harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\widetilde{I}_{2k}^{(m)}(F):= & \left(\int_{1}^{B/2}\frac{B}{y(B-y)}dy\right)\left(\idotsint_{\mathcal{R}_{k-1}}\left( \int_0^{T_m}F(x_1,\dots,x_k) \, dx_m\right)^2 dx_1 \dots dx_{m-1}dx_{m+1}\dots dx_k \right)
d_{n}^{\cal O}( R ) = \frac{1}{n !}\frac{1}{( 2 \pi )^{n}}\int d \theta_{1}\ldots d \theta_{n}F_{n , n}^{\cal O}( \theta_{n}, \ldots , \theta_{1}| \theta_{1}, \ldots , \theta_{n}) e^{- E_{n}R}\,
\begin{array}{l}x^2y^3dx=d(\frac19x^3y^3-\frac{1}{18a}x^2y^3-\frac{2}{189a^2}xy^3-\frac{2}{1701a^3}y^3) +(\frac{2}{567a^3}+\frac{2}{63a^2}x+\frac{1}{6a}x^2-\frac13x^3)ydH+[(\frac{4}{189a^2}+\frac{2}{9a}x+\frac43x^2)H -\frac{2}{567a^3}x+(\frac{4}{567a^3}-\frac{8}{189a^2})x^2+(\frac{2}{27a^2}-\frac{5}{18a})x^3+(\frac{4}{9a}-\frac13)x^4]ydx.\end{array}
\mathcal{D}(L_a) :=\{f\in\mathcal{D}(L_{\max})\ \mid\W(f,g;a)=0 g\in \mathcal{D}(L_{\max})\}, \mathcal{D}(L_b) :=\{f\in\mathcal{D}(L_{\max})\ \mid\W(f,g;b)=0 g\in \mathcal{D}(L_{\max})\}.
\theta_{N+3}~\equiv ~\frac{N}{\lambda_0}~ \phi_0 + \frac{1}{2 L}\sum_a \int_{-L}^L d x^- (~ \psi_R^{a \dag}\psi_L^a + \psi_L^{a \dag}\psi_R^a ~) \approx 0 ~.
\int_{0}^{2 \pi}d \sigma X^{i}( \sigma ) \hat{P}_{i}( \sigma ) = x_{0}^{i}\hat{p}_{0}^{i}+ \frac{g_{i j}}{2 \sqrt{\alpha^{\prime}}}\sum_{n = 1}^{\infty}\left\{\left( \chi_{n}^{i}\alpha_{- n}^{j}+ \bar{\chi}_{n}^{i}\tilde{\alpha}_{- n}^{j}\right) + \left( \bar{\chi}_{n}^{i}\alpha_{n}^{j}+ \chi_{n}^{i}\tilde{\alpha}_{n}^{j}\right) \right\}~ .
\frac{\partial Y}{\partial x}=\left(\frac{A_3}{x^2}+\frac{A_2}{x}+A_1+A_0 x \right)Y , \frac{\partial Y}{\partial t_1}=\left( x B_{11}+B_{10}\right)Y ,\frac{\partial Y}{\partial t_2}=\left( -\frac{A_3}{t_2x}+B_{20}\right)Y ,
25\sum_{k = 1}^n{( - 1)^{k - 1}F_{mk}{}^4}= \frac{{( - 1)^{n - 1}L_{4mn + 2m}+ L_{2m}}}{{L_{2m}}}&+ \frac{{4\left\{{( - 1)^{n(m - 1)}L_{2mn + m}- L_m}\right\}}}{{L_m}}&+ 3\left\{{( - 1)^{n - 1}+ 1}\right\}\,,
\prod_{j=1}^{kd_1}\prod_{s=1}^{j-1}\prod_{i=1}^{d_1-1}\prod_{t=1}^{i}\begin{bmatrix}2 2\end{bmatrix}_{1,\frac{-2j-2s+2ki+2kt+2}{kd_1}-1}\begin{bmatrix}2 2\end{bmatrix}_{1,\frac{-2j-2s+2ki+2kt+2}{kd_1}+1},
f ( x^{2}) = \frac{1}{2}\int_{0}^{x^{2}}d x{'^{2}}\int_{0}^{1}d \alpha \Big [ d_{1}( \alpha^{2}x^{' 2}) + \alpha^{2}x{'^{2}}d_{2}( \alpha^{2}x^{' 2}) \Big ] ,
\left( \begin{pmatrix}1 - \overline{z_1}0 0 \end{pmatrix}, \ 0 \right), \ \ \ \left( \begin{pmatrix}0 0 1 - \overline{z_1}\end{pmatrix}, \ 0 \right), \ \ \left( 0 , \ \begin{pmatrix}1 - \overline{z_1}0 0 \end{pmatrix}\right)\cdot v_0, \ \ \ \left(0, \begin{pmatrix}0 0 1 - \overline{z_1}\end{pmatrix}\right) \cdot v_0
\int\, \prod_{A=1,2,3,4}d\nu^A\, \exp\big[\sqrt8\pi ig^{-1}{\rm tr}_{k'}\,\chi_{AB}(\nu^A)^t\nu^B\big]=2^{6kN-6{k'}^2}(\pi/g)^{4kN-4{k'}^2}\left({\rm det}_{4 k'}\chi\right)^{N'/2- k'}\ .
\bar\delta_+ : = \bar\partial_+ + \bar\partial^+ : \Omega^{p,q}(\bar L) \to \Omega^{p+1, q}(\bar L) \bar\delta_- : = \bar\partial_- + \bar\partial^- : \Omega^{p,q}(\bar L) \to \Omega^{p, q+1}(\bar L)
l_1(a) =(\kappa-1)[a^{2(\kappa-1)}-a^{\kappa-2}-a^\kappa+1]-\kappa[a^{2(\kappa-1)}-2a^{\kappa-1}+1] &=-a^{2(\kappa-1)}-(\kappa-1)a^{\kappa-2}-(\kappa-1)a^\kappa+\kappa-1+2\kappa a^{\kappa-1}-\kappa &=-a^{2(\kappa-1)}-(\kappa-1)a^{\kappa}+2\kappa a^{\kappa-1}-(\kappa-1)a^{\kappa-2}-1.
