harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\langle 1, \gamma_{1},\dots,\gamma_{k-1}\rangle_{g,k,\beta}=\begin{cases}(\gamma_1,\gamma_2), \mbox{if}\g=0, k=3, \beta=0; 0, \mbox{if}\ 3g-3+k\geq 1\ \mbox{or}\ \beta\neq 0.\end{cases}
\|f -{\mathsf{C}}_{d_x,d_y}[f]\|_1 = \int_D \left|f(x,y) -{\mathsf{C}}_{d_x,d_y}[f](x,y)\right|dxdy \le 4 \sum_{i = d_x+1}^{\infty}\sum_{j = d_y+1}^{\infty}\left|c_{i,j}\right|,
0_{\sum_{i=1}^{n}q_i}\in \{p: p=-y^* + x^* - g({x^*})+\overline{W}^{\rm T}\lambda^* ,\,x^*=P_{\Omega}(y^*),\,g({x^*})\in\partial{f}({x^*})\}, 0_{nm}= d-\overline{W}x^*-Lz^* ,x^*=P_{\Omega}(y^*), 0_{nm}= L\lambda^*.
\Phi_s(y) = \Phi_{s + t}(z) \stackrel{\eqref{eq:thmaboutrelationbetweenboxdimensionandRokhlindimension-t-4}}{\in}\Phi_{\left[ - \left( \frac{1}{2}- 4\varepsilon \right) 8L , \left( \frac{1}{2}- 4\varepsilon \right) 8L \right]}( \Delta^{(l, \sigma)}_1 \cap K' ) \; ,
\nabla_{\mu}e_{\nu}^{\! ~ ~ A a}\equiv \partial_{\mu}e_{\nu}^{\! ~ ~ A a}- \Gamma_{\mu \nu}^{\rho}e_{\rho}^{\! ~ ~ A a}+ \omega_{\mu ~ ~ B}^{\! ~ ~ A}e_{\nu}^{\! ~ ~ B a}+ \omega_{\mu \! ~ ~ b}^{\! ~ ~ a}e_{\nu}^{\! ~ ~ A b}= 0 ,
|U_3| < \frac{3 A'}{\sqrt{\pi m}}e^{5m}\int_{Q}^{\infty}\exp\left( -\left(\frac{(\log x -M)}{2\sqrt{m}}\right)^2 \right) dx =\frac{6A'}{\sqrt{\pi}}e^{5m}\int_{6\sqrt{m}}^{\infty}\exp(-y^2) ~e^{2\sqrt{m}y+M}~dy = \frac{6A'}{\sqrt{\pi}}e^{5m+M+m}\int_{6\sqrt{m}}^\infty e^{-(y-\sqrt{m})^2}dy < 4 m e^{22m}e^{-25m}< e^{-2m}.
a_\tau ={\sqrt{\pi}\Gamma(\tau+1) \over 2^{\tau+1\over 2}\Gamma\!\left({\tau+2 \over 2}\right)}, \qquad b_\tau ={\sqrt{2}\Gamma\!\left({\tau+2 \over 2}\right) \over \Gamma\!\left({\tau+1 \over 2}\right)}.
u(c_{2})=\frac{\gamma}{\delta}\chi_{3\lambda}\left( c_{2}^{+}\right) ,u^{\prime}(c_{2})=\frac{\gamma}{\delta}\left\{\chi_{3\lambda}^{\prime}\left( c_{2}^{+}\right) -\frac{\lambda}{\delta}\chi_{3\lambda}\left( c_{2}^{+}\right) \right\}.
\mathbb{E}\left[\exp\left(r\sum_{i=1}^n \left( f_i(Z_i) - \mathbb{E}\left[f_i(Z_i)\right] \right)\right) \right] \leq e^{\kappa' n r^2},\quad0 < r \leq \left(5\sqrt{2}L(d\nu+12d^2\nu^2)^{1/2}\right)^{-1}.
U_{D T R S}={\frac{N}{g}}t r \left({\frac{1}{2 \cosh ( \beta ) ( 1 + c^{*})}}S^{2}+{\frac{1}{2 \cosh ( \beta ) ( 1 - c^{*})}}D^{2}+ S^{3}/ 3 + S D^{2}\right)
\frac{\mathrm{d}}{\mathrm{d}z}p + \frac{\mathrm{d}}{\mathrm{d}\bar{z}}\bar{p}= \frac{\mathrm{d}}{\mathrm{d}z}\frac{\partial{\cal L}}{\partial v}+ \frac{\mathrm{d}}{\mathrm{d}\bar{z}}\frac{\partial{\cal L}}{\partial \bar{v}}= \partial_{\varphi}{\cal L}\, .
K(\lambda ;\zeta ,\kappa ,\tau )=\frac{1}{\sinh \zeta}\left( \begin{array}{cc}\sinh (\lambda -\eta /2+\zeta ) \kappa e^{\tau}\sinh (2\lambda -\eta ) \kappa e^{-\tau}\sinh (2\lambda -\eta ) \sinh (\zeta -\lambda +\eta /2)\end{array}\right) \in (\mathcal{H}_{0}\simeq \mathbb{C}^{2}),
\dfrac{|J_{l}|}{N^{2l}}\leq \frac{3! \N^{2l-3}+ N \ |J_{l-1}|}{N^{2l}}\leq \frac{3! \N^{2l-3}+ N \ (2N^{2l-4})}{N^{2l}}= \frac{8}{N^{3}}.
\sigma_i \displaystyle\frac{{\rm d}{{\cal E}_i}}{{\rm d}t}= \sigma_i \left( \left(w_i-\frac{1}{2}|\boldsymbol{v}|^2_i\right) \displaystyle\frac{{\rm d}{\theta_i h_i}}{{\rm d}t}+\left< \boldsymbol{v}_i, \displaystyle\frac{{\rm d}{\theta_ih_i\boldsymbol{v}_i}}{{\rm d}t}\right> \right),
K_{t,s}(\omega,\omega') = \int_\mathbb{R}dq (2\pi)^{-1}\mathrm{e}^{\mathrm{i}q(\omega'-\omega)}\sum_{n=1}^\infty (-1)^{n-1}\mathrm{e}^{n^3t/24 -\gamma_t n\left( q^2+\frac{1}{2}(\omega+\omega')-s \right)}.
