harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
L_{e f f}^{( 1 )}= \frac{2 \pi \kappa \sigma_{1}}{q_{1}}\sum_{p_{1}}\dot{R}_{p_{1}}^{i}v_{i}^{( 2 )}( \vec{R}_{p_{1}}) + ( \; 1 \leftrightarrow 2 \; ) \; \; .
V_{n,k}\prod_{i=1}^{k}W_{m_{l}}U_{n-m_{l}}=\sum_{j=0}^{\infty}\binom{k+j}{j}U_{n}^{j}W_{1}^{j}\sum_{\underline{z}\in\left\{0,1\right\}^{k}}V_{n+1,k+j}\prod_{i=1}^{k}W_{m_{i}+z_{i}}U_{n+1-m_{i}-z_{i}},
V_{{\rm CM}}(D):=\left( \begin{array}{cc}e^{-D^- D^+}e^{-\frac{1}{2}D^- D^+}\left( \frac{I- e^{-D^- D^+}}{D^- D^+}\right) D^- e^{-\frac{1}{2}D^+ D^-}D^+ I- e^{-D^+ D^-}\end{array}\right),
\delta^2 \geq \int_x^{x_*}(e^{\phi(u)/2}- 1)^2 \, du \geq \frac{1}{4}\int_x^{x_*}\phi^2(u) \, du \geq \frac{\phi^2(x)}{4}\int_x^{x_*}\biggl(\frac{x_* - u}{x_*- x}\biggr)^2 \, du = \frac{x_*-x}{12}\phi^2(x) &\geq \frac{\phi^2(x)}{48},
\Omega\cap V_y =\rho_y\bigl({\mathbb{R}}^{n-1}\times(0,+\infty)\bigr)\cap V_y, \Sigma\cap V_y =\rho_y\bigl({\mathbb{R}}^{n-1}\times\{0\}\bigr)\cap V_y, \bigl({\mathbb{R}}^n\setminus\overline{\Omega}\bigr)\cap V_y =\rho_y\bigl({\mathbb{R}}^{n-1}\times(-\infty,0)\bigr)\cap V_y.
K_{\mathrm{Airy}}(\pi;u,v)= \int_{\mathcal{C}_{L}}\frac{dz}{2i\pi}\int_{\mathcal{C}_{R}}\frac{dw}{2i\pi}\frac{1}{z-w}e^{\frac{1}{3}w^3 -\frac{1}{3}z^3}e^{vz-uw}\prod_{k = 1}^{m}\frac{z-\pi_{k}}{w - \pi_{k}}
\left(e^{-\nu\mathrm{m}}\right)^{-1}\partial e^{-\nu\mathrm{m}}\phi =\left(e^{-\nu\mathrm{m}}\right)^{-1}(-\nu e^{-\nu\mathrm{m}}\phi+e^{-\nu\mathrm{m}}\phi') =-\nu\phi+\phi',
E(\overline F_{s_{\lambda}}^{\theta,\overline p}/(\mathfrak S_{M(\lambda)}\cap \mathfrak S_{M(\theta\star p)}))= E(X(A_{r_1-1})/\mathfrak S_{\mu_1})\times \dots \times E(X(A_{r_k-1})/\mathfrak S_{\mu_k}).
\|v_{l}\|_{W^{1-\frac{1}{p},p}(\partial\Omega)}\le C\|v_{l}\|_{L^{2}(\Omega)}^{\theta_{2}}\|v_{l}\|_{W^{2,p}(\Omega)}^{1-\theta_{2}},\qquad\theta_{2}=\frac{\frac{1}{d}}{\frac{2}{d}-\frac{1}{p}+\frac{1}{2}}\in(0,1).
P^{0,1}\mu_1P^{1,0}\bar\mu_2P^{0,1}{\rm ad}\nu_i\Delta^{-1}_0\star{\rm ad}\nu_j\star P^{0,1}\nu_i =i P^{0,1}\mu_1P^{1,0}\bar\mu_2P^{0,1}{\rm ad}\nu_i\Delta^{-1}_0{\rm ad}\nu_i\star \nu_j
I_n(p_{\phi})={1\over(n-1)!}{\Gamma(n\epsilon)\over\epsilon^{n-1}}{\Gamma(1-\epsilon)^{n+1}\over\Gamma(2-(n+1)\epsilon)}\times{1\over (1-2\epsilon)\cdots(1-n\epsilon)}\times{1\over [p_{\phi}^2]^{n\epsilon}}
\left< \frac{i}h [-h^2\Delta+V(x)-E(h), \chi_\alpha(x_n)hD_n] u_h,\, u_h \right>_{L^2(\Omega_\Gamma)}= \int_{\Sigma_{\Omega_\Gamma}}\left( 2\chi_\alpha'(x_n) \xi_n^2-\chi_\alpha(x_n)\partial_{x_n}(R+V) \right) \, d \mu+o(1) = I_1-I_2+o(1)
L_{+}^{C}= \left( \begin{array}{l l}{( q t^{- 1})^{S_{3}}}&{\Omega S^{-}}\\{0}&{( q t )^{- S_{3}}}\\ \end{array}\right) \ , L_{-}^{C}= \left( \begin{array}{l l}{( q t )^{- S_{3}}}&{0}\\{- \Omega S^{+}}&{( q t^{- 1})^{S_{3}}}\\ \end{array}\right) \ ,
R_h(x_0,\dots,x_{\rho(X)},y_0,\dots,y_{\rho(X)}) = \sum_{\beta=1}^{B_h}r_{h\beta}\prod_{i=1}^{k_{h\beta}}x_{v(h,\beta, i)}\prod_{j=1}^{\tilde k_{h\beta}}y_{w(h,\beta, j)},
\langle{\cal O}\rangle_R =\sum_{n=0}^{\infty}\frac{1}{n!}\frac{1}{(2 \pi)^n}\int \left[\prod_{i=1}^n d\theta_i g(\theta_i,R) e^{-m R \cosh\theta_i}\right]{\cal F}^{\cal O}_{2n}(\theta_1,\ldots,\theta_n)\,\,\,,
\frac{2{\rm Li}_{(k_1, k_2,\ldots, k_r)}(1-(ab)^{-t})}{(a^{-t}+b^{t})^r}e^{rxt}=\sum_{n=0}^{\infty}{E}^{(k_1, k_2,\ldots, k_r)}_n(x;a,b)\frac{t^n}{n!}.