2 i ( U^{\mathrm{T}}W )_{I J}+ ( U^{\mathrm{T}}V )_{I}{}^{K}\, F_{K J}+ F_{I K}\, ( Z^{\mathrm{T}}W )^{K}{}_{J}- \textstyle{\frac{i}{2}}F_{I K}\, ( Z^{\mathrm{T}}V )^{K L}\, F_{L J}
\left ({A(q,p) \over 4}\right )_{N=8}= \pi \left({|(\sum\limits_i |Z_i|^4 -2\sum\limits_{i>j}|Z_i|^2 |Z_j|^2 + 8|Z_1 Z_2 Z_3 Z_4|})| \right)^{1/2}\ ,
S_0 = S_1 y^1 + S_2 y^2 = \left[ 2w^1{w^1}_{\substack{x_1}}+ w^2({w^1}_{\substack{x_2}}+{w^2}_{\substack{x_1}}) \right] y^1 + \left[ w^1({w^1}_{\substack{x_2}}+{w^2}_{\substack{x_1}}) + 2w^2{w^2}_{\substack{x_2}}\right] y^2,
\Psi(x,y)=\frac{1}{2\pi i}\int_{\sigma-iT}^{\sigma+iT}\zeta(s,y)\frac{x^s}{s}ds+O\left( \frac{x^{\sigma}\zeta(\sigma,y)}{T}+ \frac{\Psi(x,y) \min\{\log T,(1-\alpha)^{-1}\}}{T^{\alpha(x,y)}}\right).
Y_{I,I'}(\theta) = \langle T b_I^{(\theta)}, b_{I'}^{(\theta)}\rangle, I, I'\in\mathcal{D}_{\leq n},\I\neq I',\ \theta\in\{\pm 1\}, Z_I(\theta) = \langle T b_I^{(\theta)}, b_I^{(\theta)}\rangle - \sum_{K\in\mathcal{B}_I}\langle Th_K, h_K\rangle, I\in\mathcal{D}_{\leq n},\ \theta\in\{\pm 1\}.
\begin{aligned}\Delta H^\pm_{C_R}=0, \mbox{in}A^\pm, H^\pm_{C_R}=0, \mbox{on}\partial A^\pm\cap \partial A, H^\pm_{C_R}=1, \mbox{on}C_R, \partial_r H^+_{C_R}+\partial_r H^-_{C_R}=0, \mbox{on}C_R.\end{aligned}
(A,B,C,D) = \left( 2^5\cdot5^2\cdot7 , \; 2^{10}\cdot5^4 , \;{-2^{13}}\cdot5^5\cdot(9a^2-236) , \; 2^{20}\cdot5^5\cdot(a^2-4)^2 \right) \ .
\widetilde{c}_{1i}(r-2)+\widetilde{c}_{1i}(r+2)-\widetilde{c}_{2i}(r)=0\ \i=1, 2, &\widetilde{c}_{2i}(r-1)+\widetilde{c}_{2i}(r+1)-\widetilde{c}_{1i}(r-1)-\widetilde{c}_{1i}(r+1)=0 \ \i=1, 2.
E_s := C([0,T], H^{s+6}_x) \cap C^1([0,T], H^{s+3}_x) \cap C^2([0,T], H^s_x), F_s := \{(g_1, g_2) : g_1 \in C([0,T], H^{s+6}_x) \cap C^1([0,T], H^s_x), g_2 \in H^{s+6}_x \}
{\ddot{w}}+ \frac{1}{2 g}\frac{\partial g}{\partial w}{\dot{w}}^{2}+ \biggl ( \frac{\dot{\lambda}}{\lambda}+ \frac{1}{g}\frac{\partial g}{\partial \sigma}\dot{\sigma}\biggr ) \dot{w}+ \frac{1}{2 g}\frac{w + 1}{( w - 1 )^{3}}= 0 ,
U = | \phi |^{2}\left\{\sigma_{1}\beta_{1 1}( | \phi |^{2}- c^{2}) + \sigma_{2}\beta_{1 2}( | \chi |^{2}-{c^{\prime}}^{2}) \right\}^{2}+ | \chi |^{2}\left\{\sigma_{1}\beta_{2 1}( | \phi |^{2}- c^{2}) + \sigma_{2}\beta_{2 2}( | \chi |^{2}-{c^{\prime}}^{2}) \right\}^{2},
\sum_{j=1}^\infty \lambda_j^{2\beta\left( \frac{\sigma}{2}+n+\tau + \frac{1}{2}-\gamma \right)}\widetilde{\lambda}_j^{-\alpha}\eqsim_{\text(\alpha,\beta,\gamma,\sigma,n,\tau)}\sum_{j=1}^\infty j^{\frac{4}{d}\left[ \beta\left( n+\tau + \frac{1+\sigma}{2}\right) -\beta\gamma -\frac{\alpha}{2}\right]},
\frac{X_k}{X_k + Y_k}= \frac{\frac{X_k}{2x_k}\frac{2x_k}{2k}}{\frac{X_k}{2x_k}\frac{2x_k}{2k}+ \frac{Y_k}{2y_k}\frac{2y_k}{2k}}\to \frac{x}{x + y}k \to \infty.
\nabla\rho^\varepsilon(x,t) = \varepsilon^{-1}\partial_r\eta_0(r)\nu+\nabla_\Gamma\eta_0(r)+\partial_r\eta_1(r)\nu +\varepsilon\{rW\nabla_\Gamma\eta_0(r)+\nabla_\Gamma\eta_1(r)+\partial_r\eta_2(r)\nu\}+O(\varepsilon^2).
