harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\frac{c_{\gamma+\alpha}^2}{c_{\gamma}^2}=\frac{\gamma_2+1}{\gamma_2+\alpha_2+1}\cdot\frac{\int\limits_{0}^{1}r^{2\gamma_1+2\alpha_1+1}e^{-(2\gamma_2+2+2\alpha_2)\phi(r)}dr}{\int\limits_{0}^{1}r^{2\gamma_1+1}e^{-(2\gamma_2+2)\phi(r)}dr}&=\frac{\gamma_2+1}{\gamma_2+\alpha_2+1}\int\limits_{0}^{1}\Phi_{2\gamma_1+1,2\gamma_2+2}(r)g_{\alpha}(r)dr
\mathcal{V}\big(\bar \lambda, \bar \mu\big) = \tau_{\lambda^{(1)}}(\rho_{0}^+)s_{\lambda^{(1)}/\mu^{(1)}}(\rho_1^-)s_{\lambda^{(2)}/\mu^{(1)}}(\rho_1^+) \dots s_{\lambda^{(n)}/\mu^{(n-1)}}(\rho_{n-1}^+)s_{\lambda^{(n)}}(\rho_n^-),
|\partial_{\xi}\left(\psi_{v_{2}}(\xi,\lambda_{1}(t))-\psi_{v_{2}}(\xi,\lambda_{2}(t))\right)|\leq \sup_{x \in [\frac{\lambda_{0}(t)}{2},\frac{1}{2}]}|\partial_{12}\psi_{v_{2}}(\xi,x)| |\lambda_{1}(t)-\lambda_{2}(t)|
|\phi(y)|\leq \frac{|\tilde{P}_{x_2-y}(\theta_{y+1})|}{|\tilde{P}_{x_2-x_1+1}(\theta_{x_1})|}\prod_{k=x_1}^y |\cos(\pi\theta_k)| \cdot |\phi(x_1-1)|+ \frac{|\tilde{P}_{y-x_1}(\theta_{x_1})|}{|\tilde{P}_{x_2-x_1+1}(\theta_{x_1})|}\prod_{k=y}^{x_2}|\cos(\pi\theta_k)|\cdot |\phi(x_2+1)|.
\kappa_2(y) = \lim_{N \to \infty}\left(2H_{2N + 1}- 2\ln\left(2N + 1\right) - H_{N}+ \ln{\frac{N}{\epsilon_2}}+ 2\ln\left(\frac{2N + 1}{N}\right)\right) + \ln{\lvert y \rvert}&= \gamma + \ln{\left(\frac{4\lvert y \rvert}{\epsilon_2}\right)}.
f_{T T T}= 4 \pi^{2}\frac{U -{\bar{U}}}{( T -{\bar{T}})^{2}}\int d^{2}\tau{\bar{F}}({\bar{\tau}}) \sum_{P_{L}, P_{R}}P_{L}{\bar{P}}_{R}^{3}q^{P_{L}^{2}/ 2}{\bar{q}}^{P_{R}^{2}/ 2}.
\partial\Lambda = \{\boldsymbol{s}\notin \Lambda :\exists_{\boldsymbol{t}\in \Lambda}\ \boldsymbol{s}\in B_{\boldsymbol{t}}\}\cup \{\boldsymbol{t}\in \Lambda :\exists_{\boldsymbol{s}\in \Lambda^c}\ \boldsymbol{t}\in B_{\boldsymbol{s}}\setminus B_{\boldsymbol{s+\boldsymbol{1}}}\}
\|(z-T)^{-1}e_m\|_p^p=\|\sum\limits_{i=0}^{m}\frac{1}{z^{i+1}}T^ie_m\|_p^p =\|\frac{e_m}{z}+\sum\limits_{i=1}^{m}\big(\frac{1}{z^{i+1}}\prod\limits_{j=0}^{i-1}w_{m-j}\big)e_{m-i}\|_p^p &=\frac{1}{|z|^p}+\sum\limits_{i=1}^{m}\Big|\frac{1}{z^{i+1}}\prod\limits_{j=0}^{i-1}w_{m-j}\Big|^p.
d s^{2}= \left( \frac{l}{u}\right)^{2}\, d u^{2}+ \left( \frac{u}{l}\right)^{2}d x^{+}d x^{-}+ \frac{l^{2}}{4}\, ( d \theta^{2}+ \sin^{2}\theta d \phi^{2}) + d s^{2}( M^{6}) ,
\psi_F \circ f^*\psi_G = \sigma_L^{-1}\circ v_F \circ p_F \circ \sigma_K \circ b_F \circ b_F^{-1}\circ \sigma_K^{-1}\circ f^\ast v_G \circ f^\ast p_G \circ f^\ast \sigma_I \circ f^\ast b_G \circ b_F.
\partial_t^\alpha \psi(t_1) \approx b_{1,0}(\psi(t_1) - \psi(t_0)), \partial_t^\alpha \psi(t_n) \approx \sum_{k=1}^{n-1}(b_{n,k-1}- b_{n,k}) \psi(t_k) + b_{n,n-1}\psi(t_n) - b_{n,0}\psi(t_0) , n=2,\cdots, N,
{\cal Z}= \sum_{\alpha}\mathrm{T r}( q^{H_{\overline{{{\alpha V}}}}^{L}}~{\overline{{q}}}^{H_{\overline{{{\alpha V}}}}^{R}}e^{2 \pi i \alpha s}\xi ({\overline{{{\alpha V}}}},{\bf n}) \eta ({\overline{{{\alpha V}}}},{\cal N}_{\overline{{{\alpha V}}}},{\bf n}) ) ~ .
