harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
x^2y^2 = 1-q^{-2}(q+q^{-1})\nu_z + q^{-4}\nu_z^2, y^2z^2 = 1-q^{-2}(q+q^{-1})\nu_x + q^{-4}\nu_x^2, z^2x^2 = 1-q^{-2}(q+q^{-1})\nu_y + q^{-4}\nu_y^2,
W_t = -|\lambda_1| e^{-|\lambda_1|t}|x|^{\xi(d+\alpha)}w+ h \left( w N(x,w) + \int (w(x+y,t)-w(x,t)) \overline{K}(x,y) dy \right) &= W \left(N(x,w) -\frac{|\lambda_1| e^{-|\lambda_1| t}|x|^{\xi(d+\alpha)}}{M+e^{-|\lambda_1| t}|x|^{\xi(d+\alpha)}}\right)+ \int (W(x+y,t)-W(x,t)) \overline{K}(x,y) dy + \bar{I},
\int_{0}^{B_{max}}(1-F(y)) \mathrm{d}y = \frac{A_1 +A_2}{2}= 2\times\frac{1}{2}\times\frac{A_1 +A_2}{2}\leq 2\times{\sup_{0\leq x \leq B_{max}}{x (1-F(x))}}.
\psi^{T}( \rho , w ) \equiv{\frac{1 + \rho}{\sqrt \rho}}\exp \left({\frac{\ln^{2}\rho}{2 \ln w}}\right) \prod_{n = 1}^{\infty}{\frac{( 1 + w^{n}\rho ) ( 1 + w^{n}/ \rho )}{( 1 - w^{n})^{2}}}.
( \bar{\varepsilon}_{+}^{a})^{\dagger}( \bar{\varepsilon}_{-}^{a}) = 0 ; ( \bar{\varepsilon}_{+}^{a})^{\dagger}( \bar{\varepsilon}_{+}^{a}) = ( \bar{\varepsilon}_{-}^{a})^{\dagger}( \bar{\varepsilon}_{-}^{a}) = 1 .
T =\frac{1}{2}tr(\dot{X}\dot{X}^{t})=\frac{1}{2}(|\mathbf{\dot{x}}_{1}|^{2}+|\mathbf{\dot{x}}_{2}|^{2}) & =\frac{1}{2}\dot{\rho}^{2}+\frac{\rho^{2}}{8}(\dot{\varphi}^{2}+\dot{\theta}^{2})+\frac{1}{2}\rho^{2}\dot{\alpha}^{2}-\frac{1}{2}\rho^{2}(\cos\varphi)\dot{\alpha}\dot{\theta},
{\rm D}\vartheta_c (y)^*=\left (\begin{array}{c}-c^{-1}Y+c^{-1}{\rm D}\theta_c(F(x_c(y))+Y/c))^* h(x_c(y)) -c^{-1}\lambda+c^{-1}\Pi_{K^*}(\lambda-c g(x_c(y)))\end{array}\right )\, , y =(Y,\mu,\lambda) \in \mathbb{B}_{\delta_0}(\overline{y})\, .
R_m(\tfrac{\beta}{2}(\tfrac{\beta}{2}+1);0) =\prod_{l=1}^{m}(\tfrac{\beta}{2}(\tfrac{\beta}{2}+1)-l(l-1)) =\prod_{l=1}^{m}\big((\tfrac{\beta}{2}+l)(\tfrac{\beta}{2}-l+1)\big) =(\tfrac{\beta}{2}-m+1)_{2m},
I=\int d^{4}x d^{4}y J^{\mu a}(x)\check G^{ab}_{\mu}(x-y)\eta^{b}(y)+\frac 1 2 \int d^{4}x d^{4}y A^{\mu a}(x)\check{\Gamma}^{ab}_{\mu\nu}(x-y)A^{\nu b}(y) + \cdots
p^{\underline{{a}}}={\frac{1}{e ( \tau )}}( \dot{x}^{\underline{{a}}}- i \dot{\bar{\theta}}\Gamma^{\underline{{a}}}\theta ) ={\frac{1}{e ( \tau )}}( \dot{x}^{\underline{{a}}}- i \tilde{\psi}\chi^{\underline{{a}}}) .
\sum_{n=1}^\infty \left | \int^\infty_0 f(\mu ) \frac{1}{\mu -n}\big(1 - \psi (n^2 - \mu^2) \big) d\mu \right |^2 & = \sum_{n=1}^\infty \left | \left ( \int^{n-1}_0 + \int^{n+1}_{n-1}+ \int^\infty_{n+1}\right ) f(\mu ) \frac{1}{\mu -n}\big( 1 -\psi (n^2 - \mu^2) \big) d\mu \right |^2 &\; \leq I_1 + I_2 + I_3 \, .
\langle X_{i},X_{j}\rangle_{a,b,c}=a\delta_{ij},\quad\langle Y_{i},Y_{j}\rangle_{a,b,c}=b\delta_{ij},\quad\langle X_{i},Y_{j}\rangle_{a,b,c}=\langle Y_{i},X_{j}\rangle_{a,b,c}=c\delta_{ij}.
\hat{\cal M}=e^{\hat{\varphi}}\left(\begin{array}{cc}|\hat{\lambda}|^{2}\hat{C}^{(0)}& \hat{C}^{(0)}1 \end{array}\right)\, ,\hspace{1cm}\hat{\cal M}^{-1}=e^{\hat{\varphi}}\left(\begin{array}{cc}1 -\hat{C}^{(0)}& -\hat{C}^{(0)}|\hat{\lambda}|^{2}\end{array}\right)\, ,
4z^2\bar{z}+4z\bar{z}^2+(2it-s)z^2-6sz\bar{z}-(s+2it)\bar{z}^2 +2z(a+2p-2ir)+2\bar{z}(a+2p+2ir)-4d=0 4z^2\bar{z}-4z\bar{z}^2+(it-2s)z^2-6itz\bar{z}+(2s+it)\bar{z}^2 +2z(a+2q+2ir)-2\bar{z}(a+2q-2ir)-4if=0
(1)\,\,\,\,\,J^{l}_{\underline{a}}\varphi:J^{l}(b)\otimes\widehat{O}_{M^{q}_{\varphi},\underline{a}}\rightarrow\bigoplus_{i=1}^{q}J^{l}(a_{i})\otimes\widehat{O}_{M^{q}_{\varphi},\underline{a}}.
