harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
{\cal O}_{1 / 2}^{[ m , 0 , 0 ]}= [ W^{1}( \theta_{2 , 3 , 4}, \bar{\theta}^{1}) ]^{m}\quad \mathrm{a n d}\quad{\cal O}_{1 / 2}^{[ 0 , 0 , m ]}= [ \bar{W}_{4}( \theta_{4}, \bar{\theta}^{1 , 2 , 3}) ]^{m}.
\eta_n= \begin{cases}\dfrac{(1)_n(-j)_n(a+c)_n(a-c-j)_n}{(-n)_n(n-N)_n}, begin{align*}1.1em] \dfrac{\alpha(1)_n(-j)_j(1)_{n-j-1}(a+c)_n(a-c-j)_n}{(-n)_n(n-N)_{N-n}(1)_{2n-1-N}}. \end{cases}
A'(t) = 2 || u_t||^2_{L^2(\mathbb{H}^n)}+ 2{\rm Re}\langle u_{tt},u \rangle + 2b{\rm Re}\int_{\mathbb{H}^n}\overline{u}u_t dx &= 2 || u_t||^2_{L^2(\mathbb{H}^n)}- 2 I(u), \,\,\, 0\leq t <T_1.
0 = \frac{\delta_L}{\delta C^{\alpha a}}\mathbf{S}_m^a(\Gamma_m) =- \mathbf{\tilde Q}_m^a \frac{\delta_L}{\delta C^{\alpha a}}\Gamma_m,\qquad0 = \frac{\delta_L}{\delta C^{\alpha a}}\mathbf{D}_A(\Gamma_m) =\mathbf{\tilde Q}_A \frac{\delta_L}{\delta C^{\alpha a}}\Gamma_m.
\lim_{r\to0}\lim_{k\to+\infty}\frac{1}{2\pi}\int_{B_r(x_0)}\vert x\vert^{2\alpha_{k,1}}V_k(x)e^{au_k}dx=\frac{a\beta_0\left(\beta_0-4\right)}{4\left(1-a\right)},
\frac{d^{2}}{d\xi^{2}}y(\xi)+\bigg(\alpha+\frac{\beta +1}{\xi}+\frac{\gamma +1}{\xi-1}\bigg)\frac{d}{d\xi}y(\xi)+\bigg(\frac{\mu}{\xi}+\frac{\nu}{\xi-1}\bigg)y(\xi)=0
S_{1 0}= \int \left( \frac{1}{4}\hat{R}- \frac{1}{2}\partial_{M}\hat{\phi}\partial^{M}\hat{\phi}- \frac{1}{1 2}e^{- 2 \hat{\phi}}\hat{H}_{M N P}\hat{H}^{M N P}\right) \sqrt{- \hat{g}}d^{1 0}\hat{x},
\prod_{i=1}^{m^\prime_a}\psi^{(a)}_2({y^\prime}_i^{(a)})\prod_{i=1}^{n_a}\overline{\psi}^{(a)}_1(x_i^{(a)}\!+ \hat{\tau})\prod_{i=1}^{m_a}\psi^{(a)}_2(y_i^{(a)}\!+ \hat{\tau})\prod_{i=1}^{n^\prime_a}\overline{\psi}^{(a)}_1({x^\prime}^{(a)}_i)\Big \rangle \; .
\begin{aligned}(\widetilde\sigma(x)-\widetilde\sigma(y))^2 = (\widetilde\sigma(x)-\widetilde\sigma(z)+\widetilde\sigma(z)-\widetilde\sigma(y))^2 \leq (1+\delta_{x,y})\cdot (\widetilde\sigma(x)-\widetilde\sigma(z))^2+\Bigl(1+\frac{1}{\delta_{x,y}}\Bigr)\cdot (\widetilde\sigma(z)-\widetilde\sigma(y))^2,\end{aligned}
\sum\limits_{k=1}^d{|x_k|^3}\prod\limits_{l\neq k}|x_l|\leq \sum_{\substack{(a_1,\dots,a_m) \in\{|p_1|,|q_1|,\dots,|p_d|,|q_d|\}{d+2}}}\prod_{m=1}^{d+2}a_m\leq(|p_1|+|q_1|+\dots+|p_d|+|q_d)^{d+2}
I_\lambda(\omega) =\frac{\lambda}{4}\displaystyle\int_{\mathbb{R}^N}a(x)|\omega(x,0)|^2dx+\left(\frac{1}{2}-\frac{1}{2_\alpha^*}\right)\displaystyle\int_{\mathbb{R}^N}b(x)|\omega(x,0)|^{2_\alpha^*}dx &\geq\frac{\lambda}{4}\displaystyle\int_{\mathbb{R}^N}a(x)|\omega(x,0)|^2dx &\geq-\lambda C\|\omega\|_X^2 &\geq-\lambda C.
=\log(1+\frac{1}{a})^{\lambda'}-\log(1+a)\log(1+\frac{1}{a})^{\lambda'-1}+(1-\lambda')\int_1^a \frac{\log(1+\frac{1}{b})^{\lambda'-1}}{1+b}+ \frac{\log(1+\frac{1}{b})^{\lambda'-2}\log(1+b)}{b(1+b)}
\left(\begin{array}{c}L M-gxg^{-1}{\bf 1}_B\end{array}\right)=\left(\begin{array}{cc}\tilde{g}0 0 \rho\cdot g \end{array}\right) \left(\begin{array}{c}L M-x{\bf 1}_B\end{array}\right) g^{-1}\cdot \rho^{-1}
\rho^\kappa_a\partial_\kappa (E_{\lambda \mu}\rho^\mu_c)= \rho^\kappa_a\partial_\kappa((\rho^+)^d_\lambda\rho^\nu_d E_{\nu \mu}\rho^\mu_c)=\rho^\kappa_a\partial_\kappa((\rho^+)^d_\lambda)\rho^\nu_d E_{\nu \mu}\rho^\mu_c+\rho^\kappa_a(\rho^+)^d_\lambda\partial_\kappa(\rho^\nu_d E_{\nu \mu}\rho^\mu_c).
