harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\langle\sigma (z_1) \ldots \sigma(z_n) \rangle_{D_a}= \big \langle (-1)^{\sum_{j=1}^n \tilde N_{D_a}(z_j)}\big \rangle_{D_a}= \exp\Big [\lambda \tilde \mu \big((-1)^{\sum_{j=1}^n N_{\gamma}(z_j)}-1\big) \Big] = \exp\Big[ -2\lambda\tilde \mu_{D_a}\big(\gamma: \textstyle \sum_{j=1}^n N_{\gamma}(z_j) \big) \Big] ,
d(p)^\top\tau^i(p) \neq 0,p\in\textrm{int}Q,\,\,i=1,2,3,4, d(p)^\top\tau^i(p) = 0,p\in v_jv_{j+1},\,\,i=1,2,3,4,\,\textrm{and}j,j+1\neq i,
\Pr\left\{\bigcup_{i=m+1}^{m+n}\left\{\sum_{\ell=1}^{i-m}\tilde X_{\ell}> \sum_{\ell=1}^i E_\ell \right\}\right\}\le \frac{2\sqrt{\ln 2}}{n}+ \frac{e^{0.4}}{n\ln n}.
\tilde{\gamma}_{\sigma, k}:=\begin{cases}\frac{n}{4(1-\sigma)}-\frac{k}{2(1- \sigma)}\ \ 0 \le \sigma \le \frac{1}{2}, & \frac{n}{2}-k \ \ \sigma = \frac{1}{2}, & \frac{n}{4\sigma}-\frac{k}{2\sigma}\ \ \frac{1}{2}\le \sigma \le 1, \end{cases}
|\psi(t)\rangle=[1-\frac{i}{\hbar}\int_{t_{0}}^{t}dt_{1}H(t_{1})-\frac{1}{\hbar^{2}}\int_{t_{0}}^{t}dt_{1}\int_{t_{0}}^{t_{1}}dt_{2}H(t_{1})H(t_{2})+...]|\psi(t_{0})\rangle.
B(\varepsilon{x_k},{\delta_0}{a_\varepsilon}) \subset T_{\varepsilon ,k}= \varepsilon{x_k}+ a_\varepsilon T_{0}\subset \subset B(\varepsilon{x_k},{\delta_1}{a_\varepsilon}) \subset \subset B(\varepsilon{x_k},{\delta_2}{a_\varepsilon}) \subset B(\varepsilon{x_k}, \delta_3\varepsilon) \subset \varepsilon{Q_k},
\sin^{2}\theta g_{1}(\nabla^{^{M_1}}_{Z}W,X) =g_{2}((\nabla\pi_{\ast})(Z,\omega\phi W),\pi_{\ast}(X))+g_{1}(\mathcal{T}_{Z}\omega W,\mathcal{B}X) &-g_{2}((\nabla\pi_{\ast})(Z,\omega W),\pi_{\ast}(X))+g_{1}(W,\varphi Z)\eta(X),
R^{\pm}( \zeta ) = \left( \begin{array}{c c c c}{a^{\pm}( \zeta )}&{}&{}&{d^{\pm}( \zeta )}\\{}&{b^{\pm}( \zeta )}&{c^{\pm}( \zeta )}&{}\\{}&{c^{\pm}( \zeta )}&{b^{\pm}( \zeta )}&{}\\{d^{\pm}( \zeta )}&{}&{}&{a^{\pm}( \zeta )}\\ \end{array}\right) .
D^{1-\nu}Z_s = \frac{1}{\Gamma(\nu)}\frac{d}{ds}\int_0^s \frac{Z_v}{(s-v)^{1-\nu}}dv &= \frac{1}{\Gamma(\nu)}\frac{d}{ds}\int_0^s\int_0^v \frac{1}{(s-v)^{1-\nu}}dZ_udv &= \frac{1}{\Gamma(\nu)}\frac{d}{ds}\int_0^s\int_u^s \frac{1}{(s-v)^{1-\nu}}dvdZ_u &=\frac{1}{\Gamma(\nu+1)}\frac{d}{ds}\int_0^s (s-u)^\nu dZ_u.
P_{(2^r)}(1,q,q^2,\dots;q^{2n+\delta}) =\sum_{r\geq r_1\geq\dots\geq r_n\geq 0}\frac{q^{r^2-r+r_1^2+\cdots+r_n^2+r_1+\cdots+r_n}}{(q)_{r-r_1}(q)_{r_1-r_2}\cdots(q)_{r_{n-1}-r_n}(q^{2-\delta};q^{2-\delta})_{r_n}}
\begin{gathered}\bigl(C3_{(1/2;\,1/2;\,0)},C3_{(1/2;\,0;\,1/2)}\bigr),\;\bigl(C3_{(1;\,0;\,0)},C3_{(1;\,0;\,0)}\bigr),\;\bigl(C3_{(0;\,0;\,0)},C3_{(0;\,0;\,0)}\bigr), \bigl(C2_{(1;\,1;\,0)},C2_{(1;\,0;\,1)}\bigr),\;\bigl(C2_{(1;\,0;\,0)},C2_{(1;\,0;\,0)}\bigr),\;\bigl(C2_{(0;\,0;\,0)},C1_{(1)}\bigr),\;\bigl(C1_{(0)},C1_{(0)}\bigr).\end{gathered}
\left\langle \Psi_{1}\left|\right.\Psi_{2}\right\rangle=\sum_{\alpha, \beta \in \Omega}\frac{1}{N(\alpha)N(\beta)}\int \sum_{n, m =1}^{N(\alpha)}\sum_{k, l =1}^{N(\beta)}{\overline{f_{1}}}^{\alpha\beta}_{nk|ml}{f_{2}}^{\alpha\beta}_{nk|ml}P_{\lambda}dq^{1}\ldots dq^{n}.