(\,u\,|\,v\,)_{M'}:=\int_{M'}\langle\,u\,|\,v\,\rangle dv_{M'},\ \u, v\in\Omega^{0,q}_c(M'), &(\,u\,|\,v\,)_{M}:=\int_M\langle\,u\,|\,v\,\rangle dv_{M'},\ \u, v\in\Omega^{0,q}_c(M),
y^-_0(z)=K_{1-2\Lambda}(2)-\frac{K_{2\Lambda}(2)}{1-2\Lambda}\left(z^{1-2\Lambda}-1\right), &y^-_{n+1}(z)=y^-_0(z)+{1\over 2\Lambda-1}\int_{z_0}^z\left[1-\left({t\over z}\right)^{2\Lambda-1}\right]y^-_n(t)dt.
\Re \left \langle \begin{pmatrix}\varphi \psi \end{pmatrix}, B(s) \begin{pmatrix}\varphi \psi \end{pmatrix}\right \rangle_\Gamma \geq \beta\, \min(1,|s|^2) \frac{\Re s}{|s|^2}\left(\|\varphi\|^2_{-1/2,\Gamma}+ \|\psi\|^2_{1/2,\Gamma}\right)
{\cal L}= \partial_{+}\sigma \partial_{-}\sigma + \partial_{+}\pi \partial_{-}\pi - \frac{1}{2}( \partial_{\bot}\sigma )^{2}- \frac{1}{2}( \partial_{\bot}\pi )^{2}- \frac{1}{2}\mu^{2}( \sigma^{2}+ \pi^{2}) - \frac{\lambda}{4}( \sigma^{2}+ \pi^{2})^{2},
\dot{V}\mid_{(\ref{dyn:err:im})}\leq \sum_{i=1}^n-[\Delta_i(\bar z_i)- c_i\phi_{1i}(\hat z_i)]||\hat z_i||^2-\sum_{i=1}^n\lambda \bar k_i^2 - \sum_{i=1}^n[2K\rho(e_{vi})-3-\delta_i \tilde\phi_i(e_{vi})]e_{vi}^2+\Xi_2
\Omega_k = P_0 + V_k \Omega_{k-1}- \sum_{n=1}^{k-1}V_n(\Omega_n - \Omega_{n-1}) = P_0 + V_k \Omega_{k}- \sum_{n=1}^{k-1}V_n(\Theta_n - \Theta_{0})\Omega_{n-1}.
z =W_Y(z)_--2W_Y(z)_++3G_Y^X(z)_+\ , U'(z) =W_{(1)}(z)_-+G_X^Y(z)_+\ , U'(z) =W_{(1)}(z)_-+G_{X_3}^{X_1+X_2}(z)_++z\ , U'_+\left(z+t_2/t_3\right) =W_{(2)}(z)_-+G_{X_1+X_2}^{X_3}(z)_++W_{(2)}(-z)\ .
|\Psi_{(+)a}>=|0>{\bf M}_a+\frac{1}{2}b_i^{\dagger}b_j^{\dagger}|0>{\bf M}_a^{ij}+\frac{1}{24}\epsilon^{ijklm}b_j^{\dagger}b_k^{\dagger}b_l^{\dagger}b_m^{\dagger}|0>{\bf M}_{ai}
\mathbb P[A\cap \mathcal E_2] \leq \mathbb P\left[\bigl\{\sum_{|\alpha|\leq C\delta L}|a_{\alpha}|^2 \geq \frac 12 e^{\delta L}\bigl\}\right]\leq \sum_{|\alpha|\leq C\delta L}\mathbb P\left[ |a_{\alpha}|^2\geq\frac{e^{\delta L}}{2 M_L}\right] &= M_L e^{-(\frac{e^{\delta L}}{2 M_L})}\leq e^{-e^{\frac{\delta}2 L}}\ .
X^2+X+k+1+\frac{(C_1^2+C_0C_2)(C_2^2+C_1C_3)}{(C_1C_2+C_0C_3)^2}=\, \Bigl(X+\frac D{C_1C_2+C_0C_3}\Bigr)\Bigl(X+1+\frac D{C_1C_2+C_0C_3}\Bigr)
\mathcal{M}_k = \bigoplus_{\iota}\mathcal{M}_{k,\iota}\;;\;\mathcal{V}_k = \bigoplus_{\iota}\mathcal{V}_{k,\iota}\;;\; \breve{\mathcal{V}}_k = \bigoplus_{\iota}\breve{\mathcal{V}}_{k,\iota}.
\mu^{s,N}_{HP}(dx) = const \times \Delta^2_N(x)\prod_{j=1}^{N}(1+ix_j)^{-s-N}(1-ix_j)^{-\bar{s}-N}dx_j &=const \times \Delta^2_N(x)\prod_{j=1}^{N}(1+x^2_j)^{-\Re(s)-N}e^{2\Im(s)Arg(1+ix_j)}dx_j.
\zeta_1^{(1)}= \frac{1}{t}\ln(a_2), \, \zeta_1^{(2)}= \frac{1}{t}\ln(|b|), \, \zeta_2^{(2)}= \frac{1}{t}\ln(a_1a_2),\, \varphi_1^{(2)}={\rm Arg}(b).