P_{k,\ell,p}(j) = \binom{k+j-2}{j}\sum_{i=0}^{\ell-2}\binom{i+j-1}{i}\Big( \frac{p}{p-1}\Big)^{\ell-i-1}+ \binom{\ell+j-2}{j}\sum_{i=0}^{k-2}\binom{i+j-1}{i}\Big( \frac{p}{p-1}\Big)^{k-i-1}- \binom{k+j-2}{j}\binom{\ell+j-2}{j}.
m_\beta \ = \ \frac{1}{2}- \frac{\hat b(1)}{\hat\sigma(1)^2}+ \sqrt{\bigg(\frac{1}{2}- \frac{\hat b(1)}{\hat\sigma(1)^2}\bigg)^2 + \frac{2\beta}{\hat\sigma(1)^2\inf_{u\in U}h(u)}}\ > \ 1.
[H,H]=\Bigl\{\bigl((\bold{0};\widehat{\bold{x}}B^{G}\widehat{\bold{y}}^{T}),(\bold{0};\widehat{\bold{z}}B^{G}\widehat{\bold{w}}^{T})\bigl)\colon (\bold{x},\bold{z}), (\bold{y},\bold{w})\in H\Bigr\}.
S_{, i j}^{G}= -{\cal G}_{i j}^{- 1}+ \left\{\begin{array}{c}{{k}}\\{{i \j}}\\ \end{array}\right\}{\cal E}_{k}+ \frac{\delta \chi^{\alpha}}{\delta \Phi^{i}}c_{\alpha \beta}\frac{\delta \chi^{\beta}}{\delta \Phi^{j}}.
Y^{(d)}(\Theta,\Upsilon_{\!\Xi}x) =\frac{\partial^d}{\partial\Theta^d}Y(\Upsilon_{(\Theta+\Xi)}x) =\frac{\partial^d}{\partial\Theta^d}\left[T(\Upsilon_\Xi^{-1})\,Y(\Upsilon_{\!\Theta}x)\right] &=T(\Upsilon_\Xi^{-1})\,\frac{\partial^d}{\partial\Theta^d}Y(\Upsilon_{\!\Theta}x)=T(\Upsilon_\Xi^{-1})\,Y^{(d)}(\Theta,x).
J_{1}( i , j , 0 , 0 ; m ) = C^{( 4 )}\;_{3}F_{2}\left( \left. \begin{matrix}{{- i ,}}&{{- j ,}}&{{- i - j - \frac{D}{2},}}\\{{\quad \frac{- i - j}{2},}}&{{\quad \frac{1 - i - j}{2}}}\\ \end{matrix}\right| \frac{t}{4 m^{2}}\right)
\gamma_{q,L}(\nu)=\frac{\pi}{2}\Delta\nu_{c}\tilde{\gamma}_{q}(\nu)=\frac{(\Delta\nu_{c}/2)^{2}}{(\Delta\nu_{c}/2)^{2}+(\nu-\nu_{q})^{2}}=\frac{(\Delta\nu_{c})^{2}}{(\Delta\nu_{c})^{2}+4(\nu-\nu_{q})^{2}}.
\int_{\Omega}\varphi^{\prime}\overline{w}\; \d x + \int_{\Omega}\nabla \varphi \cdot \overline{\nabla w}\; \d x = 2 \int_{\Omega}d^{\prime}\cdot \overline{(d w)}\; \d x + 2 \sum_{k , l = 1}^3 \int_{\Omega}\partial_k d_l \overline{\partial_k (d_l w)}\; \d x - 2 \sum_{k , l = 1}^3 \int_{\Omega}\partial_k d_l \overline{(\partial_k d_l) w}\; \d x.
P_{\theta, n}= \left[ P_{\theta, n-1}\, p_\theta(2,2) + \widetilde{P}_{\theta, n-1}\, p_\theta(1,2) \right] \, f_{\theta}^{(2)}(Y_n) ; \widetilde{P}_{\theta, n}= \left[ P_{\theta, n-1}\, p_\theta(2,1) + \widetilde{P}_{\theta, n-1}\, p_\theta(1,1) \right] \, f_{\theta}^{(1)}(Y_n)
\tilde{H}= \frac{\sqrt{\alpha^{\prime}}}{2}g^{\mu \nu}( P_{\mu}- \tilde{Q}_{_{E}}{\cal A}_{\mu}- \tilde{Q}_{_{M}}\tilde{\cal A}_{\mu}) ( P_{\nu}- \tilde{Q}_{_{E}}{\cal A}_{\nu}- \tilde{Q}_{_{M}}\tilde{\cal A}_{\nu}) + \frac{\sqrt{\alpha^{\prime}}}{2 k^{2}}({\tilde{Q}_{_{E}}}^{2}+{\tilde{Q}_{_{M}}}^{2}) ,
=\left({\begin{array}{*{20}c}E_{1}+{\left( z-z_{0}\right)[E_{1}\left( A-zI \right)E_{1}]}^{-1}0 0 E_{2}+{\left( z-z_{0}\right)[E_{2}\left( A-zI \right)E_{2}]}^{-1}\end{array}}\right).
