harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
C=\left(\begin{array}{cc}\sqrt{2{\mathcal{B}}}0 0 \sqrt{2{\mathcal{B}}}\end{array}\right)\hspace{0.1cm},\hspace{0.15cm}D=\left(\!\begin{array}{cc}\!\sqrt{\frac{2}{{\mathcal{B}}}}\! 0 0 \!\sqrt{\frac{2}{{\mathcal{B}}}}\!\end{array}\!\right)\hspace{0.1cm},\hspace{0.15cm}\Delta=\frac{1}{2}\left(\begin{array}{cc}1 0 0 1\end{array}\right)\hspace{0.1cm}.
5 \Big( N_{D_5}\big(D^2\big)+N_{D_5}\big((5D)^2 \big) \Big) + 2=N_{F_5}\big((-1)^2 5^3 D^2\big)+ N_{F_5}\big((-1)^2 5^5 D^2\big)+N_{F_5}\big((-1)^2 5^7 D^2\big)\;.
\int_0^\infty \Psi(Z) (\Psi w)'(Z) \frac{dZ}{Z}= \int_0^\infty (\Psi w)(Z) (\Psi w)'(Z) \frac{dZ}{Zw(Z)}=-\frac{1}{2}\int_0^\infty (\Psi w)^2(Z) \frac{d}{dZ}\left(\frac{1}{Z w(Z)}\right) dZ.
\begin{pmatrix}A_{\mathrm{c}\mathrm{c}}- A_{\mathrm{R}\mathrm{c}}^{\top}R + R^{\top}A_{\mathrm{R}\mathrm{c}}B_0^{\top}B_0 0 \end{pmatrix}\begin{pmatrix}U_\mathrm{c}P_0 \end{pmatrix}= \begin{pmatrix}F_\mathrm{c}- R^\top F_\mathrm{R}0 \end{pmatrix},
p_{r}^{( k )}( \mu ) = \frac{( - 1 )^{r - 1}}{( r - 1 ) ! ( k - r ) !}\cdot \frac{1 - \mu}{( 1 - \mu ) ( r - 1 ) + 1}\cdot \left( \frac{2 - \mu}{1 - \mu}\right)_{k - 1}.
\begin{aligned}\int_{\mathbb{H}^n}(v^{1/m}-\Tilde{v}^{1/m})_+\leq \int_{\mathbb{H}^n}(g-\Tilde{g})(v-\Tilde{v})\leq \int_{\mathbb{H}^n}(g-\Tilde{g})_+. \end{aligned}
\bar{\rho_{\rm s}}(\zeta)\;\approx\;4\beta\left[ 1+k'^2\left(\ln{4\over k'}+{1\over 2}\right)+O(k'^2)\right]\left[ \zeta \ln{1+\zeta\over 1-\zeta}+k'^2\left( \ln{4\over k'}+{1\over 2}\right){\zeta^2\over 1-\zeta^2}+O(k'^2)\right]\;,
T_{(i_1, \ldots, i_m),(j_1, \ldots, j_m)}(u) &= \sum_{w \in \mathfrak{S}_m}(-1)^{l(w)}T_{i_{w(1)}, j_1}(u) T_{i_{w(2)}, j_2}(u-\hbar) \cdots T_{i_{w(m)}, j_m}(u - \hbar(m-1)).
H_1 (\{1,2,3\})=- |K_1 (\{\{1,2,3\}\})|+|K_1 (\{\{1,2\},\{3\}\})| +|K_1 (\{\{2,3\},\{1\}\})| + |K_1 (\{\{1,3\},\{2\}\})|-|K_1 (\{\{1\},\{2\},\{3\}\})| =- 1+1+1-1=0.
\begin{aligned}G_a^n(z,w) := \langle x,y,s_2,\ldots,s_{n-1}\mid s_2=\lbrack y,x\rbrack,\ (\forall_{i=3}^n)\s_i=\lbrack s_{i-1},x\rbrack,\s_n=1,\ \lbrack y,s_2\rbrack=s_{n-1}^a, (\forall_{i=3}^{n-1})\ \lbrack y,s_i\rbrack=1,\x^3=s_{n-1}^w,\y^3s_2^3s_3=s_{n-1}^z,\ (\forall_{i=2}^{n-3})\s_i^3s_{i+1}^3s_{i+2}=1,\s_{n-2}^3=s_{n-1}^3=1\ \rangle,\end{aligned}
\lim_{\left\vert D\right\vert \rightarrow 0,D=\left\{t_{k}\right\}_{k=0}^{n}\subset \left[ 0,T\right]}\beta_{0}\left(g_{0},g_{0,t_{1}}\right) \beta_{t_{1}}\left(g_{t_{1}},g_{t_{1},t_{2}}\right) \cdots \beta_{t_{n-1}}\left(g_{t_{n-1}},g_{t_{n-1},T}\right) g_{s,t}:=g_{s}^{-1}g_{t}
P\left( \left\|S(t)(u_0^\omega,u_1^\omega)\right\|_{L^q_t(I; L^r_x)}>\l\right) <\left(\frac{C|I|^{\frac 1q}p^{\frac 12}\left(\|u_0\|_{L^2}+\|u_1\|_{\dot{H}^{-1}}\right)}{\l}\right)^p &=e^{-p}=\exp\left(-c\frac{\l^2}{|I|^{\frac2q}\left(\|u_0\|_{L^2}+\|u_1\|_{\dot{H}^{-1}}\right)^2}\right).
\begin{vmatrix}1 X Y X^2 Y^2 XY 1 x_1 y_1 x_1^2 y_1^2 x_1y_1 1 x_2 y_2 x_2^2 y_2^2 x_2y_2 \vdots 1 x_5 y_5 x_5^2 y_5^2 x_5y_5 \end{vmatrix}
\min\left\{\nu_{\pi}\left(\binom{n}{i}(v_1^A)^{n-i}\left( \pi - \pi^q\right)^{i}(t_1^A)^i\right): i=1, \dots ,n\right\}= \min\left\{\nu_{\pi}\left( \binom{n}{i}\right) + i\nu_{\pi}\left( \pi - \pi^q\right): i=1, \dots ,n\right\}= \min\left\{e\cdot\nu_p\left( \binom{n}{i}\right) + i : i=1,\dots ,n\right\}.