\bar{\rho}_N(z_N) = P(H_N(z_N),\N) \frac{e^{\sum_i \lambda_i F_{i,N}(z_N)}}{\int_{\Gamma_N}e^{\sum_i \lambda_i F_{i,N}(z_N')}\delta(H_N(z_N') - H_N(z_N))\dz_N'}.
\displaystyle{\frac{\phi^{(1)}(a,b+1,c|u,v,w+1/4)}{\phi^{(1)}(a,b-1,c|u,v,w+1/4)}}= \displaystyle{\frac{\phi^{(0)}(a,b+1,c|u-\eta,v-\eta,w+1/4)}{\phi^{(0)}(a,b-1,c|u-\eta,v-\eta,w+1/4)}}.
C_{121}(N)=\Re\left[ -z_0 \sqrt{2 e^{\pi i z_0}\left(e^{\pi i z_0}+ (-1)^N \right)}\frac{w_0^{-N/2}}{N^2}\right] +O\left( \frac{|w_0|^{-N/2}}{N^3}\right).
\|e_F^{\mathcal{N}}\| \leq \bar{C}\exp\{\mathcal{N}\tau_F\}\left(\tau_F\sum^{\mathcal{N}}_{n=1}\|e_C^n\|^2+ \sum^{\mathcal{N}}_{n=1}\tau_F\|(R)^{n}\|\right) \leq \bar{C}\left(\tau_C^4+\tau_F^2+h_1^2+h_2^2\right),
\Gamma\left(s-r+j\right) \mathbf{b}_i^t\left(\mathbf{\Psi}_s^{-1}-\mathbf{\Psi}^{-1}\right)\mathbf{a}_{s,j}\underset{s\to 0}{=}\frac{\left(-1\right)^{r+1-j}}{\left(r-j\right)!}\mathbf{b}_i^t\mathbf{\Psi}^{-1}\tilde{\mathbf{\Psi}}\mathbf{\Psi}^{-1}\mathbf{a}_j+o(s).
q \circ f = \sum_{1 \leq i,j \leq n}|f_{ij}|^2 = \sum_{1 \leq i,j \leq n}f_{ij}\overline{f}_{ij}= &= \sum_{1 \leq i,j,k \leq n}e_{ki}f e_{jk}\overline{f}_{ij}\in \sum_{1 \leq i,j,k \leq n}e_{ki}I e_{jk}\overline{f}_{ij}\subseteq I
\hat{\lambda}\left[ \hat{\lambda}^4 - \frac{2}{45}\frac{101 \theta_{11}+ 16 \theta_{22}}{\theta}\hat{\lambda}^2 - \frac{96}{25}\frac{q_1}{\rho \theta^{3/2}}\hat{\lambda}+ \frac{1}{15}\frac{53\theta_{11}^2 - 16 \theta_{11}\theta_{22}+ 8 \theta_{22}^2}{\theta^2}\right] = 0.
X^{\alpha,i}_t=X_0^i+\int_0^t b\big(s, X^{\alpha,i}_{s\wedge\cdot},\varphi^{N}_s ,\alpha^i_s \big) \mathrm{d}s+\int_0^t\sigma \big(s,X^{\alpha,i}_{s\wedge\cdot},\varphi^{N}_s ,\alpha^i_s \big) \mathrm{d}W^i_s+\int_0^t\sigma_0 \big(s,X^{\alpha,i}_{s\wedge\cdot},\varphi^{N}_s, \alpha^i_s \big) \mathrm{d}B_s,\; t\in[0,T],
H_{p}={{Q}\over{|\vec{z}-\vec{z}_{0}|^{7-p}}},\ \ \ \H_{W}=1+{{Q_{W}}\over{(x^{2}+{{4Q}\over{(p-5)^2}}|\vec{z}-\vec{z}_{0}|^{p-5})^{{p^2-8p+19}\over{2(p-5)}}}}.
\Gamma^{( 2 )}= \frac{n_{6}}{g_{s}( 2 \pi )^{6}}V_{6}\int d t \left( - \frac{3 \epsilon^{2}M^{2}}{4 r^{2}}v^{2}- \frac{\sqrt{3}\epsilon M^{2}}{r^{2}}v^{4}- \frac{M^{2}}{r^{2}}v^{6}\right) .
\hat{\rm P}^s_{\rm R}= \frac{1}{2}\hbar \int dx({\psi}^{\dagger s}_{\rm R}(-i \partial_x){\psi}^s_{\rm R}-{\psi}^s_{\rm R}(i \partial_x){\psi}^{\dagger s}_{\rm R})- \int dx \hat{\rm E}{\partial_1}A_1 .
\mathcal{H}^{{c,d,\nu}}(\mathcal{G},\mathcal{C})=\big( \mathcal{H}(\mathcal{G},\mathcal{C}),\otimes ,\big( \mathbf{1,}\widehat{\operatorname{Id}_{\mathbf{1}}}\big) ,\overline{a}^{c,d,\nu},\overline{l}^{c,d,\nu},\overline{r}^{c,d,\nu}\big)
d_0=3,\d_{2^{n-1}}=-1,\d_{2^{n-2}}=d_{3\cdot2^{n-2}}=1, d_{2^{n-3}}=d_{7\cdot2^{n-3}}=1+\sqrt{2}d_{3\cdot2^{n-3}}=d_{5\cdot2^{n-3}}=1-\sqrt{2}.
\Box^{(0)}_{t,b,m}N_{t,m}+S_{t,m}=I\ \ \mbox{on $L^2_m(X)$}, &N_{t,m}\Box^{(0)}_{t,b,m}+S_{t,m}=I\ \ \mbox{on ${\rm Dom\,}(\Box^{(0)}_{t,b,m})$}.
