harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
l(a)l(b)v = l(a \cdot b) v, \qquad\!\!\! r(b)r(a)v = r(a \cdot b) v, \qquad\!\!\! r(b)l(a)v = l(a)r(b) v, \qquad\!\!\! \forall a, b \in A, \, v \in V.\!
\begin{aligned}\nabla_n \varPhi (x_0, t_0) \leq \, - \eta (x_0, t_0) \nabla_{nn}(x_0, t_0) + \nabla_n H [u] (x_0, t_0) + C \leq \, \Big(- \eta (x_0, t_0) + \int_0^1 H_{p_n}[u^s] (x_0, t_0) ds\Big) \nabla_{nn}u (x_0, t_0) + C. \end{aligned}
\frac{\ell(f_1,\chi)\cdot \ell(f_2,\chi^{-1})}{\langle f_1,f_2\rangle}=\frac{\Lambda_F(2)\cdot\Lambda(1/2,\Pi,\chi)}{2\Lambda(1,\Pi,{\rm ad})}\cdot\prod_v\beta_v(f_{1,v},f_{2,v}),f_1,f_2\in\pi,
2E(u_0)=2E(u(t)) =\|(-\Delta)^{\frac{s_1}{2}}u(t)\|_2^2+\|(-\Delta)^{\frac{s_2}{2}}u(t)\|_2^2 -\frac{2}{p}\|u\|_p^p & \geq X(t)-\frac{2C_{N,p,s_1}}{p}\|u(t)\|_2^{p-\frac{N(p-2)}{2s_1}}\|(-\Delta)^{\frac{s_1}{2}}u(t)\|_2^{\frac{N(p-2)}{2s_1}}& \geq X(t)-KX(t)^{\frac{N(p-2)}{4s_1}}.
a_{n+1}= 4 \frac{(2 n+1) \left(2 n^2+2 n+1\right)}{(n+1)^3}a_n -16 \frac{n \left(4 n^2+1\right)}{(n+1)^3}a_{n-1}+8 \frac{(2 n-1)^3}{(n+1)^3}a_{n-2}.
P_{n}\left( \tau^{-1}\left( A\right) \right) =\frac{1}{n}\sum_{k=0}^{n-1}P\left( \tau^{-k-1}\left( A\right) \right) =\frac{n+1}{n}\cdot \frac{1}{n+1}\sum_{k=0}^{n}P\left( \tau^{-k}\left( A\right) \right) -\frac{1}{n}P\left( A\right) & =\frac{n+1}{n}P_{n+1}\left( A\right) -\frac{1}{n}P\left( A\right) .
I (s, t) =\frac{3^9}{2^{12}}\int_0^1 \frac{dx}{x^2}\;\frac{1-\frac{121}{729}x^2}{(1-\frac{25}{729}x^2)^{13}}\;\exp \left(-\frac{1}{2}p_iQ^{(1)}_{ij}p_j\right)\,.
g_{{\rm SO}(M)}\big(X^h-L^{-1}(X)^{h'},Y^{h'}\big) =g_{{\rm SO}(M)}\big(X^h-L^{-1}(X)^h-\xi_{L^{-1}(X)}^{\ast},Y^h+\xi_Y\big) \hphantom{g_{{\rm SO}(M)}\big(X^h-L^{-1}(X)^{h'},Y^{h'}\big)}{}=g\big(X-L^{-1}(X),Y\big)-{\bf B}\big(\xi_{L^{-1}(X)},\xi_Y\big)=0
\begin{array}{rcl}\nu = \left(E^\prime_{\rm e-vib}-E^{\prime\prime}_{\rm e-vib}\right)/hc = (E_0^\prime-E_0^{\prime\prime})/hc+(G^\prime_{q^\prime}-G^{\prime\prime}_{q^{\prime\prime}})=\nu_e+\nu_{v}~,\end{array}
L_{M}=-\frac{1}{4}e^{2\kappa\Psi}F_{\mu\nu}^{a}F^{\mu\nu,a}-\frac{1}{2}\partial_{\mu}\Psi\partial^{\mu}\Psi-\frac{1}{2}D_{\mu}\Phi^{a}D^{\mu}\Phi^{a}-e^{-2\kappa\Psi}V(\Phi^{a})-\frac{1}{2}m^2\Psi^2
\rho_{\pm}:= \frac{1}{2}(u \pm \sqrt{u^2 + 4}), c_{\pm}:= \frac{\pm 1 \pm (k + \sqrt{D})\rho_\pm}{\sqrt{4+u^2}}, c'_{\pm}:= \frac{\pm 1 \pm (k - \sqrt{D})\rho_\pm}{\sqrt{4+u^2}}.