\zeta_{A}( s ) = \sum_{j = - \infty}^{+ \infty}\sum_{n}\left[ \left( \omega_{j}^{2}+ \sigma_{n}+ m^{2}- e^{2}\mu^{2}\right)^{2}+ 4 e^{2}\mu^{2}\omega_{j}^{2}\right]^{- s}\; .
\lim_{s\to 0}\frac{\psi_{ss}-\psi_t+\sqrt{(\psi_{ss}-\psi_t)^2+4\varphi\psi_{s}\psi}}{2\psi_s\psi}(s,t)=\frac{\psi_{sss}(0,t)}{\psi_{s}^2(0,t)},
\div \beta \otimes \beta = - (\nabla_{e_1}\beta) (e_1) \beta - \beta(e_1) \nabla_{e_1}\beta - (\nabla_{e_2}\beta )(e_2) \beta - \beta(e_2) \nabla_{e_2}\beta = (\div \beta) \beta - \nabla_{\beta^\sharp}\beta,
P=\frac{2\int_{0}^{\pi/2}cos^{2}\theta sin\theta d\theta}{\int_{0}^{\pi}sin\theta d\theta}\times nmv_{rms}^{2}=\frac{1}{3}nmv_{rms}^{2}=\frac{2}{3}n\langle E_{kin}\rangle=nk_{B}T
\sum_{k=1}^2 \sum_{\mathbf{i}\in I_{kn}}\theta_n(\mathbf{i})^2 = \mathrm{o}(\lambda_n) \mbox{and}\quad\sum_{\mathbf{i}\in I_{3n}}\theta_n(\mathbf{i})^2 = \sigma^2_0\lambda_n \bigl(1+\mathrm{o}(1)\bigr),
\int_{[t,T]}D_{s-}dX_s + \frac12 \int_{[t,T]}\Delta X_s \gamma_s dX_s & = \frac12 \left( \gamma_T^{-1}D_T^2 - \gamma_t^{-1}d^2 - \int_t^T D_s^2 \left( d\gamma_s^{-1}+ \gamma_s^{-1}d[R]_s - 2\gamma_s^{-1}dR_s - 2 d[\gamma^{-1},R]_s \right) \right) .
\mathbf{b}\left( x\right) =(\left( -1\right)^{0}b_{0}\left(x\right) ,\left( -1\right)^{1}b_{1}\left( x\right) \dots,\left(-1\right)^{m}b_{m}\left( x\right) )^{\operatorname{tr}}\mathbf{a}\left( x\right) =(a_{0}\left( x\right) ,a_{2}\left(x\right) ,a_{4}\left( x\right) ,\dots,a_{2m}\left( x\right) )^{tr}.
J_s(t) = J(t) J(s)^{-1}= J(s)J(s)^{-1}+ \int_s^t \nabla_x b(r, \omega, X(r)(\omega)) J(r)J(s)^{-1}dr &+ \int_s^t \nabla_x \sigma(r, \omega, X(r)(\omega)) J(r) J(s)^{-1}dW(r) = I_d + \int_s^t \nabla_x b(r, \omega, X(r)(\omega)) J_s(r)dr + \int_s^t \nabla_x \sigma(r, \omega, X(r)(\omega)) J_s(r) dW(r).
\left[\mbox{Prod}\left(\mathbf{H},\mathbf{H}^{\top^{\left(m-1\right)}},\cdots,\mathbf{H}^{\top^{k}},\cdots,\mathbf{H}^{\top^{2}},\mathbf{H}^{\top}\right)\right]_{i_{1},\cdots,i_{m}}=\begin{cases}\begin{array}{cc}n \mbox{if}\:0\le i_{1}=\cdots=i_{m}<n 0 \mbox{otherwise}\end{array}.\end{cases}
\frac{\pi^{3/2}}{2G\Lambda}=1+\frac{\sqrt\pi(eE)^{1/2}}{2\Lambda}\Bigg( \mbox{Re}\Bigg[ \zeta \big(\frac12,1-i\frac{\mu_E^2}{2}\big)\Bigg]-\mbox{Im}\Bigg[ \zeta \big(\frac12,1-i\frac{\mu_E^2}{2}\big)\Bigg]\Bigg) +O\Big(\frac{m^2}{\Lambda^2}\Big).
{{\partial\bar{W}}\over{\partial\theta}}= N\prod_{i,x}\delta(\bar{K}^{4i}(\bar{x}))\delta(\bar{K}\raisebox{0.97ex}{\mbox{\scriptsize{$4i$}}}\raisebox{-0.36ex}{\mbox{\scriptsize{$,\!\theta$}}}(\bar{x}))\prod_{x}\delta(\bar{t}_{4}(\bar{x}))\delta(\bar{t}_{4,\theta}(\bar{x})){{\partial W}\over{\partial\theta}}\,,
u_k := h_k \log\left( \frac{f_k f_{k+2}}{f_{k+1}^2}\right) + \left( \frac{g_k^2}{f_k}- 2 \frac{g_k g_{k+1}}{f_{k+1}}+ \frac{g_{k+1}^2}{f_{k+2}}\right).
\dfrac{d}{dt}\mathtt{I}_{t_0}^R(t) =-\dot z(t)\int \big(u^2+u_x^2\big)\Psi'+\int u^2u_x^2\Psi'+\int \{p*(3uu_x^2+2u^3)\}u\Psi'+\int \{p_x*u_x^3\}u\Psi' =-\dot z(t)\int \big(u^2+u_x^2\big)\Psi'+\mathrm{J}_1+\mathrm{J}_2+\mathrm{J}_3.