C_{s}= \Bigg\{e^{\frac{1}{\overline{\gamma}_{M}}}\operatorname{Ei}\bigg(\frac{1}{\overline{\gamma}_{M}}\bigg) - e^{\frac{1}{\overline{\gamma}_{W}}}\operatorname{Ei}\bigg(\frac{1}{\overline{\gamma}_{W}}\bigg)\Bigg\}^{+}.
\mathbb{E}\left[\sup_{\mathbf{x}\in\mathcal{X}}\bigl|\breve{y}^{(t)}\bigr|\right]=\mathbb{E}\left[\bigl|\breve{y}^{(t)}\bigr|\right] =\mathbb{E}\left[\sqrt{(\breve{y}^{(t)})^{2}}\right] \leq\sqrt{\mathbb{E}\left[(\breve{y}^{(t)})^{2}\right]}\leq\sqrt{\frac{\sigma_{1}^{2}}{t}}
f(x)=\phi_3^6(x)+(6\delta x+21)\phi_3^5(x)+(65\delta x+125)\phi_3^4(x)+(256\delta x+338)\phi_3^3(x) +(474\delta x+468)\phi_3^2(x)+(42\delta x+324)\phi_3(x)+89-m+144\delta x.
de^{i\theta_0}:T_x^{1,0}X\rightarrow T^{1,0}_{e^{i\theta_0}\circ x}X, de^{i\theta_0}:T_x^{0,1}X\rightarrow T^{0,1}_{e^{i\theta_0}\circ x}X, de^{i\theta_0}(T(x))=T(e^{i\theta_0}\circ x).
\tilde{\sigma_{0}}=2(\mathit{e}-1)\big((2-2\alpha+\beta)(\mathit{e}+1) &+\sqrt{((\mathit{e}+1)(2-2\alpha+\beta))^2+4(\mathit{e}-1)(1+3\mathit{e}-2\alpha-2\alpha\mathit{e}-\beta-\beta\mathit{e})}\big)^{-1}
h_{i j}d q^{i}d q^{j}= \frac{1}{4 V^{2}}d V^{2}+ \frac{1}{4 V^{2}}\left[ d \sigma + \mathrm{i}( \bar{\xi}d \xi - \xi d \bar{\xi}) \right]^{2}+ \frac{1}{V}d \xi d \bar{\xi}\ .
\sum_{y < m \leq M}\frac{1}{m^{s_1}(m+n)^{s_3}}= \frac{M^{-s_1+1}(M+n)^{-s_3}}{1-s_1-s_3}+\frac{y^{-s_1+1}(y+n)^{-s_3}}{s_1+s_3-1}&+\frac{ns_3}{s_1+s_3-1}\int_y^M \frac{du}{u^{s_1}(u+n)^{s_3+1}}+ O \left( \frac{1}{y^{\sigma_1}(y+n)^{\sigma_3}}\right).
\det_r\sigma =\frac{q}{p}\sigma_{11}\circ\sigma_{12}+\sigma_{11}\circ\sigma_{22}-\frac{1}{p}\sigma_{21}\circ\sigma_{12}, \det_l\phi =\frac{q}{p}\phi_{21}\circ\phi_{11}+\phi_{22}\circ\phi_{11}-\frac{1}{p}\phi_{21}\circ\phi_{12};
q(i,k,l,m) \ =\ \frac{T_{jk}}{T_{ij}}\,q(i,j,l,m) \ -\ \frac{T_{jl}}{T_{ij}}\,q(i,j,k,m) \ +\ \frac{T_{jm}}{T_{ij}}\,q(i,j,k,l)\ \in\I.
S_{i j}= g^{A B}{\cal R}_{\mu \nu}\eta_{A}^{\mu}\eta_{B}^{\nu}g_{i j}- \frac{1}{2}g^{A B}{\cal R}_{\mu \nu \rho \sigma}\eta_{A}^{\mu}\eta_{B}^{\sigma}Z_{, i}^{\nu}Z_{, j}^{\rho}
F_{_{\!S}}^{-1}= \frac{\cosh\frac{1}{2}\sigma\otimes\cosh\frac{1}{2}\sigma+2\xi\,v_{+}\,e^{-\frac{1}{2}\sigma}\otimes v_+\,e^{-\frac{1}{2}\sigma}}{\cosh\frac{1}{2}\Delta_{_{\!J}}(\sigma)}\,
\boldsymbol{D}\left( h_{2m+1}^{+}+\mathbf{e}_{1}\mathbf{e}_{2}h_{2m+1}^{-}\right) =0, \boldsymbol{D}\left( h_{2m}^{+}+\mathbf{e}_{1}\mathbf{e}_{2}\frac{N\kappa-\kappa+m}{2\left( N\kappa+m+1\right)}h_{2m}^{-}\right) =0.
G(z_2,\bar{z}_1z_2 + \bar{z}_1^2) = \bar{F}_1(-\bar{z}_1-z_2)F_2(z_2,\bar{z}_1z_2 + \bar{z}_1^2)+{\bigl(\bar{F}_1(-\bar{z}_1-z_2)\bigr)}^2 .
\log | \mathcal{L}(w,\chi_D) \cdot \ldots \cdot \mathcal{L}(wu^{l-1},\chi_D) | \leq \frac{6g}{N+1}+\mathcal{M}(w,g) & + \sum_{d(P) \leq N}\sum_{j=0}^{l-1}\frac{a_j(d(P)) \chi_D(P) d(P)}{\sqrt{|P|}}\cos( (\theta+j \beta)d(P)) .
Q_{jk}\leq\frac{1}{(\sigma_j+1)^{\tau_j}}\int_{D(w_0;\textsf{d})}\frac{d\xi d\eta}{|\eta|^{\tau_j}|\zeta-w_1|^q|\zeta-w_0|^q}=\frac{1}{(\sigma_j+1)^{\tau_j}}\int_{D(0;\textsf{d})}\frac{d\xi d\eta}{|\eta_0+\eta|^{\tau_j}|\zeta-(w_1-w_0)|^q|\zeta|^q},
a(w,z) = \frac{wr(w)}{\rho(\gamma(w)) - \rho(\gamma(z))}+ \frac{\overline{zr(z)}}{\overline{\rho(\gamma(w))}- \overline{\rho(\gamma(z))}}+ \mbox{smooth}&= \frac{w}{w-z}\left(\frac{r(w)(w-z)}{\rho(\gamma(w)) - \rho(\gamma(z))}- \frac{\overline{r(z)(w-z)}}{\overline{\rho(\gamma(w))}- \overline{\rho(\gamma(z))}}\right) + \mbox{smooth}.