G(\vartheta )=\frac{1}{2\pi}\int^{+\infty}_{-\infty}dk\, e^{ik\vartheta}\frac{\sinh \frac{\pi (p-1)k}{2}}{2\sinh \frac{\pi pk}{2}\, \cosh \frac{\pi k}{2}}\, ,\, p=\frac{\pi}{\gamma}-1
\sum_{n \leq at_3}\frac{E(s_1,s_3;at_3,n,M)}{n^{s_2}}+\sum_{at_3 < n \leq N}\frac{E(s_1,s_3;n,M)}{n^{s_2}}&-B_2-B_3-B_4-C_2-C_3-C_4 & \ll \begin{cases}t_3^{-\sigma_1-\sigma_3}(\sigma_2>1) t_3^{-\sigma_1-\sigma_3}\log t_3 (\sigma_2=1) t_3^{1-\sigma_1-\sigma_2-\sigma_3}(\sigma_2<1). \end{cases}
\frac{1}{2}\int \frac{d k^{\prime}}{2 \pi}\, \frac{1}{\sqrt{k^{2}+ 4}}f ( k^{\prime}, x^{\prime}) f ( - k^{\prime}, x^{\prime}) = - \frac{\left( 1 - \coth^{2}2 ( x^{\prime}- x_{0}) \right)}{2 \pi}.
p_{d}^u(i,j) = P_i(\nu_1 > u) I(i,j) + \int_0^u \lambda(i,i) e^{\lambda(i,i) v}\left( \sum_{l \in d - \{i\}}\frac{\lambda(i,l)}{\lambda(i,i)}p_{d}^{u-v}(l,j) \right) dv
c_i(u):=\frac{1}{3(f^i(u))^2}\left(A^{ii}_{2,3;2}-u^iA_{2,3;1}^{ii}+\sum_{k\not =i}\frac{(A^{ki}_{1,2;2}-u^iA^{ki}_{1,2;1})^2}{f^k(u)(u^k-u^i)}\right),
\sup_{a,b>0}\frac{4(1+ab)(1-e^{-2t}) - 3 + 2(2e^{-2t}-1) (a+b)}{a^2 b^2}begin{align*}1ex] \ge \sup_{a>0, b=1/a}8(1-e^{-2t}) - 3 + 2(2e^{-2t}-1) (a+1/a) = +\infty.
x^{\mu}- y^{\mu}+ \frac{2 i}{( 1^{1}2_{4})}\xi_{2 / 4}\sigma^{\mu}\bar{\theta}^{1}, \qquad \xi_{2 / 4}\equiv [ ( 1^{2}2_{4}) \theta_{2}+ ( 1^{3}2_{4}) \theta_{3}+ ( 1^{4}2_{4}) \theta_{4}]
\aligned|F(x,u)| =|F(x,u)-F(x,0)|= \int_{0}^{1}|f(x,\eta u)ud\eta &\leq \frac{\varepsilon}{2}|u|^{2}+\frac{c(\varepsilon)}{p}|u|^{p}&\leq c_{1}|u|^{2},\endaligned
{\bar{\theta}}^{\dot{2}}= Y ( x ){\bar{\theta}}^{\dot{1}}, \quad{\bar{\theta}}^{\dot{\alpha}}={\bar{\theta}}^{\dot{1}}{\bar{\psi}}^{\dot{\alpha}}, \quad{\bar{\zeta}}^{\dot{2}}= Y ( x ){\bar{\zeta}}^{\dot{1}}, \quad{\bar{\zeta}}^{\dot{\alpha}}={\bar{\zeta}}^{\dot{1}}{\bar{\psi}}^{\dot{\alpha}}.