F ( x^{-}, x^{+}) = \sum_{i}\left|{\frac{L}{\pi}}\langle 0 | T^{+ +}( K ) | i \rangle \right|^{2}\left({\frac{x^{+}}{x^{-}}}\right)^{2}{\frac{M_{i}^{4}}{8 \pi^{2}K^{3}}}K_{4}\left( M_{i}\sqrt{2 x^{+}x^{-}}\right) \, ,
\mathfrak{A}\cdot\mathcal{I}_F=\prod_{\mathfrak{p}\in\mathbb{P}_F,\e(\mathfrak{p})>1}\,(\prod_{i=1}^{g(\mathfrak{p})}\,\mathfrak{P}^{\frac{e_i}{e(\mathfrak{p})}})^{v(\mathfrak{p})}\cdot\mathcal{I}_F
\begin{aligned}\int_{B\setminus B_a}\exp \big( \alpha_n|u_j|^{\frac n{n-1}+ |x|^\alpha}\big) dx =|B\setminus B_a| + o(1)_{j \searrow 0}&= \int_{B\setminus B_a}\exp \big( \alpha_n|u_j|^{\frac n{n-1}}\big) dx + o(1)_{j \searrow 0}\end{aligned}
[\bar{b}_n \mapsto \bar{a}_n](G) \cup [\bar{b}_n \mapsto \bar{a}_n](H) \cup [\bar{b}_n \mapsto \bar{a}_n](I) = [\bar{b}_n \mapsto \bar{a}_n](G \cup H \cup I) &=[\bar{b}_n \mapsto \bar{a}_n](\mathbb{N}) = \{\bar{a}_n\}
\bigcap_{n=1}^{|s|}\bigcap_{k_1,k_2 \in s(n)}\{\tau_{k_1}= \tau_{k_2}\}\cap \left \{\tau_{s(n,1)}> t_{n}\right\}\bigcap_{n_1 \neq n_2 , n_1,n_2 \le |s|}\{\tau_{s(n_1,1)}\neq \tau_{s(n_2,1)}\},
K(t_1,\ldots,t_k)=\begin{cases}\psi_1(t_1)\ldots \psi_k(t_k)\ \hbox{for}\ \t_1<\ldots<t_k ~ ~ 0\ \hbox{otherwise}\end{cases},\ \ \ \t_1,\ldots,t_k\in[t, T],\ \ \ \k\ge 2,
f_{\beta_1,\alpha_{2,2}}= f_{\beta_1,\beta_2}\circ f_{\beta_2,\alpha_{2,2}}= f_{\beta_1,\beta_2}\circ f_{\beta_2,\alpha_{1,1}\vee \tau_2 \vee \beta_2}= f_{\beta_1,\alpha_{1,1}\vee \tau_2 \vee \beta_2}& = f_{\beta_1,\alpha_{1,1}}\circ f_{\alpha_{1,1}, \alpha_{1,1}\vee \tau_2 \vee \beta_2}.
{\bf P}\left( \hat{a}_{a_{1}}^{\dagger}\ldots \hat{a}_{a_{s}+ 1}^{\dagger}A_{\quad \quad b_{1}\ldots b_{s}}^{a_{1}\ldots a_{s + 1}}\hat{a}^{b_{1}}\ldots \hat{a}^{b_{s}}\right) = \hat{a}_{a}^{\dagger}K^{a}
\sum_{n=0}^{N}\frac{(q^{-N})_{n}q^{(N+1)n+n}(bq/a)_{n}(at)^{n}}{(bq^2)_{n}(tq^2)_{n}}=\frac{(1-tq)}{(1-tq^{N+1})}F_N(a, bq; tq).
\tau_{r}(\lambda_{}) = (1-f_N(0)) \int_0^\infty \frac{\mathrm{e}^{-\frac{x \sigma^2}{\rho_{\mathrm{o}}}}}{x+1}\mathcal{L}_I \left( \frac{x}{\rho_{\mathrm{o}}}\right) \mathrm{d}x,
\max_{\{\delta_{k,\ell}\}_{{\mathcal{K}_{\ell}}}}~~{\sum_{k\in \mathcal{K}_{\ell}}\delta_{k,\ell}P_{k,\ell}\alpha_{k,\ell}N_k}\mathrm{s.t.}~~ -\sum_{k\in \mathcal{K}_{\ell}}(N_k-1)\log_2(1-\delta_{k,\ell}) = B_{\ell}& 0\leq \delta_{k,\ell}< 1,~~\forall k \in \mathcal{K}_{\ell},
\{\Gamma_{A}, \Gamma_{\widetilde{B}}\}=\frac{1}{2\mu_y}\Big(\{\{\Gamma_{A}, \Gamma_{B}\}, \Gamma_{\{x,y\}}\}-\{\Gamma_{A},\Gamma_{\widetilde{B}\setminus \{y\}}\}-2\mu_x \{\Gamma_{A}, \Gamma_{B\setminus \{y\}}\}\Big).
J_{k}(n)=n^{k}\prod_{p|n}(1-\frac{1}{p^{k}})=n^{k}(\frac{p_{1}^{k}-1}{p_{1}^{k}})(\frac{p_{2}^{k}-1}{p_{2}^{k}})\cdot\cdot\cdot(\frac{p_{\omega(n)}^{k}-1}{p_{\omega(n)}^{k}}).
\frac{dp_{1}}{dt}=\frac{q_{1}q_{2}}{r^{2}}\hat{r}+\frac{q_{1}q_{2}}{r^{2}}\frac{1}{2m_{1}m_{2}c^{2}}\{p_{1}(\hat{r}\cdot p_{2})+p_{2}(\hat{r}\cdot p_{1})-\hat{r}[p_{1}\cdot(1+3\hat{r}\hat{r})\cdot p_{2}]\}
D_{u_k}^HD_{u_{k-1}}^H\ldots D_{u_1}^HF=D_{t_1}^{H,\sharp\{u_i:u_i\in[0,t_1],~i=1,\ldots,k\}}\ldots D_{t_J}^{H,\sharp\{u_i:u_i\in(t_{J-1},t_J],~i=1,\ldots,k\}}F,
(a^\ast_x\psi)_n := \ \frac{1}{\sqrt{n}}\sum^n_{j=1}\delta(x-x_j) \psi_{n-1}(x_1, \ldots, \widehat x_j, \ldots, x_n) (a_x\psi)_n := \ \sqrt{n+1}\psi_{n+1}(x, x_1, \ldots, x_n).