\theta \circ (\theta_1 \circ (\theta_{1,1}, \ldots, \theta_{1,k_1}), \ldots, \theta_n \circ (\theta_{n,1}, \ldots,\theta_{n,k_n})) ={}(\theta \circ (\theta_1, \ldots, \theta_n)) \circ (\theta_{1,1}, \ldots, \theta_{1,k_1}, \ldots, \theta_{n,1}, \ldots, \theta_{n,k_n}),
4x^4 w^2 (D+12)^2 (D+8)(D+4)T - (D+6)DT + 12x^4(D+12)((D+4)T)^2 = V \left( 2x^4w^2(D+12)(D+8)(D+4)T - \frac{1}{2}(D+6)T + 6x^4((D+4)T)^2 \right).
\mathbf{h}_{k_ll}^\mathrm{v}= \left[\bar{h}_{1k_ll}\,\, \bar{h}_{\left(M_\mathrm{h}+1\right)k_ll}\,\, \cdots \,\, \bar{h}_{\left(M_\mathrm{h}+M_\mathrm{v}-1\right)k_ll}\right]^\mathrm{T},
\int x^{2}arcsch(ax)dx=\frac{x^{3}arcsch(ax)}{3}-\frac{1}{6a^{3}}arcoth\sqrt{\frac{1}{a^{2}x^{2}}+1}+\frac{x^{2}}{6a}\sqrt{\frac{1}{a^{2}x^{2}}+1}+C
\begin{array}{c}Z_{N}(+\infty )=N\pi +\chi_{\infty}\left( N_{H}-2N_{S}-M_{C}-2\theta (p-1)M_{W}\right) + +\pi \, \textrm{sign}(p-1)\, M_{W}+\displaystyle\int dx\, G(x)\,{\cal Q}_{N}(+\infty )+\alpha .\end{array}
\left[ - \left( 1 - \frac{2 M}{\rho}\right) \frac{d^{2}}{d \rho^{2}}- \frac{2 ( \rho - 4 M ) ( 2 \rho - 3 M )}{\rho^{2}( \rho - 3 M )}\frac{d}{d \rho}+ \frac{8 M}{\rho^{2}( \rho - 3 M )}\right] H_{1}( \rho ) = \lambda H_{1}( \rho ) ,
\sum_{3r_0+2r_1+r_2=3d_1-d_2+\lambda}{d_1+1 \choose r_0}{d_1+1-r_0 \choose r_1}{d_1+1-r_0-r_1 \choose r_2}\frac{(d_2+3)!}{2^{r_2}6^{r_3}\lambda!}{n_1 \choose d_1+1}{n_2 \choose d_2+3},
\begin{cases}-\partial_{tt}v_{4,s}+ \partial_{rr}v_{4,s}+ \frac{1}{r}\partial_{r}v_{4,s}- \frac{v_{4,s}}{r^{2}}=0 v_{4,s}(s,r) = 0 \partial_{t}v_{4,s}(s,r) = v_{4,c}(s,r)\end{cases}
F(x)=\frac{3\pi\mu_{0}}{2}M^{2}R^{4}L^{2}\frac{1}{z^{4}}=\frac{3\mu_{0}}{2\pi}M^{2}V^{2}\frac{1}{z^{4}}=\frac{3\mu_{0}}{2\pi}m_{1}m_{2}\frac{1}{z^{4}}
\sum_{\ell = 0}^{\infty}\frac{( 2 q + 2 \ell ) ( 2 q + \ell - 1 ) !}{\ell !}J_{q + \ell}( z )^{2}= \frac{( 2 q ) !}{( q ! )^{2}}\left( \frac{z}{2}\right)^{2 q}
D_{v}[D_{q}(f_{i-1}) -_{p}^{-1}\circ g_{i-1}\circ_{q}] = D_{v}[D_{0_{q}}(_{p}^{-1}\circ f_{i-1}\circ_{q}) -_{p}^{-1}\circ g_{i-1}\circ_{q}] = D_{0_{q}}(_{p}^{-1}\circ f_{i-1}\circ_{q}) -D_{v}(_{p}^{-1}\circ g_{i-1}\circ_{q}) .
{\cal L}_{C S}^{\prime}= \frac{\kappa}{2}\epsilon^{\mu \nu \rho}( A_{\mu}^{a}\partial_{\nu}A_{\rho}^{a}+ \frac{1}{3}f^{a b c}A_{\mu}^{a}A_{\nu}^{b}A_{\rho}^{c}) ,
R^{p c}= 3 q^{2}[ 2 \epsilon_{\lambda \sigma}^{\gamma \delta}\epsilon_{\omega \nu}^{\sigma \eta}H_{\gamma \delta}^{\nu}H_{\eta}^{\lambda \omega}+ \epsilon_{\nu \lambda}^{\rho \eta}\epsilon_{\alpha \beta}^{\lambda \nu}H_{\sigma}^{\alpha \beta}H_{\rho \eta}^{\sigma}]
\frac{n(n-1)(n-2)}{j}\left[y_j(1)-\frac{1}{j+1}\sum_{i=1}^{j+1}h_{j+1}(i)\right] = (n-j-1)(n-j)-j^2-(1-2n)j-(n-2)b_n-2-u_n = n^2-n-(n-2)b_n-2-u_n < n^2-n-(n-2)b_n-2-(b_n-2)(n-2)\;\;\mbox{(by (\ref{ae6}))}= (n-2)(n+3-2b_n)<0\;\;\mbox{(by (\ref{ae8}))}.