\chi_{r , s}^{( p , p^{\prime})}( q ) = \frac{1}{( q )_{\infty}}\sum_{n = - \infty}^{\infty}\left( q^{p p^{\prime}n^{2}+ ( r p^{\prime}- s p ) n}- q^{( p n + r ) ( p^{\prime}n + s )}\right) ,
\sum_{\substack{C \in \mathcal{M}_i C | f^{\infty}}}\frac{1}{|C|^2}= \frac{1}{2\pi i}\displaystyle \oint_{\Gamma_1}\frac{1}{q^{2i}u^{i+1}\prod_{P|f}(1-u^{d(P)})}\, du,
\mathcal{Q}_n( x_{[n]})=\rho_n \Big( \mathfrak{Q}_n( x_{[n]}, (\xi_{m})_{m \in C(n)}) \Big)= \displaystyle \sup_{p \in \mathcal{P}_n}\; \sum_{m \in C(n)}\; p_m \Phi_m \mathfrak{Q}_n( x_{[n]}, \xi_{m})
Z_{1,R}= Z_1 \cup_Y Y_{[0,R]}, Z_{2,R}= Z_2 \cup_Y Y_{[-R,0]}, & Z_{1,\infty}= Z_1 \cup_Y Y_{[0,\infty)}, Z_{2,\infty}= Z_2 \cup_Y Y_{(-\infty,0]},
\bar{A}_{k}^{a}= \frac{2}{L}\vartheta \: \delta^{a 3}\delta_{k 0}+ \frac{2}{L}\vartheta \: \epsilon^{a 3 b}\: \bar{\varphi}_{k}^{b}+ \mathrm{i}k \: \frac{2 \pi}{L}\bar{\varphi}_{k}^{a},
F_T(x)=\frac{(1 - x)^2 (1 - 4 x + 6 x^2 - 5 x^3 + x^4) C(x) - 1 + 6 x - 14 x^2 + 15 x^3 - 8 x^4 + x^5}{x (1 - 3 x + x^2) (1 - x + x^3)}\,.
\begin{array}{r c l}{{\cal L}_{\mathrm{g . f .}}}&{=}&{\displaystyle - \frac{1}{\xi}\mathrm{T r}( f f ) ,}\\{f}&{=}&{\partial^{\mu}G_{\mu}+ i g \xi ( \phi^{\dagger}t^{A}\phi_{0}- \phi_{0}^{\dagger}t^{A}\phi ) t^{A}.}\\ \end{array}
L_1^+(\Lambda_R(\chi^N)) = X_1^+(I_R^N) - C_1^+(\Lambda_R(\chi^N)) + \delta \Delta_1(\Lambda_R(\chi^N)), S_1(\Lambda_R(\chi^N)) + L_1^-(\Lambda_R(\chi^N)) = X_1^-(I_R^N) + C_1^-(\Lambda_R(\chi^N)) + \delta \Delta_1(\Lambda_R(\chi^N)).
{\dot x}^i=v^i, \quad\;\;{\dot \theta}^I=\hat{\rm v}^J L^I_J -{\dot x}^i \Lambda^I_i, \quad\;\;{\dot p}_i=\frac{\partial R^\mu}{\partial x^i}- \Lambda^I_i \frac{\partial R^\mu}{\partial \theta^I}- \mu_a B_{ij}^a{\dot x}^j, & p_i = \frac{\partial R^\mu}{\partial v^i}, \quad\frac{\partial R^\mu}{\partial \hat{\rm v}^a}=0.
W_{A B C}= \frac{1}{2}T_{\underline{{A}}\ \underline{{B}}\dot{A}u n d e r l i n e{C}}^{\dot{A}}, W_{\dot{A}\dot{B}\dot{C}}= - \frac{1}{2}T_{\underline{{\dot{A}}}A \underline{{\dot{B}}}\underline{{\dot{C}}}}^{A}
\| \Sigma^{\kappa}_{[0,\hat{q}]}-\Sigma^{\kappa}_{[\hat{q},1-\hat{q}]}\|_{\textrm{F}}=\inf_{q\in (0,0.5)}\| \Sigma^{\kappa}_{[0,q]}-\Sigma^{\kappa}_{[q,1-q]}\|_{\textrm{F}}.
\|a\in b\| = \{p{e_0}{e_1}: \exists c \; \langle e_0, c\rangle \in b \; \wedge \; e_1\in \| a=c\|)\}\|a= b\| = \{e_0 : \forall \; \langle n,c\rangle \in a \; e_0{n}\in \|c\in b\|\}\wedge \{e_1 : \forall \; \langle n, c\rangle \in b \; e_1{n}\in \|c\in a\|\}
d_{(\psi,a)}\mathcal{F}(\phi, b) = \lim_{t\to 0}\frac{\mathcal{F}(\gamma_t) - \mathcal{F}(\psi,a)}{t}&= \left( Q^+\phi + \lim_{t\to 0}\frac{\pi^{-}(a_t\cdot\psi_t - a\cdot\psi)}{t},\,\, d^*b,\,\, d^+b-\lim_{t\to 0}\frac{\rho^{-1}(\mu(\psi_t) - \mu(\psi))}{t}\right).