S_{boundary}=-\frac{\sqrt{2}}{\kappa_{5}^{2}}\int_{M_{4}^{(1)}}{\sqrt{-g}V^{-1}\alpha^{(1)}}- \frac{\sqrt{2}}{\kappa_{5}^{2}}\int_{M_{4}^{(2)}}{\sqrt{-g}V^{-1}\alpha^{(2)}}
\epsilon_{i}^{a}(\theta )+\Omega_{k}\ast L_{i}^{b}(\theta -\sigma_{ba})=\sum_{j=1}^{k-1}I_{ij}^{su(k)}\,\Omega_{k}\ast (\epsilon_{j}^{a}+L_{j}^{a})(\theta )\;,\quad a\neq b\,.
\eta^i(\sigma,t) \equiv - \frac{1}{4M}\sum_{n>0}\frac{\epsilon}{\tau}\, W^{'}(n+\frac{\epsilon}{\tau})\Biggl[ \frac{c_n^i}{\sqrt{n}}\, x_n^+ +\frac{\tilde{c}_n^i}{\sqrt{n}}\, x_n^- + h.c. \Biggr] \, ,
{\Theta}^{n+1}_h :=\overline{\left\{(p, [\ell_1], [\ell_2], [F])\in \widetilde{\mathcal{P}}\times \mathbb{P}(S_d) \left| \begin{array}{l}\ell_1\neq \ell_2 1 \leqslant i \leqslant 2, \ell_i\subset X_F X_F \cdot \ell_i \geqslant hp \end{array}\right.\right\}}.
\zeta^{\partial B_0}[\omega](y) :=\left(\int_{X\setminus B_0}G^{X\setminus B_0}(x,y)\omega(x)+\int_{B_0}G^{B_0}(x,y)\omega(y)\right)d_{x\in X}\zeta^{\partial B_1}[\eta](y) :=\left(\int_{\partial B_0\setminus B_1}G^{B_0\setminus B_1}(x,y)\eta(x)+\int_{B_1}G^{B_1}(x,y)\eta(x)\right)d_{x\in \partial B_0}
\varTheta_\lambda^\epsilon:=\bigl\{f:[0,\tau]\to\mathbb{R}~ |~ f(t)\geq \epsilon,\; \forall t \in[0,\tau]\ & \big|f(t)-f(s)\big|\leq(L-\epsilon)|t-s|,\forall t,s\in[0,\tau] \bigr\}.
w_{n+1}-w_{n}=A_{q}\Theta_{n+1,q}+\Theta_{n+1,q}\;\sum_{j=q}^{n}\frac{h_{j}}{\alpha_{j}\Theta_{j+1,q}}=A_{q}\Theta_{n+1,q}+\sum_{j=q}^{n}\frac{h_{j}}{\alpha_{j}}\;\Theta_{n+1,j+1}.
\vert\partial^{\beta_0}_x\nabla P_t\varphi(x)\vert=\vert\mathbb{E}[\partial^{\beta_0}_x(\nabla\varphi(X_t(x))]\vert=\vert\sum_{\vert\alpha_0\vert\leq\vert\beta_0\vert}\mathbb{E}[(\partial^{\alpha_0}\nabla\varphi)(X_t(x))\mathbf{P}_{\alpha_0}(x)]\vert\leq\Vert\nabla\varphi\Vert_{k,\infty}\sum_{\vert\alpha_0\vert\leq\vert\beta_0\vert}\mathbb{E}\vert\mathbf{P}_{\alpha_0}(x)\vert,
\hat{M}_{0}(z)=\displaystyle{\left(\begin{array}{ccc}\hat{w}(z) +\textstyle{\frac{1}{2}}\hat{\lambda}(z) 0 0 ib{\cdot\gamma}\hat{w}(z)-\frac{1}{2}\hat{\lambda}(z) \sqrt{2}\hat{\rho}{}^{b}(z) -\sqrt{2}\hat{\bar{\rho}}_{a}(z) 0 \hat{t}{}_{a}{}^{b}(z) \end{array}\right)}\,.
