harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
({\rm vol}(\alpha \wedge v_1\wedge v_4\wedge v_5)v_0+v_1)\wedge (v_0\wedge \alpha + v_1\wedge v_2 \wedge v_3) &=v_1\wedge v_0 \wedge \alpha +vol(\alpha \wedge v_1\wedge v_4\wedge v_5) v_0\wedge v_1\wedge v_2 \wedge v_3.
G_{i_1i_2i_3}^{j_1j_2j_3}=\delta^{j_1k_1}\delta^{j_2k_2}\delta^{j_3k_3}\int_0^\infty (e^{-y\tilde \gamma})_{i_1k_1}(e^{-y\tilde \gamma})_{i_2k_2}(e^{-y\tilde \gamma})_{i_3k_3}dy.
\tau ={\pi_0^r (C_{10}-C_{00}) \over \pi_1^r (C_{01}-C_{11})}\,, k_0 =\pi_0^t\sqrt{\pi_1^r\over\pi_0^r}\sqrt{(C_{10}-C_{00})(C_{01}-C_{11})}\,, k_1 =\pi_1^t\sqrt{\pi_0^r\over\pi_1^r}\sqrt{(C_{10}-C_{00})(C_{01}-C_{11})}\,.
-\frac{S_{E}}{\hbar}-W= -\frac{\pi q}{\hbar B}-\frac{2\pi}{qB}\, \biggl\{\sum\frac{a_m}{(q{\cal M})^{2m}}\biggr\}+\cdots =-\frac{\pi q}{\hbar B}\biggl(1+\sigma \frac{\hbar}{q^2}\biggr)+\cdots.
p_1 = \frac{a t_{-d}\left( 1 + \frac{{\rm sgn}(a)}{\Gamma \left( \theta^+_{\chi}\right) \cos \left( \theta^+_{\chi}\right)}\right)}{\tilde{a}c t_d \left( 1 + \frac{1}{\tan \left( \theta^+_{\chi}\right)}\right)}\;,
p_{1}\, = \, - \, \frac{\partial L}{\partial \dot{x}}\, - \, \dot{p}_{2}, \quad p_{2}\, = \, - \, \frac{\partial L}{\partial \ddot{x}}\, - \, \dot{p}_{3}, \quad p_{3}\, = \, - \, \frac{\partial L}{\partial \stackrel{\ldots}{x}}\,{,}
\frac{16}{(\lambda_{0}(t)+e_{1}(t))^{2}}\int_{t}^{\infty}dx (\lambda_{0}''(x)+e_{1}''(x)) K(x-t,\lambda_{0}(t)+e_{1}(t)) &-\frac{16}{(\lambda_{0}(t)+e_{2}(t))^{2}}\int_{t}^{\infty}dx (\lambda_{0}''(x)+e_{2}''(x)) K(x-t,\lambda_{0}(t)+e_{2}(t))
d_n^{2s+4-i}a_{n,i,k}= \sum ~~ \frac{d_n^{h}H^{(h)}(t)}{h!}\cdot \prod_{u=1}^{s+2}\frac{d_n^{f_{u}}F_{1/4}^{(f_{u})}(t)}{f_{u}!}\cdot \prod_{u=1}^{s+2}\frac{d_n^{f_{u}^{\prime}}F_{3/4}^{(f_{u}^{\prime})}(t)}{f_{u}^{\prime}!}\cdot \prod_{u=1}^{2s+4}\frac{d_n^{g_u}\left((t+k)G(t)\right)^{(g_u)}}{g_u !}\big|_{t=-k}\in \mathbb{Z},
{\cal B}^{a}(x) = \left( \begin{array}{c}C^{a}(x) \rule{0pt}{16pt}D^{a}(x) \end{array}\right) \, , \hspace{1cm}{\cal F}^{a}(x) = \left( \begin{array}{c}\chi^{a}(x){\overline \chi}^{a}(x) \rule{0pt}{16pt}\lambda^{a}(x) \rule{0pt}{16pt}{\overline \lambda}^{a}(x) \rule{0pt}{16pt}\end{array}\right) \, ,
(f_{\lambda},f_{\lambda^{\prime}})=\frac{2^{D-1}\pi^{\frac{D}{2}}\Gamma(\frac{N}{2})}{(r')^2 - r^2}\lim_{\sigma\rightarrow \infty}(1-\cosh^{2}\sigma)W[P_{-\frac{1}{2}- ir}^{\mu}(\cosh \sigma),P_{-\frac{1}{2}+ ir^{\prime}}^{\mu}(\cosh \sigma)]\:,
\begin{aligned}p^2_p(x,y)= \int_y^\infty \bigg[(u^0_e+u^0_p)\partial_x v^0_p+u^0_p\partial_x v^1_e+(v^0_p+v^1_e)\partial_y v^0_p &-(h^0_e+h^0_p)\partial_x g^0_p-h^0_p\partial_x g^1_e-(g^0_p+g^1_e)\partial_y g^0_p-\nu \partial_y^2 v^0_p\bigg](x,\theta)\mathrm{d}\theta.\end{aligned}
\begin{aligned}\frac{\alpha^d \left(m^{d+1}+m^{-d-1}\right)+ \alpha^{-d}\left(m^{d-1}+m^{-d+1}\right)}{m^{d}+m^{-d}}&~{}= \frac{(\alpha^{d}m + \alpha^{-d}m^{-1})m^d +( \alpha^{d}m^{-1}+ \alpha^{-d}m )m^{-d}}{m^{d}+m^{-d}}\geq 1.\end{aligned}
\sum_{n \geq 1}\frac{S_f(n) \overline{S_g(n)}}{n^{k-1}}e^{-n/X}= C X^{\frac{3}{2}}+ O_{f,g,\epsilon}(X^{\frac{1}{2}+ \epsilon}) \sum_{n \geq 1}\frac{S_f(n) S_g(n)}{n^{k-1}}e^{-n/X}= C' X^{\frac{3}{2}}+ O_{f,g,\epsilon}(X^{\frac{1}{2}+ \epsilon})
\theta_{3}[exp(-\pi)]=\theta_{3}[q(\frac{1}{2}\sqrt{2})]=\sqrt{2\pi^{-1}K(\frac{1}{2}\sqrt{2})}=2^{-1/2}\pi^{-1/2}\beta(\frac{1}{4})^{1/2}=2^{-1/4}\sqrt[4]{\pi}{\Gamma(\frac{3}{4})}^{-1}
\mu:= \mathbb{E}\bigg[\sum_{j \in [\epsilon n_i]}W'_i \bigg] = \epsilon \sum_{j \in [m_i]}f(x_{i,j}) = \epsilon q_i \sigma^2:= \frac{1}{n_i}\sum_{j\in [m_i]}(f(x_{i,j})- \epsilon q_i/n_i )^2 + \sum_{j\in [n_i]\setminus [m_i]}( 0 -\epsilon q_i/n_i)^2.