-\ddot{\tilde{\chi}}+ \tilde{\chi}'' - \frac{1}{r}\tilde{\chi}'- m_{eff}^2 \tilde{\chi}+ \kappa^2 \frac{q^2c_1^2}{m_H^2}K_0^2(k_0 r)\cos(2\omega_0t + 2 \delta) \tilde{\chi}=0,
\left. + \left(b_1 \left(\kappa_s + \kappa_t \right) +b_2\kappa_s + \frac{\Lambda}{3}b_3\right)\dot{\phi}^2+\frac{\Lambda}{3}b_4 \ddot{\phi}+ c_1 \dot{\phi}^4 +c_2 \dot{\phi}^2\ddot{\phi}+c_3 \ddot{\phi}^2 \right] +(\mbox{s.t.})\;,
\frac{\prod_{{j}=1}^{k}\big({j}+\alpha\cdot(\mathbf a +\mathbf 1)\big)z^{\mathbf a}\phi_{\mathbf a,\mathbf b}(z^\prime)}{\pi^k}\int_{\mathbb B^k}\frac{(1-\|\eta\|^2)^{\alpha\cdot(\mathbf a+\mathbf 1)}\eta^{\mathbf c}}{(1-\langle w,\eta\rangle)^{1+k+\alpha\cdot (\mathbf a+\mathbf 1)}}dV(\eta).
\nabla\Psi\cdot u=\frac{\partial\Psi}{\partial r}\cdot\frac{1}{r^{2}sin\theta}\frac{\partial\Psi}{\partial\theta}+\frac{1}{r}\frac{\partial\Psi}{\partial\theta}\cdot(-\frac{1}{rsin\theta}\frac{\partial\Psi}{\partial r})=0.
|\nabla\nabla_{x}^{p}{\mathbf{u}}|^{2}\leq|\nabla^{2}\nabla_{x}^{p-1}{\mathbf{u}}|^{2},p\geq1,\qquad|\nabla\nabla_{x}^{p}{\mathbf{u}}|^{2}=|\nabla{\mathbf{u}}|^{2},p=0.
\hat{u}^{j}={}\big(\mathcal{G}_3(\xi,t)-\epsilon|\xi|^2\mathcal{G}_1(\xi,t)\big)\hat{u}^{j}_0 +\kappa|\xi|\mathcal{G}_1(\xi,t)\hat{\sigma}^j_0, \hat{\sigma}^j={}-\frac{\alpha}{2}|\xi|\mathcal{G}_1(\xi,t)\hat{u}^{j}_0 +\big(\mathcal{G}_2(t,\xi)+\epsilon|\xi|^2\mathcal{G}_1(\xi,t)\big)\hat{\sigma}^j_0,
T=\frac{1}{2}\sqrt{|\begin{matrix}x_{A}&x_{B}&x_{C}\\y_{A}&y_{B}&y_{C}\\ 1&1&1\end{matrix}|^{2}+|\begin{matrix}y_{A}&y_{B}&y_{C}\\z_{A}&z_{B}&z_{C}\\ 1&1&1\end{matrix}|^{2}+|\begin{matrix}z_{A}&z_{B}&z_{C}\\x_{A}&x_{B}&x_{C}\\ 1&1&1\end{matrix}|^{2}}.
|c_{i,j}^{(k,l)}| \le \dfrac{1}{4}\left(\dfrac{1}{i}(|c_{i-1,j}^{(k+1,l)}|+|c_{i+1,j}^{(k+1,l)}|)+\dfrac{1}{j}(|c_{i,j-1}^{(k,l+1)}|+|c_{i,j+1}^{(k,l+1)}|)\right) & \le\dfrac{1}{4}\left(\dfrac{8V_{k,l}}{\pi^2ij}+\dfrac{8V_{k,l}}{\pi^2ij}\right)=\dfrac{4V_{k,l}}{\pi^2ij},~~~~i,j\ge 1.
\!\!\!\!\!\!\! \mathcal{R}_{\rm d}= \int\limits_{0}^{\infty}\int\limits_{0}^{R_{\rm M}}\frac{f_R(r_o)B_{\rm d}^{\alpha{\rm D}}}{(g+1) \ln (2)}\exp \left(- \frac{N_o g}{P_{\rm d}r_o^{-\eta}}\right) \times \mathcal{L}_{I_{\rm d}}\left(\frac{g}{P_{\rm d}r_o^{-\eta}|\mathcal{I}_{\rm d}(\alpha)|^2}\right) dr_o \dg ,
(\delta^{n}\omega)(X_{1},X_{2}, . . ., ,X_{n+1}) =[X_{1},\omega(X_{2}, . . ., X_{n+1})]_{L}&\,+\sum_{i=2}^{n+1}{(-1)^{i}[\omega(X_{1}, . . ., \widehat{X_{i}}, . . ., X_{n+1}),X_{i}]_{R}}&\,+\sum_{1\leq i<j\leq n+1}{(-1)^{j+1}\omega(X_{1}, . . ., X_{i-1},[X_{i},X_{j}],X_{i+1}, . . ., \widehat{X_{j}}, . . ., X_{n+1})},
\frac{1}{n^{2p}}\mathbb{E}\left[ \frac{\left\| T_{n+1}\right\|^{2p}}{\gamma_{n}^{2p}}\right] \leq \frac{1}{c_{\gamma}^{2p}}\frac{1}{n^{2p -2p\alpha}}\frac{K_{1}}{(n+1)^{p\alpha}}& = o \left( \frac{1}{n^{p}}\right) ,
\boldsymbol Y :=\{\underline{Y_1},\underline{Y_2},\underline{Y_3},Y_1 Y_2,Y_1 Y_3, Y_2 Y_3\}, \boldsymbol Z :=\{Z_1,\underline{Z_1Z_2},\underline{Z_1Y_1 Y_2 Y_3}, Z_1Z_2Y_1 Y_2 Y_3\}, \boldsymbol Z' :=\{Z_2,\underline{Z_1Z_2},\underline{Z_2Y_1 Y_2 Y_3}, Z_1Z_2Y_1 Y_2 Y_3\}.