{}^{M\!L\!\!}{B}_{p}(x,y)=\int_{0}^{1}t^{x-1}(1-t)^{y-1}E_{\alpha, \beta}^{\gamma}\left(-\frac{p}{t(1-t)}\right)dt. (Re(p)>0, Re(x)>0, Re(y)>0, Re(\alpha)>0)
\|(a_{ij})\|^2 =r((a_{ij})^*(a_{ij})) &\leq \|(Z\oplus\ldots \oplus Z) (a_{ij})^*(a_{ij}) (Z^{-1}\oplus\ldots \oplus Z^{-1})\| &\leq \|(Za_{ji}^*Z^{-1})\| \|(Za_{ij}Z^{-1})\| &\leq \|(Y \phi(a_{ij}) Y^{-1})\|^2 &\leq \|(a_{ij})\|^2
\lim_{m_1\rightarrow 0}\Bigl| \langle (\bar\psi_1 \psi_1 \rangle_A|_{h=0}\Bigr|= \lim_{m_2\rightarrow 0}\Bigl|\langle (\bar\psi_2 \psi_2 \rangle_A|_{h=0}\Bigr|
( \bar{S}\bar{R})_{o o^{\prime}}={\frac{1 6 \, i^{o + o^{\prime}+ 2}}{\pi^{2}o^{\prime}}}\sum_{e = 2}^{\infty}{\frac{e^{3}}{( e^{2}- ( o^{\prime})^{2}) ( e^{2}- o^{2})}}= \pm \infty \, .
\langle F^+(t_i),\varrho_\delta((x,\xi)-\theta_{t_{i-1},t_i}(\cdot,\cdot))\rangle_{z,\sigma}=\langle F^+(t_i,\pi_{t_{i-1},t_i}(\cdot,\cdot)),\varrho_\delta((x,\xi)-(\cdot,\cdot))\rangle_{z,\sigma}
\widetilde{\gamma}_n =\gamma_n, \, n\geq1; \widetilde{\beta}_n=\beta_n, \, n\geq3; \beta_{2n+1}=\beta_3, \, n\geq1; \gamma_{2n}=\gamma_2, \, n\geq1; \beta_{2n}=\beta_2, \, n\geq1; \gamma_{2n+1}=\gamma_3, \, n\geq1; &\gamma_{n}\neq 0; a_2\neq 0.
\textrm{Pr}^{\textrm{OFDMA}}\left\lbrace \textrm{FDF=1,FDN=0}\right\rbrace=e^{-\frac{d_{\textrm{BF}}^{\alpha}\sigma_{\textrm{F}}^{2}\theta}{\lambda_{\textrm{BF}}P_{\textrm{F}}}\left(2^{\frac{R}{\theta}}-1\right)}-e^{-\frac{d_{\textrm{BF}}^{\alpha}\sigma_{\textrm{F}}^{2}(1-\theta)}{\lambda_{\textrm{BF}}P_{\textrm{N}}}\left(2^{\frac{R}{1-\theta}}-1\right)}.
\omega_g^{-1}\circ R^H(g,h) = (\eta^H_g\lambda^H_h \cdot \eta^H_h - \eta^H_{gh}) \cdot 0_{h^{-1}}= (\eta^H_g\lambda^H_h \cdot \eta^H_h - \eta^H_{gh}) \cdot [(\eta^H_h)^{-1}- (\eta^H_h)^{-1}] = \eta^H_g\lambda^H_h \cdot \eta^H_h \cdot (\eta^H_h)^{-1}- \eta^H_{gh}\cdot (\eta^H_h)^{-1}= \eta^H_g\lambda^H_h - \eta^H_{gh}\cdot (\eta^H_h)^{-1}
\prod_{u\le p<z}\left(\frac{(p-1)^2}{p(p-2)}\right) \leq \prod_{p\geq u}\left(\frac{(p-1)^2}{p(p-2)}\right) =\frac{\prod_{p>2}\left(\frac{(p-1)^2}{p(p-2)}\right)}{\prod_{2<p<u}\left(\frac{(p-1)^2}{p(p-2)}\right)}\le 1.00754,
\int_{\Omega}\Bigl[\Bigl(\sum_{i=1}^k U_i\Bigr)^6-\sum_{i=1}^k U_i^6\Bigr]=\sum_{i=1}^k\int_{B_{\rho}(\zeta_i)}E_i+\sum_{i=1}^k\int_{\mathcal{O}_\rho}E_i,
\tilde{S}= \frac{\beta}{2}\sum_{i}( \nabla_{\mu}\theta_{i}- A_{i \mu}- 2 \pi n_{i \mu})^{2}+ \frac{1}{2 e^{2}}\sum_{i}( \epsilon_{\mu \nu \lambda}\nabla_{\nu}A_{i \lambda}- 2 \pi N_{i \mu})^{2},
T_{m}-\frac{(1-q^{2m}) (1-y^2 q^{2m-2})}{(1-y q^{2 m-1}) (1-y q^{2m})}T_{m-1}=\frac{q^{2 m-1}(q-y)}{\left(1-y q^{2 m-1}\right) \left(1-y q^{2 m}\right)}.
\frac{d}{t}I(t) \leq \frac{2}{\gamma-1}(2-3(\gamma-1))(t+1)\int_{A(t)}Px &+18\left(\beta+\frac{2}{3}\alpha\right)\frac{\delta}{\gamma}\int_{A(t)}\rho^\gamma x+18\left(\beta+\frac{2}{3}\alpha\right)\frac{\gamma-\delta}{\gamma}|A_{0}|.
Y^{(1)}(t) = \int_0^t \mathrm{d}\tau\h_\vartheta(\tau), Y^{(2)}(t) = -\frac{1}{2}\int_0^t \mathrm{d}\tau_1 \int_0^{\tau_1}\mathrm{d}\tau_2\ [h_\vartheta(\tau_1), h_\vartheta(\tau_2)],
\delta^{(2)}= \min_{1\leq i \leq n}\frac{[(2\Omega_2+\langle A\Omega_1 \rangle - \vert M_{\Omega_2}\vert - \vert N_{\Omega_2}\vert)Ve]_i}{[2(\Omega_3 + \vert N_{\Omega_1}\vert)Ve]_i}+1.
