right)\left(\nu+2\sqrt{\nu +\frac 32}+\frac 52\right)}\end{align}
	\begin{align}\alpha_\phi^{ij}(x;y)=\alpha^{ij}(x)+\nabla_r\alpha^{ij}(x)\,y^r+\left(\frac12\,\nabla_r\nabla_s\alpha^{ij}(x)-\frac16\,R^{[i}_{rks}(x)\alpha^{j]k}(x)\right)\,y^ry^s+\cdots,\end{align}
	\begin{align}X(t)=\sum\limits_{k \in \mathbb{Z}}\xi_{0k}a_{0k}(t)+\sum\limits_{j=0}^{\infty}\sum\limits_{l \in \mathbb{Z}}\eta_{jl}b_{jl}(t),\end{align}
	\begin{align}\bar L = L + \partial _{\mu } F^{\mu } \,\,\, , \end{align}
	\begin{align}\sum_{\{\Gamma_{i}\}} n_{i}\left ({\cal R'} \Gamma_{i}\right).\end{align}
	\begin{align}F= \frac{1}{\tilde r} \sigma(\alpha)^{-5}\ d{\rm Vol}_{S^5} =\tilde r^4 12^{-5/2} \sigma^{-8}\sin^4\alpha\ d\alpha\wedge d\Omega_{(4)} \,.\end{align}
	\begin{align}\vec{\beta}=\vec{\beta}_{(n'_{1},n_{1})(n'_{2},n_{2})(n'_{3},n_{3})}=\sum_{a=1}^{3}(\frac{1-n'_{a}}{2}\alpha_{-}+\frac{1-n_{a}}{2}\alpha_{+})\vec{\omega}_{a}\end{align}
	\begin{align}ds^{2}=-d\tau ^{2}+\tau ^{2}dx_i dx^i \end{align}
	\begin{align}\varphi _{4}=\partial _{1}A_{1}=0. \end{align}
	\begin{align}\epsilon \{ \nabla \times \dot{\bf v} + \nabla\times (({\bf v} \cdot \nabla) {\bf v}) \} = \Delta (\nabla \times {\bf v}),\end{align}
	\begin{align}{ }^{\varepsilon}A_{\mu} \equiv A_{\mu}+\nabla_{\mu} \varepsilon ,\end{align}
	\begin{align}\frac{1}{T_f}=\frac{1}{\tilde{\omega}}\ln\left[1+\frac{1}{\bar{n}_{\rm eq}}\right]\;.\end{align}
	\begin{align}m_n \simeq \left(n + \frac{1}{4} \pm \frac{4 M_{5}^{3}}{\pi W}\right) \pi k e^{-Rk\pi }\; .\end{align}
	\begin{align}U(l)=T_l {\rm exp}\, \left(\int_0^l \eta(l')dl'\right)\,.\end{align}
	\begin{align}\ln \sqrt{b'(t)}=\sigma(b),~~~\frac{b''(t)}{2b'(t)^2}=\sigma'(b),\end{align}
	\begin{align}R^2_g\ \sim\\frac{1}{m^2_\gamma}\, L^2 =\frac{1}{m^2_\gamma} \left( \ln \frac{m_\gamma}{m_q}\right)^2\, .\end{align}
	\begin{align}v_k^{''} + \left(k^2 - (z^{''}/z)\right)~v_k =0 \; .\end{align}
	\begin{align}\big( A_{j_1} \cdot B_{j_2} \big)_{j_3} := \left\{ \big( A_{j_1} \cdot B_{j_2} \big)_{j_3,m_3} \Big\| \; m_3=-j_3,\dots,+j_3 \right\}\end{align}
	\begin{align}\Gamma: R(\lambda, \mu) \equiv R_0(\lambda,\mu) + \sum_j R_j(\lambda,\mu) H_j = 0\end{align}
	\begin{align}{\cal P}=({\hat{\xi}}_{i}ad{\lambda}_{i})^2 \end{align}