M_{\alpha_{21}}(\varepsilon_1e_5) = \langle \alpha_{21}, \rho(\varepsilon_1\alpha_{12})\rangle\otimes \varepsilon_2\alpha_{21}\varepsilon_1^2-\langle \alpha_{21}, \rho(\varepsilon_1^2\alpha_{12})\rangle\otimes \varepsilon_2\alpha_{21}\varepsilon_1 = -\varepsilon_2\alpha_{21}\varepsilon_1.
\mathcal{M}(u)=\left(\frac{\bar{R}}{n+1}\mathcal{E}(u)-\mathcal{E}^{\mbox{Ric}\ensuremath{\omega_{0}}}(u)\right)+H_{\mu_{0}}(\omega_{u}^{n}),\,\,\,\,\,\bar{R}:=\frac{nc_{1}(X)\cdot[\omega_{0}]^{n-1}}{[\omega_{0}]^{n}}
d s^{2}= -{\frac{( r^{d}- r_{1}^{d}) ( r^{d}- r_{2}^{d})}{C^{2 d}}}d t^{2}+{\frac{r^{2 d - 2}C^{2}}{( r^{d}- r_{1}^{d}) ( r^{d}- r_{2}^{d})}}d r^{2}+ C^{2}d \Omega_{D - 2}^{2}
\left( T^{2}\frac{\partial}{\partial T}+ \int d x \frac{e^{2}}{2}\sum_{a = 1}^{N^{2}}\left( \frac{\delta}{\delta A^{a}( x )}\right)^{2}\right) \left< A \right| e^{- H / T}\left| A^{g}\right> = 0 ,
\sum_{\{\alpha_1,\beta_1\}\in\mathcal{S}_1(N)}\left(e^{\frac{1}{2}(\ell_{\alpha_1}(\rho)+\ell_{\beta_1}(\rho))}-1\right)^{-1}+\sum_{\{\alpha_2,\beta_2\}\in\mathcal{S}_2(N)}\left(e^{\frac{1}{2}(\ell_{\alpha_2}(\rho)+\ell_{\beta_2}(\rho))}+1\right)^{-1}=\frac{1}{2}.
\hat{\cal H}\prod_{j = 1}^{r}( B_{f_{j}}^{+})^{n_{j}}e^{W}= \left({\cal E}_{0}+ \omega \sum_{k = 1}^{r}n_{k}f_{k}\right) \prod_{j = 1}^{r}( B_{f_{j}}^{+})^{n_{j}}e^{W}.
\langle R\mathbf{r}\rangle_{s;k}=\sum_{\ell\geq 1}R_{\ell;k-1}\mathbf{r}_{s-\ell}, \langle \mathbf{r}^\dag R\rangle_{s;k}=\sum_{\ell\geq 1}\mathbf{r}^\dag_{s-\ell}R_{\ell;k-1}, \langle \mathbf{r}^\dag R\mathbf{r}\rangle_{s;k}=\sum_{p+q+t=s}\mathbf{r}^\dag_{p}R_{q;k-2}\mathbf{r}_{t}.
\hat{\mu}^p_{\Lambda, \epsilon}(dt) = \frac{1}{Z'_{\Lambda,\epsilon}(\alpha)}\det[D_{\Lambda,\epsilon}(t)]^{-1/2}\prod_{jk \in E(\mathbb{T}_N)}\Big(\exp(-e^{t_{jk}}+t_{jk})f_\alpha(e^{t_{jk}})dt_{jk}\Big),
\varphi_{n,c}^{\alpha}(\cos(\frac{\theta}{\lambda}))=(-1)^nB_n\left(\cos(\frac{\theta}{\lambda})\right)^{\alpha+1/2}P_n^{(0,\alpha)}\left(\cos(\frac{2\theta}{\lambda})\right)+\gamma_n^{\alpha+1/2}O(\frac{c^2}{n})
S [ y , \psi ] = \frac{k}{2 \pi}\int d^{2}z \left( \partial y \bar{\partial}y - \partial \psi_{1}\bar{\partial}\psi_{1}- \partial \psi_{2}\bar{\partial}\psi_{2}- 2 \cosh{2 y}\partial \psi_{2}\bar{\partial}\psi_{1}\right)
\frac{a_{\mathrm{I R}}}{a_{\mathrm{U V}}}= \frac{c_{\mathrm{I R}}}{c_{\mathrm{U V}}}= \frac{2 7}{3 2}, ~ ~ a_{\mathrm{U V}}= c_{\mathrm{U V}}, ~ a_{\mathrm{I R}}= c_{\mathrm{I R}}
J_{1}(t_{0}^{\alpha}u(t_{0}\cdot)) <\mathop{\lim}\limits_{n\rightarrow\infty}\Big(\frac{1}{2}\|\nabla\overline{u}_{k}\|_{2}^{2}+\frac{1}{2}\|\overline{u}_{k}\|_{2}^{2}+\frac{1}{2}B(\overline{u}_{k})-\frac{1}{2p}\|\overline{u}_{k}\|_{2p}^{2p}\Big) &=\widetilde{E}_{1},
\delta_0 := \inf_{i \neq j}\inf_x \sin^2 \left( \pi ( \rho_i (x) - \rho_j (x) ) \right) , \mbox{and}G ( s , t ) := \left( s^{-1}\left( \frac{t^2}{1 - t}+ \frac{t -s}{2}\right) \right)^2
\begin{cases}\||x'|^{\frac{1}{2}}|x|^{\frac{1}{2}}u^{c,\gamma}\|_{L^{\infty}(\Omega_{e^{-1}})}\leq K(c,\gamma), \||x|u^{c,\gamma}\|_{L^{\infty}(\mathbb{R}^{3}\setminus\Omega_{e^{-1}})}\leq K(c,\gamma),\end{cases}
\int_0^t \mu x_{\ell - 1}(u) |e^{\int_u^t G_{\ell}(v) \: dv}- e^{s \ell (t-u)}| \: du \leq (s \eta + \mu)t \int_0^t \mu x_{\ell - 1}(u) e^{s \ell (t-u)}\: du = (s \eta + \mu) t x_{\ell}(t).