\Lambda(t_0,x_0) = \{(t,x) \in (0,T) \times \Omega ;\, 0< t < t_0, |x-x_0| < t_0- t\}= \bigcup_{t\in (0,t_0)}\left( \{t \}\times (B_{t_0-t}(x_0)\cap \Omega) \right)).
\mathbf{G}^{\theta^{k+1}_1(\mathbf{u})}_{k,l}( t,X^{k,\theta^{k+1}_1(\mathbf{u})}_t ) - \mathbf{G}^{\theta^{k+1}_0(\mathbf{u})}_{k,l}( t,X^{k,\theta^{k+1}_0(\mathbf{u})}_t )= \mathbf{\bar{G}}^{\mathbf{u}}_{k+1,l}( t, X^{k,\theta^{k+1}_1(\mathbf{u})}_t - X^{k,\theta^{k+1}_0(\mathbf{u})}_t ) .
\left( \mathcal{X}^{K\textnormal{th,eq}}\left(q,\varphi_{q,z}^{-1}(Q)\right)P_{q,z}\right)_{li}= \left( \delta_q \right)^l \Lambda_i^{-\ell_q(Q)}\sum_{d \geq 0}\frac{1}{z^{d(N+1)}}\frac{(1-q)^{d(N+1)}Q^d}{\left(q\Lambda_0 \Lambda_i^{-1}, \dots q, \dots ,q\Lambda_N \Lambda_i^{-1};q \right)_d}
- \sum_{j = 0}^{\infty}\frac{1}{j !}( 8 \pi i K )^{- j}\left( \frac{P}{H}\right) e^{2 \pi i c_{0}}\partial_{\epsilon}^{( 2 j )}\left. \left[ e^{2 \pi i \epsilon \sum_{i = 1}^{3}\frac{\mu_{i}}{p_{i}}}\left( \frac{1}{2 i \sin 2 \pi \epsilon}- \frac{1}{4 \pi i \epsilon}\right) \right] \right|_{\epsilon = 0}.
A_d = \frac{q^{-\frac{1}{2}\chi( d, d )}}{(q^{-1})_d}x_d, \;\;\;\;\;\; \;\;\;\;\;\;\;A_e^{\sigma}= \frac{q^{-\frac{1}{2}\mathcal{E}(e)}}{(q^{-1})^{\sigma}_e}\xi_e.
\int_{P}\Big(\sum\limits_{v=v_{P}}^{\infty}\sum\limits_{h\in \mathbb{Z}^{n}}2^{v(\alpha \left( y\right) +n/2)q(y)}D_{v,h,P}^{q(y)}\chi_{v,h}(y)\Big)^{1/q(y)}dy\lesssim 1.
A_{N}= 2\pi \,\sqrt{l}\, \left(\frac{1}{\pi}\right)^{{\textstyle \frac{k+1-D}{2}}}\Gamma\left(\frac{D-3-k}{2}\right)\left(\frac{1}{Y^2}\right)^{{\textstyle \frac{D - 3 - k}{2}}}
\Gamma_{+}^{2}=\frac{1}{(2\pi)^{2}}\int d t ~d^{2}{\bf x}\int d^{2}{\bf p}~\left[\frac{-e^{2}}{2}\pi\delta(p_{1}, p_{2})\epsilon_{ij}a_{j}\dot{a}_{i}\right]~ e^{i{\bf p}\cdot{\bf x}},
\sum\limits_{\widetilde{\mathbf{r}}\in \widetilde{R}}\frac{1}{N^{d-m}}\sum\limits_{\mathbf{n}\in \mathbb{Z}^{d-u}}\Big(\prod\limits_{j=1}^d f_j(\xi_j(\mathbf{n}) + \widetilde{\mathbf{r}}_j) \Big)F_{\widetilde{\mathbf{r}}}(\mathbf{n})G_{\widetilde{\mathbf{r}}}( \pi_{m-u}PL\Xi(\mathbf{n})).
\displaystyle\sum\limits_{\substack{n=0 n}}^{n_{12}+j}Q_n^{\mu_1}(x) Q_{n_{12}+j-n}^{\mu_2}(y) \langle n, n_{12}+j-n |n_{12},j\rangle = Q_{n_{12}}^{\mu_{12}}(z) \displaystyle\sum\limits_{\substack{n=0 n}}^{j}Q_n^{\mu_1}(x) Q_{j-n}^{\mu_2}(y) \langle n, j-n | 0,j \rangle
\nu^{{\pi^{*,\infty}}}(0)=\frac{1+2^{\mu_0+\Delta{C}^{\infty}}}{1+2^{\mu_0+\mu_1+2\Delta{C}^{\infty}}+2^{\mu_0+1+\Delta{C}^{\infty}}},~\nu^{{\pi^{*,\infty}}}(1)=\frac{2^{\mu_0+\Delta{C}^{\infty}}(1+2^{\mu_1+\Delta{C}^{\infty}})}{1+2^{\mu_0+\mu_1+2\Delta{C}^{\infty}}+2^{\mu_0+1+\Delta{C}^{\infty}}}.