\Vert G(\psi_{qr})-G^m(\psi_{qr})\Vert^2 =\Vert G(\psi_{qr})\Vert^2-\Vert G^m(\psi_{qr})\Vert^2 & =\Vert G(\psi_{qr})\Vert^2-\Vert F^m(\psi_{qr})\Vert^2 & =1-\Vert \psi_{qr}^m\Vert^2
\begin{aligned}\tfrac{\chi(\sqrt{|V_\shortparallel|^2 + |V_\perp' - \zeta_\perp|^2})}{(|V_\shortparallel|^2 + |V_\perp' - \zeta_\perp|^2)^{\frac{1 - \gamma}{2}}}\leq \tfrac{2^{1 - \gamma}}{(1 + |V_\shortparallel|^2 + |V_\perp' - \zeta_\perp|^2)^{\frac{1 - \gamma}{2}}}\,. \end{aligned}
\Delta K (p_{1j}^{2}; m_{0},M,\epsilon) = \frac{-i}{32 \pi^{4}}\frac{1}{M^{2 \epsilon}}\left[ \Delta I (\frac{p_{12}^{2}}{M^{2}},\frac{m_{0}}{M},\epsilon) + \Delta I (\frac{p_{13}^{2}}{M^{2}},\frac{m_{0}}{M},\epsilon) + \Delta I (\frac{p_{14}^{2}}{M^{2}},\frac{m_{0}}{M},\epsilon) \right].
h_{j+1}^{j+1}= h_{j+1}^{1}+ h_{j+1}^{2}+ \cdots + h_{j+1}^{j}+ h_{j+1}^{j+1}\geq j + 1 - y_{j+1}+ 1 \geq j + 1 - (j + 1) + 1 = 1,
\mathsf{T}_{\mathbf{\Phi}_{g}}\mathcal{M}_g = \{\mathbf{T}_{g}\in \mathbb{C}^{2\bar{M}\times \bar{M}}: \Re\{\mathbf{\Phi}_{g}^H\mathbf{T}_{g}\}= \mathbf{0}_{\bar{M}}\}, \forall g \in \mathcal{G}.
\phi(\sigma_{k-1}\cdots \sigma_{\ell}; \vec{z}) u(r^{k_{r}}, \ldots , 1^{k_{1}})=\prod_{\begin{subarray}{c}i \in J_{p}i>\ell \end{subarray}}\frac{f(z_{i}, z_{\ell})}{f(z_{\ell}, z_{i})}\prod_{i \in J_{p+1}\cup \cdots \cup J_{1}}^{\curvearrowleft}\!\!\! Y_{i-1}(z_{\ell}, z_{i}) \, u(r^{k_{r}}, \ldots , 1^{k_{1}}).
[x^n]G(x) \prod_{1\le i\le \ell}\sum\nolimits_{j\ge 1}j^{p_i}C(x^j) \sim \frac{G(q_n) \prod_{1\le i\le \ell}\sum\nolimits_{j\ge 1}j^{p_i}C(q_n^j)}{\sqrt{2\pi \sum\nolimits_{j\ge 1}jq_n^{2j}C''(q_n^j)}}\cdot q_n^{-n}.
\varphi'(r) =-\dfrac{2 \beta R r}{(r^2+\alpha^2)^{\beta R+1}}\varphi''(r) =-(-\beta R-1)\dfrac{4\beta R r^2}{(r^2+\alpha^2)^{\beta R+2}}-\dfrac{2 \beta R}{(r^2+\alpha^2)^{\beta R+1}}.
\widetilde{\mathbb{C}}^{\textsuperscript{n-1}}= \left\{\Gamma =\{\lambda_{2},\ldots,\lambda_{n}\}: \Gamma =\overline{\Gamma}\right\}\setminus \left\{\{\mu_{1},\overline{\mu}_{1},\ldots,\mu_{m},\overline{\mu}_{m}\}: \m\right\}.
\begin{cases}L'_e(U^{a, \pm}+V^\pm_{n+\frac{1}{2}}, \Psi^{a, \pm}+S_{\theta_n}\Phi^\pm_{n})\delta \tilde V^\pm_n=f^\pm_n, ~\Omega_X, B'_{n+\frac{1}{2}}(\delta \tilde V^+_n, \delta \tilde V^-_n, \delta \phi_n)=g_n, \,\,\{y=0\}\delta \tilde V^\pm_n=0, \delta \phi_n=0, \,\, x\le 0, \end{cases}
\underline{\underline{C_{i}}}=[\begin{matrix}a_{i}&a_{i}-2e_{i}&b_{i}&0&0&0\\a_{i}-2e_{i}&a_{i}&b_{i}&0&0&0\\b_{i}&b_{i}&c_{i}&0&0&0\\ 0&0&0&d_{i}&0&0\\ 0&0&0&0&d_{i}&0\\ 0&0&0&0&0&e_{i}\\ \end{matrix}]
[S,A]_{1}= [\Delta'+\tau^{2}|\nabla'\phi|^{2},-2\tau\nabla'\phi\cdot\nabla'-\tau\Delta'\phi],{}[S,A]_{2}= \bigg[\partial_{n+1}^{2}+\tau^{2}(\partial_{n+1}\phi)^{2}+\frac{1-4s^{2}}{4}x_{n+1}^{-2},-2\tau\partial_{n+1}\phi\partial_{n+1}-\tau\partial_{n+1}^{2}\phi\bigg].
\|U_n-V_n(t)\|_{H^s}\geq (1-e^{-nt})\|n^{-s}\cos(nx)\|-\sqrt{2\pi}(n^{-1}-n^{-3}) = \sqrt{\pi}(1-e^{-nt}) \Big(\frac{1+n^2}{n^2}\Big)^{s/2}-\sqrt{2\pi}(n^{-1}-n^{-3}).
