harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
f(\eta)=\sum^{\infty}_{m=-\infty}a_m\,e^{i\pi m\eta/\chi},\;\;a_m=\sum^{\infty}_{n=-\infty}{\cal C}^{\chi}_{mn}\,a_n,\;\;{\cal C}^{\chi}_{mn}=\frac{2\varsigma\wp}{\pi\sqrt{3}}\,\frac{1}{2\chi}\left(\widehat{O}e^{i\pi n\sigma/\chi}, e^{i\pi m\eta/\chi}\right),
T_3 =R_2+S_2+R_1S_1 =R_2+S_2-R_1^2 T_2 =R_3+R_2S_1+R_1S_2 =R_3-R_1R_2+R_1S_2 T_1 =-R_3R_1+R_2S_2 T_0 =R_3S_2.
M_2(H_7)-M_2(H_8) =\sum\limits_{j=1}^{t}d_{H_7}(v)d_{H_7}(v_{j})+d_{H_7}(v_0)d_{H_7}(v_1) &-\sum\limits_{j=3}^{t}d_{H_8}(v)d_{H_8}(v_{j})-d_{H_8}(v)d_{H_8}(v_{1})-d_{H_8}(v_0)d_{H_8}(v_2) &=\sum\limits_{j=3}^{t}d_{H_7}(v_{j})+d_{H_7}(v)+(d_{H_7}(v)-2)d_{H_7}(v_2)+5>0.
\exp(-\Delta_n\widehat \beta_n) = \max\Big\{1+\frac{\sum_{q = 1}^{\widehat{\lambda}_n}\mathrm{\mathrm{sgn}}(\Delta^n_{\mathcal{I}_n(q)}X) \big(\Delta^n_{\mathcal{I}_n(q)+1}X + 2\Delta_n\sum_{j=1}^{q-1}\Delta^n_{\mathcal{I}_n(j)}X\big)}{\sum_{q = 1}^{\widehat{\lambda}_n}|\Delta^n_{\mathcal{I}_n(q)}X|}, \Delta_n\Big\}
P = \exp \Big [ -{\frac{1}{8 \pi \sigma^{2}}}+ 2 \pi \sigma r_{s t a t i c}^{2}\beta_{R N}\widetilde f^{1 / 2}- q^{2}\beta_{R N}\Big ({\frac{1}{r_{+}}}-{\frac{1}{r_{s t a t i c}}}\Big ) \Big ]
\kappa_{t}\leq \Big(\sup_{m\geq 1}\frac{\bar{\alpha}_{m}}{m}\Big)\frac{\sum\nolimits_{k=0}^{t}k3^{k}}{\sum\nolimits_{k=0}^{t}3^{k}}=\alpha^{\ast}\frac{\sum\nolimits_{k=0}^{t}k3^{k}}{\sum\nolimits_{k=0}^{t}3^{k}}\leq(\alpha^{\ast}\chi (t))t,
\langle \Lambda | ( \cdots ) | \Lambda \rangle = 2 \mathrm{t r}[{\cal H}( \cdots ) ] , \quad{\cal H}={\frac{1}{2}}\mathrm{d i a g}\left({\frac{N - 1}{N}},{\frac{- 1}{N}}, \ldots ,{\frac{- 1}{N}}\right) .
\psi(x,u) = f(\lambda(A_{g_u}(x_0))) = (A_{g_u})_{nn}(x_0)f\Big(\lambda\Big(\frac{1}{(A_{g_u})_{nn}(x_0)}A_{g_u}(x_0)\Big)\Big) & \geq (A_{g_u})_{nn}(x_0)\,\frac{1}{C_1}.
b_{\scriptscriptstyle{l-1}}\,a^l=-\pi^{-2}\,(-1)^m\,i^{l}\, e^{i \delta_{\scriptscriptstyle{l}}}\,\Delta_{\scriptscriptstyle{l}}\,( 1 + \mbox{O}(\Delta_{\scriptscriptstyle{l}}^2) )\,.
g(z,u)=exp(\sum_{d|k}(u^{d}\frac{z^{d}}{d}-\frac{z^{d}}{d})+log\frac{1}{1-z})=\frac{1}{1-z}exp(\sum_{d|k}(u^{d}\frac{z^{d}}{d}-\frac{z^{d}}{d})).
\bar{L}= 3 \epsilon \phi_{0}^{2}[ \dot{H}+ 2 H^{2}] - 1 2 ( c_{1}+ c_{2}+ 3 c_{3}) [ ( \dot{H}+ H^{2})^{2}+ H^{4}] - 1 2 ( c_{2}+ 6 c_{3}) [ ( \dot{H}+ H^{2}) H^{2}]
\sqrt{2\pi T}\int_{-\infty}^{\infty}\frac{S(e^y,\chi_\phi)}{e^y}\exp\Big(\lambda y-\frac{T}{2}y^2\Big) dy &= \sqrt{2\pi T}\exp\Big(\frac{\lambda y_0}{2}\Big) \int_{-\infty}^{\infty}\Big(\frac{S(e^{y_0},\chi_{\phi})}{e^{y_0}}+ O\Big(\frac{1+|y-y_0|^{\frac 13}}{y_0^{\frac 13}}\Big) \Big) \exp\Big(-\frac T2(y-y_0)^2\Big) dy.