( [ \hat{D}]_{q})_{i j}: = [ \ell_{i}]_{q}\delta_{i j}, \quad \quad ( [ I^{\omega}]_{q})_{i j}: = \left[ \sum_{n = 1}^{\ell_{i}}\hat{I}_{\omega^{n}( i ) j}\right]_{q}, \quad i , j \in \hat{\Delta}^{\omega}
J_{{\frac{p}{\bar{n}}}}( x ) ~ \sim ~{\frac{\operatorname{c o s}({\frac{p}{\bar{n}}}\operatorname{t a n}\beta - n \beta - \pi / 4 )}{({\frac{p}{\bar{n}}}\pi \operatorname{t a n}\beta )^{( 1 / 2 )}}}
\|R^{\#}_y\|^2_{\#}= \|P R^{\#}_y\|^2 -\Big\langle \frac{\alpha}{|x|}\Big\rangle_{R^{\#}_y}-e_{\alpha}\|R^{\#}_y\|^2 + \|R^{\#}_y\|^2_* &\geq (1- \varepsilon_2^{-1}\alpha) \|P R^{\#}_y\|^2 - (\varepsilon_2 \alpha + e_{\alpha}) \|R^{\#}_y\|^2+ \|R^{\#}_y\|^2_*,
M=\frac{1}{\sqrt{Z}}(lim_{x_{1}^{0}\rightarrow-\infty}-lim_{x_{1}^{0}\rightarrow+\infty})\int d^{3}x_{1}e^{ip_{1}\cdot x_{1}}\langle\beta out|\overline{\Psi}(x_{1})\gamma^{0}u_{p_{1}}^{s_{1}}|\alpha^{\prime}in\rangle,
[\begin{matrix}N_{0}N_{0,1}N_{0,2}\cdot\cdot\cdot N_{0,n-1}N_{1,0}N_{1}N_{1,2}\cdot\cdot\cdot N_{1,n-1}\vdots\vdots\vdots\vdots N_{n-1,0}N_{n-1,1}N_{n-1,2}\cdot\cdot\cdot N_{n-1}\end{matrix}]
\lambda_{m-i}+ \left \lfloor \frac{k(m-(2i+1))}{2}\right \rfloor \geq 1+ki+ \left \lfloor \frac{k(m-(2i+1))}{2}\right \rfloor & = 1 + \left \lfloor \frac{k(m-1)}{2}\right \rfloor,
C_{5}=\sum_{\substack{1\leq i\leq p-1 \bigl(\frac{i}{p}\bigr)\ne \bigl(\frac{a}{p}\bigr)}}(p^{e-2d-1}+p^{m-d-1}) w_{c}^{p^{e-2}+(\frac{-1}{p})p^{m+d-1}}\prod_{\substack{1\leq j\leq p-1}}w_{j+c}^{p^{e-2}-\bigl(\frac{j^{2}-4ai}{p}\bigr)p^{m+d-1}}
\phi \left( r e^{i \theta^0}, \, \theta^1, \, \theta^2, \sigma \right) = \left( \sqrt{2 \log(r \epsilon )}\, e^{i \left( p\theta^0 + a \theta^2 \right)}, \, \theta^1, \, q\theta^0 + b\theta^2, \sigma \right),
G_{\kappa}=1-\frac{\alpha+\kappa}{\alpha+\frac{1}{2}\kappa}\frac{\Gamma(\frac{1}{\kappa}-\frac{1}{2\alpha})}{\Gamma(\frac{1}{\kappa}+\frac{1}{2\alpha})}\frac{\Gamma(\frac{1}{2\kappa}+\frac{1}{2\alpha})}{\Gamma(\frac{1}{2\kappa}-\frac{1}{2\alpha})}
\partial_{\underline i}\epsilon_I -{\textstyle\frac{i}{4}}e^{2\phi}\bigl (\partial_{\underline i}a\bigr )\epsilon_I+{\textstyle\frac{1}{2}}\sqrt 2 e^{-U-\phi}\biggl (F_{0i}^{+(1)}\alpha_{IJ}+ i F_{0i}^{+(2)}\beta_{IJ}\biggr )\gamma^0\epsilon^J = 0\, .
\frac{\partial^2 F}{\partial \kappa^i \partial \kappa^j}= n^{-1/r}(1-r) \left(\sum_{l}\kappa_l^r \right)^{\frac{1}{r}- 2}\kappa_i^{r-2}( \kappa_i \kappa_j^{r-1}- \sum_{l}\kappa_l^r \delta_{ij}).
\sum_{k = 1}^n\theta(n_s(\lambda))(\theta(n_s(\lambda_s)^{-k}n_s^2\lambda_s^{k - 1})(\lambda_s^{-k+1}\cdot c') + \theta(n_s(\lambda_s)^{-k}\lambda_s^k)(\lambda_s^{-k}\cdot c_{n_s})).
\begin{cases}x \frac{\partial Y}{\partial x}= x^{-3}\begin{bmatrix}x^3+ x^2+y y^2 -1 x^3 + x^2 -y \end{bmatrix}Y y \frac{\partial Y}{\partial y}= y^{-2}\begin{bmatrix}y^2 -2 y -6 y^3 -2 y -3 y^2 -2 y -6 \end{bmatrix}Y \end{cases}
\gamma_{i}=\frac{1}{p^i}\,s+\frac{t+s\,p}{p}\,\sum_{j=0}^{i-1}\,\frac{1}{p^j}=\frac{1}{p^i}\,s+\kappa\,\left(1-\frac{1}{p^i}\right),i\in\mathbb{N}.