\begin{aligned}\|G(\cdot, u)-G(\cdot, v)\|_{L^q(\ell^2)}\lesssim \|(|u|^{h-1}+|v|^{h-1})|u-v|\|_{L^q}&\stackrel{(i)}{\lesssim}(\|u\|_{L^{hq}}^{h-1}+\|v\|_{L^{hq}}^{h-1})\|u-v\|_{L^{hq}}&\stackrel{(ii)}{\lesssim}(\|u\|_{H^{\phi,q}}^{h-1}+\|v\|_{H^{\phi,q}}^{h-1})\|u-v\|_{H^{\phi,q}}.\end{aligned}
\mbox{Problem 2:}~~\max_{\boldsymbol{P}}~~ \mathbb{E}\left[\sum_{j=1}^K \ln\left(1+\frac{h_{j}P_j}{\sigma^2+\sum_{i=1,i\neq j}^K h_{i}P_i}\right)\right], \mbox{s.t.}~~ P_i\ge 0, ~\forall i &P_i\le P_i^{av},~\forall i &\sum_{i=1}^K g_iP_i\le I.
\left( \, \langle \, T_{--}\, \rangle + G ( z^{+}, z^{-}) \, \right) \, ( \dot{z}^{-})^{2}= \left( \, \langle \, T_{+ +}\, \rangle + G ( z^{-}, z^{+}) \, \right) \, ( \dot{z}^{+})^{2}\, .
\sum_{j\in \mathbb{Z}}2^{js}\left\|[u\cdot \nabla ,\dot{\Delta}_j]v\right\|_{L^2}\lesssim \|\nabla u\|_{\dot{B}_{2,1}^{\frac{n}{2}}}\|v\|_{\dot{B}_{2,1}^{s}}, \s= \frac n2, \ -1+\frac n2
\begin{array}{l c l}{\{p^{n}, f \}_{\theta}}&{=}&{\sum_{k = 0}{\theta}^{2 k}f^{( 2 k + 1 )}p^{n - 2 k - 1}}\\{\{p^{- n}, f \}_{\theta}}&{=}&{- \sum_{k = 0}{\theta}^{2 k}c_{2 k + n}^{2 k + 1}f^{( 2 k + 1 )}p^{- 2 k - n - 1}}\\ \end{array}
S_{e f f}= \int{d^{5}\sigma d \tau [ \mu \dot{C}+ \mu_{\rho}\dot{C}^{\rho}+ P^{\mu \nu}\dot{B}_{\mu \nu}+ \Pi_{a}\dot{X}^{a}+ \hat{\mu}^{i}\dot{\hat{C}}_{i}+ \widehat{\mu}\dot{\widehat{C}}+ \frac{1}{\sqrt{W}}\hat{\delta}\mu ]}
\mathop{\operatorname{res}}\limits_{s = 0}\zeta (s)\zeta (\beta (s - z))\frac{{{x^s}}}{s}= - \frac{1}{2}\zeta ( - \beta z), \mathop{\operatorname{res}}\limits_{s = 1}\zeta (s)\zeta (\beta (s - z))\frac{{{x^s}}}{s}= \zeta (\beta (1 - z))x,
f_{\gamma_{SCHCN}}(\gamma) =\mathcal{L}^{-1}\left[ \mathcal{L}_{\gamma_{SCHCN}}(s)\right] &=\sum_{i=1}^{N_c}\frac{\alpha_i}{(i-1)!}\gamma^{i-1}e^{-\lambda_{eq}\gamma}+ \beta_0 e^{-\lambda_{0}\gamma}+\sum_{j=1}^{N-N_c}\beta_j e^{-(1+\frac{j}{N_c})\lambda_{eq}\gamma}.
\liminf_{n\to\infty}\left(\|[S_n]\|^{-1}\cdot I_n\right) \geq\liminf_{n\to\infty}\int\bigwedge_{j=1}^m(\pi_j)^*\left(\widehat{T}^k\right)\wedge\frac{[S_n]}{\|[S_n]\|}&\geq \int\bigwedge_{j=1}^m(\pi_j)^*\left(\widehat{T}^k\right)\wedge S.
\hat{V}- V = \frac{6^{\frac{1}{3}}}{4}q_{I}( \hat{G}^{I J}- G^{I J}) q_{J}= - \frac{1}{8 \cdot 6^{\frac{1}{3}}}t^{2}\: ( q_{I}G^{I 2})^{2}\rightarrow 0 \ , \qquad t^{2}\rightarrow 0 .
\lvert f_1+f_2\rvert_S =\min\{\lVert h\rVert_S\mid f_1+f_2\sim h\}&\leq \min\{\lVert h_1+h_2\rVert_S\mid f_1\sim h_1, f_2\sim h_2\}&\leq \lVert g_1+g_2\rVert_S &\leq \max(\lVert g_1\rVert_S,\lVert g_2\rVert_S) &=\max (\lvert [f_1]\rvert_S, \lvert [f_2]\rvert_S).
\begin{aligned}\frac{d}{dt}\Big( -dr^1 \wedge dr^2 \wedge dr^3 + dr^1 \wedge \omega + dr^2 \wedge \Re (\Omega) + dr^3 \wedge \Im (\Omega) \Big) = 2^{\frac{2}{3}}\cal{L}_{\nabla_{(g_4)}(|\Omega|_\omega^{-\frac{2}{3}})}\Big( 2dr^1 \wedge \omega - dr^2 \wedge \Re (\Omega) - dr^3 \wedge \Im (\Omega) \Big).\end{aligned}
gain(S) = N \log_2 N - \mu_S N \log_2 \mu_S N + p_S N \log_2 p_S N &-\sum_{a_i \in \{S\}}p_{a_i}N \log_2 p_{a_i}N &+ \sum_{a_i \in \{S\}}p'_{a_i}\mu_S N \log_2 p'_{a_i}\mu_S N
R_if(x) =\partial_{x_i}\widetilde{\mathcal{O}}^{-1/2}f(x) =-e^{-|x|^2}(\partial_{x_i}-2x_i)(\mathcal{O}+2d\,I_d)^{-1/2}(e^{|\cdot|^2}f(\cdot))(x) &= -e^{-|x|^2}(\partial_{x_i}-2x_i)\mathcal{O}^{-1/2}\Lambda (e^{|\cdot|^2}f(\cdot))(x).