{\rm Tr}\;SP_+[\vartheta_{a}^{(1)},P_-][\vartheta_a^{(2)},P_-]\cdots[\vartheta_a^{(2n+1)},P_-] &=-{\rm Tr}\;SP_+[\vartheta_{a}^{(1)},P_+][\vartheta_a^{(2)},P_+]\cdots[\vartheta_a^{(2n+1)},P_+]
\begin{array}{l}{{3 ( 3 , 2 )_{\frac{1}{6}}+ 3 ( \overline{{{3}}}, 1 )_{\frac{1}{3}}+ 3 ( \overline{{{3}}}, 1 )_{- \frac{2}{3}}+ 1 2 ( 1 , 2 )_{- \frac{1}{2}}+ 9 ( 1 , 2 )_{\frac{1}{2}}}}\\{{\ \ \ + 1 2 ( 1 , 1 )_{1}+ 9 ( 1 , 1 )_{0}+ 9 ( 1 , 1 )_{- 1}.}}\\ \end{array}
\big\langle \psi_0 \big(\cdot,( 1 + \epsilon_{R(u)}) u \phi \big), \phi^* \big\rangle_m = \big\langle \kappa (x)\big( 1 + \epsilon_{R(u)}(x)\big )^{\gamma(x)}u^{\gamma(x)}\phi(x)^{\gamma(x)}, \phi^*(x) \big\rangle_{m(dx)}\stackrel[u\to 0]{}{\sim}\langle u^{\gamma(x)}, \kappa (x)\phi(x)^{\gamma(x)}\phi^*(x) \rangle_{m(dx)}.
~[(I+R_k)1](x) = ~1+i\int_{-\infty}^x e^{ik(x-y)}e^{i\int_y^x u(t)~dt}u(y)~dy = ~ 1-\frac{u(x)}{k}+\frac1k\int_{-\infty}^x e^{ik(x-y)}\left(e^{i\int_y^x \int u(t)~dt}u(y)\right)_y~dy = ~1-\frac{u(x)}{k}+O\left(\frac1{|k|^2}\right).
M^2 = - g^4 C_2(R) C(R') \int_q \int_{k55}{N(k,k^5,{\hat k}^5,q)\over (k^2)(k^2 - (k^5)^2)(k^2 -({\hat k}^5)^2) (q^2-m_1^2)((q-k)^2-m_2^2)}
G_{D}^{(n)}({\bf r''},{\bf r'}; E) =\sigma^{n}\,\left[G_{D}^{(0)}({\bf 0},{\bf 0}; E) \right]^{n-1}\,G_{D}^{(0)}({\bf r}'',{\bf 0}; E) \,G_{D}^{(0)}({\bf 0},{\bf r}' ; E) \; ,
\int_{S^2}\alpha_H^{(2)}( \tilde \nabla \tilde X^i) = 0 \int_{S^2}\alpha_H^{(2)}( \epsilon_{pqi}\tilde X^q \tilde \nabla \tilde X^i) = 0 \int_{S^2}\alpha_H^{(3)}( \epsilon_{pqi}\tilde X^q \tilde \nabla \tilde X^i) = 0.
\varphi^\sharp\left(\begin{array}{c c}a b 0 d \end{array}\right) = \left(\begin{array}{c c c}a^{2 \, p^{e_1}}0 b^{2 \, p^{e_1}}0 1 b^{p^{e_1}}\, d^{p^{e_1}}0 0 d^{2 \, p^{e_1}}\end{array}\right) (\, e_1 \geq 0 \, ) .
f=\left(\begin{array}{c}f_{0}f_{1}f_{2}\end{array}\right) = \left(\begin{array}{c}a_{0}x_{0}y_{0}+a_{1}x_{2}y_{1}+a_{2}x_{1}y_{2}a_{1}x_{1}y_{0}+a_{2}x_{0}y_{1}+a_{0}x_{2}y_{2}a_{2}x_{2}y_{0}+a_{0}x_{1}y_{1}+a_{1}x_{0}y_{2}\end{array}\right)
\left[ \frac{\partial^{2}}{\partial \xi^{2}}+ 2 \kappa \delta \frac{s}{c}\frac{\partial}{\partial \xi}- \left\{\frac{1}{\kappa^{2}}- \kappa^{2}\delta ( 1 + \delta ) \right\}\frac{s^{2}}{c^{2}}+ ( \varepsilon + \kappa^{2}\delta ) \right] \Psi ( \xi ) = 0 ,
|h_1(r,t)| \leq \| d_0 F^{-1}\| (|\xi| + |g(t))|+|R_1(r,t)|) \leq C_2 ( |\xi| +|t| + \sigma (|r|+|t|)) \leq C_2 (\delta + 2\lambda_0 \varepsilon +\sigma \varepsilon ).
\begin{cases}\displaystyle\frac{\partial G}{\partial{\bf u}_a}({\bf u})-\sum\limits_{d=1}^a\sigma_{da}({\bf u}){\bf u}_d-\sum\limits_{d=a+1}^p\sigma_{ad}({\bf u}){\bf u}_d={\bf 0},\,\,\,\forall a\in \{1,...,p-1\}\displaystyle\frac{\partial G}{\partial{\bf u}_p}({\bf u})-\sum\limits_{d=1}^p\sigma_{dp}({\bf u}){\bf u}_d={\bf 0},\end{cases}
S_{\alpha}\geq \sum\limits_{k\in I_N}\frac{c_{\gamma+\alpha}^{2}}{c_{\gamma}^2}\geq\sum\limits_{k\in I_{N}}\frac{N-k+1}{N-k+\alpha_2+1}\int\limits_{0}^{1}\Phi_{2k+1,2(N-k)+2}(r)g_{\alpha}(r)dr\geq \sum\limits_{k\in I_N}\lambda_{\alpha}=|I_{N}|\lambda_{\alpha}.
F_{\mu \nu}= \partial_{\mu}( \hat{\phi}^{a}W_{\nu}^{a}) - \partial_{\nu}( \hat{\phi}^{a}W_{\mu}^{a}) -{\frac{1}{g}}\epsilon^{a b c}\hat{\phi}^{a}( \partial_{\mu}\hat{\phi}^{b}) ( \partial_{\nu}\hat{\phi}^{c})
F_N(p,q) := \frac{1}{2}\sum_{- N \leq k \leq N}\log\left( 1 + \left(\frac{p + k}{q}\right)^2 \right) - \frac{1}{2}\int_{- N - 1/2}^{N + 1/2}\log\left( 1 + \left( \frac{p + k + \phi}{q}\right)^2 \right) .\mathrm{d}\phi
\Xi_{[s]}(R_x)=R_{x^{(0)}}\otimes R_{\Lambda_x^{(-)}}\otimes R_{\Lambda_x^{(+)}}\quad\Xi_{[s']}(R_{x'})=R_{x'^{(0)}}\otimes R_{\Lambda_{x'}^{(-)}}\otimes R_{\Lambda_{x'}^{(+)}}.