\sum_{k=0}^\infty \frac{(aq;q^2)_k(bq;q^2)_kq^{k^2+k}}{(q^2;q^2)_k(abq^4;q^4)_k}=\frac{(aq^3;q^4)_\infty(bq^3;q^4)_\infty}{(q^2;q^4)_\infty (abq^4;q^4)_\infty},
a \frac{d}{d a}\lambda_{i}^{( j ) N}= \gamma_{i}^{( j )}\lambda_{i}^{( j ) N}+ \sum c_{\vec{n}, r , s}\alpha^{n_{1}}\left( \frac{\alpha}{\epsilon}\right)^{n_{2}}\left( \alpha \ln ( \frac{a}{z_{h}}) \right)^{n_{3}}\left( \frac{a}{z_{h}}\right)^{\gamma_{1}^{( 0 )}r + \tilde{\gamma}s}.
\mathfrak{E}(s)=\sum_{q_1\in\mathcal{Q}_1}\;\sum_{1\leq n\ll N}\Bigl|\sum_{q_2\in\mathcal{Q}_2}\;\frac{\phi(q_2)}{q_2^{3/2-s}}K_{q_1}(n\ell\bar{q}_2^3)K_{q_2}(n\ell\bar{q}_1^3)\Bigr|^2.
f(-1+h-a) = 2 + a - h + \sqrt{1-b^2/a^2}(h-a) + \sqrt{b^2 - h^2 b^2/a^2}, f(-1+h+a) = \sqrt{(a - 2 +h)^2}+ \sqrt{1-b^2/a^2}(h+a) + \sqrt{b^2 - h^2 b^2/a^2}.
\rho\frac{\partial^{2}\xi}{\partial t^{2}}=\frac{\partial}{\partial x}(P_{1}(\chi)\frac{\partial\xi}{\partial x})+\frac{\partial}{\partial x}(P_{2}(\chi)\frac{\partial^{2}\xi}{\partial x\partial t})-P_{3}(\chi)-F_{0}\frac{\partial\chi}{\partial x}
G_{1 2 , 3 1}^{\delta}( p_{3 0}^{\prime}, p_{2 0}) \, = \,{\frac{- 2 i \pi}{( P_{0}- S ) - ( p_{3 0}^{\prime}- E_{3}) + i \epsilon}}\, \beta_{3}\beta_{1}\, \delta ( p_{2 0}- E_{2}) .
\frac{1}{T}\sum \limits_{k=0}^{T-1}\|\hat{g}_{k+1}\|^{2}_q \leq 12\sigma_{q,\kappa}^{1+\kappa}c^{1 - \kappa}\log\left( \frac4\delta\left[ \log\left(\sqrt{T}\right)+2\right]^2\right) + \frac{20}{T}c^2\log\left(\frac{12}{\delta}\right).
H_{nm}^{\{{\bf R}\}}(0,0)=\frac{\mu_{nm}!|n-m|!(-1)^{\frac{m+n}2}}{2^{\frac{|m-n|}2}\left(\frac{|m-n|}2\right)!}\left[R_{11}^nR_{22}^m\left(r^2-1\right)^{\mu_{mn}}\right]^{\frac12}C_{\mu_{mn}}^{\frac{|m-n|+1}2}\left(\frac r{\sqrt{r^2-1}}\right).
\sup_{s}\, \langle X_{s}, e_{1}\rangle \ = \sup_{s}\left\{\langle X_{s}- \vec{v}s , e_{1}\rangle +v_{1}s \right\}& \leq \max \left\{\sup_{s}\left\{v_{1}s+ \delta s \right\}, \sup_{s}\left\{v_{1}s + \mathcal{R}_{\delta}\right\}\right\}& \leq \mathcal{R}_{\delta},
T_{\underline{{A}}\underline{{B}}\dot{A}}^{\dot{A}\ \dot{D}}H_{\dot{D}\underline{{C}}}= - \frac{i}{1 2}\left( \nabla_{\underline{{A}}}G_{\underline{{B}}}^{\dot{D}}\right) \overline{{R}}\chi_{\underline{{C}}\dot{D}}
R_q(a,n) =\bigg\{\frac{q^5(q^{-1};q^2)_{(n+1)/2}^2}{(q^{-1};q^2)_2(q^2/a,aq^{2};q^2)_{(n+1)/2}}-\frac{q^{(n+7)/2}}{(q^{n-1};q^2)_2}\bigg\}begin{align*}5pt] \quad\times\frac{(1-aq^n)(a-q^n)}{(1-a)^2}-\frac{q^{(n+7)/2}}{(q^{n-1};q^2)_2}.
\sum_{\substack{r_1,\ldots,r_{m-1},r_{m+1},\ldots,r_k (r_i,VP_f)=1}}\Bigl(\prod_{i=1}^k\frac{\mu(r_i)^2\varphi(r_i)^2}{g(r_i)r_i^2}\Bigr)(F^{(m)}_{r_1,\ldots,r_k})^2.