\sum_{m=1}^M\psi_\mu(m,\chi)=E(\chi)\frac{M^{\mu+1}}{\mu(\mu+1)}-\sum_\rho \frac{M^{\rho+\mu}}{\rho+\mu}\cdot\frac{\Gamma(\rho)\Gamma(\mu)}{\Gamma(\rho+\mu)}&+\frac{M^\mu}{\mu}\cdot\frac{L'}{L}(1,\overline{\chi^\ast})+O\left(M(\log2q)(\log M)\right)
\begin{bmatrix}0 0 0 3(I-P_2^*P_2)\end{bmatrix}- \omega\begin{bmatrix}S_{111}-S_{211}^*P_1 S_{112}-S_{221}^*P_2 S_{121}-S_{212}^*P_1 S_{122}-S_{222}^*P_2\end{bmatrix}- \beta^2\begin{bmatrix}S_{211}-S_{111}^*P_1 S_{212}-S_{121}^*P_2 S_{221}-S_{112}^*P_1 S_{222}-S_{122}^*P_2\end{bmatrix}\, \geq 0
B=\left(\begin{array}{ccccccc}t_1 t_2 \ldots t_u 0 \ldots 0 \ell_{1,2}\ell_{2,2}\ldots \ell_{u,2}\ell_{u+1,2}\ldots \ell_{n-1,2}\ell_{1,3}\ell_{2,3}\ldots \ell_{u,3}\ell_{u+1,3}\ldots \ell_{n-1,3}\end{array}\right),
B_{\delta,r}\leq 2 \mathop{\int_X\int_X}_{\{Cd(x,y) \cdot g (x) \geq \delta/2 , d(x,y) \leq r \}}\frac{\delta^p}{\rho(x,y) d(x,y)^{p}}\, d \mu(x) \, d \mu(y) & = 2 \int_X \int_{d(x,y) \geq r_1(x)}\frac{\delta^p}{\rho(x,y) d(x,y)^{p}}\, d \mu(y) \, d \mu(x),
\begin{cases}\frac{|P+\nabla\widetilde{u}_0 (x )+\nabla_y \widetilde{u}_1 (x,y ) |^2}{2}+V (x,y )=\ln \widetilde{m}_1 (x,y )+\widehat{H}(x,P+\nabla\widetilde{u}_0 (x ) ), -\div_y \big(\widetilde{m}_1 (x,y ) (P+\nabla \widetilde{u}_0 (x )+\nabla_y \widetilde{u}_1 (x,y ) )\big )=0.\end{cases}
u\approx u^{N}(x_{1},...,x_{d};k_{1},...,k_{p})=\sum_{i=1}^{N}{X_{1}}_{i}(x_{1})\cdot\cdot\cdot{X_{d}}_{i}(x_{d})\cdot{K_{1}}_{i}(k_{1})\cdot\cdot\cdot{K_{p}}_{i}(k_{p}),
\sum_{r_j\le R}\Big(\frac{1}{r_j}-\frac{r_j}{R^2}\Big)\cos \theta_j & = \frac{1}{2\pi}\int_1^R \Big(\frac{1}{y^2}-\frac{1}{R^2}\Big)\log\big| f(iy)f(-iy)\big|\, dy + \frac{1}{\pi R}\int_{-\frac\pi2}^{\frac\pi2}\log | f(Re^{it})|\,\cos t\, dt +A(R) \,,
2 \pi^{2}\int_{0_{( V_{4})}}^{l}\partial_{r}[ \overline{{{\Psi}}}\Psi ]_{r}r^{3}d r = \left< [ \overline{{{\Psi}}}\Psi ] \right>_{l}6 \pi^{2}\int_{0}^{l}r^{2}d r = 1
\delta_{H}={\frac{N^{3 / 4}2^{- 1 / 4}}{5 \sqrt 3 \pi^{3}}}{\frac{g_{s}( M_{s}L )^{1 / 2}}{\beta^{1 / 4}( 2 \pi R M_{s})^{9 / 2}}}= 1 . 9 1 \times 1 0^{- 5},
Y_\alpha(M^n,g)\geq J_{g,\alpha}(u) \geq Y_\alpha(\mathbb{S}^n,g_0)-C\left(\frac\delta\lambda\right)^{-n}+A\cdot \frac{\int_{B_\delta\times B_\delta}|x-y|_g^{\alpha-2}f_\lambda(x)f_\lambda(y)dxdy}{\|u\|_{L^{\frac{2n}{n+\alpha}}(M^n)}^2}.
{\frac{- i \mu^{2}}{8 H_{5}\pi V}}( 2 \pi )^{3}\delta^{3}( \vec{k}+ \vec{k}^{\prime}) \int d t | t | \int \int d u d u^{\prime}e^{i ( u - u^{\prime}){\frac{k_{y}}{H_{5}}}}e^{- i ( \mathrm{c o s h}u + \mathrm{c o s h}u^{\prime}) m t}e^{- \epsilon | t |},
\begin{array}{l l l}{{R}}&{{=}}&{{8 e^{- 2 \rho}\partial_{+}\partial_{-}\rho = 8 e^{- 2 \rho}{\left( 1 -{\frac{1}{2 4}}\, N e^{2 \Phi}\right)}^{- 1}\partial_{-}\Sigma}}\\{{\}}&{{=}}&{{4{\left( 1 -{\frac{1}{1 2}}\, N e^{2 \Phi}\right)}^{- 1}\left( 4 e^{- 2 \rho}\, \partial_{-}\Phi \, \Sigma + \lambda^{2}\right) \; ,}}\\ \end{array}
S_{e f f}= \int d^{3}x \left\{\frac{| m |}{1 6 \pi}\mathrm{t r}( \partial_{\mu}\hat{n})^{2}+ i \frac{\mathrm{s g n}( m )}{4 \pi}\epsilon^{\mu \nu \lambda}a_{\mu}\partial_{\nu}a_{\lambda}\right\},
\tilde{{\rho}}^{(1)}=\rho_i^{(1)}({\zeta},\theta)+{\frac{1}{\pi}}\left({\frac{\sqrt{2}\mu\sin{2\mu\pi}}{\cos^2\mu\beta-\sin^2\mu\pi}}\right){\sqrt{1-\frac{\xi}{\kappa_0}}}+{O\left(1-\frac{\xi}{\kappa_0}\right)},
\lim_{v \to \infty}\dfrac{r_{2^v}^{\operatorname{Bad-I}}(m)}{2^{3v}}= \lim_{v \to \infty}\dfrac{2 r_{2^{v-1},Q^{\prime}}^{\operatorname{Good}}\left(\dfrac{m}{2}\right)}{2^{3v}}&=\dfrac{r_{2^{7},Q^{\prime}}^{\operatorname{Good}}\left(\dfrac{m}{2}\right)}{2^{23}}\: .