\alpha I(X_2^1;Y_3^1) + \bar{\alpha}I(X_2^2;Y_3^2) =I(X_2;Y_3|U_{2,Q},Q) + I(U_{2,Q};Y_3|Q) \leq I(X_2;Y_3|U_{2,Q},Q) + I(U_{2,Q},Q;Y_3)=I(X_2;Y_3).
\lvert A_{N}f(y) \rvert \leq \lvert A_{N}f(y) - A_{N}f(x) \rvert + \lvert A_{N}f(x) \rvert \leq C''\cdot N^{-\gamma}\cdot \lVert f \rVert_{Lip}+ \lvert A_{N}f(x) \rvert.
{\bf{w}}_i^{{\mathrm{ZF}}}= \frac{{\left({{\bf{I}}-{\bf{\tilde H}}_i^H{\bf{\tilde H}}_i}\right){\bf{v}}_i^H}}{{\left\|{\left({{\bf{I}}-{\bf{\tilde H}}_i^H{\bf{\tilde H}}_i}\right){\bf{v}}_i^H}\right\|}}.
\hskip.1in h_{\widetilde H}(x,t) & = \sum_k \sum_{(q,m)\in\Gamma}b_{(q,m),k}\,\widetilde Hg(t - (q-k)T)\,e^{-2\pi i m(q-k)/L}& \hskip1in \widetilde \Phi_{(q,m)}(t,(x-t)+(q-k)T).
z_1w_1+z_2w_2 = 1 S w_1\,dz_1+w_2\,dz_2+z_1\,dw_1+z_2\,dw_2 = 0S w_1\,dz_1+w_2\,dz_2 = 0 H z_1\,dw_1+z_2\,dw_2 = 0 H
(F\odot G)(z)=\frac{1-a_{2}b_{2}z^{2}}{1-a_{1}b_{1}z+(a_{2}b_{1}^{2}+a_{1}^{2}b_{2}-a_{2}b_{2})z^{2}-a_{1}a_{2}b_{1}b_{2}z^{3}+a_{2}^{2}b_{2}^{2}z^{4}}.
-\frac{d}{ds}{\displaystyle\sum\limits_{n=1}^{\infty}}\frac{1}{f_{n}^{s}}\sim\frac{1}{\ln(\varphi)s^{2}}+\frac{1}{24}\left(6\ln(5)-2\ln(\varphi)-\frac{3\ln(5)^{2}}{\ln(\varphi)}\right) +\ln(c) & \sim\frac{1}{\ln(\varphi)s^{2}}+\ln(0.8992126807...)
\ll X^{1-2\delta}\iint_{|v-t| \le U}|\zeta^k(\tfrac12+\delta-it)\zeta^k(\tfrac12+\delta+iv)|(tv)^{\delta k-1}dv \, dt &\ll U X^{1-2\delta}\int_{Y-U}^{X+U}|\zeta(\tfrac12+\delta+it)|^{2k}\frac{dt}{t^{2-2\delta k}}\ll X^{1-2\delta+\eta}Y^{2\delta k-1+\epsilon},
\mathcal{J}^+ u(z,s):=\left\{( p,\tau )\in \mathbf{R}^n \times \mathbf{R}\ \left| \begin{array}{c}\exists \phi \in C^1 (\mathbf{R}^n \times (0,T)) \ \mbox{such that}(p,\tau)=(\nabla \phi,\partial_t \phi)(z,s) \ \mbox{and}\displaystyle\max_{\mathbf{R}^n \times (0,T)}(u-\phi)=(u-\phi)(z,s) \end{array}\right. \right\}.
A_{s,l}[i][j] = \frac{\rho_s[i]\rho_u[j]}{D[i][j](T_{s,l}[i]+\rho_s[i])\ln2}> 0, A_{u,l}[i][j] = \frac{\rho_s[i]\rho_u[j]}{D[i][j](T_{u,l}[j]+\rho_u[j])\ln2}> 0,
\lim_{\rho \rightarrow 0^{+}}\frac{\mathcal{H}^{n}\left((0, \tau) ) + E_{2}) \cap B_{\rho}( ( \bar{z},w))\right)}{\omega_{n}\rho^{n}}\leq \frac{\omega_{n-1}}{\omega_{n}}\lim_{\rho \rightarrow 0^{+}}\frac{\mathcal{H}^{1}\left(( \bar{z}, \bar{z}+ \rho) \cap \left\{r_{\ell}> \frac{\epsilon}{2}\right\}\right)}{\rho}=0,
\left\{\begin{array}{c}g_{\rho\sigma}g_{\mu\nu}\g_{\alpha\beta}\end{array}\right\}=\frac{1}{4}g^{\mu\nu}\delta^{\alpha\beta}_{\rho\sigma}+\frac{1}{4}g^{\alpha\beta}\delta^{\mu\nu}_{\rho\sigma}-\frac{1}{2}\delta^{\mu(\alpha}_{\rho\sigma}g^{\beta)\nu}-\frac{1}{2}\delta^{\nu(\alpha}_{\rho\sigma}g^{\beta)\mu}+\frac{1}{n-2}g_{\rho\sigma}C^{\mu\nu,\alpha\beta}.