\sum_{a=0}^{\infty}\frac{d_k(2^a)g_k(2^a)}{2^a}= 1+\sum_{a=1}^{\infty}\frac{d_k(2^a)}{2^a}\left(\frac{(-1)^a}{2}+O\left(\frac{1}{2^k}\right)\right) &= 1+\left(\frac12+ O\left(\frac{1}{2^k}\right)\right)\left(\sum_{a=0}^{\infty}\frac{d_k(2^a)(-1)^a}{2^a}-1\right)= \frac12+ O\left(\frac{2^k}{3^k}\right).
u(t)= (1-t)H_{1}(u(1/4))+tL_{1}(v(3/4))+G_1(u)(t)+ \int_{0}^{1}k_1(t,s)g_1(s)f_1(s,u(s),v(s))\,ds, v(t)= H_{2}(v(1/3))+tL_{2}(u(2/3))+G_2(v)(t)+\int_{0}^{1}k_2(t,s)g_2(s)f_2(s,u(s),v(s))\,ds,
\Omega_{s}= - \mathrm{d}x^{a}\wedge \mathrm{d}p_{a}+ \Omega_{\mathrm{m}}\qquad \Omega_{\mathrm{m}}= \frac{s}2 \frac{\epsilon^{a b c}p_{a}\mathrm{d}p_{b}\wedge \mathrm{d}p_{c}}{( - p^{2})^{3 / 2}}\qquad ( p^{2}< 0 ) \, ,
\left( \sum_{i_1, \dots,i_N=1}^{m_1, \dots,m_N}| \Phi(x_{i_1}^1, \dots, x_{i_N}^N)|^p\right)^{1/p}\leq K \prod_{j=1}^N \Vert (x_{i_j}^j)_{i_j = 1}^{m_j}\Vert_{p}^w.
\|V_{21}f\|^2\leq b_{21}^2\|H_C f\|^2,b_{21}^2:=\frac{2p_0^{2}}{d^2}+\frac{1}{d}\sum_{|\alpha|=1}p_{1,\alpha}^{2}+2\sum_{|\alpha|=2}p_{2,\alpha}^{2}.
\begin{array}{r c l}{\overline{{K}}( z + 1 )}&{=}&{- g \overline{{K}}( z ) g ,}\\{\overline{{K}}( z + \tau )}&{=}&{- h \overline{{K}}( z ) h \times \exp \left\{- 2 \pi \sqrt{- 1}\left( z + \frac{\tau}{2}+ c \right) \right\},}\\ \end{array}
D_{r,z}u(t,x) = \sum_{j\geq 1}\sum_{n\geq j}I_{n-1}(f_j^{(n)}\big(\cdot,r,z,t,x)\big) = \sum_{j\geq 1}I_{j-1}\big(f_{j-1}(\cdot,r,z)\big) \sum_{n\geq j}I_{n-j}\big(g_{n-j}(\cdot,r,z,t,x)\big).
\mathcal{A}\colon D(\mathcal{A}) \subset \mathcal{H}\to \mathcal{H}, (\varphi, \psi)' \mapsto\begin{pmatrix}-\mathrm{div}\, (\nabla \varphi + \mathrm{rot}' \psi) -\mathrm{rot}(\nabla \varphi + \mathrm{rot}' \psi)\end{pmatrix}
x_{k+1}=Px_k+Qx_{k-1}&=\frac{x_0}{\lambda_1-\lambda_2}\left[\left(B-a\lambda_2\right)\lambda_1^{k-2}(P\lambda_1+Q)-\left(B-a\lambda_1\right)\lambda_2^{k-2}(P+Q\lambda_2)\right].
\mu_{1}\|(y,p)\|_{1,\frac{1}{2}}\leq \sup_{0 \not= (z,q) \in (H^{1,\frac{1}{2}}_{0,per}(Q_T))^2}\frac{\mathcal{B}((y,p),(z,q))}{\|(z,q)\|_{1,\frac{1}{2}}}\leq \mu_{2}\|(y,p)\|_{1,\frac{1}{2}},
\int (\nabla_\xi h, \mathcal{L}\nabla_\xi h)\varrho\,m_{0}=-\int \left(\sigma\nabla_\xi\partial_k h,\nabla_\xi\partial_k h\right)\varrho\, m_{0}- \int \Psi(x,\xi) \left|\nabla_\xi h\right|^2 \varrho\, m_{0}.
K=H\otimes I=\frac{1}{\sqrt{2}}[\begin{matrix}1&1\\ 1&-1\end{matrix}]\otimes[\begin{matrix}1&0\\ 0&1\end{matrix}]=\frac{1}{\sqrt{2}}[\begin{matrix}1&0&1&0\\ 0&1&0&1\\ 1&0&-1&0\\ 0&1&0&-1\end{matrix}]
\mathrm{Res}_{s=1/2}\mathrm{I}_k=\int_{K}Z_{>d}(f_{\mathrm{Res}_{s=1/2}kM(w,s)\rho_s},k\varphi^{U_k,\psi_k},k{\varphi'}^{U_k,\psi_k^{-1}},-s)\, dk.
\langle \bar{P}\rangle_{0}= \kappa{\frac{\partial}{\partial \bar{t}}}\bar{F}_{0}={\frac{\partial}{\partial \bar{t}}}\Bigl ( \bar{t}t_{s}+ F_{0}( t_{s}) \Bigr ) = t_{s}= \bar{t}^{1 - \bar{\gamma}}\ ,
\sum_{\substack{p_{i_{l_0}-1}\in I p_{i_{l_0}-1}= a \hbox{mod}p_{i_{l_0}}}}\frac{1}{p}\leq \sum_{\substack{n \in I n = a \hbox{mod}p_{i_{l_0}}}}\frac{1}{n}\ll \frac{\log X}{\exp(\sqrt{\log X})}\ll \exp( - (1+o(1)) \sqrt{\log X})
8 3 1 0 > 8 3 0 1 > 8 1 3 0 > 8 1 0 3 > 8 0 3 1 > 8 0 1 3 > 3 8 1 0 > 3 8 0 1 > 3 1 8 0 > 3 0 8 1 > 3 0 8 1 > 3 0 1 8 >
SINR_{k,q,p}^n = \frac{\rho \frac{1}{[\mathbf{C}_{k,q,p}]_{q,q}}\alpha_{k,q,n}^2}{1+\sum^{P}_{m=n+1}\rho \frac{1}{[\mathbf{C}_{k,q,p}]_{q,q}}\alpha_{k,q,m}^2},
\frac{y K(\ell) J(m_0)}{m_0}\prod_{\substack{1\leq j \leq k 2 \mid a_j}}\left( 1- \frac{c(p_j)}{p_j}\right)\sum_{\substack{r\geq 1 (r,m)=1}}\mu(r) \frac{c(r^2)}{r^2}.