( \Lambda^{2})_{R}= \nu^{2}\exp \Bigg \{- 2 \int^{g ( \nu )}\frac{d g^{\prime}}{\left[ \beta ( g^{\prime}) \right]_{R}}\Bigg \}= \nu^{2}\exp \left\{- \frac{( 4 \pi )^{2}}{\beta_{0}g ( \nu )^{2}}\left[ 1 +{\cal O}( g^{2}) \right]_{R}\right\},
x'_1 = x_1 + t x_2, y'_1 = \frac{1}{2}\left( y_1 +\frac{1}{t}y_2 \right), x'_2 = y_1 - \frac{1}{t}y_2, y'_2 = - \frac{1}{2}(x_1 - t x_2), x'_3 = s \left( x_4 + \frac{1}{t}x_3 \right), y'_3 = \frac{1}{2s}(y_4 + t y_3), x'_4 = - \frac{s}{J}(y_4 - t y_3), y'_4 = \frac{J}{2s}\left( x_4 - \frac{1}{t}x_3 \right),
\Delta_{s,\tau}P=\frac{t}{2}\Delta_{J\mathfrak{k}}P=P\frac{t}{2}\Delta_{J\mathfrak{k}}\;\;C_{c}^{\infty}(K_{\mathbb{C}}),\quad\;\Delta_{J\mathfrak{k}}:=\sum_{j=1}^{d}\tilde{Y}_{j}^{2}.
\int f(y)\, dM^N_{(X_t^{N,\tau})^k}(y)=\sum_{i=1}^N \biggl[f\bigl((1-\tau)X_{0,t}^k(\lambda_i^k)+\tau(X_t^{N})_i^k(\hat\lambda)\bigr)-\int f \,d\mu_{k,t}^*\biggr]\forall\,f \in C_c(\mathbb R).
[ \hat{B}^{0 i}, \hat{Q}^{a}] = i \varepsilon^{a b}\hat{\pi}_{( C ) b}^{i}, [ \hat{\pi}_{( B ) 0 i}, \hat{Q}^{a}] = - i \varepsilon^{a b}( \partial_{i}\hat{\pi}_{( C ) 0 b}+ \hat{A}_{i}\wedge \pi_{( C ) 0 b}) ,
\hat{d}_y(h(\infty),h(z))=\hat{d}_y(\infty,h(z)) \geq\frac{1}{4}\cdot\frac{1}{1+d(h(y),h(z))}&\geq\frac{1}{4}\cdot\frac{1}{1+MC\,d(y,z)}=\frac{1}{4(1+M)}.
a_n(t) =\frac{(-1)^nt^{(n-1)r}\sqrt{\pi}\sqrt{t^{2r}-a_1^2}}{2^n(n-\frac{1}{2})!}D^n\left(t^{-2nr+r}\left(t^{2r}-a_1^2\right)^{n-\frac{1}{2}}\right), b_n(t) =\frac{(-1)^n(n+1)b_0t^{(n+1)r}\sqrt{\pi}}{2^{n+1}(n+\frac{1}{2})!\sqrt{t^{2r}-a_1^2}}D^n\left(t^{-2nr-r}\left(t^{2r}-a_1^2\right)^{n+\frac{1}{2}}\right),
\frac{\partial}{\partial u}g(z,u)|_{u=1}=\frac{\sum_{d|k}z^{d}}{1-z}exp(\sum_{d|k}(u^{d}\frac{z^{d}}{d}-\frac{z^{d}}{d}))|_{u=1}=\frac{\sum_{d|k}z^{d}}{1-z}.
\overline{e}(k,i+1)=\frac{1}{\sqrt{(1-\overline{e}_{b}^{2}(k,i))(1-\overline{\delta}_{D}^{2}(k,i))}}[\overline{e}(k,i)-\overline{\delta}_{D}(k,i)\overline{e}_{b}(k,i)]
\int_{\Lambda}\prod_{a = 1}^{N}\left[ \prod_{i = 1}^{n_{a}}d^{2}x_{i}^{( a )}d^{2}y_{i}^{( a )}\right] \left\langle \prod_{b = 1}^{N}\prod_{j = 1}^{n_{b}}\left[ : e^{+ i 2 \sqrt{\pi}\varphi^{( b )}( x_{j}^{( b )})}:_{Q^{0 , S}}: : e^{- i 2 \sqrt{\pi}\varphi^{( b )}( y_{b}^{( b )})}:_{Q^{0 , S}}\right] \right\rangle_{Q^{0 , S}}
\phi(f_{1},f_{2}) =f_{1}(1)\overline{f_{2}}(1)+\frac{2}{\beta}\left(f_{1}^{\prime}(1)\overline{f_{2}}(1)+f_{1}(1)\overline{f_{2}}^{\prime}(1)\right) +\left( 1+\frac{4}{\beta^{2}}\right) f_{1}^{\prime}(1)\overline{f_{2}}^{\prime}(1) & \quad\quad+\int_{-1}^{1}f_{1}^{\prime\prime}(x)\overline{f_{2}}^{\prime\prime}(x)(1+x)^{\beta+2}dx & =0.
T^{n}[\begin{matrix}\vec{f}^{n-1}\\ 0\\ \end{matrix}]=[\begin{matrix}&&&t_{-n+1}\\ &T^{n-1}&&t_{-n+2}\\ &&&\vdots\\t_{n-1}&t_{n-2}&...&t_{0}\\ \end{matrix}][\begin{matrix}\\ \vec{f}^{n-1}\\ \\ 0\\ \\ \end{matrix}]=[\begin{matrix}1\\ 0\\ \vdots\\ 0\\ \epsilon_{f}^{n}\end{matrix}].