ES_{n+1}^{2l}\le c_l \bigg(\sum_{j=1}^n\big( \sum_{r=2}^{2l}A_{2l}^r\|S_j\|_{2l}^{2l-r}+A_{2l}^2\big \|S_j\|_{2l}^{2l-2}+ A_{2l}^2\|S_{j-1}\|_{2l}^{2l-2}\big) \bigg) &\le c_l \bigg(\sum_{j=1}^n\sum_{r=2}^{2l}A_{2l}^r \|S_j\|_{2l}^{2l-r}\bigg)
(1-2\rho_{jm}\cos(\beta_{j,m}(i_{j},i_{m}))+\rho_{jm}^{2})(1-2\rho_{jm}\cos(\beta_{j,m}(i_{j},-i_{m}))+\rho_{jm}^{2}) =w_{2}(\cos(\alpha_{j}),\cos(\alpha_{m})|\rho_{jm}).
\lim_{D \rightarrow \infty}(2 \cos(\tfrac{\pi}{3}+ \tfrac{\pi}{2D}))^D = \lim_{D \rightarrow \infty}\exp(D \log(1- \tfrac{\sqrt{3}\pi}{2 D}+ O(D^{-2}))) = \exp(-\tfrac{\sqrt{3}\pi}{2}) = 0.0658\dots.
\frac{2}{N^{2 \alpha + 1}}\int_{0}^{\infty}\d x x^{2 \alpha}\left. P_{2 n}^{( \alpha )}( \frac{x}{N}) \right|_{n = N t / 2}\left. P_{2 m}^{( \alpha )}( \frac{x}{N}) \right|_{m = N t^{\prime}/ 2}= h_{2 n}^{( \alpha )}\frac 2 N \delta ( t - t^{\prime})
\Upsilon_{a \alpha}( \vec{k}) = \left[ \begin{array}{c}{{\Upsilon_{1 1}}}\\{{\Upsilon_{2 1}}}\\{{\Upsilon_{1 2}}}\\{{\Upsilon_{2 2}}}\\ \end{array}\right] = \left[ \begin{array}{c}{{c_{k \uparrow}^{\vphantom \dagger}}}\\{{c_{- k \downarrow}^{\dagger}}}\\{{c_{k \downarrow}^{\vphantom \dagger}}}\\{{- c_{- k \uparrow}^{\dagger}}}\\ \end{array}\right] .
\frac{|a|^2|\beta_0|^2}{1+|a|^2}+\frac{\sum_{j=0}^{\infty}\left|\overline{a}\beta_j+a\beta_{j+1}\right|^2}{(1+|a|^2)^2}< \frac{|a|^2|\beta_0|^2}{1+|a|^2}+\frac{2\sum_{j=1}^{\infty}|a|^2|\beta_j|^2}{(1+|a|^2)^2}+\frac{2\sum_{j=0}^{\infty}|a|^2|\beta_j|^2}{(1+|a|^2)^2}\leqslant \frac{4|a|^2}{(1+|a|^2)^2}.
g(x) - x = (s_1s_2\cdots s_k(x) - s_2\cdots s_k(x)) + (s_2s_3\cdots s_k(x) - s_3 \cdots s_k(x)) + \ldots + (s_k(x) - x) & = (s_1 - 1)(x_1) + (s_2 - 1)(x_2) + \ldots + (s_k - 1)(x_k)
\int_{C}F(r)\cdot dr^{\perp}=\int_{a}^{b}[\begin{matrix}P(x(t),y(t))\\Q(x(t),y(t))\end{matrix}]\cdot[\begin{matrix}y^{\prime}(t)\\ -x^{\prime}(t)\end{matrix}]dt=\int_{a}^{b}(-Qdx+Pdy).
\ell_{i}= \ell_{\left \lfloor \frac{N}{2}\right \rfloor +i}= \frac{2i}{N}, \i \in \left [1: \left \lfloor \frac{N}{2}\right \rfloor \right], & r_{i}= r_{\left \lfloor \frac{N}{2}\right \rfloor +i}= \frac{N-2i+2}{N}, \i \in \left [1:\left \lfloor \frac{N}{2}\right \rfloor \right], & N \ \ \ell_N = L, \r_N = \frac{1}{N},
\int_\Omega \left(\frac{1}{2}\rho|u|^2+P(\rho)\right)(\tau)+\int_0^\tau\int_\Omega \mathbb S(\nabla u)\cdot\nabla u &\leq \int_\Omega \left(\frac{1}{2}\rho_0|u_0|^2+P(\rho_0)\right)+\int_0^\tau\int_\Omega \rho f\cdot u.
[F_{i}, E_{\xi_{i}}] = [F_{i}, -E_{i}E_{\xi_{i + 1}}+ q^{-1}E_{\xi_{i+1}}E_{i}] & = \frac{K_{i}- K_{i}^{-1}}{q - q^{-1}}E_{\xi_{i + 1}}- q^{-1}E_{\xi_{i + 1}}\frac{K_{i}- K_{i}^{-1}}{q - q^{-1}}.
\underset{z\in \Omega}{\sup}\left | f^{(l)}(z) \right | \leqslant \underset{z\in \Omega}{\sup}\left | f^{(\alpha)}(z) \right|\cdot ((\Omega))^{a-l}+\sum_{k=l}^{a-1}\left | f^{(k)}(z_o) \right | \cdot ((\Omega))^{k-l}.
\Psi^*_{f_\rho}(\mu_\phi(\tilde{f_\alpha}))= \sup_t \left[ (\alpha - \rho) t - (\alpha - \rho) \frac{t}{\sqrt{n}}+ \frac{\rho-\alpha}{n}t - \frac{(1-\rho)\rho}{2n}t^2 + O \left( \frac{1}{n^{3/2}}\right) \right].