S_{+}(z) = S_{-}(z)\begin{pmatrix}0 s_{j}-s_{j}^{-1}0\end{pmatrix}, z \in (-x_{j},-x_{j-1}), \, j = 1,...,m+1, & S_{+}(z) = S_{-}(z)\begin{pmatrix}1 0 s_{j}^{-1}e^{\pm \pi i \alpha}e^{-2\sqrt{r}g(z)}1\end{pmatrix}, z \in \gamma_{j,\pm}, \, j = 1,...,m+1,
t=\frac{\overline{X}-\mu}{S/\sqrt{n}}=\frac{\frac{1}{n}(X_{1}+\cdot\cdot\cdot+X_{n})-\mu}{\sqrt{\frac{1}{n(n-1)}[(X_{1}-\overline{X})^{2}+\cdot\cdot\cdot+(X_{n}-\overline{X})^{2}]}}\sim t_{n-1}.
\begin{aligned}\phantom{.}[\varphi(a_j),\varphi(a_k)] =[b_j,b_k] \equiv \sum_{i=1}^3 \mu_i^{(j,k)}b_i \equiv \sum_{i=1}^3 \mu_i^{(j,k)}\varphi(a_i) \equiv \varphi \left( \sum_{i=1}^3 \mu_i^{(j,k)}a_i \right) \equiv \varphi\left( [a_j,a_k] \right) \pmod{\ell^t}.\end{aligned}
\delta = - \frac{V_{p}}{( 2 \pi )^{p}}\omega_{1}\prod_{i = 1}^{l}\frac{1}{\omega_{i}^{2}}\int_{0}^{\infty}\frac{d s}{s}\left( \frac{\pi}{2 s}\right)^{\frac{p}{2}}e^{- b^{2}s}\frac{( \cos ( 4 \omega_{1}s \sinh \epsilon ) - 4 \cos ( 2 \omega_{1}s \sinh \epsilon ) + 3 )}{\cosh \epsilon ~ \sin ( 2 \omega_{1}s \sinh \epsilon )}
\frac{\partial^2 \varphi_L (t)}{\partial t^2}+ 3\Bigl(\frac{\partial a(t)/ \partial t}{a(t)}\Bigr) \frac{\partial\varphi_L (t)}{\partial t}+ \frac{\delta^2 V(\hat{\phi})}{\delta\hat{\phi}^2}\varphi_L (t) = 0.
|T(\alpha)|\ll X^{n-5}L^{4}\int_{0}^{1}\sum_{|\mathbf{v}|\le X}\big|\mathcal{K}_{0,\mathbf{v}}(\alpha,\gamma)\big|\prod_{j=1}^4\Phi\big(g_j(\mathbf{v}\alpha)\big)\Phi(\gamma)d\gamma,
(\gamma,\mathcal{B})\in Hom(\mathcal{K},\mathcal{E})\oplus Hom(\mathcal{H},\mathcal{D})/A\cdot\mathcal{A}=T_{(q,\mathcal{A})}(Quot(\mathcal{H},P)\times\mathbb{P}(Hom(\mathcal{H},\mathcal{D})^{\vee}))
E \Big[ \big\vert N_t^{\frac{\beta}{2}t + y}\big\vert^2 \Big] = E \big\vert N_t^{\frac{\beta}{2}t + y}\big\vert + 2 \int_0^t \Big[ E \big\vert N_{t-s}^{\frac{\beta}{2}t + y}\big\vert \Big]^2 \frac{\partial}{\partial s}\Big( 2 \Phi(\beta \sqrt{s}) \mathrm{e}^{\frac{\beta^2}{2}s}\Big) \mathrm{d}s
P_1 = (a_2,a_3,a_4,\,a_2a_1,a_3a_1), P_2 = (a_3,a_4,\,a_2,a_3a_1), P_3 = (a_1,a_2,\,a_3,a_4,\,a_3a_2), P_4 = (a_1,a_2,\,a_3,a_4a_2), P_5 = (a_1,a_2,a_3,\,a_4a_2,a_4a_3).
q[u(\sqrt{u^{4}-u^{2}+1}-u^{2}+1)]^{3}=exp\{-\pi\frac{K^{\prime}[u(\sqrt{u^{4}-u^{2}+1}-u^{2}+1)]}{K[u(\sqrt{u^{4}-u^{2}+1}-u^{2}+1)]}\}^{3}=
G_{n}^{r}= \frac{1}{( \gamma + ( r + 1 )^{2})^{n}}+ \sum_{k = 1}^{[ ( n - 1 ) / 2 ]}\ \sum_{m = k + 1}^{n - k}{\binom{m - 1}{k}}\frac{1}{( \gamma + ( r + 1 )^{2})^{m}}\, s u m_{i_{1}= 1}^{\infty}\ldots \sum_{i_{k}= 1}^{\infty}\delta_{i_{1}+ \ldots + i_{k}, n - m}G_{i_{1}}^{r + 2}\ldots G_{i_{k}}^{r + 2}
\psi_{i,0}f_{j,l}=q^{-a_{i,j}}f_{j,l}\psi_{i,0},\ [h_{i, k}, f_{j,l}]=-d^{-km_{i,j}}\gamma^{|k|/2}\frac{[ka_{i,j}]_q}{k}f_{j,l+k}\ (k\ne 0).
r_1=(\frac{-t_0}{1}\ominus\frac{-1}{0})\bullet\frac{1}{0}-(\frac{1}{0}\ominus\frac{1}{1})\bullet\frac{1}{0}-(\frac{1}{1}\ominus\frac{1}{2})\bullet\frac{1}{0}-\cdots-(\frac{1}{-t_1-1}\ominus\frac{1}{-t_{1}})\bullet\frac{1}{0}=t_{1}-1.
\sum_{I=2}^N \delta_{JI}\Phi^{(I)}_0 =\delta_{J1}\left\{\frac{\sqrt{N}}{2\sqrt{\pi}}\left[\frac{\theta}{N}- \arcsin\left(\frac{e^2}{2mc}\sqrt{\frac{N}{\pi\!+\!gN}}\Phi^{(1)}_0 \right) \right] - \sqrt{\frac{\pi}{\pi\!+\!gN}}\Phi^{(1)}_0\right\}\; ,
{A_{\theta}}^{a}= \frac{1 - H_{2}}{e}{v_{\varphi}}^{a}\ ,{A_{\varphi}}^{a}= - \frac{n}{e}\sin \theta \left( H_{3}{v_{\tilde{r}}}^{a}+ ( 1 - H_{4}){v_{\theta}}^{a}\right) \ .