\langle \epsilon(g_1),W(v_g,-1) \epsilon(g_2) \rangle =e^{-\lVert v_g\lVert^2/2+\langle v_g,g_2 \rangle}\langle \epsilon(g_1),\epsilon(v_g-g_2) \rangle &=e^{-\lVert v_g\lVert^2/2+\langle v_g,g_2 \rangle+\langle g_1,v_g \rangle-\langle g_1,g_2 \rangle},
S^{(2)}(E_{p}/\mathbb{Q})=\begin{cases}(1,\pm1)E_{p}[2]\cup(1,\pm p)E_{p}[2]\cong(\mathbb{Z}/2\mathbb{Z})^{4}, p\equiv1\mod8, (1,\pm1)E_{p}[2]\cong(\mathbb{Z}/2\mathbb{Z})^{3}, p\equiv5\mod8.\end{cases}
\int_a^b \varphi(x) (~^{AB}~_{a}I^\alpha \psi)(x)dx=\int_0^1 (1-x)~^{AB}{~_0}I^{1/2}x=\int_0^1 (1-x)[\frac{x}{2}+\frac{2x^{3/2}}{3\sqrt{\pi}}] dx=\frac{1}{12}+\frac{8}{105\sqrt{\pi}},
\hat{I}= - \frac{1}{4}\int d^4 x \; [F^{\mu \nu}F_{\mu \nu}-\frac{1}{2}\theta^{\alpha \beta}F_{\alpha \beta}F^{\mu \nu}F_{\mu \nu}+ 2 \theta^{\alpha \beta}F_{\alpha \mu}F_{\beta \nu}F^{\mu \nu}] + O(\theta^2) \;,
\hat m_{i,k-1}\hat m_{i+1,k}-\hat m_{i+1,k-1}\hat m_{ik}=m_{i,k-1}\left(m_{i+1,k-1}+m_{i+1,k}\right) -m_{i+1,k-1}\left(m_{i,k-1}+m_{ik}\right) &=m_{i,k-1}m_{i+1,k}-m_{i+1,k-1}m_{ik}=1
0\,\longrightarrow\,{\rm ad}(E_G)\otimes T^*M\,\longrightarrow\,{\rm At}(E_G)\otimes T^*M\, \stackrel{\eta\otimes_{T^*M}}{\longrightarrow}\, TM\otimes T^*M\,=:\,{\rm End}(TM)\,\longrightarrow\, 0\, .
S = \frac{1}{16 \pi}\int \mbox{d}^{4}x \sqrt{- g}\left[ R - 2 \left( \nabla \phi \right)^{2}- \mbox{e}^{-2 \phi}F^{2}- \frac{1}{12}\mbox{e}^{- 4 \phi}H^2\right] \; .
\begin{aligned}\mathcal{H}^+_k(N;\eta,m) := \{f\in \mathcal{H}^*_k(N): L(s,{\rm sym}^m f)\neq 0, s \in \mathcal{S}\}, \mathcal{H}^-_k(N; \eta, m) := \mathcal{H}^*_k(N)\setminus \mathcal{H}^+_k(N;\eta,m),\end{aligned}
\left\langle \sigma_2(\textbf{a}), \sigma_2(\textbf{b})\right\rangle_\sim =\sum_{i=1}^{\ell-1}\sigma_2(\textbf{a}_i)\widetilde{\sigma_2(\textbf{b}_i)}&=\left(\sum_i a_{i1}\overline{b_{i1}},\ldots,\sum_ia_{is}\overline{b_{is}},\sum_i a_{i1}'b_{i1}'', \sum_ia_{i1}''b_{i1}',\ldots,\sum_ia_{it}'b_{it}'', \sum_ia_{it}''b_{it}'\right).
F(x) = \frac{1}{2\pi i}\int_{(1)}\frac{\Gamma\left( \frac{s}{2}+ \frac{1}{4}\right)\Gamma\left(- \frac{s}{2}+ \frac{1}{4}\right)}{\Gamma^2\left(\frac{1}{4}\right)}\frac{G(s)}{s}x^{-s}ds.
\left| \E^{\frac{n}{2}}_{y,M}\circ E^{\frac{n}{2}}_{x,M}\frac{G' \cdot (x^2+y^2)^{N'}}{(x^2+y^2+z^2)^{\alpha}}\ \right| \leqslant C \log (x^2+y^2+z^2) \Big|_{x=0}^{x=\frac{n}{2}}\Big|_{y=0}^{y=\frac{n}{2}}\leqslant C \log n.