u=\frac{tw^1+w^2}{t^2+1}+\frac{t(x+\mu)}{t^2+1}-\frac{y}{t^2+1}, v=\frac{-w^1+tw^2}{t^2+1}+\frac{ty}{t^2+1}+\frac{x-\mu}{t^2+1},
\rho= \, \, a_1 e^{123}+ a_2 e^{125}+ a_3 e^{126}+ a_4 (e^{135}-e^{124}) + a_5 e^{136}+ a_6 e^{234}+ a_7 (e^{134}+ e^{235}) + a_8 e^{236}begin{align*}2pt] + a_9 (e^{256}- e^{146}) + a_{10}(e^{246}+ e^{356}) + a_{11}e^{456},
\frac{1}{2}\frac{d\|\zeta^{h_1,h_2}(t)\|^2}{dt}=\langle Df(X^x(t)).\zeta^{h_1,h_2}(t),\zeta^{h_1,h_2}(t)\rangle+\langle D^2f(X^x(t)).(\eta^{h_1}(t),\eta^{h_2}(t)),\zeta^{h_1,h_2}(t)\rangle &\le -(\gamma+\epsilon)\|\zeta^{h_1,h_2}(t)\|^2+\frac{1}{4\epsilon}\mathcal{N}_{q}(f)^2(1+\|X^x(t)\|^{2q})e^{-4\gamma t}\|h_1\|^2\|h_2\|^2.
F(x,t_1(x))\geq F(x,t_2(x))\geq\dots\geq F(x,t_j(x)); F(x,t_j(x))=0j>N(x); N(x)\geq j t_{j+1}(x)\notin\{t_1(x),\dots,t_j(x)\}; F(x,t_{j+1}(x))=\sup\bigl\{F(x,t)\colon t\notin\{t_1(x),\dots,t_j(x)\}\bigr\}.
\alpha_{k+1}=P(k+1)-\sum_{i=0}^k\alpha_i\binom{k+1}{i}\vert\alpha_{k+1}\vert \leq 1+\sum_{i=0}^k 2^i i!\binom{k+1}{i}&=2^{k+1}(k+1)!\left(\frac{1}{2^{k+1}(k+1)!}+\sum_{i=0}^k\frac{1}{2^{k+1-i}(k+1-i)!}\right) &<2^{k+1}(k+1)!\left(\frac{1}{8}+e^{1/2}-1\right)<2^{k+1}(k+1)!
a_{m+k}= \log(N(\star_{i=1}^{m+k}H,\alpha^{mN+kN+1}) ) \le \log( N(\star_{i=1}^{m}H,\star_{i=1}^{m}\beta) N(\star_{i=1}^{k}H, \star_{i=1}^{k}\beta) ) &=\log( N(\star_{i=1}^{m}H,\star_{i=1}^{m}\beta)+ \log (N( \star_{i=1}^{k}H,\star_{i=1}^{k}\beta) )=a_{m}+a_{k}.
\delta_{h}S = \int_{{\cal M}^{1 + 1}}\left({\frac{1}{2}}\hat{E}^{I}\wedge \hat{E}^{--}i_{\delta}f^{+ + I}-{\frac{1}{2}}\hat{E}^{I}\wedge \hat{E}^{--}i_{\delta}f^{+ + I}+ m j_{1}\wedge \hat{E}^{i}i_{\delta}f^{i}\right)
\alpha_{(\kappa)\hat\imath}= (\underbrace{0,\dots,2(D-\kappa-3)\beta_{D-\kappa-1},-2(D-\kappa-1)\beta_{D-\kappa},0,\dots,0}_{k=D-3 \rm~ terms}\,;\underbrace{0,\dots, 0}_{q\rm ~terms}) \, ,
\tilde S=\pi t \left[ \left(m_i+ C_{ijk}n^j p^k\right)^2 +\left(n^i \right)^2 \left(p^i\right)^2 -\left(n^i p^i\right)^2\right] + 2\pi i m_i \left(\tau_1 n^i+\omega_1 p^i\right) ,
\frac{1}{2}S(n,r)\binom{n}{j+u}(-1)^{j+u+r-1}\binom{j+u+r-1}{j+u}\sum_{l=0}^{r-1}(-1)^l\binom{2n-j-u+l}{l}\cdot \cdot\binom{2(j+u+r-1-l)}{j+u+r-1-l}\binom{n-j-u}{j+u+r-l-1}\bigl{(}\frac{1}{2}S(n,n-j-u+l+1)\bigr{)}
\lambda \{x \in X \ ;\ \theta(x) \geq \varepsilon \}\leq a^* \lambda \{(g,x') \in U^2 \times \Sigma \ ;\ \theta(gx') \geq \varepsilon \}& \leq a^* \lambda \{(g,x') \in U^2 \times \Sigma\ ;\ \theta_r'(gx') \geq \varepsilon \}& = \int_{\theta_r' \geq \varepsilon}d\lambda(x) \kappa(x)
u_i=u_N^{(i)}+u_i^{(N+1)},\ \ \u_N^{(i)}=\sum_{p=0}^N\frac{a_p^{(i)}(s,\tau,y')}{(ik)^p}e^{ik(s-s_0')},\ \s_0'=s_0-T_1,
\begin{aligned}\left|\mathbb{E}^{u^\circ}\left[\mu \Phi(x_K) \right] - \mathbb{E}^{u}\left[\mu \Phi(x_K) \right] \right| \leq e_\Phi (\hat{\phi}^Kd_0+s\sum_{i=1}^{K-1}\hat{\phi}^i). \end{aligned}
S(\xi,\eta;u):= \lim_{N\to\infty}\frac{1}{N}\ln\left[\binom{k-1}{l}\binom{L}{l}t^{L-2l}(t^2-2\Delta t +1)^{l}H_N^{(k)}\right] = \ell(\xi)-\ell(\eta)-\ell(\xi-\eta)+\ell(u)-\ell(\eta)-\ell(u-\eta)-2\eta\ln t \qquad\qquad+\eta\ln(t^2-2\Delta t +1)+\lim_{N\to\infty}\frac{1}{N}\ln\left[H_N^{(\xi N)}\right].