\frac{\partial A}{\partial q_{j}}=\frac{\partial A_{j}}{\partial q_{j}}=-\frac{1}{\alpha}p^{\alpha}_{j}q^{-\alpha}_{j}+\frac{1}{\alpha}\;\;;\;\;\frac{\partial B}{\partial q_{j}}=\frac{\partial B_{j}}{\partial q_{j}}=0
\psi\left( L(0^M, 0^N) \right) = \psi\left( (1-q)^{-1}d_- L(0^{M-1}1, 0^{n-1}1 \right) &= (1-q)^{-1}\psi\left( d_- L(0^{M-1}1, 0^{n-1}1 \right) &= (1-q)^{-1}p(0^{M-1}1, 0^{N-1}) &= p(0^M, 0^N)
\Phi(E_{kl})=\sum_{i=1}^{nm}P_{k}v_{i}(P_{l}v_{i})^{*}=\sum_{i=1}^{nm}V_{i}e_{k}e_{l}^{*}V_{i}^{*}=\sum_{i=1}^{nm}V_{i}E_{kl}V_{i}^{*}.
\left( \Sigma^{\mu \nu}\right)^{\rho \tau \alpha \beta}= \frac 1 2 \left[ \eta^{\mu \rho}\eta^{\nu \alpha}\eta^{\tau \beta}+ \eta^{\mu \tau}\eta^{\nu \beta}\eta^{\rho \alpha}+ \left( \alpha \leftrightarrow \beta \right) - \left( \mu \leftrightarrow \nu \right) \right] \; \, ,
\pi^{i j}= \mathrm{d i a g}\left( \frac{1}{2}e^{- 2 \mu}\pi_{\mu}, \frac{1}{4}e^{- 2 \lambda}\pi_{\lambda}, \frac{1}{4}e^{- 2 \lambda}s i n^{- 2}\theta \pi_{\lambda}\right)
Y'(t)+ a\,Y^{\alpha}(t) \geq b + \delta + c\Big(1+\frac{1}{t^{\gamma}}\Big)\sup_{\tau \leq s \leq t}Y^{\alpha_o}(s) begin{align*}0.1cm]X'(t)+ a\,X^{\alpha}(t) \leq b + c\Big(1+\frac{1}{t^{\gamma}}\Big)\sup_{\tau \leq s \leq t}X^{\alpha_o}(s) \,,
\int_{-\alpha}^{\zeta}{\left({\alpha^{2}-\tau^{2}}\right)^{1/2}d\tau}=\int_{-\sigma}^{x}{\left\{{-f\left({\sigma,t}\right)}\right\}^{1/2}dt}=\int_{-\sigma}^{x}{\left({\frac{\sigma^{2}-t^{2}}{1-t^{2}}}\right)^{1/2}dt}.
\begin{aligned}\|\langle D\rangle^sT_{\langle D\rangle^\sigma u}f\|_{L^\infty}\lesssim \|f\|_{B^{s+\sigma,\infty}_2}\|u\|_{B^{0,\infty}_2},\sigma>0 \|\langle D\rangle^s\Pi[f,\langle D\rangle^\sigma u]\|_{L^\infty}\lesssim \|f\|_{B^{s+\sigma,\infty}_2}\|u\|_{B^{0,\infty}_2},\sigma\geq0.\end{aligned}
a^{3}( z ) ={\frac{6 \pi G A}{( z_{-}- z_{+})^{5}}}( C - z )^{2}={{\frac{3}{4}}\pi^{3}\zeta_{R}^{\prime}( - 4 ) G_{5}}{\frac{( z_{0}- z )^{2}}{( z_{-}- z_{+})^{5}}},
4\|\nabla\omega_{(g,b)}\|^2_{L^2}=\lambda(g,b)-\int_M (R-\frac{1}{12}|H|^2)\omega_{(g,b)}^2 dV_g &\leq \sup_{M}(R-\frac{1}{12}|H|^2) - \inf_{M}(R-\frac{1}{12}|H|^2)\leq C .
\begin{cases}\| \sum_{l \leq j \epsilon}T_a^{j,l}\|_{L^r \to L^{r'}}\lesssim_{\epsilon}2^{jm + jn(\frac{2}{r}-1)}\|T_a^{j, l}\|_{L^r\to L^{r'}}\lesssim_{\epsilon}2^{10n(m-n)(j+l)}, l>j\epsilon\end{cases}
\begin{cases}-u^3_{10}u^2_{01}+u^3_{01}u^2_{10}-u^3_{10}u^1_{01}+u^3_{01}u^1_{10}=0\,, -u^4_{10}u^3_{01}+u^4_{01}u^3_{10}-u^3_{10}u^1_{01}+u^3_{01}u^1_{10}=0\,.\end{cases}
G_{2 m - 1 , 2 n}= R_{2 m - 1 , 2 n}= \frac{4 n ( - 1 )^{n + m}}{\pi ( 2 m - 1 )}\left( \frac{1}{2 m - 1 + 2 n}- \frac{1}{2 m - 1 - 2 n}\right) .
\sharp T_K(\varepsilon,\eta, d,B)=\frac{(\varepsilon-\eta)K^2}{2\alpha^2 d^2}B^{2-\frac{1}{r}}+O_{\varepsilon,\eta}\left(\frac{K^2B^{2-\frac{1}{r}}}{N d^2}\right)+O_{\sigma}\left(\frac{K^\sigma B^\sigma N}{d^\sigma}\right).
R_C =\{s(t) g s(t)^{-1}w^{-1}\mid g\in S_N, t\in S_Q, w\in N, [s(t)g s(t)^{-1}]=i(w)\in G\}\tilde{R}_Q = \{s(r) h^{-1}\mid r\in R_Q, h\in N, [s(r)]=i(h)\in G\}.
\bigg(\sum_{k=1}^mR^h(\tau(\varphi),d\varphi(e_k))d\varphi(e_k)\bigg)^{\top}= 0,\quad\mbox{and}\bigg(\sum_{k=1}^mR^h(\tau(\varphi),d\varphi(e_k))\,d\varphi(e_k)\bigg)^{\perp}= \sum_{j,k=1}^mh(\tau(\varphi),B_{\varphi}(e_j,e_k))\,B_{\varphi}(e_j,e_k).