Z : = \int{\cal D}A_{\mu \nu}{\cal D}\Lambda_{\mu}{\cal D}C_{\mu}{\cal D}\bar{C}_{\mu}{\cal D}d{\cal D}\bar{d}{\cal D}N{\cal D}P{\cal D}B^{( 1 )}\delta ( \partial^{\nu}A_{\mu \nu}- \partial_{\mu}N ) \exp \left[ i \int d^{4}x{\cal L}_{t o t}^{\prime}\right] ,
\psi_r(t)= e^{-itH_0}(\psi(0)-\Omega_\alpha^*\psi(0))+(-i)\int_0^tds e^{-i(t-s)H_0}\mathcal{N}(x,t,\psi(s)) = (-i)\int_0^tds e^{-i(t-s)H_0}\mathcal{N}(x,t,\psi_r(s)) +(-i)\int_0^tds e^{-i(t-s)H_0}\left(\mathcal{N}(x,t,\psi(s))-\mathcal{N}(x,t,\psi_r(s))\right) =: S_0(x,t)+S_1(x,t)+S_2(x,t).
+ \overline{{\psi}}_{+}e^{\mu}{}_{a}\gamma_{+}^{a}( i \partial_{\mu}+ i \omega_{+ \mu}+ q V_{\mu}) \psi_{+}+ \overline{{\psi}}_{-}e^{\mu}{}_{a}\gamma_{-}^{a}( i \partial_{\mu}+ i \omega_{- \mu}+ q V_{\mu}) \psi_{-}
D_X([a,b]) + D_X^*(i_bi_a\varphi) = [a,D_X(b)] + i_{D_X(b)}i_a\varphi - [b,D_X(a)] - i_{D_X(a)}i_b\varphi + D_{[\rho(b),X]}(a) - D_{[\rho(a),X]}(b).
\tilde{\phi}(x) := - 4^{-\frac{1}{3}}\left( \frac{|x-y|^{\frac{4}{3}}}{(t_{0}- t')^{\frac{1}{3}}}+ \frac{|x-z|^{\frac{4}{3}}}{(t''-t_{0})^{\frac{1}{3}}}\right) \cdot
var_{P}\left[ \psi_{P,\mathbf{a}}(\mathbf{G,B};\mathcal{G})\right] =E_{P}\left[ var_{P}(\psi_{P,\mathbf{a}}(\mathbf{G,B};\mathcal{G})\mid \mathbf{A},Y,\mathbf{G})\right] +var_{P}\left[ E_{P}(\psi_{P,\mathbf{a}}(\mathbf{G,B};\mathcal{G})\mid \mathbf{A},Y,\mathbf{G})\right] .
E ={\frac{T}{2 \pi}}\rho_{0}^{2}\int_{1}^{\infty}d y \left[{\frac{y^{n / 6}( y - \lambda )^{\delta / 4}}{\sqrt{y^{2 n / 3}( y - \lambda )^{\delta}- ( 1 - \lambda )^{\delta}}}}-{\frac{1}{\sqrt{y}}}\right]
\left(\int \sup_{k \leq n}\left |\sum_{i=1}^k (f\circ T_{\gamma}^{i}(x) -\nu_{\gamma}(f))- \sum_{i=1}^k Z_i (x,y) \right|^2 \! \! \! \nu_{\gamma}(dx) dy \! \right)^{\frac 1 2}\! \! \! = o (n^{\max(\gamma,1/4)}(\log n)^{\eta}) \, ,
Z_{PT}=\prod_\beta \left( \prod_{j=1}^\infty \left(1+(-1)^{j+1}q^j Q^\beta\right)^{jn^0_\beta}\right. & \left. \cdot\prod_{g=1}^\infty\prod_{k=0}^{2g-2}\left(1+\left(-1\right)^{g-k}q^{g-1-k}Q^\beta \right)^{\left(-1\right)^{k+g}n^g_\beta{2g-2 \choose k}}\right),
II'' =\sum_{n=p-2}^{2p-5}q^n\sum_{j=1}^{2p-4-n}(n-2p+2j+3)j +\sum_{n=0}^{p-3}q^n\sum_{j=1}^{n+1}(2j-n-1)j =\sum_{n=p-2}^{2p-5}q^n \frac16(2p-n-3)(2p-n-4)(2p-n-5) +\sum_{n=0}^{p-3}q^n\frac16(n+1)(n+2)(n+3) \ge0.
P_{\mu \nu \rho \sigma}= \frac{k_{\mu}k_{\nu}k_{\rho}k_{\sigma}}{k^{4}}, \ \ \M_{\mu \nu \rho \sigma}= \frac{1}{2}\left( L_{\mu \rho}P_{\nu \sigma}+ L_{\mu \sigma}P_{\nu \rho}+ L_{\nu \rho}P_{\mu \sigma}+ L_{\nu \sigma}P_{\mu \rho}\right) , \ \ \m^{2}= \frac{\alpha \Lambda}{2}.