R_{\alpha_0, j}^i R_{\beta_0}^j -(-1)^{\epsilon_{\alpha_0}\epsilon_{\beta_0}}R_{\beta_0, j}^i R_{\alpha_0}^j =- R_{\gamma_0}^i F_{\alpha_0 \beta_0}^{\gamma_0}- M^{ij}_{\alpha_0 \beta_0}S_{{\rm cl}, j},
\sum_{i=A}^{A+B-1}(i-1)\binom{i+n-2-A}{n-2}=(A-1)\sum_{i=0}^{B-1}\binom{i+n-2}{n-2}+\sum_{i=0}^{B-1}j\binom{i+n-2}{n-2}&= (A-1)\binom{B+n-2}{n-1}+\sum_{i=0}^{B-1}i\binom{i+n-2}{n-2}.
k_{i}^{\pm}(u)k_{j}^{\pm}(v)^{-1}=k_{j}^{\pm}(v)^{-1}k_{i}^{\pm}(u), k_{i}^{\pm}(u)^{-1}k_{j}^{\pm}(v) =k_{j}^{\pm}(v)k_{i}^{\pm}(u)^{-1}, k_{i}^{\pm}(u)^{-1}k_{j}^{\pm}(v)^{-1}=k_{j}^{\pm}(v)^{-1}k_{i}^{\pm}(u)^{-1},
\frac{\partial^2 \widetilde{R}_{1,k}}{\partial\alpha^2}=-\frac{2\tau}{T}\cdot\frac{1}{ln2}\cdot\frac{1}{\alpha+g}+\big(1-\frac{\tau}{T}+\frac{\alpha\tau}{T}\big)\cdot\frac{1}{ln2}\cdot\frac{1}{(\alpha+g)^2}&=\Big(\frac{1}{ln2}\cdot\frac{1}{\alpha+g}\Big)\Big(-\frac{2\tau}{T}+\big(1-\frac{\tau}{T}+\frac{\alpha\tau}{T}\big)\cdot\frac{1}{(\alpha+g)}\Big).
\mathbf{v}_{k}(z):=\begin{cases}(A_{0}^{(k)},\ldots,A_{p-k-1}^{(k)},-\sum_{j=k}^{p}a_{0}^{(j)}A_{j-k}^{(k)},\ldots,-\sum_{j=1}^{p}a_{k-1}^{(j)}A_{j-1}^{(k)}), 0\leq k\leq p, begin{align*}0.5em](-a_{k-p}^{(p)}A_{0}^{(k)},-\sum_{j=p-1}^{p}a_{k-p+1}^{(j)}A_{j-p+1}^{(k)},\ldots,-\sum_{j=1}^{p}a_{k-1}^{(j)}A_{j-1}^{(k)}), k\geq p+1,\end{cases}
\begin{array}{c}L=g_{\alpha\beta}(dX^{\alpha}\cdot dX^{\beta})+2g_{\alpha a}(dX^{\alpha}\cdot dX^a)+g_{ab}(d\Phi^a \cdot d\Phi^b)+ +b_{\alpha\beta}dX^{\alpha}\wedge dX^{\beta}+2b_{\alpha a}dX^{\alpha}\wedge d\Phi^a + b_{ab}d\Phi^a\wedge d\Phi^b\end{array}
|B_j|_2^2=\int_{y_j}^{y_j+a_j}\int_{y_j}^{y_j+a_j}|B_j(u,v)|^2dudv = \int_{y_j}^{y_j+a_j}\int_{y_j}^{y_j+a_j}O\left(\dfrac{n^2}{(1+n|u-v|)^2}\right)dudv = \int_{y_j}^{y_j+a_j}O\left(\int_{\mathbb{R}}\dfrac{n^2}{(1+n|u-v|)^2}du\right)dv = \int_{y_j}^{y_j+a_j}O\left(n\right)dv=O(na_j)=O\left((\ln n)^{1/2}\right),
\beta(\lambda-\lambda_1 z_1-\lambda_2 z_2)=\int_0^{\infty}\sum_{k=0}^{\infty}\frac{(\lambda(q z_1+p z_2)t)^k}{k!}\cdot e^{-\lambda t}dF_{\beta}(t)=\beta(\lambda)+\sum_{k=1}^{\infty}b_k (q z_1+p z_2)^k,
\sigma_{\lambda \mu}( I_{1}, I_{2}, I_{3}, I_{4}) ={\frac{[ 2 I_{3 4}+ \lambda + \mu ] ! [ I_{3 4}- I_{1 2}+ \lambda ] !}{[ 2 I_{3 4}+ 2 \lambda ] ! [ I_{3 4}- I_{1 2}+ \mu ] !}}\left[{\lambda \atop \mu}\right] \ .
v_n(t)\leq \left(\dfrac{a}{\sqrt{M}}\right)^{n-1}\left(\dfrac{C_{\varphi}}{\sqrt{M}}\sum_{k=0}^{n-2}\dfrac{\sqrt{M}^kC_f^k(T-t)^k}{a^kk!}\binom{n-2}{k}+\dfrac{C_{\varphi}+C_0T}{\sqrt{M}}\sum_{k=1}^{n-1}\dfrac{\sqrt{M}^kC_f^k(T-t)^k}{a^kk!}\binom{n-2}{k-1}\right)
\mathbb{E}^V \theta \left[\frac{N}{\pi}\int_I \tilde{G}_{\mu\nu}(z)q[\operatorname{Tr}(\mathcal{X}_E* \vartheta_{\tilde{\eta}})(Q_1)]dE\right]-\mathbb{E}^V\theta \left[\int_I x(E) q(y(E))dE\right]=o(1),
\mathrm{T a b l e I . \, C l a s s i f i c a t i o n \, o f \,}Q_{\alpha}^{a}\mathrm{\, a n d \, Z}_{\mu_{1}\cdots \mu_{p}}^{a b}\mathrm{\, u n d e r \, 1 1 D \, ( o r \, 1 2 D ) \, a n d \,}K \mathrm{.}
\langle i | P^{-}| j k \rangle = \left[ ( p_{i}^{-}- p_{j}^{--}p_{k}^{-}) \sqrt{\frac{J_{j}J_{k}}{J_{i}}}C_{i j k}\right] p_{i}^{+}\delta ( p_{i}^{+}- p_{j}^{+}- p_{k}^{+})
\xi^{a i}= \psi^{a i}\sqrt{2 \frac{\partial^{2}A}{{( \partial \phi^{i})}^{2}}}, \quad \bar{\xi}_{a}^{i}= \bar{\psi}_{a}^{i}\sqrt{2 \frac{\partial^{2}A}{{( \partial \phi^{i})}^{2}}},
J_{\nu}^{(1)}\left(z;q\right)\left(\frac{z}{2}\right)^{-\nu}=\frac{\sqrt{\log q^{-1}/\left(2\pi\right)}}{\left(q;q\right)_{\infty}}\int_{-\infty}^{\infty}q^{\alpha^{2}/2}\left(\frac{z^{2}q^{1/2}}{4q^{\alpha}},q\right)_{\infty}A_{q}\left(q^{\alpha+\nu+1/2}\right)d\alpha.