\mathcal{I}_{0}(z)=\sum\limits_{k=0}^\infty \frac{(z^2/4)^k}{(k!)^2}\textrm{and}\mathcal{I}_{1}(z)=\frac{z}{2}\sum\limits_{k=0}^\infty \frac{(z^2/4)^k}{k!(k+1)!}\,\textrm{for any}z\in \mathbb{R}\,.
d s^{2}= - ({\frac{r^{2}}{b^{2}}}+ 1 -{\frac{G M}{r^{2}}}) d t^{2}+{\frac{d r^{2}}{({\frac{r^{2}}{b^{2}}}+ 1 -{\frac{G M}{r^{2}}})}}+ r^{2}d \Omega^{2}.
\xi^{2, 1}(\mathbf{a}) =s_6s_4s_5s_3s_4(\alpha_2) + s_6s_4s_5s_3s_4s_2s_3s_1(\alpha_2)=\alpha_1+2\alpha_2+2\alpha_3+2\alpha_4+\alpha_5+\alpha_6, \xi^{2, 2}(\mathbf{a}) =s_6s_4s_5s_3s_4s_2s_3(\alpha_1)\cdot s_6s_4s_5s_3s_4s_2s_3s_1(\alpha_2)=(\alpha_1+\cdots+\alpha_4)(\alpha_1+\cdots+\alpha_6).
\nabla_{E_a}^{\perp\psi}H^{\psi}=\frac{p}{p+q}\nabla^{\perp}_{E_a}H_1+\frac{q}{p+q}B^j(E_a,H_2),A^{\psi}_{H^{\psi}}E_a=\frac{p}{p+q}A_{H_1}E_a
[[r,r]](\widetilde{X}^{m}\otimes \widetilde{X}^{n}\otimes \widetilde{X}^{p})=[r_{21},r_{31}](\widetilde{X}^{m}\otimes \widetilde{X}^{n}\otimes \widetilde{X}^{p})+[r_{21},r_{32}](\widetilde{X}^{m}\otimes \widetilde{X}^{n}\otimes \widetilde{X}^{p}),
\overline{\nabla}_{X}Y=\nabla_{X}Y+B(X,Y)N+D(X,Y)W, \overline{\nabla}_{X}N=-A_{N}X+\tau(X)N+\rho(X)W, \overline{\nabla}_{X}W=-A_{W}X+\phi(X)N, \nabla_{X}PY = \nabla^{*}_{X}PY + C(X,PY)E, \nabla_{X}E =-A^{*}_{E}X -\tau(X) E,
\|F_v-F_v^{(0)}\|_Y \leq \|F_v- F_v^{(N+1)}\|_{Y}+ \sum_{k=0}^{N}\left\|F_v^{(k+1)}- F_v^{(k)}\right\|_{Y}&\leq \|F_v- F_v^{(N+1)}\|_{Y}+ \sum_{k=0}^N \alpha^k \|F_v^{(1)}-F_v^{(0)}\|_{Y}&\leq \|F_v- F_v^{(N+1)}\|_{Y}+\frac{\alpha}{1-\alpha}\frac{C}{\omega_1}.
\begin{aligned}|\gamma^{\epsilon}(s)|^{2}\leq M^2\left(\rho^{2}+|Y^{\epsilon}(s-)|^{2}\right) &\leq\frac{M^2}{\big(1- \epsilon\lambda(\epsilon)M\big)^{2}}\cdot\left(\rho^{2}+|\tilde{Y}^{\epsilon}(s)|^{2}\right) &\leq 2M^{2}\left(\rho^{2}+|\tilde{Y}^{\epsilon}(s)|^{2}\right) ,\end{aligned}
A_{D}= \frac{{\cal N}_{D}}{2 \pi^{2}}\int_{0}^{1}d \nu \int_{0}^{1}\frac{d q}{q}\left( \frac{\sin^{2}( \pi \nu )}{\pi^{2}}\right) \left( \frac{- \ln ( q )}{2 \pi^{2}}\right)^{( k + 1 - D ) / 2}\exp \left[ ( Y^{2}+ 2 l \sin^{2}( \pi \nu ) ) / \ln ( q ) \right]
u_{n-1}d^{n}_{M(\alpha)}=\begin{pmatrix}id_{B_{n-1}}h_{n-2}0 id_{A_{n-2}}\end{pmatrix}\begin{pmatrix}d^{n}_B -\alpha_{n-1}0 -d^{n-1}_{A}\end{pmatrix}=\begin{pmatrix}d^{n}_B -\alpha_{n-1}-h_{n-2}d^{n-1}_{A}0 -d^{n-1}_{A}\end{pmatrix}.
E\left(\left(g_0-{\textstyle\sum_{g\in F}}\,x_gg\right)^*\left(g_0-{\textstyle\sum_{g\in F}}\,x_gg\right)\right)=I+{\textstyle\sum_{g\in F}\,\alpha_{g^{-1}}(x_g^*x_g)}\geq I
\hat{S}= \int d^{10}\hat{x}\sqrt{|\hat{g}|}\,\left\{e^{-2\hat{\phi}}\left[ \hat{R}-4(\partial\hat{\phi})^{2}\right] +{\textstyle\frac{1}{2\cdot 7!}}e^{2\hat{\phi}}\hat{\tilde{H}}{}^{2}\right\}\, ,
\partial_{x_1}f(x_1, y_1,x_2,y_2) =0 &\textit{or,}\,\,\frac{x_1 -x_2}{(x_1 -x_2)^2+(y_1 -y_2)^2}+\frac{x_2(1- x_1 x_2 -y_1 y_2) -y_2 (x_1 y_2 -x_2 y_1)}{(1- x_1 x_2 -y_1 y_2)^2+(x_1 y_2 -x_2 y_1)^2}\ =0 &\textit{or,}\,\, x_1 + 0\cdot y_1 + \big(f(\underline{z})^2 (1- x_1 x_2 -y_1 y_2) -1\big)x_2 + \big(-f(\underline{z})^2 (x_1 y_2 -x_2 y_1)\big)y_2=0.