\int_X |\varphi_\varepsilon| (\beta + \varepsilon \omega + dd^c \varphi_\varepsilon)^n =\int_X |\psi_\varepsilon + M_\varepsilon| e^{\psi_\varepsilon + M_\varepsilon}f\omega^n &\leq \int_X (-\psi_\varepsilon) e^{\psi_\varepsilon +M}f \omega^n + \int_X M e^Mf \omega^n.
S=\int_{-1}^{1}2\pi r(x)\frac{dl}{dx}dx=\int_{-1}^{1}2\pi r(x)\sqrt{1+(\frac{dr}{dx})^{2}}dx=\int_{-1}^{1}2\pi\sqrt{1-x^{2}}\frac{1}{\sqrt{1-x^{2}}}dx=4\pi
\textbf{I}\leq \frac{2}{s(1-s)^2}\frac{1}{(r^2 -1)^3}\sum_{n=0}^{\infty}\left(\frac{|z|^2}{r_{s}}\right)^{n}= \frac{2}{s(1-s)^2}\frac{1}{(r^2 -1)^3}\frac{r_{s}}{r_{s}- |z|^2}.
f_4(n,p,q,c,d) = (p+q+c+d+3)(p+q+c+d+4) \quad\cdot [c^3(p+q+c+d+2)+(p+c+1)(3c^2+3c+1)(n-c-d)] + (p+c+1)(p+c+2)(n-c-d)(n-c-d-1) \quad\cdot [3(p+q+c+d+4)(c+1)+(p+c+3)(n-c-d-2)]
d(x_n,x_{n+p}) \leq d(x_n,x_{n+1})+d(x_{n+1},x_{n+2})+\dots+d(x_{n+p-1},x_{n+p}) &\leq c^n\cdot d(x_0,x_1)+c^{n+1}\cdot d(x_0,x_1)+\dots+c^{n+p}\cdot d(x_0,x_1) &=c^n\cdot \dfrac{1-c^p}{1-c}\cdot d(x_0,x_1),
D^u_nD^v_n [(v_1-u_n)^2\ldots (v_{n-1}-u_n)^2(v_n-u_n)u_n^k] = D^u_nD^v_{n-r}[(v_1-u_n)^2\ldots (v_{n-r-1}-u_n)^2(v_{n-r}-u_n)\cdots(v_n-u_n)u_n^k] \equiv D^u_{n-r}D^v_{n-r}[(v_1-u_{n-r})^2\ldots (v_{n-r-1}-u_{n-r})^2(v_{n-r}-u_{n-r})u_{n-r}^k].
\frac{(1 - \frac{c_n}{n})^{2 n}}{(1 - \frac{c_n}{n}) - (1 - \frac{c_n}{n})^{n}}=\frac{e^{-2 c}}{1 - e^{- c}}\Bigl(1 +\frac{c}{2 n (1 - e^{- c})}\bigl[2 - c e^{-c}- 2 c\bigr]\Bigr)
A_{nh}=\frac{1}{2\pi}\int_{-\pi}^\pi\frac{(1+u)^2(1-u)^3(u^2+3u+1)^{n-1}}{(1-u^h)(1-u^{h+1})u^{n-h+1}}\,d\varphi = \frac{1}{2\pi}\int_{-\pi}^\pi \frac{g(u)\exp(nf(u))}{(1-u^h)(1-u^{h+1})}\,d\varphi
\Sigma_{c,a,e,d,g_1,g_2,g_3,g_4}\ll (X Q L aedg_1g_3g_4 )^{\varepsilon}\left( \frac{L Q}{c}\right) \mathop{\sum_{S_5>0}\sum_{S_6>0}}_{S_6\leq S_5}\frac{S_6^{\varepsilon}}{S_5^{j_1-\varepsilon}}\bigg(1+\frac{XcQ^{\vartheta-1}}{L}\bigg)^{j_1-1}\times \Big( Q^{\vartheta}LS_5 +aeL^2S_5^2 \Big).
d s^{2}= - f ( r , \theta ) d t^{2}+ \frac{m ( r , \theta )}{f ( r , \theta )}( d r^{2}+ r^{2}d \theta^{2}) + \frac{l ( r , \theta )}{f ( r , \theta )}r^{2}\sin^{2}\theta d \varphi^{2}
R_2\geq \beta_2, &\log\frac{|\sigma^{2}{\bf I}_2+\sum_{j=2}^{3}{\bf G}_{1j}{\bf Q}_j{\bf G}_{1j}^{T}|}{|\sigma^{2}{\bf I}_2+{\bf G}_{13}{\bf Q}_3{\bf G}_{13}^{T}|}\geq 2\beta_2, &\log|\sigma^{2}{\bf I}_2+\sum_{j=2}^{3}{\bf G}_{1j}{\bf Q}_j{\bf G}_{1j}^{T}|-\log|\sigma^{2}{\bf I}_2+{\bf G}_{13}{\bf Q}_3{\bf G}_{13}^{T}|\geq 2\beta_2.