\tilde{\Theta}_\pm A\varphi =\tilde{\Theta}_\pm \gamma^{(2n+1)}(P_{\rm F}-D_aP_{\rm F}D_a)\varphi &=-\gamma^{(2n+1)}\tilde{\Theta}_\pm (P_{\rm F}-D_aP_{\rm F}D_a)\varphi=-A\tilde{\Theta}_\pm \varphi
\dim_{E}\mathrm{Hom}_{G}(I_{\sigma_{i}}, P(\chi)) = \dim_{E}\mathrm{Hom}_{G}(I_{\sigma_{i}}\otimes_{\mathfrak{Z}_{\Omega}}\kappa(\mathfrak{m}), P(\chi)) = \dim_{E}\mathrm{Hom}_{G}(P(\chi)^{\oplus n_{\sigma_{i}}}, P(\chi)) = n_{\sigma_{i}}.
H=\sum_{i=1}^N\lbrace \sqrt{m_i^2+{({\vec p}_i-e_i\vec A({\vec x}_i,x^o))}^2}+V({\vec x}_i,x^o)\rbrace +{1\over 4}\int d^3z [{\vec \pi}^2(\vec z,x^o)+{\vec B}^2(\vec z,x^o)],
H = \sum_{i=0}^{d}\frac{a^i q^{id}(A-\theta_0 I )( A-\theta_{1}I ) \cdots (A-\theta_{i-1}I)}{(q^2;q^2)_i}, H^{-1}= \sum_{i=0}^{d}\frac{a^{-i}q^{-id}(A-\theta_0 I )( A-\theta_{1}I ) \cdots (A-\theta_{i-1}I)}{(q^{-2};q^{-2})_i}.
\sum_{m,m';\,m\pm m'{\rm even}}A_M^{B,m}\biggl\{\widetilde\Pi_{MN}^{+BC}(m,\frac{m-m'}{2})+\widetilde\Pi_{MN}^{-BC}(m,\frac{m+m'}{2})\biggr\}A_N^{C,m'}.
\left({\begin{array}{*{20}c}\tilde{A}-z(I-P) \left( I-P \right)AP PA(I-P) P\left( A-zI \right)P \end{array}}\right)\left({\begin{array}{*{20}c}X Y Z W \end{array}}\right)=\left({\begin{array}{*{20}c}I-P 0 0 P \end{array}}\right),
P_{0,2}^{(\Lambda,N)}(\tilde{\boldsymbol{\phi}}) :=\frac{\partial}{\partial \phi_2}P_0^{(\Lambda,N)}(\tilde{\boldsymbol{\phi}}) &=\frac{1}{2\pi i}\oint dz \frac{1}{z-H_0^{(\Lambda,N)}(\tilde{\boldsymbol{\phi}})}J^{(2)}(\ell;\tilde{\boldsymbol{\phi}})\frac{1}{z-H_0^{(\Lambda,N)}(\tilde{\boldsymbol{\phi}})}.
\Vert f \Vert_{U^{d}(\mathbb{R})}^{2^{d}}= \int\limits_{(x,\mathbf{h}) \in\mathbb{R}^{d+1}}\prod\limits_{\boldsymbol{\omega}\in \{0,1\}^d}\mathcal{C}^{\vert \boldsymbol{\omega}\vert}f(x+\sum\limits_{i=1}^d h_i\omega_i) \, dx\,dh_1\cdots dh_d
\left\|\Pi_{V_2}|_{W_1}\right\|^2_{HS}=\sum_{k=1}^{\dim W_1}\left\|\Pi_{V_2}\phi_{1k}\right\|^2\leq \sum_{k=1}^{\dim W_1}\rho_0^2\|\phi_{1k}\|^2=\rho_0^2\dim W_1,
\mathrm{P}^{out}_{m}= 1 - \mathrm{P}( |h_m|^2>\psi_m^* ) =\int^{\psi_m^*}_{0}\frac{M!\left(F_{|\tilde{h}|^2}(x)\right)^{m-1}\left(1-F_{|\tilde{h}|^2}(x)\right)^{M-m}f_{|\tilde{h}|^2}(x)}{(m-1)!(M-m)!}dx,
T_{s}= \frac{1}{4}\left( \! \! \! \begin{array}{c c c c c c c c c}{{2}}&{{0}}&{{- 2}}&{{- 4}}&{{2}}&{{2}}&{{3}}&{{- 1}}&{{- 2}}\\{{- 1}}&{{3}}&{{- 2}}&{{2}}&{{- 1}}&{{- 1}}&{{0}}&{{- 1}}&{{1}}\\{{0}}&{{6}}&{{- 6}}&{{0}}&{{0}}&{{0}}&{{3}}&{{- 3}}&{{0}}\\ \end{array}\right)^{\! \! \sim}
\Delta^{(0)}=(\alpha_1(yH),\alpha_2(H),\alpha_3(yH)) \Delta^{(1)}=(\alpha_1(yH),\alpha_2(Hx),\alpha_3(yHx)) \Delta^{(2)}=(\alpha_1(yH),\alpha_2(Hx^2),\alpha_3(yHx^2))
\tilde{\phi}_{0}(\textbf{x}) = I \left\lbrace \textbf{x}\in \mathcal{C}^{'}_{1}\cap \mathcal{C}^{'}_{0}, (1-q(\textbf{x},1)) \dfrac{p}{\mu{(\mathcal{C}^{'}_{1})}}> (1-q(\textbf{x},0)) \dfrac{1-p}{\mu{(\mathcal{C}^{'}_{0})}}\right\rbrace &+ I \left\lbrace \textbf{x}\in \mathcal{C}^{'}_{1}- \mathcal{C}^{'}_{0}, q(\textbf{x},1) < 1\right\rbrace.