\int_{\overline{M}}\overline{g}( \mathrm{div}^{\nabla'}(T_{r}),\overline{\nabla}_{\widehat{W}}\widehat{W})dV =\int_{\overline{M}}((-1)^{r}(r+2)S_{r+2}+\sum_{i=1}^{n}\epsilon_{i}\overline{g}(\overline{R}(Z_{i},\widehat{W})T_{r}Z_{i},\widehat{W})dV
t^{18}a_6' = b + r a + r^3 \Leftrightarrow 32 t^3 a_6' = -\mu^5 \tilde{c}_6 - 2 \mu^4 \tilde{c}_4 t^2 (a_0 + y_1 t) + 32(a_0 + y_1 t)^3
L2_L \lesssim \| C \|_{L^\infty L^2}\| U^2_0 \|_{L^1 L^2}\lesssim \frac{C_1 \epsilon}{\nu}\frac{C_0 \epsilon}{\nu}= \frac{C_0}{C_1}\left( \frac{C_1 \epsilon}{\nu}\right)^2,
|f^{(k)}(z)| =\left|\sum_{i=0}^k \binom{k}{i}f_1^{(i)}(z)f_2^{(k-i)}(z)\right| &\leq \sum_{i=0}^k \binom{k}{i}|f_1^{(i)}(z)||f_2^{(k-i)}(z)| &\leq \sum_{i=0}^k \binom{k}{i}C^{f_1}_{m-k+i,i}(1+|z|)^{m-k}M^{f_2}_{k-i}=:C'_{m,k}(1+|z|)^{m-k},\mbox{for all}~m\in\mathbb{Z},
E=\{t\in(\tfrac{4}{3}r,\tfrac{5}{3}r)\colon\;\mathcal{M}Du(x_{0},t)=\infty\}\cup E_{x_{0}}\cup \big(\tfrac{4}{3}r,\tfrac{401}{300}r\big)\cup\big(\tfrac{499}{300}r,\tfrac{5}{3}r\big)
0 \in \partial f(x^{k+1})+\eta A^T(Ax^{k+1}+By^{k+1}-b-\frac{1}{\eta}\lambda^{k+1}) = \partial f(x^{k+1})+\eta A^T(Ax^{k+1}-Ax^k+\frac{1}{\eta}\lambda^{k}-\frac{2}{\eta}\lambda^{k+1})\ .
F\big(3^{-1/3}x, 3^{-2/3}t;\beta=6\big) \qquad{}= -i\kappa e^{\frac{x^3}{6}-\frac{xt}{2}}\left[ u^{-\frac{1}{2}}\left(\frac{1+q_2}{2}x -\alpha\right)\Psi_{011}(x,t) +u^{\frac{1}{2}}\Psi_{021}(x,t)\right].
\begin{aligned}u (\bar{x}) \leq \, u (x^*) +C\epsilon_0\delta^2b_{\alpha}^2+ Cb_{\alpha}\bar{x}_n + C \delta^3 b_\alpha^2 \leq \, u (x^*) + C \left(\epsilon_0 + \delta\right) \delta^2 b_\alpha^2 \leq \left(1+\frac{c_0}{3}\right) u (x^*)\end{aligned}
\pi^{-s/2}\Gamma(s/2)\zeta(s)\delta^{-s}= -\frac{1}{s}-\frac{\delta^{-1}}{1-s}+\sum_{n=1}^{\infty}G(s/2,\pi n^2\delta^2) +\delta^{-1}\sum_{n=1}^{\infty}G((1-s)/2,\pi n^2/\delta^2),
H_{G/H}=\frac{1}{2}\left( \pi_{\alpha}-\frac{i\hbar}{2}\Gamma_{\alpha}\right)g^{\alpha\beta}\left( \pi_{\beta}+\frac{i\hbar}{2}\Gamma_{\beta}\right)=\frac{1}{2}\pi_{\alpha}g^{\alpha\beta}\pi_{\beta}+V_{G/H}\,,
\| f \|_C = \int \limits_{C}\, (\, \mathfrak{J}_L ((\mathfrak{J}^L f)_x ), f_x \,) \, d\mu_{\mathfrak{G}/G_1}= \int \limits_{(G_1 x_c G_2 ) \, \cap B}\, (f_b , f_b \,) \, d\mu_{B}(b) < \infty,
D_a X_{\mu}( \delta_{\mu\nu}I - i [X_{\mu}, X_{\nu}] )^{-1}D_bX_{\nu}= ( I - [X,Y]^2 )^{-1}( D_a X_{\mu}D_b X_{\mu}+ i D_a X_{\mu}[X_{\mu}, X_{\nu}] D_b X_{\nu}).
v^1(x_2) := \left(\begin{array}{c}0 1 \end{array}\right); v^3(x_2) := \left(\begin{array}{c}\xi \gamma \end{array}\right) e^{\mathrm{i}\gamma x_2}; begin{align*}8pt] v^2(x_2) := \left(\begin{array}{c}1 - \mathrm{i}\xi x_2 \end{array}\right); v^4(x_2) := \left(\begin{array}{c}\xi - \gamma \end{array}\right) e^{- \mathrm{i}\gamma x_2}.