\frac{6d^{2}-14d+7}{4\left(d-1\right)^{2}}=\frac{3}{4}+O\left(\frac{x}{n}\right), \frac{1}{2\left(d-1\right)^{3}}=\frac{1}{2}+O\left(\frac{x}{n}\right), \frac{\left(2d^{2}-4d+1\right)}{4\left(d-1\right)^{4}d^{2}}= \frac{1}{16}+O\left(\frac{x}{n}\right).
g_{i}(x_{i}^{*})\leq0,\,\,\,\,\,\,\lambda_{i}^{*}g_{i}(x_{i}^{*})=0,\,\,\,\,\,i=1,\ldots,N, \sum_{i}^{N}\frac{\partial f_{i}(x_{i}^{*})}{\partial x_{i}}+\lambda_{i}^{*}\frac{\partial g_{i}(x_{i}^{*})}{\partial x_{i}}=0,\,\,\,\,i=1,\ldots,N.
\sum_{x\in\Bbb F_{q^2}}f(x)^{1+(q-2)q}=\, -(a^{q-1}b^{1-q}+a^{-2}b^{1-q})\sum_{0\le l\le\frac{q-1}2}(l+1)\binom{-l}{l+1}z^l\cr +2a^{-2}b^{1-q}\sum_{0\le l\le\frac{q-1}2}\binom{-l}{l+1}z^l-a^{-2}b^2\sum_{0\le l\le\frac q2}(l+1)\binom{-l}lz^l\cr +2a^{-2}b^2\sum_{0\le l\le\frac q2}\binom{-l}lz^l-a^{-2}b^2-a^{-1}.
[ \, X_{0}, \, Y_{0}\, ] = X_{1}, \, \, \, [ \, X_{0}, \, Y_{1}\, ] = - X_{1}, \, \, \, [ \, X_{1}, \, Y_{0}\, ] = - Y_{0}, \, \, \, [ \, X_{1}, \, Y_{1}\, ] = 2 Y_{1}.
R_+ = R_- v_R, v_R=I + \Delta_R \tag*{(i)$_R$}\| \Delta_R\|_{L^1 (\Sigma_R)}+\| \Delta_R\|_{L^\infty (\Sigma_R)}= \mathcal{O}\left( N^{-1}\right) , &\lim_{|z| \to \infty}R(z)=I. \tag*{(ii)$_R$}
Z_{N}( \tau ) = \frac 1{2 N}\sum_{k , l = 0}^{2 N - 1}D_{k / ( 2 N ) , l / ( 2 N )}( \tau ) = \frac 1{| \eta ( \tau ) |^{2}}\sum_{n = 0}^{4 N^{2}- 1}\left| \Theta_{n , 2 N^{2}}( \tau ) \right|^{2}.
\left|\int_{x^n_i(t)}^{x^n_{i+1}(t)}\left[\phi(t,x)-\phi(t,x^n_{i+1}(t))\right]{\rm{d}}x\right|\le\frac{{\mathrm{Lip}}[\phi]}{2}\left[x^n_i(t) - x^n_{i+1}(t)\right]^2= \frac{{\mathrm{Lip}}[\phi]}{2}\frac{\kappa_n^2}{y^n_i(t)^2},
\sum_{j,l=1}^k \langle \pi(\psi(x_j^\ast a x_l)) c \pi(b_l) \xi, \pi(b_j) \xi\rangle = \langle \pi(\sum_{j,l=1}^k b_j^\ast\psi(x_j^\ast a x_l) b_l) c \xi , \xi\rangle = \eta_{\psi_0}(a\otimes c)
\mathbb E\left[e^{s\underline{X}_{e(q)}}\right] =\prod_{k=1}^{n^-}\left(\frac{s+\vartheta_k}{\vartheta_k}\right)^{n_k}\prod_{k=1}^{N}\left(\frac{\gamma_{k,q}}{s+\gamma_{k,q}}\right)=\sum_{k=1}^{N}\frac{D^q_{k}}{s+\gamma_{k,q}}:=\psi_q^-(s),
dP(x) \equiv d\sum\limits_{i=0}^{q-2}\binom{q}{i}(lx+1)^{q-i}B_{i}-dqB_{1}x^{q-1}-d \sum\limits_{i=0}^{\frac{q-2}{2}}\binom{q}{2i}B_{2i}x^{q-2i}\pmod{4}.
iG^{[s]}(x,x')_{mn}^{ab}=\int\limits_0^\infty\,dT\int\limits_{x(0)=x',x(T)=x}Dx(t) exp\left({-1\over 4}\int\limits_0^Tdt\,\dot{x}^2\right)\Phi^{[s]}_{mn}[C^{xx'}]U[C^{xx'}]^{ab},
\frac{y}{dy}\frac{dx}{x}\frac{e^{-w\,x\,y}}{(x\,y)^v}(x\partial_x-v)F(w,v,z)=(v+y\partial_y+ (y\partial_y(x\,y))\,w)\frac{e^{-w\,x\,y}}{(x\,y)^v}F(w,v,z).
f_{\omega^{\prime}l m}^{b}= \int d \tilde{\omega}\, d \theta_{0}\, d \eta_{0}\, ( \alpha_{\omega \omega^{\prime}l m}^{b}p_{\omega}+ \beta_{\omega \omega^{\prime}l m}^{b}\bar{p}_{\omega}+ \mathrm{t e r m s ~ i n v o l v i n g ~}q_{\omega}) ,
f = z^3 (\alpha^2 x^{4}+ \alpha^2 x^{3}+ x^{2}+ 1) + z^2 (\alpha x^{4}+ x^{2}+ \alpha^2 x) &+ z (\alpha x^{4}+ x^{3}+ \alpha x^{2}+ 1) + (\alpha^2 x^{4}+ \alpha x^{3}+ \alpha x^{2}+ \alpha^2 x).