\frac{\partial\xi^{0}(\tilde{x})}{\partial\tilde{x}}=0,g_{\alpha\beta}(\tilde{x})\frac{\partial\xi^{\beta}(\tilde{x})}{\partial\tilde{t}}-\frac{\partial\xi^{0}(\tilde{x})}{\partial\tilde{x}^{\alpha}}=0.
\{e_{a,b}^{b}\mid 1\le a,b \le n,\; a\ne b \}\cup\{e_{a,b}^{c}\mid 1\le a,b,c\le n,\; a< b,\; a\ne c\ne b\}=\{e_{a,b}^{c}\mid 1\le a,b,c\le n,\; a< b\}.
(-2\gamma_1-1)^{-1}(k+2)^{2\gamma_1+1}=\int_{2}^\infty (x+k)^{2\gamma_1}dx \le \int_{2}^\infty (x+k)^{\gamma_1}x^{\gamma_1}dx \le\sum_{i=2}^\infty (i+k)^{\gamma_1}i^{\gamma_1}\le \sum_{i=2}^\infty i^{2\gamma_1}\le \int_{1}^\infty x^{2\gamma_1}dx =(-2\gamma_1-1)^{-1}.
\left \lfloor \frac{n_1+t}{q^3+1}\right \rfloor +\left \lfloor \frac{n_2+t}{q^3+1}\right \rfloor+(q^2-q) \left \lfloor \frac{t}{q^2-q+1}\right \rfloor+\left \lfloor \frac{-t q^3}{q^3+1}\right \rfloor = \begin{cases}q^2-1+(q^2-q)q-q^3=-1 t=q^3 q^2-2+(q^2-q)(q-1)-q^3+q^2-q+1=-1 t=(q^2+1)(q-1).\end{cases}
\delta S=\int_{M}d^{2}X\mbox{}\left( \frac{\partial{\cal L}}{\partial\Lambda_{A}}\mbox{}\delta \Lambda_{A}+\frac{\partial{\cal L}}{\partial\partial_{\alpha}\Lambda_{A}}\mbox{}\delta \left( \partial_{\alpha}\Lambda_{A}\right) \right) \mbox{.}
\forall x, y, z\, (J(x, y, z) -> (x \le z) \wedge (y \le z) \wedge (x \wedge y = \bot)), \forall x, y, z, w\, (J(x, y, z) \wedge (x \le w) \wedge (y \le w) -> (z \le w)), \forall x, y\, ((x \wedge y = \bot) -> \exists!_\le^{x,y}z\, J(x, y, z))
\pi\left(\dfrac{1}{2}\right) =\sum_{v=1}^{s-1}(\sigma_{v}- \sigma_{v-1})\nu_v+\left(\dfrac{1}{2}-\sigma_{s-1}\right) \nu_s &= \sum_{v=1}^{s-1}\sigma_{v}(\nu_v-\nu_{v+1})+\dfrac{1}{2}\nu_s.
\mathcal{M}_{u}(\gamma_0,0)[v]=\frac{2}{\sin^{2}\theta}\frac{\partial^{2}\overline{v}}{\partial\phi^{2}}+\frac{2}{\sin^{2}\theta}\frac{\partial^{2}\overline{v}}{\partial\psi^{2}}+\frac{\partial^{2}\overline{v}}{\partial\theta^{2}}+(2\cot\theta)\frac{\partial \overline{v}}{\partial\theta}+\left(3+\frac{(2+B)}{\sin^{2}\theta}\right)\overline{v}.
D_n(x|a_1,\dots,a_r) =\sum_{m=0}^n S_1(n,m)B_m(x|a_1,\dots,a_r) &=\sum_{j=0}^n\left(\sum_{l=j}^n\binom{n}{l}S_1(l,j)D_{n-l}(a_1,\dots,a_r)\right)x^j &=\sum_{m=0}^n\binom{n}{m}D_{n-m}(a_1,\dots,a_r)(x)_m\,,
{\mathsf W}^{(210)}_{-+}=\frac{1}{4!}h^{^{(210)}}_{ab}<\widehat{\Psi}_{(-)a}^*|B\Gamma_{[\mu}\Gamma_{\nu}\Gamma_{\rho}\Gamma_{\lambda]}|\widehat{\Psi}_{(+)b}>\Phi_{\mu\nu\rho\lambda}
\begin{array}{rcl}\vec\alpha_1 = (a_{11},a_{12},\ldots,a_{1d}) \vec\alpha_2 = (a_{21},a_{22},\ldots,a_{2d}) \multicolumn{3}{c}{\vdots}\vec\alpha_d = (a_{d1},a_{d2},\ldots,a_{dd}).\end{array}
\sup_{i,l}\frac{1}{\mu_i\mu_l}\left| \sum_j \mu_j \langle \phi_i, \psi_j \rangle \langle \phi_l, \psi_j \rangle\right|^2 = \sup_{i,l}\frac{1}{\mu_i\mu_l}\left| \sum_j \mu_j \langle \phi_i, \phi_j \rangle \langle \phi_l, \phi_j \rangle\right|^2 = \sup_{i}\frac{1}{\mu_i^2}\left| \sum_j \mu_j \langle \phi_i, \phi_j \rangle^2 \right|^2 =1.
\theta^{\prime}(x,\overline{y})\equiv \bigvee_{i=0}^{N_{\theta}}[\left| \theta(\cdot, \overline{y})\right| = i \; \& \; x=i] \vee [\left| \neg\theta(\cdot, \overline{y})\right| = i \; \& \; x=-(i+1)]
\begin{array}{l}{{( \partial b )^{*}= - q a b^{- 1}( \partial a ) ,}}\\{{( \partial c )^{*}= - \frac{1}{q^{2}}c^{- 1}( a \partial a ) ,}}\\{{( \partial d )^{*}= q ( a^{2}- 1 ) ( b c )^{- 1}( \partial a ) .}}\\ \end{array}
\int_{\mathbb{T}_\alpha}K_2^{k}(x,y)dx =\int_{\mathbb{T}_\alpha}\frac{R_+^2(\alpha k-1,y,\lambda,E)r(\lambda)}{\sum_{i=0}^{\alpha-1}f_iu_+^2(\alpha k+i,y,\lambda,E)}\left|\frac{d x}{d\lambda}\right|d\lambda &=\int_{\mathbb{T}_\alpha}r(\lambda)d\lambda &=1.