\sum_{k}{\frac{1}{\left[ r^{2}+ ( \bar{x}- k L )^{2}\right]^{\frac{n}{2}}}}={\frac{1}{r^{n}}}\sum_{k}{\frac{1}{\left[ 1 + \left({\frac{\bar{x}}{r}}- k{\frac{L}{r}}\right)^{2}\right]^{\frac{n}{2}}}}
\Gamma_{NP}= \int{d^d k \over (2 \pi)^d}A_a(k) A_b(-k) \left( -{g^2\over 2 \pi^{d \over 2}}\Gamma({d \over 2}){\tilde{k}^a\tilde{k}^b \over |\tilde{k}|^d}\right)
|\partial_{r}\left(-\frac{1}{r}\int_{t}^{\infty}ds \int_{0}^{s-t}\frac{\rho d\rho}{\sqrt{(s-t)^{2}-\rho^{2}}}\lambda''(s) \partial_{3}F_{3}(r,\rho,\lambda(s))\lambda'(s)\right)| \leq \frac{C}{t^{3}\log^{1+b}(t)}
\bar{\mu}^{\dot{A}}= \left( X^{B\dot{A}}- i \Theta^{B}\bar{\Theta}^{\dot{A}}\right) \lambda_{B}+ 2\bar{z}^{\dot{A}\dot{B}}\bar{\lambda}_{\dot{B}}- i \bar{\Theta}^{\dot{A}}\bar{\Theta}^{\dot{B}}\bar{\lambda}_{\dot{B}}
\mathrm{d e t}^{\prime}\Delta^{( \beta , \alpha )}= \prod_{n_{1}, n_{2}}^{\prime}({\frac{4 \pi^{2}}{l^{2}}}) \left[ ( n_{1}+ ( \alpha - 1 ) / 2 )^{2}+{\frac{l^{2}}{4}}( n_{2}+ ( \beta - 1 ) / 2 )^{2}) \right] \quad .
M_{2k}= \sqrt{\frac{(k+\alpha+1)(k+\alpha+\beta+1)(N-k)}{(2k+\alpha+\beta+1)(2k+\alpha+\beta+2)}}, M_{2k+1}= \sqrt{\frac{(k+\beta+1)(k+\alpha+\beta+2+N)(k+1)}{(2k+\alpha+\beta+2)(2k+\alpha+\beta+3)}},
u(\mu x_1+(1-\mu)x_2,t)\le \max\{u(\lambda_1y_1+(1-\lambda_1)z_1,t), u(\lambda_2y_2+(1-\lambda_2)z_2,t)\}\le \, \max_{i=1,2}\max\{u(y_i,t),u(z_i,t)\}<h+\varepsilon.
\langle\alpha_1 \cdot ...\cdot \alpha_{n-1}\rangle \cdot \pi_* \widehat{\omega}\geq \langle\pi^*\alpha_1 \cdot ...\cdot \pi^*\alpha_{n-1}\cdot \widehat{\omega}\rangle &=\langle\pi^*\alpha_1 \cdot ...\cdot \pi^*\alpha_{n-1}\rangle \cdot \widehat{\omega}.
\int d^{1 0}\hat{x}\sqrt{| \hat{\jmath}|}\left[{\textstyle \frac{1}{2 \cdot 5 !}}\left( \hat{G}_{( 5 )}\right)^{2}+{\textstyle \frac{1}{2 \cdot 5 !}}\left( \hat{\overline{{G}}}_{( 5 )}\right)^{2}\right] .
(a)\sum_{k=1}^{n}c_{k,n-k}^{(n)}t^k \alpha^{n-k}= (t+ \alpha)^{n-1}, (b) c_{k,n-k-1}^{(n)}= (k+1) \binom{n-1}{k+2}, 0 \le k \le n-3, (c) c_{1,0}^{(n)}= c_{0,0}^{(n)}+ (-1)^{n-1}, n \ge 3.
\left\langle l_1, m_1 \right| e^{i k \phi}\left| l_2 , m_2\right\rangle=\sum\limits_{l=|l_1-l_2|}^{l_1+l_2}\frac{(-1)^{m_1}}{2}\sqrt{\frac{(2 l_1+1)(2 l_2+1)(l+k)!}{(l-k)!}}
\langle{\vec{u}}( \vec{x}) .{\vec{u}}( 0 ) \rangle = \sum_{i = j}^{3}i S_{i j}^{F}({\vec{x}}-{\vec{x^{\prime}}}) = \int{\frac{\mathrm{d}^{3}{\vec{k}}}{( 2 \pi )^{3}}}{{\frac{e^{i{\vec{k}}. ({\vec{x}}-{\vec{x^{\prime}})}}}{\omega_{k}}}}
\overline{\overline{C}}_{X^{\ast}}(g_1, g_2) =\overline{\overline{C}}_{X^{\ast}}(\omega, \omega^{-1}g_1)^{-1}\overline{\overline{C}}_{X^{\ast}}(\omega, \omega^{-1}g_1 g_2)\overline{\overline{C}}_{X^{\ast}}(\omega^{-1}g_1, g_2).
S_{[ 1 ]}^{c}\, = \, e^{i \vartheta_{[ 1 ]}^{c}}\, \left( \begin{array}{c c}{0}{\Theta_{[ 1 ]}}{- \, \Theta_{[ 1 ]}}{0}\end{array}\right) \,{\cal K}\, \equiv \,{\cal C}_{[ 1 ]}\,{\cal K}\quad .