\small\frac{1-14\delta}{2+2\gamma}=\frac{1}{4+4\gamma}, \varsigma\in \left(0,\frac{1-14\delta}{2+2\gamma}\right),\quad\varsigma+\varsigma\gamma +7\delta-\frac12= \frac{1}{8}+\frac{1}{4}-\frac12=-\frac{1}{8},
\lambda_{11}=0, \lambda_{22}= \frac{2(n^3-n^2+r^2)}{rn}, \lambda_{33}= \cdots = \lambda_{nn}= \frac{2(n+r)(n-1)}{r}, &\lambda_{ij}= \frac{2(n+r)(n-1)}{r}, \lambda_{in}= \lambda_{ni}=\frac{2(n+r)}{n}
\|\mathcal{A}x-\mathcal{A}y\|_{\ell_2}\leq \sum_{i=1}^g\|\mathcal{A}_ix_i-\mathcal{A}_iy_i\|_{\ell_2}\leq \sum_{i=1}^g \sqrt{2\bar{k}_i(1+\delta_{\bar{k}_i})}
w_1 ({y},{\bar{y}}\mid x)=\sum_{n,m=0}^{\infty}\frac{\lambda^{1-|\frac{n-m}{2}|}}{2i\,n!m!}w_1^{\alpha_1\ldots\alpha_n}{}_,{}^{\dot{\beta}_1\ldots\dot{\beta}_m}(x){y}_{\alpha_1}\ldots{y}_{\alpha_n}{\bar{y}}_{\dot{\beta}_1}\ldots{\bar{y}}_{\dot{\beta}_m}\,
\left(2\pi i K_+^2-\frac 3 2 k_0 K_+ -\frac{k^2 k_0}{32\pi^2 K_+}\right)-\left(2\pi i K_-^2-\frac 3 2 k_0 K_- -\frac{k^2 k_0}{32\pi^2 K_-}\right)=0
\tilde{L}= \sum_{i}\tilde{\alpha}_{i}^{\prime 2}- \tilde{\alpha}_{i}^{\prime}\sum_{j}\tilde{\alpha}_{j}^{\prime}+{\frac{1}{D - 2}}\varphi^{\prime 2}+ \sum_{a}\alpha_{a}^{\prime 2}-{\frac{P^{2}}{2}}e^{\tilde{\Phi}}
a=\frac{d^{2}\rho}{dt^{2}}u_{\rho}+\frac{d\rho}{dt}\frac{du_{\rho}}{dt}+\frac{d\rho}{dt}u_{\theta}\frac{d\theta}{dt}+\rho\frac{du_{\theta}}{dt}\frac{d\theta}{dt}+\rho u_{\theta}\frac{d^{2}\theta}{dt^{2}}.
\psi_7(m) = \sum_{p_1,p_2}\bigg\{\psi(m/(p_1p_2),z) - \sum_{z \le p_3 < p_2}\psi(m/(p_1p_2p_3), z) + \sum_{z \le p_4 < p_3 < p_2}\psi(m/(p_1p_2p_3p_4), p_4) \bigg\}= \psi_{11}(m) - \psi_{12}(m) + \psi_{13}(m), .
\begin{aligned}\delta^2 = \frac{\Bigl(\Delta \pm \sqrt{\Delta}\sqrt{\Delta + 1 - R^2}- (1-R)\Bigr)^2\Bigl(\Delta \pm \sqrt{\Delta}\sqrt{\Delta + 1 - R^2}\Bigr)^2}{2 \Bigl(\Delta \pm \sqrt{\Delta}\sqrt{\Delta - 1 + R^2}+ \frac{1}{2}\bigl(1 - R^2\bigr)\Bigr)^2}\frac{1+R}{1-R}.\end{aligned}
\left\Vert t^{-{\alpha}_{1}{(\cdot )}}(\varphi_{t}\ast f)\right\Vert_{{p}_{1}{(\cdot )}}\lesssim \left\Vert t^{-{\alpha}_{0}{(\cdot)}}(\varphi_{t}\ast f)\right\Vert_{{p}_{0}{(\cdot )}}=\delta .
R_v\left( \pmb{\rho}^{(N)}\right) = \frac{P_v\left( \pmb{\rho}^{(N)}\right)}{P^{\textsc{idle}}\left( \pmb{\rho}^{(N)}\right) + \tau \left (1 - P^{\textsc{idle}}\left( \pmb{\rho}^{(N)}\right) \right)}
\lim\limits_{j\to +\infty}\frac{G(t_0)-G(t_0+B_j)}{\int_{t_0}^{t_0+B_j}c(t)e^{-t}dt}=\liminf\limits_{B\to 0+0}\frac{G(t_0)-G(t_0+B)}{\int_{t_0}^{t_0+B}c(t)e^{-t}dt}
\pi_{j}=\frac{\omega_{j}}{\sum_{i=1}^{R}\omega_{i}},\omega_{j}=\frac{f(y_{t+1}|\alpha_{t+1}^{j})f(\alpha_{t+1}^{j}|\alpha_{t}^{k})}{g(\alpha_{t+1}^{j},k^{j}|Y_{t+1})}
I: \q'(e')=q(b)+q(c)=q(a)+q(d), II: \q'(e')=q(b)+q(c)=q(a)+q(d)+1, III: \q'(e')=q(b)+q(c)+1=q(a)+q(d), IV: \q'(e')=q(b)+q(c)+1=q(a)+q(d), V: \q'(e')=q(b)+q(c)=q(a)+q(d)+1, VI: \q'(e')=q(b)+q(c)+1=q(a)+q(d)+1.