{\psi}_{m}\propto \quad < \mathrm{s o u r c e}_{a f t e r}\vert < 0 \vert \frac{( a_{i}( \vec{k}) )^{m + 1}}{\sqrt{[ m + 1 ]_{q}!}}\vert \quad a_{i}^{\dagger}g_{i}\quad \vert \frac{( a_{i}^{\dagger}( \vec{k}) )^{m}}{\sqrt{[ m ]_{q}!}}\vert 0 > \vert \mathrm{s o u r c e}_{b e f o r e}>
u=\exp{(\i\phi)}f(t,r), &u=f(t-\epsilon,r), &u=\lambda^{-2/p}f(\lambda^{-2}t,\lambda^{-1}r), &u=(1+\epsilon t)^{-2/p}\exp{(-\i\epsilon r^2/(4+4\epsilon t))}f(t/(1+\epsilon t),r/(1+\epsilon t))\quad,
\hat\tau = \frac{z_+-m}{z_+-y_+}\delta_{y_+}+ \frac{m - y_+}{z_+-y_+}\delta_{z_+}, \check\tau = \frac{z_--m}{z_--y_-}\delta_{y_-}+ \frac{m - y_-}{z_--y_-}\delta_{z_-}.
\delta(y)+\delta(y+e_i)=0, \forall i\neq n, \textrm{whenever}y\in B_0, y+e_i\in B_0; & \delta(y)+\delta(y+e_i+\overline{e_n})=1, \forall i\neq n, \textrm{whenever}y\in B_0, y+e_i+\overline{e_n}\in B_0; & \delta(y)+\delta(y+\overline{e_n})=1, \textrm{whenever}y\in B_0, y+\overline{e_n}\in B_0.
R_{j+1}\leq \widetilde C R_j\Big((m_{j,0}+ R_j)(1+m_j)+ R_j^2+ m_0\sum_{i=0}^{j-1}R_i +\big|\sum_{i=0}^{j-1}R_i\big|^2\Big) &=:\widetilde CQ_j R_j,
Z\left[ \eta \right] =Z\left[ 0\right] \exp \left\{-iT^2\sum_{n,m=-\infty}^\infty \int d\mathbf{x\,}d\mathbf{y}\eta_n^{+}\left( \mathbf{x}\right)\gamma^0S_F^{n-m}\left( \mathbf{x-y}\right) \eta_m\left( \mathbf{x}\right)\right\}.
\int_{\gamma_\lambda}\left(\frac{|f(z)|^2}{\lambda}-1\right) \frac{\partial}{\partial n}\log |z|^2\, \frac{ds}{4\pi}= -\frac1{\lambda}\int_{\gamma_\lambda}\log |z|^2\frac{\partial}{\partial n}|f(z)|^2 \, \frac{ds}{4\pi}+ \int_{E_f(\lambda)}\frac 1{\lambda}\log |z|^2\Delta |f(z)|^2\frac{dxdy}{4\pi}+\frac{|f(0)|^2}{\lambda}-1.
\hat{\hat{T}}_{\hat{\hat{\mu}}\hat{\hat{\nu}}}{}^{\hat{\hat{\rho}}}= -\left(i_{\hat{\hat{k}}_{(n)}}\hat{\hat{C}}\right)_{\hat{\hat{\mu}}\hat{\hat{\nu}}}Q^{nm}\hat{\hat{k}}_{(m)}{}^{\hat{\hat{\rho}}}\, .
({\cal C}_{m}^{k})_{A 2}= ({\cal C}_{m}^{k})_{A 3}= ({\cal C}_{m}^{k})_{A 4}- 2 ({\cal C}_{m}^{k})_{A 5}= - ({\cal C}_{m}^{k})_{A 1}\equiv - c_{m}^{k}.
( z \! - \! \zeta ) \, \exp \! \left( \sum_{n , m \geq 1}\frac{v_{n m}}{n m}z^{- n}\zeta^{- m}\right) \! = \! z \exp \! \left( - \sum_{k \geq 1}\frac{v_{0 k}}{k}z^{- k}\right) \! - \! \zeta \exp \! \left( - \sum_{k \geq 1}\frac{v_{0 k}}{k}\zeta^{- k}\right)
\mathcal{H}^{\ast}=\int d^{2}\,x\,\left[ \frac{1}{2}\,\left( \pi_{i}^{\perp}\,\pi_{i}^{\perp}+4\,m^{2}A\,_{i}^{\perp}\,A\,_{i}^{\perp}\right) +\frac{1}{2}\left( \,\epsilon^{\,i\,j}\,\partial_{\,i}\,A\,_{j}^{\perp}\right)^{2}+2\,m\,\left( A_{1}^{\perp}\,\pi\,_{2}^{\perp}-A\,_{2}^{\perp}\,\pi_{1}^{\perp}\right) \right] .
\sum_{k=0}^m\sum_{i=0}^{m-k}p(m,k,i)u^k((x+1)^i-x^i) = \sum_{k=0}^{m-1}\sum_{i=0}^{m-k-1}\left(mp(m-1,k,i) +(k+1)p(m-1,k+1,i)+kp(m-1,k,i)\right)u^k x^i.
\delta^{(s)}v_k+\frac{\sum_{ij}(\sum_l \Gamma^{k(s)}_{ij,l}c_l)v_i^{\phi^s}v_j^{\phi^s}}{(\sum_l r_l c_l)(\sum_l v_l^{\phi^s})}=0,\ \k\in \{1,\ldots,n\}.
\psi_{\alpha}( z_{1}, \ldots , z_{\frac{N - 1}{2}}) = \prod_{\gamma = 1}^{\frac{N - 1}{2}}( z_{\alpha}- z_{\gamma}) \prod_{\gamma < \delta = 1}^{\frac{N - 1}{2}}( z_{\gamma}- z_{\delta})^{2}\prod_{\gamma = 1}^{\frac{N - 1}{2}}z_{\gamma}
\kappa_4(F_n(t)) =\frac{1}{\mathfrak{s}_n^4(t)}\sum_{i=1}^nK_h(t_{i-1}-t)^4\kappa_4((B_{t_i}-B_{t_{i-1}})^2)\lesssim(nh)^2\cdot\frac{nh}{h^4}\cdot\frac{1}{n^4}=\frac{1}{nh},
U = \frac{r}{l_{\mathrm{s}}^{2}}\ , \quad g_{\mathrm{Y M}}^{2}= \frac{g_{\mathrm{s}}}{l_{\mathrm{s}}^{3}}\, \quad H - p_{1 0}= H - \frac{N}{g_{\mathrm{s}}l_{\mathrm{s}}}\ .