	\begin{align} \phi\cdot f_{uu} =\frac{(f_v\cdot f_{u})_u-(f_{uv}\cdot f_u)}v =-\frac{(f_{uv}\cdot f_u)}v =-\frac{E_v}{2v}=-\frac{2E_0+v (E_0)_v}{2}. \end{align}
	\begin{align}H = \sum_{\bf n} \frac{1}{2}(1+\sigma_{3}({\bf n}))+\kappa\sum_{{\bf n},{\bf i}} ( \sigma_{+}({\bf n}) \mu_{3}({\bf n},{\bf i}) \sigma_{+}({\bf n+i})+ h.c.)\end{align}
	\begin{align}{\cal H}^0_{\alpha=\pm \infty}={ 1 \over 2} p^2_0\end{align}
	\begin{align}\sum_{k=1,\cdots,6}({\rm Tr}\gamma_{k,6})^2 = 6 \cdot 16^2.\end{align}
	\begin{align}W[{\psi},A] = \int_{- \infty}^{\infty} dt \int_{0}^{\rm L} dx\sum_{k=1}^{2} \overline{\psi}_k ({\gamma}^{\mu} i{\partial}_{\mu} - m_k) {\psi}_k + \frac{1}{2}\int_{- \infty}^{\infty} dt \int_{0}^{\rm L} dx \int_{0}^{\rm L}dy J^0(x,t) D(x,y\|{\rm L}) J^0(y,t)\end{align}
	\begin{align}\varphi(x)=\sqrt{\frac{\lambda}{2\pi}}\,e^{i\frac{N+1}{2}x},\;\;\psi(x)=\sqrt{\frac{\lambda}{2\pi}}\,e^{-i\frac{N-1}{2}x}.\end{align}
	\begin{align}\sum_{n=1/2}^{\Lambda} ( \Delta_n(\zeta) + \Delta_n(-\zeta) ) a_0 \sim (\ln \Lambda + {\rm{finite}} ) a_0\;.\end{align}
	\begin{align}{1\over 2}\int {d^4p\over (2\pi)^4}\phi(p)(p^2+m^2)\phi(-p)K^{-1}(p^2/\Lambda^2)\end{align}
	\begin{align}\phi'=\frac{b}{(r-m)^2},\hspace{5mm}r'^2=1-\frac{b^2}{(r-m)^2},\end{align*}
	\begin{align}a^{ijkl}=a^{\sigma(i)\sigma(j)\sigma(k)\sigma(l)},\end{align}
	\begin{align}N!=\sqrt{2\pi N}N^Ne^{-N}\big( 1+\frac{1}{12N}+{\cal O}\big(\frac{1}{N^2}\big) \big) \; .\end{align}
	\begin{align}c^{111122}=4c^{111122}+2c^{110022}+2c^{001122}+c^{000022},\end{align}
	\begin{align}\gamma^{ijklqrst}(\eta_{ab})+\gamma^{ijklqstr}(\eta_{ab})+\gamma^{ijklqtrs}(\eta_{ab})=0.\end{align}
	\begin{align}\mbox{\mbox{}$^\!$} F_{\mu\nu}(x) = -\mbox{\small $\frac{1}{2}$} \epsilon_{\mu\nu\rho\sigma}F^{\rho\sigma}(x),\end{align*}
	\begin{align}S(\bar{\gamma}_n)\cdot\bar{\gamma}_{n-1}/\gamma_n-\bar{\gamma}_n\cdot S(\bar{\gamma}_{n-1}/\gamma_n)=-\bar{\gamma}_n\cdot\bar{\gamma}_{n-1}/\gamma_n+\bar{\gamma}_n\cdot\bar{\gamma}_{n-1}/\gamma_n=0.\end{align}
	\begin{align}J^{V_L}=\lambda^{(d-1)} \left(-\frac{16\pi^2 \lambda^{\varepsilon}}{15g} \right)^{\frac{d+1}{2}} [1+{\cal O}(\varepsilon,g)] \, .\end{align}