F ( x , y ) = \sum_{A = - \infty}^{N - 1}e^{-{\frac{2 \pi i A}{L}}( x - y )}={\frac{e^{-{\frac{2 \pi i ( N - 1 )}{L}}( x - y )}}{( 1 - e^{{\frac{2 \pi i}{L}}( x - y )})}},
\partial_\tau \epsilon_0 \|\phi_0\|^2_\rho + \epsilon_0(\tau) \partial_\tau \| \phi_0\|^2_{\rho}= \langle \partial_\tau \epsilon, \phi_0 \rangle_{L^2_\rho}+ \langle \epsilon, \partial_\tau \phi_0 \rangle_{L^2_\rho}.
\lambda_{112}\lambda_{qrr}[a_{(1\,p)},a_{(2\,s)}]_c=-3\xi \lambda_{112}\lambda_{122}\lambda_{qrr}\chi_{2}(g_{(2\,p)})a_{(3\,s)}a_{(3\,p)}& - \lambda_{112}\lambda_{qrr}\chi_{(2\,p)}(g_1)(1-\xi)a_{(2\,p)}a_{(1\,s)}.
\iint_{Q_T}\varphi_t (\varrho_1 - \varrho_2) + \nabla \cdot(m \nabla \varphi) (p_1-p_2) - \nabla \varphi \cdot \vec{b}(\varrho_1 - \varrho_2) + \varphi f (\varrho_1 - \varrho_2) dxdt = 0.
\mathbf{S}_{f,g}= (-1)^{\sum_{i=0}^{n-1}(k_i(f)+k_i(g))}\frac{(-1)^{nd(d-1)/2}\tau(d)^n}{d^n}\prod_{i=0}^{n-1}\zeta^{-k_i(f)k_i(g)}\sum_{\sigma\in\mathfrak{S}_{n+1}}(-1)^{l(\sigma)}\prod_{i=1}^{n+1}\zeta^{f(i)g(\sigma(i))}.
t \frac{\partial U}{\partial t}= \lambda(t,x)U +b(t,x)\frac{\partial U}{\partial x}+ R \Bigl(t,x, u_0+U, \frac{\partial u_0}{\partial x}+ \frac{\partial U}{\partial x}\Bigr) - R \Bigl(t,x, u_0, \frac{\partial u_0}{\partial x}\Bigr).
\ell' = \frac{1}{2}\big((n+p^{k-a+1}-2)p^k-(m_b+p^{k-a})p^a\big) = \frac{1}{2}\big( (n-2)p^k- m_bp^a + (p^{k-a+1})p^k -p^k\big) = \ell + \frac{1}{2}(p^{k-a+1}-1)p^k.
h(y^k) - h(y^+) + \big<w^+, By^k - By^+\big> = h(y^k) - h(y^+) + \big<B^Tw^+, y^k - y^+\big> = h(y^k) - h(y^+) - \big<\nabla h(y^+), y^k - y^+\big> = -\frac{L_h}{2}\|y^k - y^+\|^2 = -\frac{L_h\bar M}{2}\|By^k - By^+\|^2.
\chi(p;Z_{t^n}^{0}) &= S({V}_{t^n}^{0}(p^n) ) - \sum_{a^n\in \mathsf{A}^n}p^n(a^n)S({V}_{t^n}^{0}(a^n) ) &= S({V}_{t^n}^{0}(0^n) ) - \sum_{a^n\in \mathsf{A}^n}p^n(a^n)S({V}_{t^n}^{0}(0^n) ) &=0
\begin{array}{l l}{{D_{a}= \partial_{a}- i \sigma_{a \dot{a}}^{\mu}\bar{\theta}^{\dot{a}}\partial_{\mu},}}&{{}}\\{{\bar{D}_{\dot{a}}= - \partial_{\dot{a}}+ i \theta^{a}\sigma_{\dot{a}a}^{\mu}\partial_{\mu},}}&{{}}\\ \end{array}
u =\tilde{U}(\mu')(Hu)=\tilde{U}(\mu')h_2 =\tilde{U}(\mu')\circ{\it HD}(\mu\rightarrow\mu')(Hu) =\tilde{U}(\mu')\circ{\it HD}(\mu\rightarrow\mu')\circ (\tilde{U}(\mu)|_{\mathbb{HD}(\mu)})^{-1}u.
Z [ j ] = e^{\frac{i}{\hbar}\int d x^{4}{\cal{L}}_{I}( \frac{\hbar}{i}\frac{\delta}{\delta j})}Z_{0}[ j ] \textrm{w i t h}Z_{0}[ j ] = \int_{\Lambda}{\cal{D}}\eta e^{\frac{i}{\hbar}\int d x^{4}\left( \frac{1}{2}[ ( \partial \eta )^{2}- m^{2}\eta^{2}] + j \eta \right)},
\dot{H}=\frac{D}{p_{\infty}(t)+B}H-\frac{D}{\rho}(\frac{P+B}{p_{\infty}(t)+B})^{\frac{n-1}{n}}+\frac{\dot{R}}{\rho_{L}R}[\frac{p_{\infty}(t)+B}{P+B}]^{\frac{1}{n}}[\frac{2\sigma}{R}-3kp_{g}(t)]
\frac{1}{L}F_{B}(\beta,L)=\frac{\pi}{6L^{2}}+\frac{1}{\beta L}\ln{\frac{L}{\beta}}+\frac{2}{\beta L}\ln{\eta\left(i\frac{L}{\beta}\right)}\longrightarrow -\frac{\pi}{6\beta^{2}}\;.