{\frac{d^{2}f ( r )}{d r^{2}}}- \left[{\frac{l ( l + 1 ) + 2 G Q^{2}E^{2}- G^{2}s^{2}}{r^{2}}}-{\frac{2 G s E}{r}}- E^{2}\right] f ( r ) ~ = ~ 0 ~ ~ .
-\sum_{j=0}^{\infty}\frac{1}{j!}(8\pi iK)^{-j}\left(\frac{P}{H}\right)e^{2\pi ic_{0}}\partial_{\epsilon}^{(2j)}\left.\left[e^{2\pi i\epsilon\sum_{i=1}^{3}\frac{\mu_{i}}{p_{i}}}\left(\frac{1}{2i\sin 2\pi\epsilon}-\frac{1}{4\pi i\epsilon}\right)\right]\right|_{\epsilon=0}.
\xi_q^{\frac{q+1}2kp}\left(\cot\frac{k(p+1)}{q}\pi-\cot\frac{k(p-1)}{q}\pi\right)+\xi_q^{\frac{q+1}2kp^*}\left(\cot\frac{k(p^*+1)}{q}\pi-\cot\frac{k(p^*-1)}{q}\pi\right) = \xi_q^{\frac{q+1}2ks}\left(\cot\frac{k(s+1)}{q}\pi-\cot\frac{k(s-1)}{q}\pi\right)+\xi_q^{\frac{q+1}2ks^*}\left(\cot\frac{k(s^*+1)}{q}\pi-\cot\frac{k(s^*-1)}{q}\pi\right)
\min_{\mathfrak{E}}s_g= 16\pi\left(1+\sum ab\right)\left(-1+4\sum a-6\sum a^2+4\sum a^3-\sum a^4+4\sum a^3b-3\sum a^2b^2\right)\Big/ begin{align*}0.1cm] \left(1-6\sum a+15\sum a^2+\sum ab-20\sum a^3+4\sum a^2b+15\sum a^4+4\sum a^3b-3\sum a^2b^2\right. begin{align*}0.1cm] \left.-6\sum a^5+2\sum a^4b+4\sum a^3b^2+\sum a^6-6\sum a^5b+15\sum a^4b^2-10\sum a^3b^3\right)
S_{E G H}^{b o u n d}= -{\frac{1}{3 2 \pi^{2}}}\int_{\partial M}\epsilon_{a b c d}\; ( 2 \theta^{a b}\wedge R^{c d}-{\frac{4}{3}}\; \theta^{a b}\wedge \theta^{a}{}_{e}\wedge \theta^{e b})
\|2\rho_{n_{j}}-\langle{\bf k}^{*}_{n_{j}},\alpha \rangle\|_{\mathbb{T}}= \|2\rho -\langle{\bf d}_j,\alpha \rangle\|_{\mathbb{T}}& \geq \min\{\kappa,\gamma\}(1+|2{\bf d}_j|_{\eta})^{-C_{1}|2{\bf d}_j|_{\eta}^{\frac{1}{\eta+1}}}.
D_n(z) =p_n(z)+nq_n(z) &=z^{16}\! -\! 2z^{15}\! +\! z^{14}\! -\! z^{13}\! +\! z^{12}\! -\! z^{10}\! +\! 2z^9\! -\! 2z^8\! +\! 2z^7\! -\! z^6\! +\! z^4\! -\! z^3\! +\! z^2\! -\! 2z\! +\! 1 &\qquad+nz(-2z^{14}+z^{12}+z^{10}+z^9+2z^7+z^5+z^4+z^2-2),
\mathrm{length}(\mathcal{O}_Z)=\chi(L)-\chi(L\otimes I_Z)=\chi(L)-\chi(T_X)+\chi(\mathcal{O}_X(\Delta))= \chi(\mathcal{O}_X(\Delta))+\chi(\omega_X^{-1}\otimes \mathcal{O}_X(-\Delta))-\chi(T_X).
\mathfrak{S}(\{0,d\})=\begin{cases}2\prod\limits_{p>2}\left(1-\frac{1}{(p-1)^2}\right)\prod\limits_{p|d,p>2}\left(\frac{p-1}{p-2}\right) , \qquad\qquad0\qquad\qquad\qquad .\end{cases}
g_{1}(\omega V,\mathcal{T}_{U}\mathcal{B}X)+g_{1}(V,\phi U)\eta (X) =g_{2}((\nabla \pi_*)(U,\mathcal{C}X),\pi_*\omega V)-g_{2}((\nabla \pi_*)(U,X),\pi_*\omega\phi V)
g = - \kappa^{\frac{4}{3}}\left( \begin{matrix}{{- 2 ( \psi^{2}-{\frac{f}{\kappa^{2}}})}}&{{0}}&{{-{\frac{1}{\sqrt{2}}}\psi}}\\{{0}}&{{{\frac{2}{f \kappa^{2}}}}}&{{0}}\\{{-{\frac{1}{\sqrt{2}}}\psi}}&{{0}}&{{-{\frac{1}{4}}}}\\ \end{matrix}\right) .
\left( \Gamma^{M}\right)_{\alpha \beta}\left( \tilde{\Gamma}_{M}\right)^{\gamma \delta}= 2 \delta_{( \alpha}^{\gamma}\delta_{\beta )}^{\delta}+ 2 \left( \Lambda_{j}\right)_{( \alpha}^{\gamma}\left( \Lambda_{j}\right)_{\beta )}^{\delta}.
{}^{M\!L\!\!}{B}_{p}(x,y) = \sum_{n=0}^{\infty}\int_{0}^{1}t^{x+n-1}(1-t)^{y}E_{\alpha, \beta}^{\gamma}\left(-\frac{p}{t(1-t)}\right) dt =\sum_{n=0}^{\infty}{}^{M\!L\!\!}{B}_{p}(x+n,y+1)
\log BF_{12}=\log \frac{\{| \mathcal{M}_{1}\}}{\{| \mathcal{M}_{2}\}}+ \log \frac{\{| \mathcal{M}_{1}\}}{\{| \mathcal{M}_{2}\}}-\log \frac{\{| \mathcal{M}_{1}\}}{\{| \mathcal{M}_{2}\}}.