	\begin{align}\dot{C}_{1(2)}({\vec k},t) = \frac{\dot{\nu}_k}{{\nu}_k}{\cal N}^t({\vec k},t) \mp 2{\nu}_k C_{1(2)}({\vec k},t) .\end{align}
	\begin{align}\delta_F ( \left[ v \theta^6 \right] )+ \delta_B ( \left[ v^2 \theta^4 \right] )+ v^i \frac{\partial}{\partial \phi^i } \left( \epsilon L^{(4)} \theta \right) = 0 ~ ,\end{align}
	\begin{align}S_{gauge} = -\frac{1}{2\kappa^2}\int d^{10} x e^{-2\Phi}\frac{\alpha^\prime}{8} tr F^2,\end{align}
	\begin{align}X^{\alpha\beta\mu\lambda}{}_{,\lambda \alpha} = 0\end{align}
	\begin{align}\Delta_l^{TE}(i\hat \omega, a\rightarrow \infty)=\Delta_l^{TM}(i\hat \omega, a\rightarrow\infty)=\frac{1}{2}(\sqrt{\epsilon_1\mu_2}+\sqrt{\epsilon_2\mu_1}).\end{align}
	\begin{align}{\cal L}=\|\partial_{\mu}\vec{\varphi}\|^2 +\frac{i}{2} \bar{\vec{\psi}} \sigma^{\mu}\partial_{\mu} \vec{\psi} +\frac{i}{2} \vec{\psi} \bar{\sigma}^{\mu}\partial_{\mu} \bar{\vec{\psi}}.\end{align}
	\begin{align}\left\langle f(a^a_\mu) g(A^m_\nu) \right\rangle = \left\langle f(a^a_\mu) \right\rangle \left\langle g(A^m_\mu) \right\rangle\end{align}
	\begin{align}\Psi[\sigma\ ;A]=\psi[\sigma]e^{-iA{\cal E}}\end{align}
	\begin{align}G(z)=\dfrac{1}{2\pi i }\sum_{j=1}^n\int_{\gamma_j}\dfrac{Q(s)}{(s-a)^M(p(s)-z)}ds,\end{align}
	\begin{align}\chi_{\{h\}}(A)=\det_{_{\hskip -2pt (k,l)}} \bigl(P_{h_k+1-l}(\theta)\bigr)\end{align}
	\begin{align}S_{\rm CS}(A) = {k\over 4\pi} \int_{\cal M}{\rm Tr}\left( A\wedge dA + {2\over 3} A\wedge A\wedge A\right).\end{align}
	\begin{align}f[\phi]=i\int d^4x\Biggl[{\partial\cdot A^\alpha\over\lambda}(\partial\cdot A^\alpha-\eta\cdot A^\alpha)+{\bar c}(\partial\cdot{\rm D}-\eta\cdot{\rm D})c^\alpha\Biggr].\end{align}
	\begin{align}D_{i\omega }(k\xi _{1},k\xi _{2})=0. \end{align}
	\begin{align}\psi _{\pm }^{(3)}=C_1^{\prime \prime }e^{ip_{\bot }r}+C_2^{\prime\prime }e^{-ip_{\bot }r}.\end{align}
	\begin{align}V_g(\Phi) = g_n\, { \|\Phi\|^{2m}\, \Phi^{4-2m+n}\over M_{\rm P}^n} + h.c.\ ,\end{align}
	\begin{align}{ \bf (3,1)_{-x} \otimes (1,2)_{-y} \rightarrow (3,2)_{-(x+y)} } \qquad\quad { \bf (\bar{3},1)_x \otimes (1,2)_y \rightarrow (\bar{3},2)_{x+y} }\end{align}
	\begin{align}\{f\circ g\}=\sum_k{\partial f\over\partial a_k}{\partial g\over\partial a^_k}={\partial f\over\partial{\vec a}}\cdot{\partial g\over\partial{\vec a}^}.\end{align}