\mathcal{H}_{C_{n}^{( 1 )}}^{\mathit{T o d a}}={\frac{1}{2}}P^{2}+ m^{2}e^{- 2 Q_{1}}+ m^{2}\sum_{j = 1}^{n - 1}e^{Q_{j}- Q_{j + 1}}+ m^{2}e^{2 Q_{n}},
\mathrm{E}_{\theta,\varepsilon^2}\sum_{i=1}^{D}(\theta_{i}-\hat{\theta}_{\mathrm{MA},\beta,i})^2\leq \mathrm{E}_{\theta,\varepsilon^2}\sum_{d=1}^{D}\tilde{w}_{d}^{(D)}\hat{r}^{(D)}_{d}+ D\varepsilon^{2}.
S_{\alpha}(s)u_{k}=S_{\alpha}(s)\big(\frac{1}{\nu(d,\alpha)}\phi_{k}\chi_{r_{k}}\big) &=\frac{\phi_{k}}{\nu(d,\alpha)}S_{\alpha}(s)\chi_{r_{k}}&\geq \phi_{k}\chi_{r_{k}+t^{1/\alpha}}&\geq \phi_{k}\chi_{r_{k}},
~~~~\|(\partial_x^jG\ast (uv_xm-u_x vm)\varphi_N\|_{\infty}&\leq C_0(\|v_xm\|_{\infty}\|u\varphi_N\|_{\infty}+\|vm\|_{\infty}\|u_x\varphi_N\|_{\infty}) &\leq C_0K^2(\|u\varphi_N\|_{\infty}+\|u_x\varphi_N\|_{\infty}),~~j=0,1
\left[{\frac{1}{l}}\hat{\gamma}^{a}e_{a}^{\i}( x ) \Pi_{i}+ \hat{\gamma}^{\mu}( - i \partial_{\mu}+{\frac{1}{2}}{\mathcal A}_{\mu , a b}\hat{\Sigma}^{a b}) +{\mathcal O}( l ) \right] \hat{\psi}= \hat{\sf p}\ \hat{\psi}
v^{\infty}(\hat{x}) = \sqrt{\frac{2k_p}{\pi}}^{i\frac{\pi}{4}}\sum_{n \in \mathbb{Z}}i^{-n}a_n^{in\theta_x}\hat{x}+ \sqrt{\frac{2k_s}{\pi}}^{i\frac{\pi}{4}}\sum_{n \in \mathbb{Z}}i^{-n}b_n^{in\theta_x}\hat{x}^{\bot}.
\mathcal{B}_{10}((\tilde{F}, \tilde{G}, \tilde{H}), (\hat{F}, \hat{G}, \hat{H}))=\int_0^T\begin{pmatrix}R^{\tilde{F}_s, \tilde{G}_s}_{s,1}R^{\tilde{F}_s, \tilde{G}_s, \tilde{F}_s}_{s,2}R^{\tilde{G}_s, \tilde{F}_s}_{s,3}\end{pmatrix}\cdot\begin{pmatrix}\hat{F}_s \hat{G}_s \hat{H}_s\end{pmatrix}ds.
F_{\alpha\beta}=(\begin{matrix}0&\frac{E_{x}}{c}&\frac{E_{y}}{c}&\frac{E_{z}}{c}\\ -\frac{E_{x}}{c}&0&-B_{z}&B_{y}\\ -\frac{E_{y}}{c}&B_{z}&0&-B_{x}\\ -\frac{E_{z}}{c}&-B_{y}&B_{x}&0\end{matrix})
\frac{d^{2}\tilde{F}_{0}}{d \tilde{z}^{2}}+ \tilde{z}\frac{d \tilde{F}_{0}}{d \tilde{z}}+ ( 1 - \frac{1}{4 \tilde{z}^{2}}+ \frac{\mu^{2}}{\tilde{z}^{4}}) \tilde{F}_{0}= 0
\int_{\textstyle{\prod_{t=1}^m [b_{t,1}, b_{t,2}]}}\,\frac{du_1du_2\ldots du_m}{\displaystyle \sum_{t=1}^{m}\left(u_t-c_t\right)^{2s}\left(u_t-d_t \right)^{2s}}&\hskip 2cm\le M\,\Big(\max\Big\{b_{1,2}-b_{1,1},b_{2,2}-b_{2,1},\ldots,b_{m,2}-b_{m,1}\Big\}\Big)^{m-4s}.
\frac{e^{2}}{( 2 \pi )^{2}}\int \langle F_{1 2}( x ) F_{1 2}( 0 ) \rangle e^{- i p x}d x \; \; = \; \; P^{0}( 0 ) - \int_{0}^{\infty}\frac{d \rho ( \mu^{2})}{p^{2}+ \mu^{2}}\; .
\mathcal{L}_X (*_\mu P ) = \mathcal{L}_X \iota_P \mu = \mathcal{L}_X \iota_P \mu - \iota_P \mathcal{L}_X \mu + \iota_P \mathcal{L}_X \mu = \iota_{[X, P]}\mu + \iota_P (_\mu X) \mu = \iota_{\mathcal{L}_X P}\mu + (_\mu X) \iota_P \mu = *_\mu (\mathcal{L}_X P) + (_\mu X) *_\mu (P).
f = (f_1,f_2)= \left(\frac{\varphi^1-\varphi^1\circ I}{g - g\circ I},\frac{\bar\psi^1-\bar \psi^1\circ I}{\bar g - \bar g\circ I}\right)=\left(\frac{\varphi^1 - \bar \psi^2}{g + \frac{1}{\bar g}},\frac{\bar\psi^1 + \varphi^2}{\bar g + \frac{1}{g}}\right).