\left[ G_{\mathrm{r a d}}^{a}( \vec{x}) , G_{\mathrm{r a d}}^{b}( \vec{y}) \right] = i \epsilon^{a b c}G_{\mathrm{r a d}}^{c}( \vec{x}) \delta^{( 3 )}( \vec{x}- \vec{y}) \ .
(x)_n=(x;q)_n:=\prod_{i=0}^{n-1}(1-q^ix), \ \ (x)_{\infty}=(x;q)_{\infty}:=\prod_{i\ge 0}(1-q^ix),{}\ \j(x;q):=(x)_{\infty}(q/x)_{\infty}(q)_{\infty}=\sum_{n=-\infty}^{\infty}(-1)^nq^{\binom{n}{2}}x^n,
\langle \psi ({\bf{x}}^{'}, t^{'}) \psi^{\dagger}({\bf{x}}, t ) \rangle = \rho_{0}e^{- \sum_{{\bf{k}},{\bf{q}}\neq 0}f_{{\bf{k}},{\bf{q}}}^{*}({\bf{x}}^{'}) f_{{\bf{k}},{\bf{q}}}({\bf{x}}^{'})}
\Psi ( \theta ) = \left( \begin{array}{l r}{{\psi_{n , m - 1}( \theta )}}\\{{\psi_{n , m}( \theta )}}\\{{\psi_{n , m - 1}( \theta )}}\\{{\psi_{n , m}( \theta )}}\\{{\, \, \, \, \, \, \, \vdots}}\\{{\psi_{n , m - 1}( \theta )}}\\ \end{array}\right)_{( 2 k + 1 ) \times 1}
A_0 (s) = \frac{z^2}{\pi R}\frac{1}{\left(s+\frac12\right)}\left( 1 +\frac1{12 z^2}\right) - \frac{z^2}{\pi R}\left[ 1 +\frac1{6z^2}+ 2 \left( 1 + \frac1{12 z^2}\right) \log \left(\frac{z}{2 R}\right) \right.
\zeta(z,q)=\frac{1}{2q^{z}}+\frac{q^{1-z}}{z-1}+2\int^{\infty}_{0}(q^{2}+y^{2})^{\frac{-z}{2}}\sin(z\arctan\frac{y}{q})\frac{1}{e^{2\pi y}-1}dy.
\alpha_0=\frac{1}{r^{p}}, &\beta_0=y^p, &\alpha_{i+1}=\alpha_i(2-\alpha_i r^{\sigma}(x^p)) \pmod{\; p^{2^{i+1}}}, \beta_{i+1}= \beta_i - \mathcal{Q}^{\sigma}(x^p,\beta_i) s^{\sigma}(x^p, \beta_i) \alpha_i \pmod{\; p^{2^{i+1}}}.
\underline{\Theta}(u,\mathcal{O})(x)\,:=\,\inf_{M\in\mathbb{R}}\left\lbrace\exists \,P_M\,|\,P_M(x)=u(x)\;\;\mbox{and}\;\;P_M(y)\leq u(y),\;\;\forall y\in \mathcal{O}\right\rbrace.
h(\lambda) = B\left(\lambda-\frac{1}{\lambda}\right) +\int\left( \frac{1}{t-\lambda}-\frac{1}{t-1/\lambda}\right)\, d\sigma(t) & = \left(\lambda-\frac{1}{\lambda}\right) \left( 1-\sigma_{-2}+ \int \frac{d\sigma(t)}{(t-\lambda)(t-1/\lambda)}\right ) .
\int \tilde{Q}( X ) = \int e^{- K}\left[ Q \left( e^{K}( X ) \right) \right] \star \mathcal{I}= \int Q \left( e^{K}( X ) \right) \star e^{K}( \mathcal{I}) = \int Q \left( e^{K}( X ) \right) = 0 \ ,
{\cal L}= \int_{0}^{2 \pi R}\! \! \! d y \, \left[ - \frac{1}{4 g^{2}}F^{M N}F_{M N}+ \bar{\psi}\, i \Gamma^{M}\big ( \partial_{M}- i A_{M}\big ) \psi - M \eta ( y ) \bar{\psi}\psi \right] \; .
\chi_{\nu}= - \frac{i}{2}\sum_{\alpha , \beta}\left( \operatorname{l o g}{( \frac{z - z_{\alpha}}{\overline{{z}}- \overline{{z_{\alpha}}}})}- \operatorname{l o g}{( \frac{z - z_{\beta}}{\overline{{z}}- \overline{{z_{\beta}}}})}\right)
y\in KerP\to 0=\left[ \Gamma \left( \Gamma^{+}\Gamma \right)^{-1}\Gamma^{+}y,x \right]=\left( \left( \Gamma^{+}\Gamma \right)^{-1}\Gamma^{+}y,\Gamma^{+}x \right),\forall \Gamma^{+}x\in H.
\psi_{\theta}( 1 ) = e^{- i \Delta_{1}}e^{i p^{k}x_{1}^{k}}K_{\theta}( 2 , 1 ) = e^{- i \Delta_{2}^{\prime}}e^{i p^{k \prime}x_{2}^{k}}e^{i \Delta_{1}^{\prime}}e^{- i p^{k \prime}x_{1}^{k}}.