G(v) = -\frac{v^2}{2}+\frac{4 b^2 + (1-2b)^2\cdot v^2}{2\,\big(4 b^2 + (1 - 2 b)\cdot\sqrt{4 b^2 + (1 - 4 b)\cdot v^2}\big)}(v\in[-1,1]).
( - )^{a}\sum_{j = 0}^{a}( - 1 )^{j}\left( \begin{array}{c}{{j}}\\{{a}}\\ \end{array}\right) \mathrm{e x p}\left[ 2 \pi i \frac{Q}{P}( h_{e_{1}+ P e , n_{1}+ P n - \frac{P}{a}j}- h_{e_{1}, n_{1}}) \right] | e_{1}+ P e , n_{1}+ P n - \frac{P}{a}j >
\int_{0}^{\infty}e^{-s x}\hat{F}_1(x)dx=\frac{1}{s}\left(\frac{\mathbb E\left[e^{-s \overline{X}_{e(p+q)}}\right]}{\mathbb E\left[e^{-s \overline{X}_{e(q)}}\right]}-1\right), \ \s > 0.
\|\chi_d\|_{\nu_{pn}}^2 = 2^d \frac{(pn)^{\underline{d}}((1-p)n)^{\underline{d}}}{n^{\underline{2d}}}= (2p(1-p))^d \frac{\left(1-\frac{O(d^2)}{pn}\right) \left(1-\frac{O(d^2)}{(1-p)n}\right)}{\left(1-\frac{O(d^2)}{n}\right)}.
zu =\frac{1}{72}\left(P_{1}-6P_{2}+20P_{4}-25P_{5}+30P_{10}-20P_{20}\right)+\frac{1}{3}z, \frac{z}{u}=\frac{1}{72}\left(-5P_{1}+6P_{2}-4P_{4}+5P_{5}-30P_{10}+100P_{20}\right)+\frac{1}{3}z, zv =\frac{1}{72}\left(-P_{1}+4P_{4}+P_{5}-4P_{20}\right)-\frac{1}{3}z, \frac{z}{v}=\frac{1}{72}\left(P_{1}-4P_{4}-25P_{5}+100P_{20}\right)-\frac{5}{3}z.
\bar{D}^{\dot{\alpha}}J^{0}_{\alpha\dot{\alpha}}= - \frac{1}{24}D_{\alpha}\left[\frac{3N_c-N_f}{2\pi^2}{\rm Tr}\,W^2 +\gamma Z \bar{D}^2 \sum_{f}\left({Q}^\dagger_f e^{V}{Q}_f + \bar{Q}^\dagger_{f}e^{-V}\bar{Q}_f \right)\right]
\int_{DR_g(a_1,\ldots,a_n)}q([\Gamma,\gamma])=\left(\int_{DR_{g_1}(A_{L_1},-a_{L_1})}\xi_{\Gamma_1*}(\gamma_1)\right)\left(\int_{DR_{g_2}(A_{L_2},-a_{L_2})}\xi_{\Gamma_2*}(\gamma_2)\right).
a_h^+(r_h^+ u - r_h u, e_h + e_h^+) = a_h^+(r_h^+ u - u, e_h) + a_h(u - r_h u, e_h^+) &\;\;\;+ a_h^+(u - r_h u, e_h + e_h^+) - a_h(u - r_h u, e_h + e_h^+)
\frac{(p;p)_\infty (q;q)_\infty}{2}\int_{\mathbb{T}}\frac{\prod_{a=1}^6\Gamma(t_a z^{\pm 1};p,q)}{\Gamma(z^{\pm 2};p,q)}\frac{dz}{2 \pi \textup{i}z}= \prod_{1 \leq a < b \leq 6}\Gamma(t_a t_b;p,q),
(\varphi\circ\iota)(Xj_P^-(T_w^P)) = (\varphi\circ\iota)(XT_w) & = \varphi(\iota(X)\iota(T_w)) & = (-1)^{\ell(w)}\varphi(\iota(X)T_w^*) & = (-1)^{\ell(w)}\varphi(\iota(X))\sigma(T^{P*}_w) & = (-1)^{\ell(w) - \ell_P(w)}\varphi^\iota(X)\sigma^{\iota_P}(T^{P}_w).
\mathop{\sum_c \sum_d \sum_n \sum_r \sum_s}_{(rd,sc)=1}b_{n,r,s}g(c,d) e \left(n \frac{\overline{rd}}{sc}\right) \ll_{\epsilon,g_0}(CDNRS)^{\epsilon}K(C,D,N,R,S)\| b_{N,R,S}\|_2,
({\bf e}_{\underline{{a}}}, \; s k_{\underline{{a}}}, \; s k_{{\underline{{a}}}- 1}, s ) \to ( \mathrm{i}{\bf e}_{\underline{{a}}}, \; \mathrm{i}s k_{\underline{{a}}}, \; \mathrm{i}s k_{{\underline{{a}}}- 1}, ( - \mathrm{i})^{\delta_{1{\underline{{a}}}}}s ) .
c(i^{*}_{D}W^{-}_{i}(\nu^{j}))=((1-x)^{r-g_{Y}}\prod_{i=1}^{g_{Y}}(1-x-\sigma_{i}))\cdot( (1-2x)^{r'-g_{Y}}\prod_{i=1}^{g_{Y}}(1-2x+4\sigma_{i}) )