(-\lambda-Y+Z -t)(-\lambda-C-t)+Z(\lambda - B + C) = t^2+t(\lambda+Y-Z+\lambda+C)+Z(\lambda - B+C)+\lambda^2+\lambda C+\lambda Y+YC-Z\lambda-CZ = t^2 + t(2\lambda+Y+C-Z)-BZ+\lambda^2+\lambda C+\lambda Y+YC
v_1 = b_1 \sum \limits_{\mu \in \mathrm E \setminus \mathrm E_{\varrho_1}}\frac{\langle \varrho_1, \mu \rangle}{\langle \alpha^\vee, \mu \rangle}e_{-\alpha}v_\mu \; \; v_2 = -b_2 \sum \limits_{\mu \in \mathrm E \setminus \mathrm E_{\varrho_2}}\frac{\langle \varrho_2, \mu \rangle}{\langle \alpha^\vee, \mu \rangle}e_{-\alpha}v_\mu.
{\rm{t}}{{\rm{r}}_k}\left({\bf{A}}\right) = \sum\limits_{\left\{{\vec{a}}\right\}}{\prod\limits_{i = 1}^{k}{\lambda_{x,a_i}}}=\sum\limits_{\left\{{\vec{a}}\right\}}{\left|{{{\left[{\bf{A}}\right]}_{{\vec{a}},{\vec{a}}}}}\right|}
\int (x_{k})^{4}\mathcal{V}_{J}\, d^{2}x=6\int (x_{k})^{2}\partial_{j}\partial_{j}h \, d^{2}x=6\int \left(\partial_{j}((x_{k})^{2}\partial_{j}h)-2\partial_{k}(x_{k}h) +2 h \right) d^{2}x,
E(vq) - E(v)E(q) = \frac{<\{\hat{Q}, \hat{P}\}>_{\psi}}{2}\, ; \; \;\; \; \; \hat{Q}=\hat{q}-<\hat{q}>_{\psi}\, , \; \; \hat{P}=\frac{\hat{p}- <\hat{p}>_{\psi}}{m}\, ,
a\left|z;-\frac{\gamma}{2(\delta-\gamma)}\right\rangle_i = \sqrt{-\gamma}\frac{z}{\sqrt{1-|z|^2}}\left|z;-\frac{\gamma}{2(\delta-\gamma)}+\frac{1}{2}\right\rangle_i, b^{\dagger}\left|z;\frac{\delta}{2(\delta-\gamma)}\right\rangle_j = \sqrt{\delta}z\sqrt{1-|z|^2}\left|z;-\frac{\delta}{2(\delta-\gamma)}+\frac{1}{2}\right\rangle_j.
\frac{d}{dt}\frac{\partial T}{\partial\dot{x}}-\frac{\partial T}{\partial x}=F_{x}+\lambda\frac{\partial f}{\partial x},\frac{d}{dt}\frac{\partial T}{\partial\dot{y}}-\frac{\partial T}{\partial y}=F_{y}+\lambda\frac{\partial f}{\partial y}.
S_{[ 1 / 2 ]}^{c}= e^{i \vartheta_{[ 1 / 2 ]}^{c}}\left( \begin{matrix}{{0}}&{{i \Theta_{[ 1 / 2 ]}}}\\{{- i \Theta_{[ 1 / 2 ]}}}&{{0}}\\ \end{matrix}\right){\cal K}\, .
J_4 =\int_1^{m/2}\int_1^{n/2}\int_{1}^{\infty}\frac{dy_1dsdr}{(y_1^{\alpha_1}+(n-s)^{\alpha_2}+(m-r)^{\alpha_3})((y_1+k)^{\alpha_1}+s^{\alpha_2}+r^{\alpha_3})}&\leq C\int_1^{m/2}\int_1^{n/2}\int_{1}^{\infty}\frac{dy_1dsdr}{(y_1^{\alpha_1}+m^{\alpha_3})(s^{\alpha_2}+r^{\alpha_3})},
Q_0^{\alpha\beta\mu\nu}= - \frac{1}{4\pi}\int d^2{\bar{\cal S}}\int\limits_0^{\infty}d{\bar r}(\log{\bar r})\frac{\partial}{\partial{\bar r}}\left[{\bar r}\frac{\partial}{\partial{\bar r}}\Bigl({\bar r}^2{\bar \nabla}_{\bar \mu}{\bar r}\;{\bar Z}^{\bar \mu}\biggl|_{i^-}\Bigr)\right] + O[R^2_{..}]\; .
\left( \begin{array}{c}{\dot{\Phi}}{\Phi^{\prime}}\end{array}\right) \; \; \rightarrow \; \; \left( \begin{array}{c c}{c o s h \, \theta}{s i n h \, \theta}{s i n h \, \theta}{c o s h \, \theta}\end{array}\right) \; \; \left( \begin{array}{c}{\tilde{\dot{\Phi}}}{\tilde{\Phi}^{\prime}}\end{array}\right) \; \; .
\begin{array}{rcccccl}\tilde Y_e=W_t \stackrel{\beta_t}{\rightarrow}\cdots \rightarrow W_1 \stackrel{\beta_1}{\rightarrow}W_0=\tilde Y \downarrow \downarrow \downarrow \tilde X_e=V_t \stackrel{\alpha_t}{\rightarrow}\cdots \rightarrow V_1 \stackrel{\alpha_1}{\rightarrow}V_0=\tilde X\end{array}
\left( \sum_{j=N(t)-s+1}^{N(t)}X_{j}^{\ast},X_{N(t)-s}^{\ast},\sum_{j=1}^{N(t)-s-1}X_{j}^{\ast}\right) /U(N(t))\overset{D}{\rightarrow}\left( \sum_{k=1}^{s}\Gamma_{k}^{-1/\alpha},\Gamma_{s+1}^{-1/\alpha},\sum_{k=s+2}^{\infty}\Gamma_{k}^{-1/\alpha}\right)
\mathcal{W}(v)=\frac{1}{2}\|v\|_{\dot{H}^m}^2+\sum_{0\leq k<m}\frac{\lambda_k}{p_k}\int_{\mathbb{R}^n}|\nabla^kv(x)|^{p_k}\,dx\qquad(\lambda_k\geq0,\,p_k>1).