V^{\sigma}(\hat{\nu}^*, \mu^*) - V^{\sigma}(\nu^*, \mu^*) \geq \int_{\mathcal{X}}\left(\frac{\delta F}{\delta \nu}(\nu^*,\mu^*,x) + \frac{\sigma^2}{2}\log\left(\frac{\nu^*(x)}{\pi(x)}\right)\right) (\hat{\nu}^*-\nu^*)(\mathrm{d}x) + \frac{\sigma^2}{2}\operatorname{D_{KL}}(\hat{\nu}^*|\nu^*) = \frac{\sigma^2}{2}\operatorname{D_{KL}}(\hat{\nu}^*|\nu^*),
\theta_2(t) = \hat{\theta}_2(t) \Big( 1 - \mathrm{e}^{\beta |x_0(t)|}P^{x_0(t)}\big( T_0 > \alpha t \big) \Big) - \mathrm{e}^{\beta |x_0(t)| + \beta z(t)}P^{x_0(t)}\big( T_0 > \alpha t \big) \to 1
\begin{array}{lll}{\cal A}_{a+i}(u_i) \equiv \frac{\textstyle \sin((a+i)\gamma)\sin((a+i-2)\gamma)\sin(u-u_i)\sin(u+u_i)}{\textstyle \sin^2((a+i-1)\gamma)}{\cal B}_{a+i}(u_i) \equiv \frac{\textstyle\sin^2\gamma\sin((a+i-1)\gamma+u-u_i)\sin((a+i-1)\gamma+u+u_i)}{\textstyle\sin^2((a+i-1)\gamma)}{\cal C}_{a+i}(u_i) \equiv \sin(\gamma-u+u_i)\sin(\gamma-u-u_i)\end{array}
(r+2)W^rS_r=\alpha_{l_{r+1}}\,G_{r-1}(f_1,\ldots,\widehat{f}_{l_{r+1}},\ldots,f_n) +2\sum_{1\leq i_1<\ldots<i_r\leq n \atop i_1,\ldots,i_r\not=l_{r+1}}\ddot{f}_{i_1}\ldots\ddot{f}_{i_r}\,.
\eqref{eq:test function applied sum}= \sum_{t=1}^{\lfloor sT \rfloor-1}\Bar{\Theta}^{d,T}(\frac{t}{T},x)\big( f(\frac{t-1}{T})-f(\frac{t}{T}) \big)+f(\frac{\lfloor sT \rfloor-1}{T})\Bar{\Theta}^{d,T}(\frac{\lfloor sT \rfloor}{T},x)-f(0)\Bar{\Theta}^{d,T}(0,x).
M_3 = \left\{\begin{bmatrix}0 0 v^T a \end{bmatrix}\right\}\oplus \left\{\begin{bmatrix}\tilde{A}0 0 0 \end{bmatrix}\right\}\oplus \left\{\begin{bmatrix}0 u 0 0\end{bmatrix}\right\}=: \mathcal{S}_1 \oplus \mathcal{S}_2 \oplus \mathcal{S}_3,
p_k^{(1,b)}(X) = \frac{3}{2}I_2+\frac{3}{2}\sum_{t,i}I_3^{tp_i}+\frac{3}{2}(1+k-\phi(k)-1)I_1 - \frac{3}{2}(k-\phi(k))k \geq - \frac{3}{2}(k-\phi(k))k,
\sum_{i = 1}^{N - 1}\epsilon_{i}{\frac{\mu_{i}^{N - 1 - j}d \mu_{i}}{\displaystyle 2 \sqrt{h_{N}- \mu_{i}}\prod_{j = 1}^{N - 1}( \mu_{i}- h_{j})}}= \delta_{j , 1}d x , \quad 1 \leq j \leq N - 1 , \quad \epsilon_{i}= \pm 1 .
\frac{1}{\langle I_{x}^{2}\rangle\langle I_{y}^{2}\rangle-\langle I_{x}I_{y}\rangle^{2}}[\begin{matrix}\langle I_{y}^{2}\rangle&-\langle I_{x}I_{y}\rangle\\ -\langle I_{x}I_{y}\rangle&\langle I_{x}^{2}\rangle\end{matrix}].
A_{0}( s ) = \frac{z^{2}}{\pi R}\frac{1}{\left( s + \frac 1 2 \right)}\left( 1 + \frac 1{1 2 z^{2}}\right) - \frac{z^{2}}{\pi R}\left[ 1 + \frac 1{6 z^{2}}+ 2 \left( 1 + \frac 1{1 2 z^{2}}\right) \log \left( \frac{z}{2 R}\right) \right.
&\left[g_{i\bar{j}}\right]=\left[ \begin{matrix}\displaystyle 4pK 0 &0 \frac{f^{(1)}(0)}{4}\end{matrix}\right],\left[ g^{\bar{\alpha}\beta}\right]=\left[ \begin{matrix}\displaystyle \frac{1}{4pK}0 &0 \frac{4}{f^{(1)}(0)}\end{matrix}\right],
e(C(v_b) ,F(v_a) ) +e(C(v_a) ,F(v_b) ) &=e(e_i^*(Fv_b) e_j^* - e_j^*(Fv_b) e_i^* ,F(v_a) ) +e(e_i^*(Fv_a) e_j^* - e_j^*(Fv_a) e_i^* ,F(v_b) ) &=e_i^*(Fv_b) e_j^*(Fv_a) - e_j^*(Fv_b) e_i^*(Fv_a)+e_i^*(Fv_a) e_j^*(Fv_b ) - e_j^*(Fv_a) e_i^* (Fv_b ) &=0.