\dot{x}^{A A^{\prime}}= o^{A}\bar{o}^{A^{\prime}}, \, \, \acute{x}^{A A^{\prime}}= o^{A}\bar{\iota}^{A^{\prime}}+ \iota^{A}\bar{o}^{A^{\prime}}, \, \, ( \rho^{\tau}o^{A}\bar{o}^{A^{\prime}}) \dot{}={\cal F}^{A A^{\prime}},
\mathcal{U}(h,k) = -\mathcal{L}^0(h,k)+\mathcal{U}^2(h,k)+\mathcal{U}^r(h,k)+ O \big( (Xhk)^\varepsilon XC \big) + O\bigg(Q\frac{(XCHK)^{\varepsilon}}{\sqrt{HK}}\bigg).
\Vert \psi_E\Vert_{\ell^2(\Lambda_{2L, L})}\leq l^{d - 1}e^{-m''l}\sup_{x\in \Lambda_{2L_+,L_-}(x_0)}|\psi_E(x)| \leq (\tfrac{L}{100})^{d - 1}e^{-\frac{m''}{100}L}(2L_+)^\nu\Vert T_{x_0}^{-1}\psi_E\Vert_{\ell^2}\leq e^{-\frac{m}{100}L}\Vert T_{x_0}^{-1}\psi_E\Vert_{\ell^2}
{\textstyle\sum\limits_{\left( i,j\right) \in\mathbb{N}^{2}}}x_{i,j}={\textstyle\sum\limits_{i\in\mathbb{N}}}\left({\textstyle\sum\limits_{j\in\mathbb{N}}}x_{i,j}\right) ={\textstyle\sum\limits_{j\in\mathbb{N}}}{\textstyle\sum\limits_{i\in\mathbb{N}}}x_{i,j}.
\begin{aligned}\int_{-\infty}^{+\infty}\left(u_\xi,(|D_{\gamma,\xi}|^2+|z|^2)^{-1/2}|D_{\gamma,\xi}|(|D_{\gamma,\xi}|^2+|z|^2)^{-1/2}u_{\xi}\right)_{L^2_\xi}dz=\pi\|u_\xi\|^2_{L^2_\xi}\end{aligned}.
H_{1}y_{i}= - a \cdot y_{i}(a q), B_{1}y_{i}= - q a^{-1}\cdot y_{i}(a q^{-1}), \allowdisplaybreaks H_{2}y_{i}= - b \cdot y_{i}(b q), B_{2}y_{i}= - q b^{-1}\cdot y_{i}(b q^{-1}), \allowdisplaybreaks H_{3}y_{i}= - c^{-1}\cdot y_{i}(c q), B_{3}y_{i}= - c q^{-1}\cdot y_{i}(c q^{-1}).
\alpha_{i}^{\prime}={\cal{F}}( \alpha_{1}^{\prime \prime}, \ldots , \alpha_{N}^{\prime \prime}) \alpha_{i}^{\prime \prime}, \mathrm{w i t h}{\cal{F}}( \alpha_{1}, \ldots , \alpha_{N}) = \frac{\sum \alpha_{i}}{\sum \frac{\alpha_{i}}{m_{i}}}.
V = ( - 1 )^{F + 1}g{\frac{k^{4}a^{4}}{3 2 \pi^{2}}}\left({\frac{k^{2}a^{2}}{4 \pi \mu^{2}}}\right)^{- \epsilon / 2}\Gamma ( - 2 + \epsilon / 2 ) \sum_{n}x_{n}^{4 - \epsilon}.
\mathbf{T}f = \chi (D) D^{-1}\tilde{D}^{1 -\sigma}e^{\pm i t D}f; \mathbf{T}^* F = \int_{-\infty}^\infty \chi (D) D^{-1}\tilde{D}^{1 -\sigma}e^{\mp i s D}F(\cdot, s) \,ds; \mathbf{T}\mathbf{T}^* F = \int_{-\infty}^\infty \chi^2 (D) D^{-2}\tilde{D}^{2 -2\sigma}e^{\pm i(t-s) D}F(\cdot, s)\, ds;
[\rho(L_i) , \rho(L_j) ]\, =\, h(z)^{i+j}\,\cdot \left( (L +i \lambda -j ) P(L -\lambda -i-j, i) (L +j\lambda ) P(L -\lambda-j, j)\right.\, - -\, \left.(L +j \lambda -i) P(L -\lambda -i-j, j) (L +i\lambda ) P(L -\lambda-i, i) \right)\,= = \, (i-j) h(z)^{i+j}(L +(i+j) \lambda ) P(L -\lambda-(i+j), i+j)
\frac{d^k}{dx^k}p^{n,k}(x) =\left(1-\frac{k}{n}\right)\mu^{n,k}p^{n,k}(x)+\sum_{m=0}^{k-1}r_{m}^{n,k}\frac{d^m}{dx^m}p^{n,k}(x), \frac{d^k}{dx^k}v^{n,k}(x) =\left(1-\frac{k}{n}\right)^2\mu^{n,k}v^{n,k}(x)+\sum_{m=0}^{k-1}r_{m}^{n,k}\frac{d^m}{dx^m}v^{n,k}(x), \frac{d^k}{dx^k}T^{n,k}(x) =\sum_{m=1}^{k-1}r_{m}^{n,k}\frac{d^m}{dx^m}T^{n,k}(x)+\mu^{n,k}.