\sigma_{2k-2\dim{Y}- 1}(P(t)) = \sigma_{k + (k - 2\dim{Y}) - 1}(P(t)) \le Mt^{1-2\alpha}\exp{-\eta \sqrt{k - 2\dim{Y}}}+ \hat{M}\exp{-2\tilde{\eta}(t) \sqrt{k - 2\dim{Y}}}\le 2\max(Mt^{1-2\alpha}, \hat{M})\exp{-\min(\eta, 2\tilde{\eta}(t)) \sqrt{k - 2\dim{Y}}},
\frac{c_{n}}{c_{n-1,\xi_{\boldsymbol{s}k}}^{\ast\ast}}= \frac{\Gamma\left(\frac{n}{2}\right)2^{(n-3)/2}}{\Gamma(n)\Gamma\left(\frac{n-1}{2}\right)\sqrt{\pi}}\bigg\{\int_{0}^{\infty}t^{\frac{n-3}{2}}\left[(2+3\sqrt{2})\left(t+\frac{\xi_{\boldsymbol{s},k}^{2}}{2}\right)+3\right]\exp\left(-\sqrt{2t+\xi_{\boldsymbol{s},k}^{2}}\right)\mathrm{d}t\bigg\}
D^{T}w_{I}-w_{I}^{T}Aw_{I}=u^{T}\Delta w_{I}+w_{I}^{T}\Delta w_{I}-w_{I}^{T}\Delta w_{I}-w_{I}^{T}Aw_{I}& =\left( u-w_{I}\right)^{T}\Delta w_{I}+w_{I}^{T}\left( \Delta-A\right)w_{I}
- \frac{1}{4}\det \left( e_{M}^{A}\right) e^{- \phi}T r \left( G_{M N}G_{P Q}g^{M N}g^{P Q}\right) = - \frac{e}{4}e_{\stackrel{.}{0}}^{0}e^{- \phi}T r \left( G_{a b}G_{a b}- 2 G_{a 0}G_{a 0}\right)
\int_{\pi^{-1}(X_{\mathfrak{P}_{\Gamma_0}(Y)})}\left(\sum_{\gamma\in\Gamma}k_t^{p,\rho}(x^{-1}\gamma x)\right)dx=\tilde{\alpha}_{-1}(t;\rho)\log{Y}+K_{X;X_0}(t,\rho)+o(1),
G^+_c(\gamma_1,\gamma_2) = \frac{2\sqrt{J_1J_2}}{E_q(J_1,J_2)}\sum_{n_1,n_2 =0}^\infty\frac{J_1^{n_1}J_2^{n_2}}{[n_1]_q![n_2]_q!}\cos(\gamma_1 q^{2[n_1]_q}+\gamma_2q^{2[n_2]_q})\,;
R_{p,p^{\prime}}(x)=\frac{1}{4\left(1+\frac{1}{\pi p}\right)}+\frac{1}{2}\sqrt{\frac{\pi}{x}}\,\zeta_{p^{\prime}}\left(\frac{1}{2}\right)+\frac{\pi^{2}}{6x}\,\frac{1+\frac{3}{\pi p}(1+\frac{1}{\pi p})}{\left(1+\frac{1}{\pi p^{\prime}}\right)\left(1+\frac{1}{\pi p}\right)^{2}},
\frac{\mathbf{P}\Big( \frac{1}{\sigma}(\hat{\theta}_n -\theta )\sqrt{\Sigma_{k=1}^n X_{k-1}^2}> x \Big)}{1-\Phi \left( x\right)}=1+o(1)\ \ \ \textrm{and}\ \ \ \frac{\mathbf{P}\Big( \frac{1}{\sigma}(\hat{\theta}_n -\theta )\sqrt{\Sigma_{k=1}^n X_{k-1}^2}<-x \Big)}{\Phi \left(- x\right)}=1+o(1).
w^{\eta^-_i}_{\xi^-_i}(\rho_\tau)=\sum_{[\{\alpha,\beta;\epsilon\}]\in\vec{\triangle}^{\eta^-_i}_{\xi^-_i}}\left(e^{\frac{1}{2}(\ell_{\alpha}(\rho_\tau)+\ell_{\beta}(\rho_\tau))}+(-1)^{\alpha\cdot\beta}\right)^{-1}.
\begin{array}{ll}\kappa' = \kappa \left( \frac{\partial z}{\partial w}\right)^{\frac{3}{2}}\left( \frac{\partial \bar{z}}{\partial \bar{w}}\right)^{-\frac{1}{2}}, s' = s \left( \frac{\partial z}{\partial w}\right)^2 + S_w(z),\end{array}
F_{g}= \epsilon_{i j}^{1}\epsilon_{i j}^{2}k_{0}^{2}{\frac{\Gamma ( 1 - 2 k_{0}^{2}) \Gamma ( \vec{k}\cdot \vec{p}+ k_{0}^{2})}{\Gamma ( \vec{k}\cdot \vec{p}- k_{0}^{2}+ 1 )}}\ .
{\cal L}_{0}: \, \, \partial_{z}\partial_{\bar{z}}\chi_{n}= \widetilde{\mathrm{I n h}}\left( \chi_{0}, \ldots , \chi_{n - 1}\right) \, , \quad{\cal L}_{1}: \, \, \partial_{z}\partial_{\bar{z}}\chi_{n}+ m^{2}\chi_{n}= \widetilde{\mathrm{I n h}}\left( \chi_{0}, \ldots , \chi_{n - 1}\right) \, .
V_{a d d}^{\omega}= - \frac{m}{4 \pi}g^{2}\frac{3 m^{2}}{2 M^{2}}( \frac{1}{x}+ \frac{1}{x^{2}}) \phi{\bf L}\cdot{\bf S}+ \frac{m}{4 \pi}g^{2}\frac{3 m^{4}}{1 6 M^{4}}\frac{1}{x^{2}}\chi Q_{1 2}
\left\|\psi\left(\frac{\mathbf y}{\tau}\right)\right\|_{L^8(\mathcal C)}=\tau^{\frac{3}{8}}\|\psi(\mathbf x)\|_{L^8(\mathcal C)}\leq\tau^{\frac{3}{8}}\|\psi(\mathbf x)\|_{H^{1,8}(\mathcal C)}=\mathcal O(\tau^{-\frac{1}{40}}),
\|M\|_{r \ast}\; = \; \max_{\sum_{i=1}^r s_i^2=1}\sum_{i=1}^{n}\sigma_i(M) s_i \geq \|M\|_F \; \geq \; \max_{\sum_{i=1}^r s_i^2=1}\sum_{i=1}^{r}\sigma_i(M) s_i \; = \; \|M\|_r.