	\begin{align}g^{\mu,\l}_{hom,2n}= k^{n-n^2}{\prod_{i=1}^{2n-1}(i!)^{-1}}\times\det\left\{\frac{1}{\sqrt{k}}h_{\mu,\l}^{(2j+2k-3)}(0)\right\},\end{align}
	\begin{align}e^{ \varphi(z,\bar {z})}={\|{J_H^{-1}(z)}'\|^2\over({\rm Im}\,J_H^{-1}(z))^2},\end{align}
	\begin{align} \tilde{M}_{(D-1)a}:= \tilde{R}^{bc} \wedge \tilde{E}^{\wedge (D-3)}_{abc} + \ldots = J_{(D-1)a} \; ,\end{align}
	\begin{align}R^i_aZ^a_{a_1}=2y_jf^{ji}_{a_1}\ , \end{align}
	\begin{align}H^{0} (dP_9, {\cal O}_{dP_9}(9 \sigma\|_{dP_{9}} -F))=0.\end{align}
	\begin{align}\lbrack M_{1}\sigma,M_{2}\sigma]=0\,\,. \end{align}
	\begin{align}X^+(\sigma,\tau) = x^{+} + \alpha' P^+ \tau\end{align}
	\begin{align}L = {1\over 2} g_{ij}(z) \left[\dot z^i \dot z^j +i \bar \psi^i \gamma^0 D_t \psi^j \right] + {1 \over 6} R_{ijkl} (\bar \psi^i \psi^j)(\bar \psi^k \psi^l)\end{align}
	\begin{align}(E\times H)^k = (D\times B)^k\ ,\end{align}
	\begin{align}\frac{1}{2\pi i} \oint du j_u \ +\ 3. \end{align}
	\begin{align}\left\| d_{a}-d_{b}\right\| <<<1,\quad \forall a,b=1,...,n,\end{align}
	\begin{align}D^{j}_{mn}(u)=(e_{m}, T^{j}e_{n})\end{align}
	\begin{align}e^{\alpha} _{\underline {\alpha}}(x, \phi, \eta)= (f (\phi,\eta_i))^\alpha_\beta\left [ 1 -{1\over 2} x^m \Gamma_m \Gamma_r (1- \tilde \Gamma)\right]^\beta_{\underline {\alpha}} \ .\end{align}
	\begin{align}{\tilde M}_n(u) = M^{1/n}(u),\end{align}
	\begin{align}a_{ {\bf{k}} }({\bf{q}}) = \frac{1}{\sqrt{n_{ {\bf{k}} - {\bf{q}}/2 }}}c^{\dagger}_{ {\bf{k}} - {\bf{q}}/2 }M({\bf{k}}, {\bf{q}})c_{ {\bf{k}} + {\bf{q}}/2 }\end{align}
	\begin{align}f\;f^{\dagger }\rightarrow (\frac{ {\cal D} }{ {\cal D} z}\frac z{w-z})\;\;\theta(\|w\|-\|z\|).\end{align}
	\begin{align}\det(Q)^2= \lim_{n \rightarrow \infty} [ \det(P_n(Q^2))]^{-1} \end{align}
	\begin{align}c_{\epsilon}= \frac{3 - 2 \epsilon}{(4\pi)^{1-\epsilon}\Gamma(2-\epsilon)},\quad (c_0=\frac{3}{4\pi}).\end{align}
	\begin{align}M_i^{(s)}(u-\lambda)L_i^{(x)}(u)=L_i^{(y)}(u)M_i^{(t)}(u-\lambda)~.\end{align}
	\begin{align}K=-\ln\{i(S - {\bar S}) -2iG^{(1)} \} + G^{(o)},\end{align}
	\begin{align}e^{2\Phi}d(e^{-2\Phi}\phi^{\pm}) = \mp H .\end{align*}
	\begin{align}{\cal L} = R + \gamma \,( R_{A B C D} \, R^{A B C D} - 4 \, R_{A B} \, R^{A B} + R^2 ) \, , \end{align}