\partial_{m}A^{m}= i e \int \frac{d^{3}k}{( 2 \pi )^{3}2 k^{0}}\left( e^{i k x}k^{m}a_{m}^{\dagger}( \vec{k}) - e^{- i k x}k^{m}a_{m}( \vec{k}) \right)_{\displaystyle |_{k^{0}= \sqrt{\vec{k}^{2}}}}
\Gamma_0(\lambda)=1 &(\langle\mu,\mu\rangle-2i\,\langle\mu,\lambda\rangle)\Gamma_\mu(\lambda) &\quad=2\sum_{\alpha\in\Sigma^+}m_\alpha\sum_{\substack{k>1, \mu-2k\alpha\in\Lambda^+}}(\langle\mu+\rho-2k\alpha,\alpha\rangle-i\langle\lambda,\alpha\rangle)\Gamma_{\mu-2k\alpha}(\lambda)
D^1_{sh}(V,\hat{\mathcal{T}}) = \left( H^1(V,\hat{\mathcal{T}}) \rightarrow \prod_{v \in X_0^{(1)}}H^1(K_{0,v},\hat{T})\right), \mathcal{D}^2(V,\mathcal{T}) = \left( H^2_c(V,\mathcal{T}) \rightarrow H^2(K_0,T)\right),
| f (x) -P_n (x)| \le c(k,r) A\begin{cases}|x-x_0|^{m+1}\rho_n^{r -m-1}(x) \omega_\ell(f^{(r)}, \rho_n(x)) , \; m \le r-1 , |x-x_0|^{r}\omega_\ell(f^{(r)},|x-x_0|^{1/\ell}\rho_n^{1-1/\ell}(x)) , \; m=r .\end{cases}
\Phi_1 = \{(f_1,f_2) \mid f_1 = x_1,\, f_2(f_3,f_4) = x_2\}\Phi_2 = \{(f_3,f_4) \mid f_3(x_1) = x_3, \, f_4(x_2, x_3, x_6, f_5) = x_4\}\Phi_3 = \{f_5 \mid f_5(x_3) = x_5\}.
0 =(Qv)^T \xi + a - \mu - 2s = (Qv)^T x_1p_1+ a- \mu -2(\lambda-\mu) = (Qv)^T x_1p_1 + a+d_1 -(v_1+a+d_1) = v^T(x_1 Q^T p_1) -v_1.
\tilde{\mu}\Bigl((u,u,v,v) \in \mathcal{X}_{u,s}\times \mathcal{X}_{u,w}\times \mathcal{X}_{u,s}\times \mathcal{X}_{u,w}:u=v\Bigr) =\tilde{\mathbb{P}}\Big(\tilde{u}^\alpha =\tilde{u}^\beta \mbox{in}\mathcal{X}_{u,s}\mbox{and in}\mathcal{X}_{u,w}\Big)=1.
f_2(s,y):= s^{2H-1}\Big(\alpha_2\frac{1}{\tilde{\theta}^{\rho}}(-\tilde{\theta}\rho+\tilde{\theta}+1)y^{-\tilde{\theta}\rho+\tilde{\theta}}+H\tilde{\sigma}(\tilde{\theta}+1)y^{-2}\Big).
\int d \tau \, \delta M^{2}= - \frac{2 \, G_{N}}{\pi \, \ell_{s}^{2}}\int \frac{d^{D}\, k}{( 2 \pi )^{D}}\, \frac{1}{k^{2}- i \varepsilon}\int \int d^{2}\sigma_{1}\, d^{2}\sigma_{2}\, e^{i \, \ell_{s}^{2}\, k \cdot p \, ( \tau_{1}- \tau_{2})}\, W ( k , 1 , 2 ) \, ,
\mathcal{F}_{\mu \nu}= F_{\mu \nu}- [ \overline{{\theta}}\Gamma_{1 1}\Gamma_{M}\partial_{\mu}\theta ( \partial_{\nu}X^{M}- \frac{1}{2}\overline{{\theta}}\Gamma^{M}\partial_{\nu}\theta ) - ( \mu \leftrightarrow \nu ) ] .
R=\left[\begin{array}{llll}q \, \, \, \, 1 0 \, \, \lambda 1 \, \, \, \, q\end{array}\right]\quad , \quad{\cal P}=\left[\begin{array}{llll}1 \, \, \, \, 0 1 \, \, 1 0 \, \, \, \, 1\end{array}\right]\quad .
\int_0^1 |\partial_xf|^2 + (\lambda^2+n^2x^2)|f|^2 = \int_0^1 fg \leq \frac12 \int_0^1 (\lambda^2+n^2x^2)|f|^2 + \frac{|g|^2}{\lambda^2+n^2x^2}
C_v=\frac{3k\langle N\rangle}{2}\left[ 1+\frac{\langle N\rangle}{2V}\lambda_T^3\delta(q)-2\frac{\langle N\rangle^2}{V^2}\lambda_T^6\left(4\delta^2(q)-\frac{\Gamma(q)}{12}\right)+...\right],
\big\|{\mathcal{N}}u\big\|_{L^p(\partial\mathbb{R}^{n}_+)}\leq C\|f\|_{L^p(\mathbb{R}^{n-1})}\,\,\mbox{and}\,\,u\Big|^{{}n.t.}_{\partial{\mathbb{R}}^n_{+}}=f\,\,\mbox{a.e. in}\,\,{\mathbb{R}}^{n-1}.
\delta V_{\varepsilon_i}(g) = \int_{\{|\nabla u_{\varepsilon_i}|\neq 0\}}\nabla g \cdot \left(I - \dfrac{\nabla u_{\varepsilon_i}}{|\nabla u_{\varepsilon_i}|}\otimes \dfrac{\nabla u_{\varepsilon_i}}{|\nabla u_{\varepsilon_i}|}\right) \, d\mu_{\varepsilon_i}
J=\begin{pmatrix}\begin{smallmatrix}J_{n_1}(\lambda_{n_1},\overline{\lambda}_{n_1}) &\ddots & J_{n_p}(\lambda_{n_p},\overline{\lambda}_{n_p}) & J_{n_q}(\lambda_{n_q}) & \ddots & J_{n_r}(\lambda_{n_r})\end{smallmatrix}\end{pmatrix},
(\left. G_{0,1}(\omega_{1})\right\vert_{\Omega},\omega_{1})_{\Omega}=(\left. R_{0}\Delta^{2}\omega_{1}\right\vert_{\Omega},\omega_{1})_{\Omega}& =\left( \Delta^{2}\omega_{1},\left[ R_{0}\omega_{1}\right]_{\Omega}\right)_{\Omega}.