\limsup_{n \rightarrow \infty}n^{-1/2}\big \Vert \sum_{i=1}^n d_{i,N}\big \Vert_2 \leq \limsup_{n \rightarrow \infty}n^{-1/2}\big \Vert \sum_{i=1}^n Y_{i,N}\big \Vert_2 \, .
\int_0^t\phi(s)dB^H(s)=\sum_{n=1}^{\infty}\int_0^t\sqrt{\lambda_n}\phi(s)e_nd\beta^H_n(s)=\sum_{n=1}^{\infty}\int_0^t\sqrt{\lambda_n}(K_H^*(\phi e_n)(s)d\beta_n(s)
P\left(\sup_{0<r\leq T}\left|\int_0^{r}B(X_{s}^\epsilon,y)(1_{\{Y_{s}^\epsilon =y\}}-\pi(y))ds\right|>\eta\right) &\leq P(A_\epsilon^c)+P(n^\epsilon\geq K')+P(\tau_1^\epsilon\leq \rho')+P( D_\epsilon^c)\leq 4\delta,
g_{\mu\nu}g^{\alpha\beta}Q^{\kappa}_{\alpha\rho}Q^{\rho}_{\kappa\beta}(-2 +\frac{a(a-2)(5a^2 + 45a + 26)}{30(1-a)^2}) +g^{\alpha\beta}Q_{\mu\alpha\kappa}Q^{\kappa}_{\beta\nu}(\frac{-a(a -2)}{2}),
\psi'_n(x) =(\log(e^3/nx^2))^{-1/2}+ x \cdot (-1/2) \cdot(\log(e^3/nx^2))^{-3/2}\cdot (nx^2/e^3) \cdot (-2e^3/nx^3) &= (\log(e^3/nx^2))^{-1/2}(1+ (\log(e^3/nx^2))^{-1}) =h(x)(1+h(x)^2),
V_1C_1^{-1}=\left( \prod_{i=1}^N{}_i\langle 0|\right) \exp \left[ -\frac{1}{2}\sum_{\scriptstyle i,j=1\atop \scriptstyle i\neq j}^{N_1}\sum_{n,m=0}^\infty \alpha^{\mu i}_nD_{nm}\left( \Gamma V^{-1}_iV_j\right) \alpha^j_{m\mu}\right] \ ,
\gamma_{\xi}^{( \beta )}( X ) \simeq \frac{\pi^{2}}{4 5}b_{\xi}( X ){\cal T}_{\xi}^{4}( X ) - \frac{1}{1 2}c_{\xi}( X ){\cal M}_{\xi}^{2}( X ){\cal T}_{\xi}^{2}( X ) + \frac{1}{6 \pi}{\cal M}_{\xi}^{3}( X ) \,{\cal T}^{( \xi )}( X ) .
m^{2}\frac{\partial}{\partial m^{2}}\ln \mathrm{d e t}_{{\mathrm{\scriptsize{Q E D}}_{4}}}= \frac{1}{2 4 \pi^{2}}\| B \|^{2}+ 2 m^{2}\mathrm{T r}\, [ ( D^{2}+ m^{2})^{- 1}- ( P^{2}+ m^{2})^{- 1}] .
\nu_{(0, i)}^*(c, d) = \max_{S \subseteq N\colon i \in S}\frac{f_i^S - c - \big\{1 - (1-\beta) g_i^S\big\}d}{g_i^S}= (1-\beta) d + \max_{S \subseteq N\colon i \in S}\frac{f_i^S - (c+d)}{g_i^S},
\dot{\bar{\mathcal{F}}}(\omega,\phi_{\partial},\Sigma)=\{\bar{\mathcal{F}},\bar{H}\}_{D}(\omega,\phi_{\partial},\Sigma)-\int_{\Gamma}E(\frac{\delta\bar{\mathcal{F}}}{\delta\Sigma})\wedge\ast\boldsymbol{n}(d\phi),
\int_{\mathcal L_{2t}}(c,d)_p\chi_{\psi}(c) \chi_{\delta}(c)|c|^{-3/2}_p\mathbf h_p(h) dh = -p^{1+3t}\mathrm{vol}(\mathcal L_{2t}) = -p^{1+t}(1-p^{-1})^2.
\int_{\mathbb{R}^n}\varphi\Phi g dx = \iint_{\mathbb{R}^{2n}}\frac{\partial}{\partial y_j}(\varphi\Phi) d\pi &= -\iint_{\mathbb{R}^{2n}}\varphi(x)\frac{\partial\Phi}{\partial y_j}(x,y) d\pi(x,y) &= -\int_{\mathbb{R}^{n}}\varphi(x)\frac{\partial\Phi}{\partial y_j}(x,F(x))g(x) dx,
\lambda(b_{},c_{}) =\begin{pmatrix}i(b_{}-c_{}) r_{}0 \dots 0 -r_{}0 0 \dots 0 begin{align*}1ex] 0 0 \vdots \vdots \text{\Large $0_{(n-2)\times (n-2)}$}begin{align*}-.5ex] 0 0 \end{pmatrix}\in({\mathfrak g}_0/{\mathfrak h}_0)^*.
\begin{array}{l c r}{{P ( W_{2}) = ( 1 + U_{1}+ U_{2}+ U_{1}^{2}U_{2}^{3})}}\\{{+ U_{3}( a_{0}U_{1}^{2}U_{2}^{3}+ a_{1}U_{1}U_{2}+ a_{n - 1}U_{1}^{{\frac{( 5 - n )}{2}}}U_{2}^{{\frac{( 6 - n )}{2}}}) + U_{3}^{2}\sum_{i = 0}^{n - 4}a_{i + 1}U_{1}^{3 - i}U_{2}^{4 - i}.}}\\ \end{array}
\mathsf{S_b}(f)(z) &=\int_{0}^{1}f'(tz)dt &=2\pi^{-1}\int_{\mathbb{D}}f'(w)(1-|w|^{2})\left(\int_{0}^{1}\frac{dt}{(1-t\bar{w}z)^{3}}\right)dm(w).