N_{k}^{( p , 0 )}= \int_{0}^{1}d t_{1}\int_{0}^{t_{1}}d t_{2}\ldots \int_{0}^{t_{p - k - 1}}d t_{p - k}\int_{0}^{1}d s_{1}\int_{0}^{s_{1}}d s_{2}\ldots \int_{0}^{s_{k - 1}}d s_{k}\times
\begin{aligned}\sum_{2\leq n \leq x}\frac{\Lambda_{{\rm sym}^m f}(n)}{n\log n}= \log L(1,{\rm sym}^m f)+O\left(\frac{\log (kNTx)}{T}+\frac{\log (kN)\log T}{x^{\eta/4}}\right) & = \log L(1,{\rm sym}^m f)+O\big((\log (kN))^{-4/\eta+1}\big).\end{aligned}
\begin{array}{r r}{( 8 , 1 , 1 ; 1 , 1 , 8 ) ,}{({\bar{8}}, 1 , 1 ; 1 , 8 , 1 )}{( 1 , 8 , 1 ;{\bar{8}}, 1 , 1 ) ,}{( 1 , 1 , 8 ; 8 , 1 , 1 )}\end{array}
\bar{u}(t) =\bar{u}_{0}+\int_{0}^{t}\int_{\Omega}g(u)\geq\bar{u}_{0}+\int_{0}^{t}\int_{\Omega}\left( g(u(s))-g(u(t^{\prime}))\right) ds & \geq\bar{u}_{0}-L\int_{0}^{t}\int_{\Omega}u(s)ds=\bar{u}_{0}-L\int_{0}^{t}\bar{u}(s)ds\quad\forall t\in [0,T]
I_{\rho^{2}}=I_{\frac{\rho^{2}}{\mu^{2}}\mu^{2}}<\frac{\rho^{2}}{\mu^{2}}I_{\mu^{2}}&=\frac{\rho^{2}-\mu^{2}+\mu^{2}}{\mu^{2}}I_{\mu^{2}}&=\frac{\rho^{2}-\mu^{2}}{\mu^{2}}I_{\frac{\mu^{2}}{\rho^{2}-\mu^{2}}\rho^{2}-\mu^{2}}+I_{\mu^{2}}\le I_{\mu^{2}}+I_{\rho^{2}-\mu^{2}}.
\varphi_*f_*\mu_X((-\infty,t]) =\tilde G_*F_*f_*\mu_X((-\infty,t]) &=F_*f_*\mu_X(\tilde G^{-1}((-\infty,t])) &=F_*f_*\mu_X(\tilde G^{-1}((-\infty,t])\setminus\{0\}) &=F_*f_*\mu_X((0,G(t)]) &=F_*f_*\mu_X((-\infty,G(t)]) &=G(t) &=g_*\mu_X((-\infty,t]).
O(T,n):=C \left( \int_0^{1/T}\frac{\frac{1}{T}-s}{(1-t_n-s)^{3/2}}ds+ \sqrt T \int_0^{1/T}\frac{\frac{1}{T}-s}{1-t_n-s}ds +\frac{1}{T^{3/2}(1-t_n)}\right).
\Phi_{0,classical}^X(y_0, \ldots, y_m) = \sum_{\substack{k_0, \ldots, k_m\geq 0 k_0 + \cdots + k_m = 3}}\langle\gamma_0^{k_0}, \ldots, \gamma_m^{k_m}\rangle_{0,0}^X \frac{y_0^{k_0}}{k_0!}\cdots \frac{y_m^{k_m}}{k_m!}
\langle\Psi_{1}^{\prime I_{1}}(z_{1})\Psi_{2}^{\prime I_{2}}(z_{2}) \cdots\Psi_{n}^{\prime I_{n}}(z_{n})\rangle=\langle \Psi_{1}^{I_{1}}(z_{1})\Psi_{2}^{I_{2}}(z_{2})\cdots \Psi_{n}^{I_{n}}(z_{n})\rangle\,.
FE \; + \; \frac{1}{(q-q^{-1})^2}\big( qK-q^{-1}\l^{-1}K')(K-\l K') \; =\; F\big(E-\kappa s(K-\l^{-1}K')\big) \; + \; \kappa \big(qK-q^{-1}\l^{-1}K'\big)\big(sF+\kappa (K-\l K')\big),
\int_S f_{\mathcal{A'}(D)}d \mu =\mathbb{P}(\mathcal{A'}(D) \in S) \&\leq \exp \left(\epsilon d(D,D') \right)\mathbb{P}(\mathcal{A'}(D') \in S) &= \exp \left(\epsilon d(D,D') \right) \int_Sf_{\mathcal{A}'(D')}d \mu,
1 / 6 ( A A_{s y m})^{3}( Q{\bar{Q}})^{2}+ 4 ( A A_{s y m}) ( A A Q{\bar{Q}}_{a n t i})^{2}+ 6 4 ( A A A{\bar{Q}}{\bar{Q}}) ( A A A Q Q ) = \Lambda^{8}( Q{\bar{Q}})
E = -{\frac{1}{4 \pi a}}\int_{- \infty}^{\infty}d y \, e^{i y \delta}\sum_{l = 1}^{\infty}( 2 l + 1 ) x{\frac{d}{d x}}\ln [ 1 - \xi^{2}( ( s_{l}e_{l})^{\prime})^{2}] ,
H_{C T M}= - 2 \mathrm{i}\sum_{j = 1}^{\infty}j \left\{( 1 + \Gamma ) \tau_{j}^{y}\tau_{j + 1}^{x}- ( 1 - \Gamma ) \tau_{j}^{x}\tau_{j + 1}^{y}+ h ( \tau_{j}^{x}\tau_{j}^{y}+ \tau_{j + 1}^{x}\tau_{j + 1}^{y}) \right\}\nonumber
(\alpha_0 - \beta_0) \cdot \alpha_{-2}= -\frac{1}{2 \cdot 3}(a_1 - a_{-1}) + \frac{23t - 2}{3^2}(a_2 + a_{-2}) + \frac{2}{3^2}(a_3 + a_{-3}) + \frac{t}{2 \cdot 3}v_{(1,2)}-\frac{1}{3}v_{(2,3)}&-\frac{2^3}{3^2}(a_1 + a_{-1}) \cdot v_{(2,3)}+ \frac{20}{3^2}(a_2 + a_{-2}) \cdot v_{(1,3)}+ \frac{2^3}{3^2}(a_3 + a_{-3}) \cdot v_{(1,2)}
\Upsilon(g'g^{-1})(t) = (\frac{1}{2}g'(t/2)g(t/2)^{-1}, - \frac{1}{2}g'(1 - t/2)g(1 -t/2)^{-1}) &=(\tilde{g}_1'(t) \tilde{g}_1(t)^{-1}, \tilde{g}_2'(t) \tilde{g}_2(t)^{-1})=(\tilde{g}_1, \tilde{g}_2)' (\tilde{g}_1, \tilde{g}_2)^{-1}=\Omega(g)' \Omega(g)^{-1}(t).