\int dx \, b_x^* \Phi_{\varphi_t}(x) b_x = 2 \int dx \, b_x^* \Re \left\{\int dk \, \frac{1}{1 + k^2}G_x(k) \overline{\varphi_t(k)}\right\}b_x &-\int dx \, b_x^* \Re \left\{\int dk \, \frac{i k}{\pi (1 + k^2)}G_x(k) \overline{\varphi_t(k)}\right\}i \nabla_x b_x + \, .
\varepsilon^{-1}\partial_r\eta_0(r)+\partial_r\eta_1(r)+k_d^{-1}V_\Gamma\eta_0(r) +\varepsilon\{\partial_r\eta_2(r)-\nabla_\Gamma g_i\cdot\nabla_\Gamma\eta_0(r)+k_d^{-1}(\partial^\circ g_i)\eta_0(r)+k_d^{-1}V_\Gamma\eta_1(r)\}= O(\varepsilon^2).
f(9) = f(1^2+1^2+1^2+1^2+1^2+2^2) = 5+f(4) f(12) = f(3)\,f(4) &= f(1^2+1^2+1^2+1^2+2^2+2^2) = 4+2f(4) f(15) = f(3)\,f(5) &= f(1^2+1^2+1^2+2^2+2^2+2^2) = 3+3f(4) f(18) = f(2)\,f(9) &= f(1^2+1^2+2^2+2^2+2^2+2^2) = 2+4f(4) f(20) = f(4)\,f(5) &= f(1^2+1^2+1^2+2^2+2^2+3^2) = 3+2f(4)+f(9),
l_{\langle\rangle}(\mathcal C_\Omega\mathcal E^CP_\alpha^u\langle\rangle) =\langle\rangle, o_{\langle\rangle}(\mathcal C_\Omega\mathcal E^CP_\alpha^u\langle\rangle) =\vartheta(\Omega+\Omega^{\varepsilon_{\langle\rangle}}).
h_{-1-L}^{\mathcal{L}_{1,2,..,L}\otimes\mathcal{K}_{L+1}}(t) =h_{-1}^{\mathcal{K}_{L+1}}(t)h_{-L}^{\mathcal{L}_{1,2,..,L}}(t) h_{1-L}^{\mathcal{L}_{1,2,..,L}\otimes\mathcal{K}_{L+1}}(t) =h_{-L}^{\mathcal{L}_{1,2,..,L}}(t)h_{1}^{\mathcal{K}_{L+1}}(t)+h_{-1}^{\mathcal{K}_{L+1}}(t)h_{2-L}^{\mathcal{L}_{1,2,..,L}}(t)
\mathcal{S}(\tau_1,\tau_2;h) := \sum_{g \mid h}\frac{1}{g^{\tau_1}}\prod_{(p,g)=1}\Big ( 1 - \frac{f_1(p) f_2(p)}{p^{\tau_2}}\Big ) \prod_{p^{\alpha}\mid \mid g}\Big( f_1(p^{\alpha})f_2(p^{\alpha}) - \frac{f_1(p^{\alpha+1})f_2(p^{\alpha+1})}{p^{\tau_2}}\Big).
\mu(B) \widetilde g= (\sqrt{L}a_{11})^{-1/2}\ \mu_1(a_{21}/a_{11})\, \mathcal{F}^\ast \sum (\widehat{c'})_m \delta_{m/(L a_{11})}= (\sqrt{L}a_{11})^{-1/2}\ \mu_1(a_{21}/a_{11})\, \sum c'_n \delta_{n a_{11}L /L'}&= (\sqrt{L}a_{11})^{-1/2}\ \sum c'_n \, e^{2\pi i n^2 a_{21}a_{11}(L/L')^2/2}\delta_{n a_{11}L /L'}.
{\bf H}=\frac{i\omega}{2}\int d^3 p\,p^i\sum_{\lambda}\left[a^{\dagger}(\vec{p},\lambda)\stackrel{\leftrightarrow}{\partial}_{p^{i}}a(\vec{p},\lambda) +b^{\dagger}(\vec{p},\lambda)\stackrel{\leftrightarrow}{\partial}_{p^{i}}b(\vec{p},\lambda) \right]
\bar{g}_1^{\,[K,L]}(x) = \frac{1-q}{\alpha_1}x^{\lambda}\sum_{n=K}^{L}(q^n\xi )^{-1}\frac{\big(x q^{-n}/ \xi, q^{1-n}/( \beta_1 \xi ), q^{1-n}/( \beta_2 \xi ) ; q\big)_{\infty}}{\big(x q^{-n-\lambda}/ \xi, q^{-n}/ (\alpha_1 \xi),q^{1-n}/ (\alpha_2 \xi) ; q\big)_{\infty}}.
\begin{gathered}(p\circ_i q)(a_1, \cdots , a_{m+n-1}) \,\,=\,\, =\,\, p\bigl( a_1, \cdots, a_{i-1}, q(a_i, a_{i+1}, \cdots, a_{i+n-1}), a_{i+n}, \cdots a_{m+n-1}\bigr). \end{gathered}
\begin{array}{rcl}\displaystyle\mu_{g_s}(\hat x,t) = \displaystyle g_s(\hat x) + \frac{t^2}{6}\,\sum_{i=1}^{k-1}{(g_s)}_{ii}(\hat x) + \left(\frac{1}{120}\sum_{i=1}^{k-1}{(g_s)}_{iiii}+\frac{1}{36}\,\sum_{\substack{i,j=1 j>i}}^{k-1}{(g_s)}_{iijj}\right) t^4\,,\end{array}
\frac{1}{2}\langle y, g(A) e^{A2\varepsilon}x \rangle = \int_0^{\infty}\langle y, (-A) e^{A2t}g(A) e^{A2\varepsilon}x \rangle dt = \int_0^{\infty}\langle (-A^*)^{\frac{1}{2}}e^{A^*\varepsilon}e^{A^* t}y , g(A) (-A)^{\frac{1}{2}}e^{A\varepsilon}e^{A t}x \rangle dt,