m_{\mu}L_N (T_{N-1}T_{N-2}\dots T_{N-h}) &= m_{\mu}\big\{(q-q^{-1})q^{h-1}L_{N}+ \sum_{s=2}^h (q-q^{-1}) q^{h-s}q^{s-1}L_{N-s+1}+ q^h L_{N-h}\big\}&= m_\mu \big( q^h \Phi_1^+(L_N,L_{N-1}, \dots, L_{N-h}) - q^{h-2}\Phi_1^+ (L_N, L_{N-1}, \dots, L_{N-h+1}) \big).
i_{C,k}=\sum_{e=(j,k)\in B}i_e= \sum_{e\in B}\frac{1}{r_{e}}v_e=\sum_{j:(j,k)\in B}\frac{1}{r_{e}}(v_{C,j}-v_{C,k}) \Leftrightarrow \dot{v}_k =\sum_{j:(j,k)\in B}\frac{1}{c_k r_{kj}}(v_{C,j}-v_{C,k}),
E=m_{0}c^{2}\frac{dt}{d\tau}=\frac{m_{0}c^{2}}{\sqrt{1-\frac{\dot{r}^{2}(t)}{c^{2}}}}=m_{0}c^{2}+\frac{1}{2}m_{0}\dot{r}^{2}(t)+\frac{3}{8}m_{0}\frac{\dot{r}^{4}(t)}{c^{2}}+\cdot\cdot\cdot.
\|(U'(t),-U(t))\|_{(\mathcal H_{-1})^N \times (\mathcal H_0)^N}=\|L_t\circ S_t^{-1}\| \qquad\qquad\qquad \leq c(\|(\widehat{U}_0, \widehat{U}_1)\|_{(\mathcal H_0)^N\times( \mathcal H_{-1})^N}+\|H\|_{L^2(0,T; (L^2(\Gamma))^M)})
\epsilon_{RPA}({\bf{q}},{\tilde{\omega}}) = 1 + \frac{v_{{\bf{q}}}}{V}\sum_{{\bf{k}}}\frac{n_{F}({\bf{k}}+{\bf{q}}/2) - n_{F}({\bf{k}}-{\bf{q}}/2)}{{\tilde{\omega}}- \frac{{\bf{k.q}}}{m}}
{\sf W}_p^{(\sigma)}(\hat \mu_n,\nu_{\theta}) > \frac{C}{2}\frac{\xi_n}{\sqrt n}-{\sf W}_p^{(\sigma)}(\hat \mu_n,\mu) &={\sf W}_p^{(\sigma)}(\hat \mu_n,\mu) + o_{\mathbb{P}}(n^{-1/2}),
\ell_0(\bar{X}) := \frac{1}{(n+\sqrt{n})\sqrt{N}}(\sqrt{nN}-\sum_{1\le j\le n}X_j), \ell_i(\bar{X}) :=\frac{1}{(n+\sqrt{n})\sqrt{N}}(X_i+\sqrt{N}) \hbox{for}i=1,\dots, n.
\eta_{j}^{+}\eta_{j}^{-}- q^{- 2}\eta_{j}^{-}\eta_{j}^{+}= q^{4 ( L_{z})_{1}+ \cdots + 4 ( L_{z})_{j - 1}}\frac{q^{4 ( L_{z})_{j}}- 1}{q^{2}- 1}.
\psi_n^{\,\omega,t}\coloneqq (\psi_n\circ\tau_{\omega}-\psi_n)\circ\vartheta_n\circ\tau_t=\psi_n^{\,\omega,0}\circ\tau_t \varphi_n^{\,\omega,t}\coloneqq(\varphi_n\circ\tau_{\omega}-\varphi_n)\circ\vartheta_n\circ\tau_t=\varphi_n^{\,\omega,0}\circ\tau_t
\rho^m\Delta_\phi W = \left((2-m)A + c\rho_{(1)}+ O(\rho)\right)\left(e^{-Bs}-e^{-Bs_0}\right) & + \left(\rho + O(\rho^2)\right)\left(-\frac{nB}{s}+B^2+o(1)\right)e^{-Bs},
\lim_{m_{\pi}^{2}\rightarrow 0}m_{\pi}^{2}\int d^{3}\vec{x}\omega_{\pi}^{( 1 )}= - \lambda \int d^{3}\vec{x}( \varphi_{\pi}^{3}+ \varphi_{\pi}\varphi_{\sigma}^{2}+ 2 v \varphi_{\pi}\varphi_{\sigma}) \ne 0 \quad ,
\vert I_2 \vert \le cn^{-3/2}\int_{\substack{0 \le \vert y\vert \le O(n) \arg y = \pm2\pi/3}}\left\vert \left( \begin{matrix}-\Psi_{12}\Psi_{22}(y) \Psi_{12}^2(y) -\Psi_{22}^2(y) \Psi_{12}\Psi_{22}(y)\end{matrix}\right)\right\vert \vert{\rm d}y\vert = O\big(n^{-3/2}\big),
\varepsilon_\Phi(X) = - \frac{q^{1-g}h}{\zeta(2)(1-q^{-1})(q^2-1)}+ \sum_{j=1}^{2g}\frac{Z(q \gamma_j^{-1})}{Z'(\gamma_j^{-1})}\frac{\gamma_j}{\gamma_j - 1}+ \frac{1}{2 \pi i}\sum_{N = 0}^{X-1}\oint_{C_\rho}\frac{1}{u^{N+1}}\frac{Z(qu)}{Z(u)}\,du.