E(\mbox{\mathversion{bold}$\varphi$}(t)) \geq \| \mbox{\mathversion{bold}$\varphi$}_{s}(t)\|^{2}-\frac{1}{R^{2}}\left(\frac{L}{\pi}\right)^{2}\|\mbox{\mathversion{bold}$\varphi$}_{s}(t)\|^{2}=c_{0}\|\mbox{\mathversion{bold}$\varphi$}_{s}(t)\|^{2}
\left(\frac{\dot a_0}{a_0}\right)^2+\frac{K}{a_0^2}=\frac{\rho_{0M}+\rho_{\frac12M}e^{2kb}-\rho_{0M}\bar\rho_{\frac12M}e^{kb}\sinh(kb)}{3(M^3/k)\left[e^{kb}-1-\bar\rho_{\frac12M}e^{kb}(\cosh(kb)-1)\right]}.
D\xi=\sum_{\alpha=1}^{k}D(e_{\alpha}\xi^{\alpha}(e))=\sum_{\alpha=1}^{k}e_{\alpha}\otimes d\xi^{\alpha}(e)+\sum_{\alpha=1}^{k}\sum_{\beta=1}^{k}e_{\beta}\otimes\omega_{\alpha}^{\beta}\xi^{\alpha}(e).
\Theta^{ij}= 2\pi \alpha' \left (\begin{array}{cccc}0 - \frac{cy^+}{R \sqrt{c^2 + g_s^2 d^2}}0 0 \frac{cy^+}{R \sqrt{c^2 + g_s^2 d^2}}0 0 0 0 0 0 - \frac{g_s d}{R \sqrt{c^2 + g_s^2 d^2}}0 0 \frac{g_s d}{R \sqrt{c^2 + g_s^2 d^2}}0\end{array}\right ),
{\cal L}_P=-\frac12 G^{\dagger}_{\mu\nu}G^{\mu\nu}- M^2 \phi^{\dagger}_{\mu}\phi^\mu + \frac{\theta}{2}\varepsilon_{\mu\nu\rho}A^\mu \partial^\nu A^\rho +\frac{\lambda}{2}(\partial_\mu A^\mu)^2\; ,
\big | \Psi_{2 , 1}\big > = \big ( ( x \cdot x ) -{\frac{Q_{N}q^{2}}{\omega ( 1 + q^{2})}}\big ) \mathrm{e x p}_{q^{2}}\bigg [{\frac{- \omega ( x \cdot x )}{q^{N + 2}\mu}}\bigg ] \ .
\mathrm{Var}\{\mathbf{z}^H\mathbf{A}\mathbf{z}\}=\sum_{i=1}^M \frac{\lambda_i^2(\mathbf{A})}{4}\mathrm{Var}\{\chi_{i}^2(2)\}&=\sum_{i=1}^M \frac{\lambda_i^2(\mathbf{A})}{4}\left(\mathrm{Var}\{z_{i,1}^2\}+\mathrm{Var}\{z_{i,2}^2\}\right) &\overset{(u)}{=}\sum_{i=1}^M \lambda_i^2(\mathbf{A}) &\overset{(t)}{=}\mathrm{tr}(\mathbf{A}^2)\;,
m+1-n_{1}-n_{i}=\underbrace{n_{1}+\cdots+n_{1}}_{M_{1}-1\;\mathrm{times}}+\cdots+\underbrace{n_{i}+\cdots+n_{i}}_{M_{i}-1\;\mathrm{times}}+\cdots+\underbrace{n_{\ell}+\cdots+ n_{\ell}}_{M_{\ell}\;\mathrm{times}},
\tilde{a}(\delta) := d^{m/2}\sup_{x:\|x\| \leq g(\delta)}\max\biggl\{\max_{r=1,\ldots,m}q_r(\|x\|) \, , \, d^{1/2}e^{\|x\|}q_{m+1}(\|x\|+1)\biggr\}&= d^{m/2}\max\biggl\{\max_{r=1,\ldots,m}q_r\bigl(g(\delta)\bigr) \, , \, d^{1/2}e^{g(\delta)}q_{m+1}\bigl(g(\delta)+1\bigr)\biggr\}.
\arrowvert B_{1}\cup B_{2}\cup \dots \cup B_{k}\arrowvert = \sum_{m=1}^{k}(-1)^{m+1}\sum_{1\leq i_1 < i_2 < \dots <i_m \leq k}\arrowvert B_{i_1}\cap B_{i_2}\cap \dots \cap B_{i_m}\arrowvert\,.
\partial_{r}\left((1+r)^{2\gamma+1}r^{2}\psi^{2}\right) =(2\gamma+1)(1+r)^{2\gamma}r^{2}\psi^{2}&+2(1+r)^{2\gamma+1}r\psi^{2}+2(1+r)^{2\gamma+1}r^{2}\psi\partial_{r}\psi.
p_{ij}(x_i^{*},y_j^{*})=\frac{\bar{u}_{ij}^n-\sum_{\beta=1}^L\omega_{\beta}\hat{\omega}_1[\mu_1(u_{i+\frac{1}{2},\beta}^{-}+u_{i-\frac{1}{2},\beta}^{+})+\mu_2(u_{\beta,j+\frac{1}{2}}^{-}+u_{\beta,j-\frac{1}{2}}^{+})]}{1-2\hat{\omega}_1}.