\begin{aligned}\bar{y}_j = r \, \left( \sqrt{1-h^2}\cos\left(\frac{2(j-1)\pi}{k}\right), \sqrt{1-h^2}\sin\left(\frac{2(j-1)\pi}{k}\right),h\right) \underline{y}_j = r \, \left( \sqrt{1-h^2}\cos\left(\frac{2(j-1)\pi}{k}\right), \sqrt{1-h^2}\sin\left(\frac{2(j-1)\pi}{k}\right),-h\right)\end{aligned}
(-i)\tilde S(p)={p^2+ b^2-m^2+2\left(b\cdotp+m \not\!b\right)\gamma_5 - [\not\!b, \not\!{\hat p}]\gamma_5\over\left(p^2+b^2-m^2+i\varepsilon\right)^2-4\left[(b\cdotp)^2- b^2\left(m^2 -\hat p^2\right)\right]}\left( \not\!p+m+\not\!b\gamma_5\right)\ ,
\frac{d}{{ds}}X_3 (\tilde{t}, \tilde{x_1},\tilde{x_2}, \tilde{x_3}, \tilde{m}; s) =u_3 (\tilde{t}+s, X (\tilde{t}, \tilde{x_1},\tilde{x_2}, \tilde{x_3}, \tilde{m}; s)) \le -A_0.
\begin{array}{c}{- \beta \frac{\partial G}{\partial \mu}( \beta , R , \mu ) = T r \{\frac \partial{\partial \mu}\ln \left( D ( \beta , R ) - i \mu \gamma^{0}\right)_{b c}\}}\\{= T r \{- i \gamma^{0}K_{b c}( t , x ; t^{\prime}, x^{\prime}) \},}\\ \end{array}
F(t;x,y,1,u,z,v) =\frac{tx(y-yzr+z)F(t;x,y-yr+1,1,u,z,v)}{(tux+y^{-1}-tu)(y-yr+1)}&\quad-\frac{txz(ytuv(1-z)+z)}{tux+y^{-1}-tu}F(t;x,y-yr+1,1,u,1,v) &\quad+z(ytuv(1-z)+z)F(t;x,y,1,u,1,v).
\sum_{i=1}^{N-1}\epsilon_{i}\,{\mu_{i}^{N-1-j}\, d\mu_{i}\over\displaystyle 2\sqrt{h_{N}-\mu_{i}}\,\prod_{j=1}^{N-1}(\mu_{i}-h_{j})}=\delta_{j,1}\, dx,\quad 1\leq j\leq N-1, \quad \epsilon_{i}=\pm 1.
\begin{aligned}d_1 = -\rho\{(12m_1^2+12m_1m_2+4m_2^2-1)p^2-(8m_2^2-2)pq+(4m_2^2-1)q^2\}, d_2 =\rho\{(12m_1^2+12m_1m_2+4m_2^2-1)p^2-(24m_1^2+24m_1m_2+8m_2^2-2)pq & +(4m_2^2-1)q^2\}, r = \rho\{(24m_1^2+24m_1m_2+8m_2^2-2)p^2-(8m_2^2-2)q^2\},\end{aligned}
\left(\theta_{f(n)}-\theta_{f(n+1)}\right)\cdot\sum_{j=f(n)}^{f(n+1)}\lambda_j \le\frac{1+\log(n+1)}{\log(n\log n)}\cdot\frac{\log\log(e(n+1))}{\log((n+1)\log(n+1))}\le\frac{\log(e(n+1))}{\log(n\log n)}\cdot\frac{\log(1+\log(n+1))}{\log(n+1)}\le 1\cdot\frac{\log(1+\log(n+1))}{\log(n+1)}.
u(\zeta(w_{1,2N}),\dots,\zeta(w_{m,2N}))=\zeta(u(w_{1,2N},\dots,w_{m,2N}))=\zeta(U), v(\zeta(w_{1,2N}),\dots,\zeta(w_{m,2N}))=\zeta(v(w_{1,2N},\dots,w_{m,2N}))=\zeta(V),
=\sum\limits_{(M_b, \mu_b) \in \mathcal{I}^{G, \mu}_{M_b, b'}}\mathrm{Ind}^G_{P_b}(\mathrm{Mant}_{M_b, b', \mu_b}(e(J_b)LJ( \delta^{\frac{1}{2}}_{G,P_b}\otimes I^{M_b}_{M_S}(\rho))))\otimes [1][|\cdot|^{\langle \rho_G, \mu_b - \mu \rangle}],
h_{Q_1,v_1}(P_m) =m+O(1), h_{Q_1,v_2}(P_m)=O(1), h_{Q_1,v_i}(P_m)=O(1), i=3,\ldots, n, h_{Q_2,v_1}(P_m) =O(1), h_{Q_2,v_2}(P_m)=m+O(1), h_{Q_2,v_i}(P_m)=O(1), i=3,\ldots, n, h_{Q_3,v_1}(P_m) =O(1), h_{Q_3,v_2}(P_m)=O(1), h_{Q_3,v_i}(P_m)=\frac{m}{n-2}+O(1), i=3,\ldots, n.
((u_a)_*\circ \Delta_*\circ CG^{[j]})(z) =\left[(u_a)_*\Delta_*(z)\right]\cup \sum_{i=0}^j \Big[(-1)^i i!(j-i)!\times x^{[i]}y^{[j-i]}\boxtimes y^{[i]}(x-ay)^{[j-i]}\Big]\otimes \zeta_{p^r}^{\otimes -j}.
\boldsymbol{\Theta}=\left[\sigma^2,\boldsymbol{\theta}^T,\boldsymbol{\phi}^T,\boldsymbol{\gamma}^T,\boldsymbol{\eta}^T,\mathfrak{R}(\boldsymbol{\beta}^T) ,\mathfrak{I}(\boldsymbol{\beta}^T)\right]^T
\left\{\begin{bmatrix}N_1 (\lambda)^T -\widehat{N}_2 (\lambda) M(\lambda) N_1 (\lambda)^T\end{bmatrix}h_1(\lambda), \ldots , \begin{bmatrix}N_1 (\lambda)^T -\widehat{N}_2 (\lambda) M(\lambda) N_1 (\lambda)^T\end{bmatrix}h_p(\lambda) \right\},
(\Gamma^{\underline m})^{\underline\alpha}_{~~\underline\beta}(C\Gamma_{\underline m})_{\underline{\gamma\delta}}+(\Gamma^{\underline m})^{\underline\alpha}_{~~\underline\gamma}(C\Gamma_{\underline m})_{\underline{\delta\beta}}+(\Gamma^{\underline m})^{\underline\alpha}_{~~\underline\delta}(C\Gamma_{\underline m})_{\underline{\delta\beta}}=0,
\Delta ( q_{1}, q_{2}; t ) ={\frac 1{2 \pi}}~ \sqrt{{\frac \pi{i t}}}~ \exp{\left({\, \frac{i ( q_{1}- q_{2})^{2}}{4 t}}-{\frac{i ( q_{1}+ q_{2}) t}2}-{\frac{i ~ t^{3}}{1 2}}\right)}.