({\beta_i}{\tan{\phi_i}}-\tan{\phi_r})[(\gamma+1)(\beta_i-1)(1+{\beta_i}{\tan^2{\phi_i}})\sec^2{\phi_r}&+2({\beta_i}{\tan{\phi_i}}+\tan{\phi_r})(1-\tilde{b}{\beta_i})((1+{\beta_i}{\tan^2{\phi_i}})\tan{\phi_r}-\tan{\phi_i}(\beta_i-1))]=0,
T(r,M) = T\left(r,\frac{P'}{P}-\frac{Q'}{Q}+P_0\right) =m\left(r,\frac{P'}{P}-\frac{Q'}{Q}+P_0\right)+N\left(r,\frac{P'}{P}-\frac{Q'}{Q}+P_0\right) \leq S(r,P)+ S(r,Q)+O(\log r)+N\left(r,\frac{P'}{P}\right)+ N\left(r,\frac{Q'}{Q}\right)+ N(r,P_0) &=N(r,\frac{1}{P})+N(r,\frac{1}{Q})+ S(r,W) &\leq T(r,P)+T(r,Q)+S(r,W),
\boldsymbol Q_{\bar{\boldsymbol\omega}_k}(i)=\lambda^{-1}\boldsymbol Q_{\bar{\boldsymbol\omega}_k}(i-1)-\lambda^{-1}\boldsymbol t_k(i)\bar{\boldsymbol\omega}_k^H(i-1)\boldsymbol Q_{\bar{\boldsymbol\omega}_k}(i-1).
{\cal L}^{(2)}_{\rm spin}[B]={(4\pi )}^{-D}e^2\int_0^{\infty}{dT\over T}e^{-m^2T}T^{2-D}\int_0^{\infty}d\hat T \int_0^1 d u_a\,J(z,u_a,\hat T,D)\,
\Omega([u], [v]) :=\langle\Delta^*u, \bar v\rangle-\langle u,\Delta^*\bar v\rangle & =\sum_{k=1}^{2g-2}X_k(u)\left(\begin{matrix}0\ \ \ -I_3 I_3 \ \ 0 \end{matrix}\right){X_k(v)}^t\,,
\begin{aligned}W_{l}= V_{l,1,1}+V_{l,2,2}+\frac{K_{I,k}}{\Delta_k}( X_{l,1,1}+X_{l,2,2}) +\frac{V_{l,1}+V_{l,2}+V_{l,3}+V_{l,4}-V_{l,1,1}-V_{l,2,2}}{\Delta_{k}}.\end{aligned}
\left( p_{\Gamma^\circ}\right)^*\left(X_{m-i,n-m-j}\right)=\frac{A_{m-i,n-m-j+1}A_{m-i+1,n-m-j}A_{m-i-1,n-m-j-1}}{A_{m-i+1,n-m-j+1}A_{m-i-1,n-m-j}A_{m-i,n-m-j-1}}
E_{\mathrm{g a p}}\approx \frac{( 2 n + 1 ) \pi}{3^{( 1 / 4 )}8 \sqrt{\eta}}\left( 1 - \frac{\eta}{2 \sqrt{3}}\right) \ , \qquad \Gamma_{\mathrm{g a p}}\approx \frac{6 4 \sqrt{3}\eta}{\pi^{3}( 2 n + 1 )^{3}}\ , \qquad n = 0 , 1 , 2 . . .
\Big\langle \widehat{\nabla\mathrm{H}}^{\textbf{in}};\mathrm{e}^{(3)}_{\mathrm{n}_{0}}\Big\rangle_{\mathbb{L}^2(\mathrm{B})}= \nabla\mathrm{H}^{\textbf{in}}(\mathrm{z})\int_\mathrm{B}\Tilde{\mathrm{e}}^{(3)}_{\mathrm{n}_{0}}(\mathrm{x})d\mathrm{x}+ \mathcal{O}(\delta).
X_{L}=\left( c_{1}+2\psi_{I}\int T^{I}\left( t\right) dt\right) +\left(T^{I}\left( t\right) Y_{I}^{\alpha}\left( x^{\beta}\right) \right)\partial_{\alpha}+\left( a\left( x^{\beta},t\right) u+b\left( x^{\beta},t\right) +c_{2}u\right) \partial_{u},
Z(\epsilon_1,\epsilon_2,a,\mathfrak{q},\lambda)= \sum_{n=0}^\infty z^n \sum_{i=1}^{2nr}(-1)^i\mathrm{ch}\,H^i(\mathcal M(r,n),\mathcal O),z=\mathfrak{q}\lambda^{2r}e^{-\lambda(\epsilon_1+\epsilon_2)r/2},
\ddot{q}_{nn'}^{l}+\omega_{ln'}^{2}(t) q_{nn'}^{l}=2h\sum_{p=1}^{\infty}\dot{q}_{np}^{l}a_{pn'}^{l}+ \dot{h}\sum_{p=1}^{\infty}q_{np}^{l}a_{pn'}^{l}+ h^{2}\sum_{p,s=1}^{\infty}q_{np}^{l}a_{ps}^{l}a_{n's}^{l}.