[ \: \frac{1}{4}\, ( \frac{\ddot{z}^{+}}{\dot{z}^{+}}- \frac{\ddot{z}^{-}}{\dot{z}^{-}}) \, l + \frac{1}{2}\, ( \, \dot{z}^{+}\, \partial_{+}l - \dot{z}^{-}\, \partial_{-}l \, ) \: ] \, \sqrt{\frac{\dot{z}^{+}}{\dot{z}^{-}}}= F_{-}\, ,
\Delta J(X) = \sum\limits_{k=1}^D\frac{d^2J(X)}{dx_k^2}=\sum\limits_{i=1}^n\frac{-2Dw_i\left[\sum\limits_{k=1}^Dx_{ki}^2\right]^2+8w_i\left[\sum\limits_{k=1}^Dx_{ki}^2\right]^2}{\left[\sum\limits_{k=1}^Dx_{ki}^2\right]^4}=\sum\limits_{i=1}^n\frac{(8-2D)w_i}{\left[\sum\limits_{k=1}^Dx_{ki}^2\right]^2}.
\left({\frac{V_{2}}{V_{1}}}\right)^{2}={\frac{n_{1}\bar{n}_{1}}{n_{2}\bar{n}_{2}}}\ , \qquad \left({\frac{V_{3}}{V_{2}}}\right)^{2}={\frac{n_{2}\bar{n}_{2}}{n_{3}\bar{n}_{3}}}\ , \qquad \left({\frac{V_{4}}{V_{3}}}\right)^{2}={\frac{n_{3}\bar{n}_{3}}{n_{4}\bar{n}_{4}}}\ ,
\frac{\partial f(t)}{\partial t}=\sum_{k=1}^{\infty}\frac{t^{k-1}}{(k-1)!}\phi_{k,\lambda}(|z|^{2})=\sum_{k=0}^{\infty}\phi_{k+1,\lambda}(|z|^{2})\frac{t^{k}}{k!}.
P_\lambda (z_1,\ldots ,z_n,x_1,\ldots, x_l ; q ,t ,\mathbf{t})= &\sum_{\substack{\mu^{(i)}\in\Lambda_{n+i}, i=0,\ldots ,l \mu=\mu^{(0)}\preceq \mu^{(1)}\preceq \cdots \preceq \mu^{(l)}=\lambda}}P_\mu (z_1,\ldots ,z_n ; q ,t ,\mathbf{t})\prod_{1\leq i\leq l}P_{\mu^{(i)}/\mu^{(i-1)}}(x_i; q ,t ,\mathbf{t}).
0 \leq{\rm tr}_{\omega_{\epsilon}}(\sqrt{-1}\partial \bar{\partial}F_0+C_1\omega_0) \leq \nu^{-1}{\rm tr}_{\omega_0}(\sqrt{-1}\partial \bar{\partial}F_0+C_1\omega_0) =\nu^{-1}(C_1n+\Delta_{\omega_0}F_0).
\mathfrak{S}_{\sigma_{n,k,p,b}}(1) = \dim V_{\nu_{n,p,b}}^{\mathfrak{gl}(k-1)}\hphantom{\mathfrak{S}_{\sigma_{n,k,p,b}}(1)}{}= (1+p)^{{n \choose 2}}{k+n-1 \choose b}\prod_{j=1}^{n}{(p+1)(n-j+1) \over (n-j+1)(p+1)+b}\prod_{j=1}^{n}{{k+j-2 \choose (n-j+1)p+b}\over{(n-j+1)(p+1)+b-1 \choose n-j}}.
\mathbb{P}(T_{m}>X_{(n)}) =\mathbb{E}(\mathbb{P}(T_{m}>X_{(n)})|X_{(n)}) =\mathbb{E}\left(\sum_{k=0}^{m-1}e^{-sX_{(n)}}\frac{s^{k}}{k!}X_{(n)}^{k}\right) =\sum_{k=0}^{m-1}\frac{s^{k}}{k!}\mathbb{E}\left(e^{-sX_{(n)}}X_{(n)}^{k}\right),
\frac{3l^{1-n}}{4}E_1 \leq \frac{\mu_{t_0}(B_{r_0}(x_0) )}{r_0^{n-1}}= \frac{\tilde{\mu}_1(B_{\tilde{r}})}{\tilde{r}^{n-1}_0}\leq e^{\frac{1}{16}}(\sqrt{4\pi})^{n-1}\int_{B_{\frac14}}\tilde{\rho}\d \tilde{\mu}_1.
h(k,n,m){}:=2^{-\max(k/n,k/m,1)}\max\left[1,\log^{1/2}(m/k+1)\right]{}\cdot \min\left\{1 , \max\left[ \frac{\log(n/k+1)}{k}, \frac{1}{n}\right]^{1/2}\right\}.
\psi(\hat{y}-z,\hat{s})-\psi(\hat{y},\hat{s})-\varphi(\hat{y}-z,\hat{s})+\varphi(\hat{y},\hat{s}) =\psi(\hat{y}-z,\hat{s})-\varphi(\hat{y}-z,\hat{s})-\lambda' &\ge\psi(\hat{y}-z,\hat{s})-w^*(\hat{y}-z,\hat{s}) &\ge0.