\frac{2}{3}\mathbb{D}(f,\chi; x) = \frac{2}{3}\mathbb{D}(1,f\chi; x) & \leq \mathbb{D}(1, n \mapsto n^{2i\alpha}; x) &= \mathbb{D}(n \mapsto n^{-i\alpha}, n \mapsto n^{i\alpha}; x) \leq \mathbb{D}(f\chi, n \mapsto n^{-i\alpha}; x)+ \mathbb{D}(f\chi; n \mapsto n^{i\alpha}; x) &= 2 \mathbb{D}(f, n \mapsto \chi(n) n^{i\alpha}; x),
v_p(k!)=\sum_{\ell=1}^{C_{p,k}}\left \lfloor \dfrac{k}{p^{\ell}}\right\rfloor\le v_p(N_k)\le \sum_{\ell=1}^{C_{p,k}}\left(1+\left \lfloor \dfrac{k}{p^{\ell}}\right\rfloor \right)=v_p(k!)+C_{p,k},
\begin{array}{r c l}{F^{( m )}( \zeta_{1}, \cdots , \zeta_{2 N})}&{= \displaystyle \sum_{\iota_{1}, \cdots , \iota_{2 N}}}&{v_{\iota_{1}}\otimes \cdots \otimes v_{\iota_{2 N}}F^{( m )}( \zeta_{1}, \cdots , \zeta_{2 N})_{\iota_{1}, \cdots , \iota_{2 N}}}\\ \end{array}
=\sum_{n_{1}=0}^{N_{1}-1}[e^{-\frac{2\pi i}{N_{1}N_{2}}n_{1}k_{2}}](\sum_{n_{2}=0}^{N_{2}-1}x_{N_{1}n_{2}+n_{1}}e^{-\frac{2\pi i}{N_{2}}n_{2}k_{2}})e^{-\frac{2\pi i}{N_{1}}n_{1}k_{1}}
R_{1,0}(\alpha,v) =\frac{\Delta f(\nu_-,\mu_-)}{2S''(\nu_-)}, R_{2,1}(\alpha,v) =-\frac{1}{4S''(\nu_-)^2}\left(S^{(4)}(\nu_-)+2S^{(3)}(\nu_-)D\right)f(\nu_-,\mu_-), R_{3,2}(\alpha,v) =\frac5{12}\frac{S'''(\nu_-)^2f(\nu_-,\mu_-)}{S''(\nu_-)^3},
d_X(F(x,y),x) \leq d_X(F(x,y),F^{j+1}(x_0,y_0))+ d_X(F^{j+1}(x_0,y_0),x) &= d_X(F(x,y),F(F^j(x_0,y_0),G^j(y_0,x_0)))+ d_X(F^{j+1}(x_0,y_0),x) &\leq k\d_X(x,F^j(x_0,y_0))+ l\d_Y(y,G^j(y_0,x_0))+ d_X(F^{j+1}(x_0,y_0),x) &\rightarrow 0\ \j\rightarrow \infty
u{\otimes}_{A}u{\otimes}_{A}\ldots{\otimes}_{A}u ={\tau}_{{\mu}_{1}}^{( 1 )}a_{{\mu}_{1}}{\tau}_{{\mu}_{2}}^{( 2 )}a_{{\mu}_{2}}\ldots{\tau}_{{\mu}_{N}}^{( N )}a_{{\mu}_{N}}.
\gamma_{\mathbf{r}}= \prod_{i=1}^{t-1}r_i! \cdot \binom{q_{i+1}}{r_i}\cdot \binom{q_i+ \sum_{j=1}^{i-1}(q_j - 2 r_j)}{r_i}\cdot \frac{\sqrt{(q_{i+1}+ \sum_{j=1}^{i}(q_j - 2 r_j) )!}}{\sqrt{q_{i+1}!}\cdot \sqrt{(q_i+ \sum_{j=1}^{i-1}(q_j - 2 r_j))!}}.
\lambda' h(a)=\log(1/a)\log(1+\frac{1}{a})^{\lambda'-1}+(1-\lambda')\int_1^a \frac{\log(1+\frac{1}{b})^{\lambda'-1}}{b}(\frac{\log(b)+(1+b)\log(1+1/b)}{(1+b)\log(1+\frac{1}{b})})
r \int_{0}^{1}d\beta \int_{t}^{\infty}ds \int_{B_{s-t}(0) \cap B_{\frac{s}{2}}(-\beta x)}\frac{dA(y)}{(s-t)}\frac{|v_{4,c}(s,|\beta x+y|)|}{|\beta x+y|}\leq \frac{C r}{t^{2}\log^{3b+2N-1}(t)}
H^{\pm}\left(p,t\right) =\underset{c\in\mathcal{C}}{}\left\langle e^{-\left(T-t\right)A}B\left(T-t\right)c,p\right\rangle -\underset{d\in\mathcal{D}}{}\left\langle e^{-\left(T-t\right)A}D\left(T-t\right)d,p\right\rangle .
\Bigl\lVert \sup_{j_0 \geq s+k_0}\Bigl\lvert \sum_{j = j_0}^{\infty}T_{\mathcal N(j,s)}B_{j-s}\Bigr\rvert\, \Bigr\rVert_2 \lesssim 2^{-s/5}\lvert I_0\rvert^{1/2}.