\operatorname*{l i m}_{\beta \rightarrow \infty}\; \operatorname*{l i m}_{m \rightarrow 0^{-}}\langle{\bar{\psi}}\psi \rangle_{\beta}= \operatorname*{l i m}_{\beta \rightarrow \infty}\; \operatorname*{l i m}_{m \rightarrow 0^{+}}\langle \bar{\psi}\psi \rangle_{\beta}= \frac{e B}{4 \pi}\; .
B_T:=\Big\{u\in C(]-\infty,-T];Z), &\left\| A_j u\right\|_{L^q_TL^r_yL^2_x}+ \left\| A_j u\right\|_{L^\infty_TL^2_{xy}}\le 4C \|A_ju_-\|_{L^2_{xy}}, j\in \{0,1,2\}, & \left\| A_j u\right\|_{L^p_T L^k_yL^2_x}\le 2 \left\| A_j e^{-itH}u_-\right\|_{L^p_T L^k_yL^2_x},j\in \{0,1\}\Big\}.
Q = Q_0+ Q_1; Q_0 ={\begin{pmatrix}0 0 0\cr q_{2,1}0 0 0 q_{3,2}0\end{pmatrix}, \Q_1=\begin{pmatrix}q_{1,1}q_{1,2}q_{1,3}\cr 0 q_{2,2}q_{2,3}0 0 q_{3,3}\end{pmatrix}}.
\begin{array}{c}{\zeta_{\nu}( s ) = \displaystyle{- \frac{1}{\pi}}\Re \left\{x^{1 - s}e^{\displaystyle{- \imath \frac{\pi}{2}s}}\displaystyle{\int_{0}^{\infty}}( y - \imath )^{- s}\displaystyle{\frac{{\Delta_{\nu}^{T E}}^{\prime}\left( \imath x ( y - \imath ) \right)}{\Delta_{\nu}^{T E}\left( \imath x ( y - \imath ) \right)}}\d y \right\},}\\ \end{array}
\sum\limits_{j=1}^{n+1}A_{ij}y_{j}=\left( \sum\limits_{j=1}^{n}c_{ij}x_{j}\mathbf{f+}A_{i,n+1}y_{n+1}\right) \mathbf{=}\left( \lambda x_{i}\right)\mathbf{f=}\lambda \left( x_{i}\mathbf{f}\right) =\lambda y_{i}
|E_1(x)| \geq C(x \mid y) + \log(1-\epsilon) - O(1) |E_2(y)| \geq C(y \mid x) + \log(1-\epsilon) - O(1) |E_1(x)| + |E_2(y)| \geq C(x,y) + \log(1-\epsilon) - O(1).
\hat{A}_{p}v''_{p}=\frac{1}{\lambda_{p}}\hat{A}_{12}\hat{Q}^{-1}\mathbb{E}^{\tilde{\mathbb{P}}}\left[\pi\right]+\frac{1}{\lambda_{p}}\hat{A}_{12}\hat{Q}^{-1}\hat{A}_{12}^{\top}\mu'_{p}.
\mathbb{V}(R^*_{\nu}) = \mathbb{V}^{\phi}_\theta(T) - \sum_{i,j=1}^{\nu}\left( g^{(i)}(\theta)-{\rm Q}_1^i + \tau{\rm Q}_2^i \right) \left( g^{(j)}(\theta)-{\rm Q}_1^j + \tau{\rm Q}_2^j \right) J^{\phi}_{ij}.
Du\,=\,\sqrt{\rho}\,Du\,+\,\left(\frac{1}{\sqrt{\rho}}\,-\,1\right)\,\sqrt{\rho}\,Du\quad\in\,\Bigl(L^2_T\bigl(L^2\bigr)+L^1_T\bigl(L^{3/2}\bigr)\Bigr)\,\cap\,\Bigl(L^2_T\bigl(L^2+L^1\bigr)\Bigr)\,.
Z\left[ J^\mu ,\overline{\eta},\eta \right] =\int\left[ dA_\mu \right] \Im \left( A_\mu \right) \exp \left( -\left\langle\frac 14F_{\mu \nu}F^{\mu \nu}\right\rangle +\left\langle J^\mu A_\mu\right\rangle +\left\langle \overline{\eta}^{\prime}\left( i\gamma^\mu\partial_\mu \right)^{-1}\eta^{\prime}\right\rangle \right) ,
A = I_{r+1}-\left[\begin{matrix}0 -1 \delta -1 2\delta \ddots \ddots & -1 (r-1)\delta 0 r\delta \end{matrix}\right], \Delta A = \left[\begin{matrix}0 \cdots \cdots \cdots 0 \vdots \cdots \cdots \cdots \vdots 0 \cdots 0 \epsilon 0 \end{matrix}\right].
S(\hat{f};\hat{y},\hat{y}')(\hat{y}-\hat{y}') =\hat{f}(\hat{y})-\hat{f}(\hat{y}') &=(V^T\otimes W^{-*})(f(y)-f(y')) &=(V^T\otimes W^{-*})S(f;y,y')(y-y') &=(V^T\otimes W^{-*})S(f;y,y')(V^{-T}\otimes W^*)(\hat{y}-\hat{y}').