{}^{M\!L\!}\textrm{F}_{p}(a,b;c;z)= \frac{2}{B(b,c-b)}\int_{0}^{\infty}\sinh^{2b-1}v \cosh^{2a-2c+1}v(\cosh^{2}v-z\sinh^{2}v)^{-a}\times E_{\alpha, \beta}^{\gamma}\left(-p\cosh^{2}v\coth^{2}v\right)du
S_{SD}[\vec{B}_{m},V_{mn}] = \int d^{5}x \left\{-{\textstyle\frac{1}{4}}m^2 \vec{B}^{T}_{m}\vec{B}_{m}-{\textstyle\frac{1}{4}}{\cal F}_{mn}{\cal F}_{mn}-{\textstyle\frac{m}{8}}\epsilon\vec{B}^{T}_{m}\eta \mathcal{D}\vec{B}_{m}\, \right\}\, ,
\frac{\partial \bar{\alpha}( x , y ; \alpha )}{\partial \ln x}= \beta \left( \frac{y}{x}, \bar{\alpha}( x , y ; \alpha ) \right) \, \, ; \, \, \, \frac{\partial s ( x , y ; \alpha )}{\partial \ln x}= \gamma \left( \frac{y}{x}, \bar{\alpha}( x , y ; \alpha ) \right) s ( x , y ; \alpha ) ~ ,
\begin{array}{l}LG_{2(1,Z)}=Z_{R}^{1/3}Z_{L}^{2/3}LG_{2}Z_{L}^{-2/3}Z_{R}^{-1/3}, G_{2(1,Z)}=Z_{R}^{1/3}Z_{L}^{2/3}G_{2}Z_{L}^{-2/3}Z_{R}^{-1/3}. \end{array}
m(f_{n+1}(C_{f_{n+1}, \delta}\cap I)) = \sum_{i=1}^N m(f_{n+1}(C_{f_{n+1}, \delta}\cap I_i))= \sum_{i=1}^N \sum_{i' = 1}^{\tilde N_i}m(f_{n+1}(C_{f_{n+1}, \delta}\cap \tilde I_i^{i'}))\leq \sum_{i=1}^N \sum_{i' = 1}^{\tilde N_i}m(f_{n+1}(\tilde I_i^{i'}))
b_{2}= - \zeta \cdot \frac{4 y_{2}( y_{1}y_{2}+ ( y_{1}^{2}+ y_{2}^{2}) \arctan{y_{2}/ y_{1}})}{y_{1}( y_{1}^{2}+ y_{2}^{2})}+ \zeta \cdot f ( y_{1}) y_{2},
S_{F}^{\alpha}(T)=-\frac{1}{\sqrt{2 g_{F}^{\alpha}}}\int_{S_{F}^{\alpha}(y)}^{S_{F}^{\alpha}(0)}(S_{F}^{\alpha}(y)-S_{F}^{\alpha}(\eta))^{-1/2}D^{\alpha}_{F,\eta}h_{F}^{\alpha}(\eta)d_{F}^{\alpha}\eta.
\begin{aligned}\tilde{F}_0(M,y,s) &=\tilde{F}_0(M,y,s)-\tilde{F}_0(0,y,s)+\tilde{F}_0(0,y,s) =\int_{0}^{1}F_{0,ij}\left(\tau r^{k-1+\alpha}M+D^2H_{m_0},x,t\right)M_{ij}d\tau +\frac{F_0(D^2H_{m_0},x,t)-(H_{m_0})_t}{r^{k-1+\alpha}}:=G_1(M,y,s)+G_2(y,s). \end{aligned}
\|f\|_\mu^2 = \frac{2^{n-1}\Gamma(\frac{\rho+\mu}{2})\Gamma(\frac{\rho-\mu}{2})}{\pi^{n+1}\Gamma(\mu)}\cdot\int_{\overline{N}}\int_{\overline{N}}|n_1n_2^{-1}|^{2(\mu-\rho)}f(n_1)\overline{f(n_2)}\,dn_1 \,dn_2
\frac{df}{dv}(v)-\frac{\lambda}{3\kappa^2}v=\frac{\lambda}{18\kappa^2}\left\{2v(y(v)+1)+\frac{1}{v^2}\int_0^vv'^3\frac{dy}{dv}(v')dv'\right\}+\mathcal{O}(v^2),
\frac{d}{dt}H_C\circ y_n = \langle dH_C(y_n),\dot{y}_n \rangle = \frac{1}{2c_{\mathbb{H}}(C)}\langle - \Omega \dot{y}_n - \eta_n, \dot{y}_n \rangle = - \frac{1}{2c_{\mathbb{H}}(C)}\langle \eta_n, \dot{y}_n \rangle,
\langle b_{i}\ |\ab_{j}\rangle b =\langle b_{i}\ |\ab_{j}b\rangle &=\langle b_{i}\ |\bab_{j}\rangle &=\langle b^{*}b_{i}\ |\ab_{j}\rangle &=\langle b_{i}b^{*}\ |\ab_{j}\rangle &=b \langle b_{i}\ |\ab_{j}\rangle
~\frac{1}{(r+1)^2}\sum^r_{i=0}\int^\infty_0 (\cosh t)^{2i-2\mu_1-2\alpha-4}D(t)dt\|\phi\|^2 = ~\frac{1}{(r+1)^2}\sum^r_{i=0}\frac{1}{(\mu_1-i+\alpha)(\mu_1-i+\alpha+1)}\|\phi\|^2 = ~\frac{1}{(r+1)^2}\bigg(\frac{1}{\mu_2+\alpha}-\frac{1}{\mu_1+\alpha+1}\bigg)\|\phi\|^2 = ~\frac{1}{(r+1)(\mu_2+\alpha)(\mu_1+\alpha+1)}\|\phi\|^2.
\frac{\#\mathcal{P}_{L,\mathcal{A}}(x)}{\phi_L(W)}\sideset{}{'}\sum_{\substack{\mathbf{d},\mathbf{e}\in\mathcal{D}_k' d_m=e_m=1}}\frac{\lambda'_{\mathbf{d}}\lambda'_{\mathbf{e}}}{\phi_L([\mathbf{d},\mathbf{e}])},