\frac{M^2}{\cosh^2(M\omega )}[\omega^{\prime \prime}-\ddot \omega -\frac1\tau \dot \omega ]+2\frac{M^3\sinh (M\omega )}{\cosh^3(M\omega )}[\dot\omega^2-\omega^{\prime}{}^{\;2}]=2\frac{M^3\sinh (M\omega )}{\cosh^3(M\omega )}[\dot \omega^2-\omega^{\prime}{}^{\;2}],
\chi =2(n_{\frac{p-1}{2}}+Nk+k)+N(N-1)/2 &\quad+2(N-n_{\frac{p-1}{2}})+4k(k-1)/2+3k+(N+1)(2-2g) &\quad+k(4-4g)+3-5g+2g^2 &=(1/2)(2g-2-N-2k)(2g-5-N-2k), h^* =N^2/2+7N/2+2Nk+7k+2Ng+5+5g+2k^2+4kg+2g^2.
\begin{array}{rr}g=[q_1-c_1(q_2)]*[q_1-c_2(q_2)]*\cdots*[q_1-c_{\deg_{q_1}f}(q_2)]= =[q_2-d_1(q_1)]*[q_2-d_2(q_1)]*\cdots*[q_2-d_{\deg_{q_2}f}(q_1)]\end{array}
\mathfrak{d}_{kj}(z_{\ast})=\sum_{r=1}^{k-1}\mathfrak{d}_{rj}(z_{\ast})\cdot\lambda_{r}\ \ \j\in\{1,\hdots,k,m+1,\hdots,m+k\}\setminus\{j_1,\hdots,j_{k-1}\}.
: \hat{\mathrm{H}}_{\pm}: | m_{1}; m_{2}; . . . ; m_{n}; A ; \pm \rangle = \hbar \sum_{k = 1}^{n}{\varepsilon}_{m_{k}, \pm}\cdot \mathrm{s i g n}({\varepsilon}_{m_{k}, \pm}) | m_{1}; m_{2}; . . . ; m_{n}; A ; \pm \rangle
S_{e f f}= N S_{0}( \sigma^{0}) + S_{1}( \sigma^{0}) -{\frac{i}{2}}C_{2}+{\frac{1}{2}}D_{2}+ \sum_{n = 1}^{\infty}{\frac{1}{\sqrt{N}^{n}}}\Big [ i ( - 1 )^{n + 1}[{\frac{C_{n + 2}}{( n + 2 )}}+ m L^{d}\delta^{d}( 0 ){\frac{C_{n}}{n}}] +{\frac{1}{( n + 2 ) !}}D_{n + 2}\Big ]
\max_{j\in\{2,\dots,N_h\}}\Big( \big\| \rho_1^j \big\|^2 + \big\| \rho_2^j+\rho_3^j \big\|^2 + \frac{1}{h}\big\|P_h^{j-1}\rho_2^j\big\|^2 + \frac{1}{h}\big\|P_h^{j-2}\rho_3^j\big\|^2 \Big) \leq C h^2.
-\frac{i\omega}{2h}= \frac{1}{\sqrt{\lambda}}\sum_{n=1}^{\infty}\omega_n (-\lambda)^n,\;\; -\frac{i\tau}{h}= \frac{1}{\sqrt{\lambda}}\sum_{n=0}^{\infty}\tau_n (-\lambda)^n, \;\;-\frac{i\sigma}{h}= \frac{1}{\sqrt{\lambda}}\sum_{n=1}^{\infty}\sigma_n (-\lambda)^n.
\int\limits_{A}^{0}\int\limits_{\mathbb{R}^{3}_{+}}\partial_{\tau}^{l}\varphi(y,\tau)U_{i(k)}(y,\tau) dy d\tau= (-1)^{l}\int\limits_{A}^{0}\int\limits_{\mathbb{R}^{3}_{+}}\varphi(y,\tau)\partial_{\tau}^{l}U_{i(k)}(y,\tau) dy d\tau
\begin{cases}\begin{rcases}v_t+v\cdot \nabla v=-\nabla P-(0,1) &div~v=0,~~curl~v=0 \end{rcases}\Omega(t) P\equiv 0 \quad\quad\quad\quad\quad~~\quad\Sigma(t) (1,v) (t, \Sigma(t)).\end{cases}
\mathcal{L}_{i}(\phi):=(-\Delta)^{m}\phi -\sum_{i=1}^{k}e^{w_{i}}\phi=(-\Delta)^{m}\phi -\alpha_{2m}(2m-1)!\sum_{i=1}^{k}\frac{\mu_{i}^{2m}}{(\mu_{i}^{2}+|z|^{2})^{2m}}\phi,
w_{i j}^{k}= 0 , w_{a j}^{k}= 0 , w_{a b}^{k}= 0 , w_{a b}^{c}= 0 , w_{i j}^{a}= \delta_{i}N_{j}^{a}- \delta_{j}N_{i}^{a}, w_{j a}^{b}= - w_{a j}^{b}= \partial_{a}N_{j}^{b}.