\frac{\Gamma_2(z\,|\,a_1,a_2)}{\Gamma_2(z+a_1\,|\,a_1,a_2)}= \frac{a_2^{z/a_2-1/2}}{\sqrt{2\pi}}\Gamma\bigl(\frac{z}{a_2}\bigr), \frac{\Gamma_2(z\,|\,a_1,a_2)}{\Gamma_2(z+a_2\,|\,a_1,a_2)}= \frac{a_1^{z/a_1-1/2}}{\sqrt{2\pi}}\Gamma\bigl(\frac{z}{a_1}\bigr),
H_{\tau_1^*}^{\tau_2^*}=H_{\tau_1^*}\cap \mathcal{D}_{\tau_1}^{\tau_2},\quad\underline{H}_{-\tau_2^*}^{\tau_1^*}=\underline{H}_{-\tau_2^*}\cap \mathcal{D}_{\tau_1^*}^{\tau_2^*}.
B_2=\left\{\left.\frac{18+q+2\ell}{2(40+q)}\right|0\leq\ell<40+q,\ell\neq 11\right\}, B_{22}=\left\{\frac{20+q}{2(40+q)},\frac{22+q}{2(40+q)},\frac{24+q}{2(40+q)},\frac{30+q}{2(40+q)},\frac{32+q}{2(40+q)}\right\}.
X=\begin{pmatrix}(g^1)^T (g^2)^T \end{pmatrix}\begin{pmatrix}g^1 g^2 \end{pmatrix}=\begin{pmatrix}\|g^1\|^1 \langle g^1,g^2\rangle \langle g^1,g^2\rangle \|g^2\|^2 \end{pmatrix},
H = - M + \frac{1}{2 M}\left( P - \sum_{n , m = 1}^{\infty}G_{m n}p_{n}q_{m}\right)^{2}+ \sum_{n = 1}^{\infty}\left( \frac{p_{n}^{2}}{2}+ \frac{\Omega_{n}^{2}q_{n}^{2}}{2}\right)
\hat{\Gamma}_{(X_+,D_+)}:=\bigoplus_{\vec d\in (\mathbb Z)^{|I_+|},f\in \mathbb K_{+}/\mathbb L}\prod_{i\in M_0}\Gamma(1+\bar D_i-\langle D_i\cdot f\rangle)\prod_{i\in I_+,d_i<0}\frac{1}{\bar D_i-d_i}\textbf{1}_{f}[\textbf 1]_{(d_i)_{i\in I_+}}.
E=\left(\begin{array}{ccc}\partial_z\varphi 0 -\partial_1\varphi 0 \partial_z\varphi -\partial_2\varphi -\partial_1\varphi -\partial_2\varphi \frac{1+|\nabla_y\varphi|^2}{\partial_z\varphi}\end{array}\right).
\partial_{c}(e_{a_{0}}e_{a_{1}}\cdots e_{a_{k}}e_{a_{k+1}})=\sum_{r\in\{\pm1\}}r\sum_{i=1}^{k}{\rm ord}_{z-c}(a_{i}-a_{i+r})e_{a_{0}}\cdots\widehat{e_{a_{i}}}\cdots e_{a_{k+1}}.
Z_n(\lambda,T) := \lim_{m \to \infty,}Z_{m,n}\left( 1- \frac{\lambda}{m}, 1 , \frac{T}{m}\right) = \frac{e^{-\lambda}}{2}\sum_{\theta \in \{0,1\}^2}\prod_{w^n = (-1)^{\theta_2}}\left| e^{Tw + \lambda}- (-1)^{\theta_1}\right|.
\tilde{\Xi}_\pm A\varphi =\tilde{\Xi}_\pm\gamma^{(2n+1)}(P_{\rm F}-D_aP_{\rm F}D_a)\varphi &=-\gamma^{(2n+1)}\tilde{\Xi}_\pm (P_{\rm F}-D_aP_{\rm F}D_a)\varphi &=\gamma^{(2n+1)}(P_{\rm F}-D_aP_{\rm F}D_a)\tilde{\Xi}_\pm \varphi=A\tilde{\Xi}_\pm \varphi
\sin{\angle{mi(x)m'}}=\frac{\sinh{d(m,m')}}{\sinh{d(i(x),m)}}\leq \frac{\frac{\delta}{100}\cosh{d(i(\bar{x}),m)}+\frac{\delta}{100}\sinh{d(i(\bar{x}),m)}}{\sinh{(d(i(\bar{x}),m)-\frac{\delta}{250})}}\leq \frac{\delta}{20}.
S_{e f f}={\frac{1}{1 6 \pi}}\int d^{4}x \sqrt{- g}( - R +{\frac{1}{2}}{\frac{\partial_{\mu}\lambda \partial^{\mu}\bar{\lambda}}{( \mathrm{I m}\lambda )^{2}}}- \sum_{n = 1}^{N}{F_{\mu \nu}^{( n )}{}^{\star}\tilde{F}^{( n )}{}^{\mu \nu}}) ,
\mathbb{E}[Z_{xy}Z_{uv}] \leq p^2\Pr(\tau^{\kappa-1}(x,y)=\tau^{\kappa-1}(u,v)=0) &\leq p^2\exp[-(1-O(n^{\kappa-3}p^{\kappa-2}))2\mu]=p^2\exp[-2\mu+o(1)].
m_i(\mu,T,1) = k_{i+1}-\delta_\nwarrow(\mu_{i+1})+c_i(T) = k_{i+1}+\delta_\nearrow(\mu'_{i+1})+c_{i+1}(T') = m_{i+1}(\mu',T',0), m_{i+1}(\mu,T,0) = k_{i+1}+\delta_\nearrow(\mu_{i+1})+c_{i+1}(T) = k_{i+1}-\delta_\nwarrow(\mu'_{i+1})+c_{i}(T') = m_{i}(\mu',T',1),