{\cal S}_{quad}=tr\int d^4x\left( \frac{g^2}8\omega^2\chi^{*\mu \nu}\chi_{\mu \nu}^{*}-\frac{g^2}4\omega \chi^{*\mu \nu}\varepsilon_\mu \psi_v^{*}-\frac{g^2}{32}\varepsilon^\mu \varepsilon^\nu \psi_\mu^{*}\psi_v^{*}+\frac{g^2}{32}\varepsilon^2\psi^{*\mu}\psi_\mu^{*}\right) \;.
\frac{\partial}{\partial t}(\frac{\partial z}{\partial x}\cdot\frac{\partial x}{\partial t})=\frac{\partial}{\partial t}\frac{\partial z}{\partial x}\cdot\frac{\partial x}{\partial t}+\frac{\partial z}{\partial x}\cdot\frac{\partial}{\partial t}\frac{\partial x}{\partial t}.
\sum_{n = 0}^{\infty}\frac{(a;q)_{n}(b;q)_{n}}{(q;q)_{n}(c;q)_{n}}z^{n}= \frac{(b;q)_{\infty}(az;q)_{\infty}}{(c;q)_{\infty}(z;q)_{\infty}}\sum_{n = 0}^{\infty}\frac{(c/b;q)_{n}(z;q)_{n}}{(q;q)_{n}(az;q)_{n}}b^{n}, \,\,\, |z| < 1, |b| < 1, |q| < 1.
\left(\int_0^\tau \| A e^{-tA}\int_0^\tau e^{-(\tau -s) A}g(s) ds \|^2 dt \right)^{1/2}& = \left( \int_0^\tau \| A^{1/2}e^{-tA}\int_0^\tau A^{1/2}e^{-(\tau -s) A}g(s) ds \|^2 dt \right)^{1/2}&\le C \| \int_0^\tau A^{1/2}e^{-(\tau -s) A}g(s) ds \|.
\Phi\,\,\,(N \times N) = \left( \begin{array}[c]{cccccc}\alpha \beta 0 0 \cdots 0 \beta \alpha \beta 0 \cdots 0 0 \beta \alpha \beta \cdots 0 \vdots \vdots \ddots \ddots \ddots \vdots 0 0 \cdots 0 \beta \alpha \end{array}\right)
\sqrt{[X]_{\sigma+t}}- \sqrt{[X]_{\sigma}1_{\{\sigma>0 \}}}\leq \sqrt{[X]_{\sigma+t}- [X]_{\sigma}1_{\{\sigma>0 \}}}=\sqrt{[\hat{X}]_t - X_{\sigma}^2 1_{\{\sigma>0 \}}}\leq \sqrt{[\hat{X}]_t}\,
\frac{1}{2^d}\sum_\Phi h^\mathrm{Falt}_{(E,\Phi)}= - \frac{1}{2}\cdot \frac{L'(0,\chi)}{L(0,\chi )}- \frac{1}{4}\cdot \log \left| \frac{D_E}{D_F}\right| - \frac{d}{2}\cdot \log(2\pi ).
A = (\gamma_{ij}) \in \mathbb{Z}^{nk}, B = \left( \begin{array}{c}\mathbf{b_1}\vdots \mathbf{b_n}\end{array}\right) \in \mathbb{Z}^{nm}, X = \left( \begin{array}{c}\mathbf{x_1}\vdots \mathbf{x_k}\end{array}\right).
\liminf_{n\to\infty}\int_0^T\langle \mathfrak{A}_n(v^n(t)),v^n(t)\rangle\d t =-\frac{1}{2}\limsup_{n\to\infty}\|v^n(T)\|_{\L^2}^2+\frac{1}{2}\liminf_{n\to\infty}\|v_0^n\|_{\L^2}^2 &\leq -\frac{1}{2}\|v(T)\|_{\L^2}^2+\frac{1}{2}\|v_0\|_{\L^2}^2=\int_0^T \langle\mathfrak{A}_0(t),v(t)\rangle\d t.
\pi_*[C^{\rm sing}_{a,b}]= \left( \sum (-1)^k\binom{n}{k}a^{i+1}b^{j+1}- \sum (-1)^k\binom{n+1}{k}a^ib^j \right)c_1 &+\left( \sum (-1)^k\binom{n+1}{k}a^ib^{j+1}(1+(i+1)a) \right)u &+\left( \sum (-1)^k\binom{n+1}{k}a^{i+1}b^j (1+(j+1)b) \right)v
\int_0^1s^j|\sigma^{(j)}(st,s\zeta)|ds =\int_0^1|\partial_t^j(t^{2d-1}\sigma_t(s,\zeta))|ds &\leq||\partial_t^j(t^{2d-1}\varphi(T+z^2)^{-d})||_{tr}\leq C_0,
U_{r,R}(t):=\begin{cases}0{\rm if\}t\in[0,r/2[\cup[2R,+\infty[, \sum_{i=0}^{2m-1}\frac{a_i}{r^i}t^i{\rm if\}t\in[r/2,r[, 1{\rm if\}t\in[r,R[, \sum_{i=0}^{2m-1}\frac{b_i}{R^i}t^i{\rm if\}t\in[R,2R[.\end{cases}
F_{\chi}(Z) = \sum_{B = \left(\begin{smallmatrix}n r/2 r/2 m\end{smallmatrix}\right)}\left( \sum_{\substack{a \mid \gcd(n,r,m), \gcd(a,N)=1}}a^k\chi(a) c\left(\frac{4nm-r^2}{a^2}\right) \right) e^{2\pi \sqrt{-1}\mathrm{Tr}(BZ)}.
Z(X_{(z,a,b,c)}, X_{(\bar{z}, \bar{a}, \bar{b}, \bar{c})}) = X_{(z,a,b,c)}+ X_{(\bar{z}, \bar{a}, \bar{b}, \bar{c})}- \int^1_0 dt \, \sum^{+\infty}_{n=1}\frac{\left(I-M\right)^n}{n(n+1)}X_{(\bar{z}, \bar{a}, \bar{b}, \bar{c})}.