= \frac{1}{2}\times ( \frac{1}{5}- \frac{1}{7}+ \frac{1}{7}- \frac{1}{9}+ \frac{1}{9}- \frac{1}{1 1}+ \frac{1}{1 1}- \frac{1}{1 3}+ \frac{1}{1 3}- \frac{1}{1 5})
\delta \Psi = \epsilon^{2}( 1 + i \Psi \partial_{\tau}\Psi ) , \quad \delta \Phi = - i \epsilon^{2}( 2 \eta - \Psi \partial_{\tau}\Phi ) = - i \epsilon^{2}( 2 \eta - L ) , \quad L = \Psi \partial_{\tau}\Phi ={\frac{\partial_{\tau}\Phi D \Phi}{1 + \sqrt{1 - ( \partial_{\tau}\Phi )^{2}}}},
\sum_{x_1,x_2\in X}c(x_1,x_2)^2 \geq \binom{|X|}{2}\left(\frac{\sum_{x_1,x_2\in X}c(x_1,x_2)}{\binom{|X|}{2}}\right)^2 &\geq 2\cdot\frac{\left(\sum_{x_1,x_2\in X}c(x_1,x_2)\right)^2}{|X|^2}&\geq 2\cdot\frac{\left(\,\frac{1}{2}\left((2-\alpha-2\beta-\gamma)\Delta^3-6\Delta^2\right)\,\right)^2}{(2-\alpha)^2\Delta^2},
{\rm vec}({{{\bf{H}}}}) = \sum\limits_{n = 1}^{{N_{\mathrm r}}}{\sum\limits_{m = 1}^{{N_{\mathrm t}}}{\left({{{\big[{{\bf{\tilde H}}}\big]}_{nm}}+{{\big[{\hat{\bf{H}}}\big]}_{nm}}}\right)\left({\bf{u}}_{\mathrm t}^* ({{\phi_m}}) \otimes{{\bf{u}}_{\mathrm r}}({{\theta_n}})\right)}}
\begin{array}[c]{rl}-d{p}^{\theta}(t)= \{b_x(X^{\theta}(t),u^{\theta}(t))^{}p^{\theta}(t) -\beta^{0,\theta}f_x(X^{\theta}(t),u^{\theta}(t))\}dt,\t\in(t_{i-1},t_{i}), p^{\theta}(t_{i})= -\beta^{0,\theta}\Psi_x(X^{\theta}(t_n))1_{i=n}(i)-\displaystyle{\beta^{i,\theta}}{}+p(t_{i}^{+}), i=1,\ldots,n.\end{array}
\left(^{\rho}I_{a+}^{\alpha}f\right) \left( x\right) =\frac{\rho^{1-\alpha}}{\Gamma \left( \alpha \right)}\int_{a}^{x}\frac{\tau^{\rho-1}f\left( \tau \right)}{\left( x^{\rho}-\tau^{\rho}\right)^{1-\alpha}}d\tau ,
\frac{\partial \rho}{\partial \tau}= \frac{(\tau - \xi \cdot \omega)(2\tau) - (\tau^2 - |\xi|^2)}{2(\tau - \xi \cdot \omega)^2}= \frac{\tau^2 - 2\tau \xi \cdot \omega + |\xi|^2}{2(\tau - \xi \cdot \omega)^2}.
\sum_{g\in G}| \langle w, \rho(g) v \rangle |^2 = \sum_{j=1}^n \sum_{h \in H}| \langle w, \rho(g_j h) v \rangle |^2 = \sum_{j=1}^n \sum_{h \in H}| \langle w, \rho(g_j) \alpha(h) v \rangle |^2 &= |H| \sum_{j=1}^n |\langle w, \rho(g_j) v \rangle |^2,
F_{i,\nu}^{(\frac{l_i(\nu)}{e_i})}={F_{i-1,\nu}^{(\frac{l_i(\nu)}{e_{i-1}})}}^{n_i}-c_i{x_1^{(\nu_1)}}^{\alpha_1^{(i)}}{x_2^{(\nu_2)}}^{\alpha_2^{(i)}}{z^{(\langle\nu,\gamma_1\rangle)}}^{r_1^{(i)}}\cdots{F_{i-2,\nu}^{(\frac{l_{i-1}(\nu)}{e_{i-2}})}}^{r_{i-1}^{(i)}}+G_{i,\nu},
\bar{M}:= \inf \Big\{\frac{1}{r}\Big| D(u, r) \leq \frac{1}{C_0}r^{\frac12}\chi_r \leq \frac{1}{C_0}\Big\}&\leq \max \Big\{\Big( C_0 \sup_{r \leq 1}\frac{1}{r^{\alpha_0}}D(u,r)\Big)^{\frac{1}{\alpha_0}}, \inf \big\{M>0 \big| \bar{\chi}_{\frac{1}{M}}\leq \frac{M^{\frac12}}{C_0}\big\}\Big\}.
\int_{K_{\rho}\times \{-s\}}H(\Psi^2) \zeta^{g_1}\,dx \leq \int_{K_{\rho}\times \{-\tau\}}H(\Psi^2) \zeta^{g_1}\,dx &+ 2 C_{0}^{1-g_1}C_{1}^{g_1}g_{1}^{2g_1}\int_{-\tau}^{-s}\int_{K_{\rho}}h(\Psi^2)|\Psi||\Psi'|^2 G\left(\frac{|D\zeta|}{\Psi'}\right) \,dx\,dt,
|T_{1}(x)| \leq\int_{|xy^{-1}|<R}\frac{\rho(|xy^{-1}|)}{|xy^{-1}|^{Q}}|f(y)|dy = \sum_{k=-\infty}^{-1}\int_{2^{k}R\leq|xy^{-1}|<2^{k+1}R}\frac{\rho(|xy^{-1}|)}{|xy^{-1}|^{Q}}|f(y)|dy.