\lim_{\epsilon\to 0}f(\epsilon^{-1}x,\epsilon) = \textstyle \lim_{\epsilon\to 0}(1 - Nx + O(\epsilon))^{-1}(1+O(\epsilon)) (1 - N^{t+1}x^{t+1}+ O(\epsilon)) = \textstyle \lim_{\epsilon\to 0}(1 - Nx + O(\epsilon))^{-1}(1 - N^{t+1}x^{t+1}+ O(\epsilon)) \textstyle = (1 - Nx)^{-1}(1 - N^{t+1}x^{t+1}) = \sum_{j=0}^t N^jx^j.
\begin{aligned}\int_\Omega B_0 (v_0-v_1)\cdot(v_0-v_1) = \int_\Omega ( B_0 v_0\cdot v_0 -2 B_0 v_0\cdot v_1 + B_1 v_1\cdot v_1) &+ \int_\Omega (B_0-B_1)v_1\cdot v_1 = \Re\int_{\partial\Omega}(g_1-g_0)h+\int_D (B_0-B_1)v_1\cdot v_1.\end{aligned}
{\cal L}+{\cal L}_{ct}={1\over 2}(\partial_{\mu}\phi)^2 -{1\over 2}M^2\phi^2 -{\lambda\over 4!}\mu^{2\epsilon}\phi^4 +{1\over 2}A (\partial_{\mu}\phi)^2 -{1\over 2}M^2 B\phi^2 -{\lambda\over 4!}\mu^{2\epsilon}C\phi^4 -{1\over 2}(m^2 - M^2) (1+B)\phi^2
\chi_{\mathcal{L}}(u,v) = u +v\frac{e^{-\pi H_vf(u)}-\cos(\pi P_v f(u))}{\sin(\pi P_v f(u))}, \eta_{\mathcal{L}}(u,v) = 1 - v\frac{e^{\pi H_vf(u)}+ e^{-\pi H_vf(u)}-2 \cos(\pi P_v f(u)))}{\sin(\pi P_v f(u)))}.
\Theta_\nu=\prod_{i=0}^{k-1}(-q_i)^{-\nu_i}\sum_{E^{{\bf c}}_{{\bf h},p}\in{\bf B}^{(k)}_\nu}\prod_{s=1}^N\prod_{t=1}^{c_s}(1-q_{i_s}^{2t})E^{{\bf c}}_{{\bf h},p}\otimes F^{{\bf c}}_{{\bf h},p},
\widetilde{\alpha}_{t,\varepsilon}: =\widetilde{\alpha}_{t,\varepsilon}(0)=\int_0^t\int_0^tf_\varepsilon(X^{H_1}_s-\widetilde{X}^{H_2}_r)drds &=\frac{1}{2\pi}\int_{[0,t]^2}\int_{\mathbb{R}}e^{-\frac{\varepsilon}{2}p^2}e^{\iota p(X^{H_1}_s-\widetilde{X}^{H_2}_r)}dpdrds,
g(z,\alpha) = \sum_{k=0}^{\infty}\sum_{j=0}^{\infty}c_{jk}(\alpha-\alpha_0)^j (z-z_0)^k=: \sum_{k=0}^{\infty}c_k(\alpha) (z-z_0)^k \,, \frac{\partial g}{\partial u}(z,\alpha) = \sum_{k=0}^{\infty}\sum_{j=0}^{\infty}c_{jk}\, j (\alpha-\alpha_0)^{j-1}(z-z_0)^{k}=: \sum_{k=0}^{\infty}\widetilde{c}_k(\alpha) (z-z_0)^k \,.
(x_1,x_2,y_1,y_2,w)=\left(r \cos\vartheta,r\sin\vartheta, p_r \cos\vartheta -\frac{\sin\vartheta}{r}p_\vartheta, p_r \sin\vartheta + \frac{\cos\vartheta}{r}p_\vartheta,w\right).
\forall\:0\le k<n,y_{k}^{(1)}\,y_{k}^{(0)}=\mbox{Prod}_{\mathbf{P}_{k}}\left(\left(\mathbf{x}^{(1)}\right)^{\top},\mathbf{x}^{(0)}\right)\:\mbox{where}\mathbf{P}_{k}=\mbox{Prod}_{\boldsymbol{\Delta}^{(k)}}\left(\mathbf{A}^{\left(1\right)},\mathbf{A}^{\left(2\right)}\right)
J_{\lambda,\mu}(u_k,v_k)= \Big(\frac{1}{p}-\frac{1}{\alpha+\beta}\Big)\|(u_k,v_k)\|^p -\Big(\frac{1}{q}-\frac{1}{\alpha+\beta}\Big)\int_{\Omega}(\lambda|u_{k}|^q+\mu|v_{k}|^q)dx <\frac{c_{\lambda,\mu}}{2}.
\mathcal{B}_0\partial_t{\mathbf V}_{t}+\mathcal{B}_1\partial_1{\mathbf V}_{t}+\mathcal{B}_2\partial_2{\mathbf V}_{t}+(\partial_t\mathcal B_0+\mathcal{B}_3){\mathbf V}_{t}=\tilde{\mathcal F}_t\,,
T_{F}( \bar{z}) = \frac{1}{\sqrt{2}}( G^{+}( \bar{z}) + G^{-}( \bar{z}) ) = i \sum_{i = 1}^{3}\tilde{\rho}^{i}\tilde{\sigma}^{i}\tilde{\psi}^{i}+ \mathrm{h . c .}
G^{(2)}(\tau,0)=Tr[\rho\hat{E}^{-}(\tau)\hat{E}^{+}(\tau)\hat{E}^{-}(0)\hat{E}^{+}(0)]=\int P(\{\alpha_{k}\})E^{*}(\tau)E(\tau)E^{*}(0)E(0)d\{\alpha_{k}\}
\{a_{2i-1}, a_{2i}\}\subset G_{\Gamma( \mathcal{B}_{\sum_{i=1}^{2n}\varepsilon_{i}2^{i-1}})}\ \mathrm{and}\b_{2i-1}, b_{2i}\not\in G_{\Gamma( \mathcal{B}_{\sum_{i=1}^{2n}\varepsilon_{i}2^{i-1}})}.