\sum_{k=0}^{\infty}\bigg(\frac{1}{2}\bigg)^{k}\frac{(\frac{1}{2})_k^2}{(1)_{k}(b)_{k}}\bigg\{H_{2k}^{(2)}-\frac{1}{4}H_{k}^{(2)}\bigg\}=\frac{\Gamma(\frac{b}{2})\Gamma(\frac{1+b}{2})}{16\Gamma(\frac{1+2b}{4})^2}\psi\,'\bigg(\frac{1+2b}{4}\bigg).
\lambda_{a|b|c|d}^{e|f|g|h}= \sum_{(p,q)\in [a,b]\times [c,d]}\lambda_{a|p|c|q}^{e|e|g|h}\lambda_{p|b|q|d}^{e|f|h|h}&= \sum_{(p,q)\in (a,b]\times [c,d]}\lambda_{a|p|c|q}^{e|e|g|h}\lambda_{p|b|q|d}^{e|f|h|h}+ \sum_{q\in [c,d]}\lambda_{a|a|c|q}^{e|e|g|h}\lambda_{a|b|q|d}^{e|f|h|h}&=0
\delta\lambda=\frac{\mathbb{E}^{\mathbb{Q}}\left[\delta\mathcal{Y}_T^{v}-\delta\mathcal{Y}_0^{v}\right]}{T}=\frac{\mathbb{E}^{\mathbb{Q}}\left[\mathbf{y}^{\alpha_T}(V_T^v)-\bar{\mathbf{y}}^{\alpha_T}(V_T^v)\right]-\left[\mathbf{y}^i(v)-\bar{\mathbf{y}}^i(v)\right]}{T}.
\begin{aligned}\int_{\mathcal C}\vert e^{\rho\cdot \mathbf x}\vert\vert \psi(\mathbf x)\vert\mathrm d\mathbf x \leq\frac{1}{\tau^3}\|e^{-\mathbf d\cdot \mathbf y}\|_{L^{\frac{8}{7}}(\mathcal C)}\left \|\psi\left (\frac{\mathbf y}{\tau}\right )\right \|_{L^8(\mathcal C)}\lesssim \tau^{-\frac{121}{40}}, \end{aligned}
D^{\mu\nu}(x,y) = i\int \frac{d^4k}{(2\pi)^4}frac{e^{-ik_\lambda(x-y)^\lambda}}{k_\lambda k^\lambda +i\epsilon}\left[g^{\mu\nu}-\frac{\gamma k^\mu k^\nu}{k_\lambda k^\lambda+ i\epsilon}\right];
- \partial_{q}W = \frac{4 \pi \hbar}{T_{B H}}\int_{- \infty}d z \, r^{2}F \, \biggl [ \frac{\partial_{q}F}{F}\, \langle T_{\tau}^{\tau}+ T_{z}^{z}\rangle + 2 \frac{\partial_{q}r}{r}\, \langle T_{\theta}^{\theta}+ T_{\phi}^{\phi}\rangle \biggr ] + \cdots
- \frac{h^2}{2 \lambda}\int dt d^2x d^2x'\, ({\tilde J}_k - \frac{J^0_k}{\rho_0}{\tilde J}_0) (t,x) \Delta^{-1}(x-x')({\tilde J}_k - \frac{J^0_k}{\rho_0}{\tilde J}_0) (t,x')\;.
\mu_{4}=m^{4}(\frac{K_{4}(a)}{K_{0}(a)}-4\frac{K_{1}(a)K_{3}(a)}{K_{0}^{2}(a)}+6\frac{K_{1}^{2}(a)K_{2}(a)}{K_{0}^{3}(a)}-3\frac{K_{1}^{4}(a)}{K_{0}^{4}(a)})
\sum_{\pi: \pi \sim (13)(24) \textnormal{or}(1324)}\Gamma_{\pi^{-1},G}\, = \frac{2}{L^{2d}}\sum_{k_1,\ldots,k_4}\hat \gamma^{(1)}(k_1) \cdots \hat \gamma^{(1)}(k_4) \chi_{(k_1+k_2-k_3-k_4=0)}\left[ \hat g(k_2 -k_4) \hat g(k_4-k_2) - \hat g(k_2-k_4) \hat g(k_4-k_1) \right].
\int_{M}|\tilde{F}-(1-b_{t_0,B}(\Psi_1))fF^{1+\delta}|^2_{\tilde{h}}e^{v_{t_0,B}(\Psi_1)-\delta \tilde{M}}c(-v_{t_0,B}(\Psi_1)) \le \left(\frac{1}{\delta}c(T_1)e^{-T_1}+\int_{T_1}^{t_0+B}c(t)e^{-t}dt\right)\int_M \frac{1}{B}\mathbb{I}_{\{-t_0-B< \Psi_1 < -t_0\}}|fF|^2_{\tilde{h}}.
S \ll \widehat{Y}^{(n+1)/2+\varepsilon}|d|\left(\widehat{V}^n\widehat{Z}^{-n/2}+\widehat{Y}^{n/3}\right)\sum_{\substack{|r|=\widehat{Y}d\mid r}}|b_2r_3'|^{1/2}|b_1'b_2'r_3''| & = \widehat{Y}^{(n+3)/2+\varepsilon}|d|\left(\widehat{V}^n\widehat{Z}^{-n/2}+\widehat{Y}^{n/3}\right)\sum_{\substack{|r|=\widehat{Y}d\mid r}}|b_1|^{-1}|b_2r'_3|^{-1/2},
K ( y , x ; T )_{\mu \nu}^{a b}= \int_{x ( 0 ) = x , x ( T ) = y}{D x ( t ) \exp ( - \frac{1}{4}\int_{0}^{T}{d t \dot{x}^{2}}}) \mathrm{P}\exp ( \frac{i}{2}\int_{0}^{T}{d t J \cdot \omega )_{\mu \nu}}\mathrm{P}\exp ( i g \int_{0}^{T}{d t \dot{x}\cdot B )^{a b}}.