\frac{x}{\sqrt{2 \pi}}\Big(\underset{i=0}{\overset{\infty}{\sum}}\frac{(-1)^i ( x/\sqrt{2})^{2i}}{(2i+1) i!}\Big)\geq\frac{x}{\sqrt{2 \pi}}\left(\underset{i=0}{\overset{\infty}{\sum}}\frac{(-1)^i c^{2i}}{(2i+1) i!}\right) = \frac{x}{\sqrt{2}c}\left( \Phi(\sqrt{2}c) - \frac{1}{2}\right),
=(\lambda_{i}-\lambda_{k})(\frac{x^{2}}{(a^{2}-\lambda_{i})(a^{2}-\lambda_{k})}+\frac{y^{2}}{(b^{2}-\lambda_{i})(b^{2}-\lambda_{k})}+\frac{z^{2}}{(c^{2}-\lambda_{i})(c^{2}-\lambda_{k})}).
H_{R}= m_{1}+ m_{2}+ 2 \Delta + \frac{p^{2}}{2 m_{1}}+ \frac{p^{2}}{2 m_{2}}- \frac{\Delta}{2}\left( \frac{p^{2}}{m_{1}^{2}}+ \frac{p^{2}}{m_{2}^{2}}\right) + V_{0}+ V_{S D}+ V_{v^{2}}.
K_{\nu}\left({\nu z}\right) =\left({\dfrac{\pi}{2\nu}}\right)^{1/2}\dfrac{1}{\left({1+z^{2}}\right)^{1/4}}\exp \left\{-{\nu \xi+\sum\limits_{s=1}^{n-1}}\left( -1\right)^{s}{{\dfrac{\tilde{{E}}_{s}\left(p\right)}{\nu^{s}}}}\right\}\left\{{1+\eta_{n,2}\left({\nu ,z}\right)}\right\},
(\begin{matrix}S_{11}&S_{12}\\S_{21}&S_{22}\end{matrix})=(\begin{matrix}e^{i\varphi}e^{i\delta}\cdot r&e^{i\varphi}\sqrt{1-r^{2}}\\e^{i\varphi}\sqrt{1-r^{2}}&-e^{i\varphi}e^{-i\delta}\cdot r\end{matrix})=e^{i\varphi}(\begin{matrix}e^{i\delta}\cdot r&\sqrt{1-r^{2}}\\ \sqrt{1-r^{2}}&-e^{-i\delta}\cdot r\end{matrix})
\mathbf{L}_{b}^{a}=(\delta_{c}^{a}+\mathbf{l}_{c}^{a}\mathbf{m}^{2})(\delta_{d}^{c}+_{1}\lambda_{d}^{c}\mathbf{m}^{1})L_{b}^{d}, \mathbf{l}_{c}^{a}=_{2}\lambda_{c}^{a}+\lambda_{c}^{a}\mathbf{m}^{1},
\Phi_{s}( x , a^{+}) = \frac{4}{\beta}{\textrm{a r c t g}}( e^{m ( x - a^{+})}) , \qquad \qquad \Phi_{\bar{s}}( x , a^{-}) = \frac{2 \pi}{\beta}- \Phi_{s}( x , a^{-}) ,
\aligned \widehat{\mathcal U}({\rm mor};3,1;\alpha_1,\alpha_3) &=\bigcup_{\alpha_2 \in \frak A_2}\widehat{\mathcal U}({\rm mor};2,1;\alpha_1,\alpha_2)\times_{R_{\alpha_2}}^{\boxplus\tau}\widehat{\mathcal U}({\rm mor};3,2;\alpha_2,\alpha_3).\endaligned
E_{0}= - \frac{\mu_{e f f}^{2}}{\left( 2 \pi \right)^{2}}\sum_{r \, n = 1}^{\infty}\frac{1}{n^{2}}K_{2}\left( 2 \pi n \mu_{e f f}\right) + \frac{\mu_{e f f}^{3}}{2 \pi}\sum_{n = 1}^{\infty}\frac{1}{n}K_{3}\left( 2 \pi n \mu_{e f f}\right) .
\sum_{n=0}^\infty (-1)^n 4^n \int_{\mathbb{Z}_p}{\frac{x}{2}\choose n}d\mu_{-1}(x)t^n = \sum_{n=0}^\infty C_n t^n &= \sum_{n=0}^\infty \frac{1}{n+1}{2n \choose n}t^n.
\min_{\partial B(p_i, 1)}\sum\limits_{j=1}^{K}|f^i_j(x)|^2 > \frac{3}{2}\max_{B(p_i, \frac{3\delta_2}{\delta_1})}\sum\limits_{j=1}^K|f^i_j(x)|^2\geq c(n, v)>0.
S = \frac{M_{5}^{3}}{2}\int_{{\cal M}_{5}}d^{5}x \sqrt{- g}\left[ R - \frac{1}{2}\partial_{\alpha}\phi \, \partial^{\alpha}\phi - \frac{3}{2}e^{- 2 \phi}\alpha^{2}\right] - 3 \sum_{i = 1}^{2}M_{5}^{3}\alpha_{i}\int_{{\cal M}_{4}^{( i )}}d^{4}\xi_{( i )}\sqrt{- g}\, e^{- \phi}\, .
\begin{aligned}\ \ \ \ \eta^{am+1}(3z)\eta^{bm-1}(z)\ |\U(m) \equiv_m\left(\sum_{n=0}^\infty b_3(n)q^{n+\frac{m(3a+b)+2}{24}}\ |\U(m)\right)\cdot\prod_{n=1}^\infty(1-q^{3n})^a(1-q^{n})^b. \end{aligned}
\left. K^{(0)}_{\delta}(r,r',\tau-\tau',s)\right|_{{\cal C}_\beta}=K^{(0)}(r,r',\tau-\tau',s)+{1 \over 2i\beta}\int_{A}{\exp i{(\delta-\pi) \over \beta}(\tau-\tau'+z) \over \sin{\pi \over \beta}(\tau-\tau'+z)}K^{(0)}(r,r',z,s)dz~~~.