\psi (y) := \Bigl(Ty_{1,-1}+ W(y_{1,-1};-1) +{\gamma\over 2}(y_{1,-1}- y)^2 \Bigr){}\qquad- \Bigl( Ty_{1,0}+ W(y_{1,0};0) +{\gamma\over 2}(y_{1,0}- y)^2 \Bigr) \,.
\sum_{\substack{D \leq T^{1-1/H}(D,Q)=1}}\frac{\bar{f}_i(D)}{D}\ll \exp \bigg( \sum_{\substack{p \leq T p\nmid Q}}\frac{|f_1(p) + \dots + f_H(p) - f_i(p)|}{p}\bigg)= \exp \bigg( \frac{H-1}{H}\sum_{\substack{p \leq T p\nmid Q}}\frac{|f(p)|}{p}\bigg),
\tilde{Z}_{p}= \frac{1}{( \sqrt{4 \pi})^{2 6}}\int_{0}^{\infty}\frac{d t}{2 \pi}\left( \frac{2 \pi}{t}\right)^{( 2 5 - p ) / 2}e^{2 t}f^{- 2 4}( t ) \, .
f_{n}\left( \Delta x_{n}\right) =e_{n}+\Delta y_{n-1}f_{n}(u) :=\Pi_{N_{n+1}}\left( F(x_{n}+u)-F(x_{n})\right) e_{n}:=\Pi_{N_{n+1}}(\Pi_{N_{n}}-1)F(x_{n}) \Delta y_{n-1}:=\Pi_{N_{n}}(1-\Pi_{N_{n-1}})\bar{y}
\rho ( \omega ) ={\frac{1}{Z}}\sum_{m}e^{- \beta E_{m}}\sum_{n > m}\vert \langle n \vert \phi \vert m \rangle \vert^{2}\, 2 \omega \left( 1 - e^{- \beta \omega}\right) \delta ( \omega - E_{n}+ E_{m}) \, .
\frac{\lVert u_1-v(t)\rVert}{\sin(\gamma(t))}=\frac{\lVert u_1\rVert}{\sin(\pi-\theta-\gamma(t))}\Leftrightarrow{t\lVert u_1-u_2\rVert}=\lVert u_1\rVert \frac{{\sin(\gamma(t))}}{\sin(\pi-\theta-\gamma(t))}\Leftrightarrow{t\lVert u_1-u_2\rVert}=\lVert u_1\rVert \frac{{\sin(\gamma(t))}}{\sin(\theta+\gamma(t))}
\hat{C}_{l \, j \, l - j}^{l \, l_{1}\, l_{2}}= ( - 1 )^{l_{1}- j}{\frac{( l_{1}+ l_{2}- l ) !}{( l_{1}+ l_{2}+ l + 1 ) ! \Delta ( l_{1}, l_{2}, l )}}\left[{\frac{( l_{1}+ j ) ! ( l_{2}+ l - j ) !}{( l_{1}- j ) ! ( l_{2}- l + j ) !}}\right]^{1 / 2},
\chi_{1 , i}^{( 2 , 2 k + 3 )}( q ) = \chi_{1 , 2 k + 3 - i}^{( 2 , 2 k + 3 )}( q ) = \displaystyle \sum_{n_{1}\geq \cdots \geq n_{k}\geq 0}\frac{q^{n_{1}^{2}+ \cdots + n_{k}^{2}+ n_{i}+ \cdots + n_{k}}}{( q )_{n_{1}- n_{2}}\cdots ( q )_{n_{k - 1}- n_{k}}( q )_{n_{k}}},
(\mu_y \circ g_x)^*(ae_1^*+be_2^*+ce_3^*) = \mu_y^*\circ (g_x^*)^{-1}(ae_1^*+be_2^*+ce_3^*) &= \max(|a|, \; y^{-1}|b|, \; y^{-2}|{-bx+c}|)= y^{-2}|x|\nu_{y,x}
\begin{aligned}q^{-17}L_{10}(q^{16}) = \sum_{\substack{n \geq 1 -n \leq j \leq n-1}}q^{6(4n)^2 - (8j+1)^2}+ \sum_{\substack{n \geq 0 -n \leq j \leq n}}q^{6(4n+2)^2 - (8j+3)^2}+ \sum_{\substack{n \geq 1 -n+1 \leq j \leq n}}q^{6(4n)^2 - (8j-1)^2}+ \sum_{\substack{n \geq 0 -n \leq j \leq n}}q^{6(4n+2)^2 - (8j-3)^2}.\end{aligned}
I''_{h,k,n}(x) =\frac{1}{2 \pi n^{1/3}}\,\int_{- \frac{2 \pi n^{1/3}}{m+m'}}^{\frac{2 \pi n^{1/3}}{m+m''}}\exp\left[n^{2/3}\left(\frac{L_m(x)^3}{2 ( \Re L_m(x) - iz)^2}+ ( \Re L_m(x) - iz)\right)\right] \, \times \left\{\exp\left[g_{h,k}\left( x, \frac{1}{n^{1/3}}( \Re L_m(x) - iz) \right)\right]-1\right\}\,dz .
\begin{aligned}\Phi_a(x) =\int_{X}\Psi_a(y_a,y^a)\lambda(x_a-y_a,x^a-y^a)dy &=\int_{X}\Psi_a(y_a)\lambda(x_a-y_a,x^a-y^a)dy &=\int_{X_a}\Psi_a(y_a)\int_{X^a}\lambda(x_a-y_a,x^a-y^a)dy^ady_a &=\int_{X_a}\Psi_a(y_a)\int_{X^a}\lambda(x_a-y_a,-y^a)dy^ady_a &=\int_{X}\Psi_a(y_a)\lambda(x_a-y_a,-y^a)dy &=\Phi_a(x_a,0).\end{aligned}