\check\Gamma_{\alpha\beta}^{\mu\nu}\epsilon^{3,4}_{\mu\nu}= \left(k^2 + \Pi_B\right) \epsilon^{3,4}_{\mu\nu}+ 2\left(1-\frac{k^2}{(k\cdot u)^2}\right)\Pi_6 \epsilon^{5,6}_{\mu\nu}
x = (x_{j})_{j=1}^\infty \in \ell_{q,p}^{}(E) \Rightarrow \Psi_x \in \Pi_{p}(E^{*};\ell_{q}) & \Rightarrow \Psi_x \in \Pi_{s}(E^{*};\ell_{q}) & \Rightarrow \Psi_x \in \Pi_{s}(E^{*};\ell_{r}) \Rightarrow x = (x_{j})_{j=1}^\infty \in \ell_{r,s}^{}(E).
((\phi^{-1})^*(\phi^*(r_{g})\circ \sigma))^{-1}\circ (r_g\circ \psi)=(r_g\circ (\phi^{-1})^*\sigma)^{-1}\circ r_g\circ \psi=((\phi^{-1})^*\sigma)^{-1}\circ\psi.
\sum_{j=1}^{n}\binom{n}j(-1)^{j-1}\frac{a_j}{j^k}= \sum_{i_1=1}^{n}\frac1{i_1}\sum_{i_{2}=1}^{i_{1}}\frac{1}{i_{2}}\dots \sum_{i_{k}=1}^{i_{k-1}}\frac1{i_{k}}\sum_{j=1}^{i_{k}}\binom{i_{k}}{j}(-1)^{j-1}a_j.
k_{s , \zeta}={\frac{\rho}{2 \alpha^{2}}}( \lambda_{, \zeta}- \tau_{0}\tau_{, \zeta})^{2}, \ \ \k_{e , \zeta}= - \rho \gamma q_{1}q_{2}( \tau_{\zeta})^{2}, \ \ \ \tau_{0}= q_{1}+ q_{2},
\begin{aligned}Q^{(m,n)}r(\infty) = C_{m,n}\sum_{i=0}^\infty \left(\prod_{j=0}^{i-1}C_{m,n}(j)\right)^{-1}\left(A_{m,n}(i)\right)^{-1}r(i) + \alpha I_{m,n}(\infty)= = \beta K_{m,n}(\infty)\end{aligned}
\bigg\lvert \frac{\lambda_1 (be-cd)}{(b-a_1d)^2}\frac{\lambda_2 (be-cd)}{(b-a_2d)^2}\bigg\rvert < \bigg\lvert \frac{(a_2-a_1 )(be-cd)}{(b - a_1 d)(b-a_2 d)}\bigg\rvert^2 & = \bigg\lvert \frac{c-a_1 e}{b - a_1 d}- \frac{c-a_2 e}{b - a_2 d}\bigg\rvert^2
\Delta_{A|M}(\hat{\rho}_{AM})(s+t) = \Delta_{A|M}(\hat{\rho}_{AM})(s) + \Delta_{A|M}((\mathcal{N}_A(s)\otimes\mathbb{I}_M)(\hat{\rho}_{AM}))(t) &\ge \Delta_{A|M}(\hat{\rho}_{AM})(s)\;.
\sum_{i = 0}^{n_x-l_x}{\vphantom{\sum}}'\sum_{j = n_y-l_y+1}^{\infty}\left|c_{i,j}\right|\le (n_x-l_x+1)\dfrac{4V_l}{2l\pi^2}\begin{cases}\Pi_{1}[r](n_y-l_y),~~~~~~~~~\mbox{if}l=2r, \Pi_{0}[r](n_y-l_y),~~~~~~~~~\mbox{if}l=2r+1,\end{cases}
\epsilon^{\alpha \mu \nu \beta}\gamma_{5}\gamma_{\mu}\partial_{\nu}\Psi_{\beta}^{i}-{\frac{\kappa \epsilon^{i j}}{\sqrt 2}}\big [ F^{\alpha \beta}+{\scriptstyle{\frac{1}{2}}}\gamma_{5}\epsilon^{\alpha \beta \rho \sigma}F_{\rho \sigma}\big ] \Psi_{\beta}^{j}= 0 .
{\cal L}_{H E}^{( 1 )}= \sum_{k = 2}^{\infty}{\frac{( - 1 )^{k}}{8 \pi^{2}}}{\frac{2^{2 k}\vert B_{2 k}\vert m^{4}}{2 k ( 2 k - 1 ) ( 2 k - 2 )}}{\frac{B^{2 k}}{B_{\mathrm{c r}}^{2 k}}},
S_{P}^{4}= \int d^{4}x \Bigl [{\frac{1}{4}}\varepsilon^{\mu \nu \lambda \rho}B_{\mu \nu}F_{\lambda \rho}-{\frac{1}{4}}B_{\mu \nu}B^{\mu \nu}-{\frac{\mu^{2}}{2}}A_{\mu}A^{\mu}\Bigl ] \; ,
ds^2=\alpha' \tilde{F}^{1/2}\left [ \frac{u^4}{R^4}(-\tilde{f}d\tilde{x}_0^2 +d\tilde{x}^2_1) + \frac{1}{\tilde{G}}(d\tilde{x}_2^2+d\tilde{x}_3^2 ) +\tilde{f}^{-1}du^2 +u^2d\Omega_5^2\right],
[D_*^q,M_{z^q}](f) =D^q_*(z^qf)(z)-z^qD_*^q(f)(z) &=a_0\Gamma(q+1)+\sum_{n=1}^{\infty}z^{qn}a_n\left[ \underbrace{\frac{\Gamma(q(n+1)+1)}{\Gamma(qn+1)}-\frac{\Gamma(qn+1)}{\Gamma(q(n-1)+1)}}_{:=\beta_{m,q}}\right] &=a_0\Gamma(q+1)+\sum_{n=1}^{\infty}a_n\beta_{m,q}\;z^{qn}.