{\Bigl(\partial^2 -2ieF\partial-{B\over\bar T}\Bigl)}^{-1}={\Bigl(\partial^2 -2ieF\partial\Bigr)}^{-1}+{\Bigl(\partial^2 -2ieF\partial\Bigr)}^{-1}{B_{ab}\over \bar T}{\Bigl(\partial^2 -2ieF\partial\Bigr)}^{-1}+\cdots,
\sqrt{\lambda}\cot\sqrt{\lambda}=\frac{1}{2}+\nu\frac{\frac{\Gamma(1-\nu)}{\Gamma(1+\nu)}\left( \frac{kR}{2}\right)^{2\nu}+1}{\frac{\Gamma(1-\nu)}{\Gamma(1+\nu)}\left( \frac{kR}{2}\right)^{2\nu}-1}.
\prod_{j=1}^d\prod_{\mathit{v}\in S_j}\left|-\rho_\mathit{v}\left(\frac{a_2}{a_1}\right)p_2-\cdots-\rho_\mathit{v}\left(\frac{a_r}{a_1}\right)p_r+c_{\mathit{v}(1)}q\sigma_{\mathit{v}(1)}(u)+\cdots+c_{\mathit{v}(d)}q\sigma_{\mathit{v}(d)}(u)\right|_{\mathit{v}}<\frac{1}{H^\epsilon(u)q^{\frac{d}{r}+\epsilon}}.
g_{2}= A_{3}^{3}\cdot g_{2}^{( 1 )}\cdot g_{2}^{( 2 )}, \qquad g_{3}= A_{3}^{4}\cdot g_{3}^{( 1 )}, \qquad \Delta = A_{3}^{8}\cdot \Delta_{1}\Delta_{2}\Delta_{3}\Delta_{4}\Delta_{5}^{3}
+\sum_{k=0}^{\infty}\sum_{\substack{{r_1,r_2,r_3=1}r_1\ne r_2\ne r_3}}^{n}\frac{a_{r_1}b_{r_2}c_{r_3}}{(r_3-r_2)(r_1-r_2)}\left(\frac{1}{r_2+k+1}-\frac{1}{r_1+k+1}\right).
M = \textstyle{\frac{3}{2}}\mu + \sum_{i = 1}^{3}q_{i}= \textstyle{\frac{3}{2}}\left( r_{+}^{2}+ r_{+}^{4}g^{2}\prod_{i = 1}^{3}H_{i}\right) + \sum_{i = 1}^{3}q_{i}\ ,
\Psi_{g,n}= \left(\Psi_{1,0}^{\otimes g}\otimes \Psi_{0,1}^{\otimes n}\right) \circ \alpha_{g,n}\: : \: \mathcal{L}_{g,n}(H) \overset{\sim}{\rightarrow}\mathcal{H}(\mathcal{O}(H))^{\otimes g}\otimes H^{\otimes n}.
\varepsilon(h,x+y)[[\alpha(x),\alpha(y)],\alpha(h\cdot z)] +\varepsilon(x+y,z)[[h\cdot z,\alpha(x)],\alpha^2(y)] &+\varepsilon(h,y)\varepsilon(x,y+z)[[\alpha(y),h\cdot z],\alpha^2(x)]=0,
\Phi (x,\lambda ):=\frac{\sigma (x+\lambda )}{\sigma (x)\sigma (\lambda )},\mbox{$\ \ \ \ $}\b_{j}:=\prod_{k\neq j}f(x_{j}-x_{k}),\ \ \ \mbox{$\ \ \ $}(E_{ij})_{kl}=\delta_{ik}\delta_{jl}.
\frac{R}{\sqrt{p^{2}+q^{2}}}\left(1+2\left\lfloor \frac{\epsilon q^{2}R}{\sqrt{p^{2}+q^2}}\right \rfloor\right) = \frac{2q^2}{p^{2}+q^{2}}\epsilon R^{2}+\left(1-2\left\{\frac{\epsilon q^{2}R}{\sqrt{p^{2}+q^{2}}}\right\}\right)\frac{R}{\sqrt{p^{2}+q^{2}}}
\begin{bmatrix}A_{\deg A}\cdots A_0 \end{bmatrix}\begin{bmatrix}P_k P_{k-1}P_{k-2}\cdots P_k P_{k-1}P_{k-2}\cdots \ddots \ddots & P_k P_{k-1}& P_k\end{bmatrix}= \begin{bmatrix}V_{\deg V}\cdots V_k \end{bmatrix},
\lim_{P\rightarrow \infty}\lim_{L\rightarrow \infty}\frac{(T_E + T_F)\log(P)}{L}= \left(1 + \frac{1}{r}\right) \lim_{P\rightarrow \infty}\lim_{L\rightarrow \infty}\frac{T_E \log(P)}{L}&= \left(1 + \frac{1}{r}\right) \frac{K}{\min\{M,K\}},
r \cdot v =G_{i, [k]}x_{\ell,t}-G_{i, [k]}x_{\ell', t}=G_{i, [k]}(x_{\ell}-x_t)-G_{i, [k]}(x_{\ell', t}-x_t) &=G_{i, [k]}x_{\ell}-G_{i, [k]}x_{\ell'}=c_{\ell, i}-c_{\ell', i}=0
S= \sum_{i_0=0}^\infty \sum_{i_1=0}^\infty \cdots \sum_{i_{d}= 0}^\infty1_{i_0\le i_1\le \cdots \le i_d}~2^{-(i_0+i_1+\cdots +i_{d})}\left( \prod_{t=0}^k |E_{i_{\ell_t}}| \right) \left( q^{-\sum\limits_{t=1}^k (\ell_t-t)}\right),
_{\mathcal{H}}\langle f,h\rangle_{\bar{\mathcal{H}}}=\mathbb{E}[Z\cdot I(f)]=\mathbb{E}\left[Z\cdot\int_{0}^{1}f_{s}dB_{s}\right]=\int_{0}^{1}f_{s}\, \mathbb{E}[ZdB_{s}]=\int_{0}^{1}f_{s}dh_{s}.