V^TXU = \begin{bmatrix}\hat{R}^{-1}0 B \hat{R}^{-1}\end{bmatrix}, \Lambda = \begin{bmatrix}\Lambda_1 0 0 0 \Lambda_2 0 0 0 \Lambda_3 \end{bmatrix}, V_{r}= \begin{bmatrix}I_{r}0 \end{bmatrix}
\mp{R \over \sqrt{\lambda}}\left( \lambda ' - \lambda \varphi ' \right) \varepsilon_{ul}-{\bf Q}_i \gamma^i \varepsilon_{lu}+{1 \over 3}({\bf P}_{ij}\mp 2 i{\bf Q}^{ij}) \gamma^{ij}\varepsilon_{lu}= 0 ,
\sum_{s \in \mathbb{Z}}\oint \frac{dz}{2 \pi i z}(-w)^s z^{n-s}\mathcal{B}(m-s,z\alpha^{-1}q^{\frac14}) \; \mathcal{B}(m+s,z^{-1}\alpha^{-1}q^{\frac14}) =(-w)^n\mathcal{B}(m+n,w\alpha q^{\frac14}) \; \mathcal{B}(m-n,w^{-1}\alpha q^{\frac14}) \; \mathcal{B}(m+\frac12,\alpha^{-2}q^{\frac14}) \; .
\max_{|\zeta|\leq 1}G^+\circ\widehat\psi_n(\zeta) =\max_{|\zeta|\leq 1}G^+\circ\phi_n\circ h_n(\zeta) &=\max_{\zeta\in h_n(\Delta)}G^+\circ \phi_n(\zeta) &\leq \max_{|\zeta|\leq m\varepsilon}G^+\circ\phi_n(\zeta) &<1,
G_{4,4,3}(-q^3,-q^2,-1,-1;q) =\frac{1}{4}\overline{J}_{0,3}\mu(q^3) -\frac{1}{2}\overline{J}_{1,3}\phi(q) +\overline{J}_{1,3}+\overline{J}_{1,4}\Theta_{1}(q) +\overline{J}_{1,3}\Theta_{2}(q) +\frac{1}{4}\overline{J}_{0,3}\frac{J_{6,12}^2}{J_{3}^3},
( G_{i j}^{E})_{i j = 1 , \cdots , M + 1}= \left( \begin{array}{c c}{\displaystyle g_{r s}= ( \rho \cdot \cos \theta )^{\frac{2}{M}}{\bar{g}}_{r s}}&{\displaystyle 0}\\{}&{\vdots}\\{0 \cdots 0}&{{\dot{\vec{x}}}^{2}= \sin^{2}\theta}\\ \end{array}\right) \qquad .
w_{n+m}\left( x\right) =\sum_{k=0}^{m}\sum_{j=0}^{n}\sum_{i=0}^{j}\genfrac{\{}{\}}{0pt}{}{m}{k}\binom{n}{j}\genfrac{\{}{\}}{0pt}{}{j}{i}k^{n-j}\left( i+k\right) !x^{k},
\Psi ( \tau \to - \infty ) = \frac{\Gamma \left( \frac{i p}{\mu_{0}}\right) \Gamma \left( 1 - \frac{i p}{\mu_{0}}\right)}{\Gamma ( - s ) \Gamma ( 1 + s )}e^{- i p \tau}+ \frac{\Gamma \left( - \frac{i p}{\mu_{0}}\right) \Gamma \left( 1 - \frac{i p}{\mu_{0}}\right)}{\Gamma \left( - \frac{i p}{\mu_{0}}- s \right) \Gamma \left( 1 + s - \frac{i p}{\mu_{0}}\right)}e^{i p \tau}\, .
\begin{array}{l}a_0 = Vol(M, g) a_2 = \frac{1}{6}\int_M dV \,R a_4 = \frac{1}{360}\int_M dV \,(-12\Delta R+5R^2 -2 R^{ij}R_{ij}+2 R^{ijkl}R_{ijkl}) \end{array}
\frac{\sqrt{\ell}}{2\pi}\sum_{\nu>\ell}\frac{1}{\sqrt{\nu}(\nu+\ell)}\left(1 - \frac{1}{12\nu\frac{k}{\ell}}+\mathcal{O}\left(\frac{1}{\nu^2\min(1, k^2/\ell^2)}\right)\right) = \frac{\sqrt{\ell}}{2\pi}\sum_{\nu>\ell}\frac{1}{\sqrt{\nu}(\nu+\ell)}+\mathcal{O}\left(\frac{1}{\sqrt{k}}\right),
\bar{\theta}(t, \cdot) = \sum_{l = 0}^{k - 1}\frac{t^{l}}{l!}\bar{\theta}^{l}+ \int_{0}^{t}\int_{0}^{t_{1}}\dots \int_{0}^{t_{k-1}}\tilde{\theta}(\tau, \cdot) \mathrm{d}\tau \mathrm{d}t_{k-1}\dots \mathrm{d}t_{1}
\lim_{t \rightarrow -\infty}s(t) = 0 \lim_{s \searrow 0}f(s) = 0 \lim_{s \searrow 0}f_s(s) = \lim_{t \rightarrow -\infty}\frac{W}{Y^2}\lim_{s \searrow 0}g(s) > 0 \lim_{s \searrow 0}g_s(s) = 0 \lim_{s \searrow 0}h_s(s) = 0
\delta L_0 =i\Lambda^a D_- \chi_a (\varphi ,\tilde{\varphi})+\Lambda^a A_{-}^{b}\left( \partial_\alpha \chi_a \xi^{\alpha}_{b}+\partial_{\overline{\alpha}}\chi_a A_{-}^{b}\tilde{\xi}^{\overline{\alpha}}_b \right) (\varphi ,\tilde{\varphi}) .
\begin{array}{l}\theta_t^k = \rho_t \left( \mathfrak{Q}_{t}^{k-1}(x_{[n]}^{k}, \xi_t) \right)= \displaystyle \sum_{m \in C(n)}p_{k, m}\Phi_{m}\mathfrak{Q}_t^{k-1}(x_{[n]}^k, \xi_{m}),\;\;\;\beta_t^k = \displaystyle \sum_{m \in C(n)}p_{k, m}\Phi_{m}\pi_{k, m},\end{array}