u_{k+1}(t) = S_{-1}(t) u_{k}(\tau) + \int_{0}^{t}\mathcal{A}_{-1}S_{-1}(t - s) \big(u_{k}(s) + \gamma_{k}(s)\big) \mathrm{d}s + \int_{0}^{t}S_{-1}(t - s) f_{k}(s) \mathrm{d}s.
\chi_{[-L,0]^d}\{h(g(H^{R_{\sigma}})_{\Lambda_L})-f^{R_{\sigma}}_0\}\quad= U^*_L\chi_{[0,L]^d}\{h(g( H^{T_L R_\sigma})_{[0,2L]^d})-f_{0}^{T_LR_\sigma}\}U_L ,
\mathrm{wt}(T^{(k+1)}) = \mathrm{wt}(S_{1}\cdots S_{n-l+k+1}T^{(k)}) & =(\underbrace{\lambda_{l-k-1},\ldots,\lambda_{l}}_{k+2},\underbrace{0,\ldots,0}_{n-l},\lambda_{l-k-2},\ldots,\lambda_{1},0,\ldots).
\mu(g,1) = W\left(e^{-\frac{f+c}{2}}, g, 1 \right)=-n-\frac{n}{2}\log (4\pi)+\int_{M}\left\{\left(R+|\nabla f|^2 \right) + f+c\right\}(4\pi)^{-\frac{n}{2}}e^{-f-c}dv,
\frac{\partial \tilde g_{\mathfrak c}}{\partial s}=\pm\frac{1}{g_{\mathfrak c}|\nabla V(\mathfrak c)|}\Big[-\frac{\partial g_{\mathfrak c}}{\partial x_1}\frac{\partial V}{\partial x_2}+\frac{\partial g_{\mathfrak c}}{\partial x_2}\frac{\partial V}{\partial x_1}\Big] \,,
\begin{pmatrix}1 0 0 0 0 \zeta^{4}\zeta^{-40}7 0 \zeta^{-8}\zeta^{44}1 0 \zeta^{28}\zeta^{32}7\end{pmatrix}, \begin{pmatrix}1 0 0 0 0 7 \zeta^{40}\zeta^{44}0 \zeta^{32}7 \zeta^{40}0 \zeta^{28}\zeta^{32}7\end{pmatrix},
\int_{\mathbb{R}^{nd}}\bigl|\widehat{\zeta}_{\tau}\left(y\right)\bigr|dy\leq\int_{\mathbb{R}^{nd}}\int_{\mathbb{R}^{nd}}\left|\widehat{w}_{\tau}\left(y-v\right)\right|\left|\widehat{\chi}\left(v\right)\right|dvdy=\left\Vert \widehat{\chi}\right\Vert_{1}\left\Vert \widehat{w}_{\tau}\right\Vert_{1}.
{\cal L}_{P V}= \int d^{4}\theta [ Q_{+}^{r e g ~ \dagger}e^{V}Q_{+}^{r e g}+ Q_{-}^{r e g ~ \dagger}e^{V}Q_{-}^{r e g}] + ( \int d^{2}\theta \Phi \Lambda_{U V}Q_{+}^{r e g}Q_{-}^{r e g}+ \mathrm{h . c .}) .
S_{3}=(\begin{matrix}1&0&0&0&0&0&0&0\\ 1&1&0&0&0&0&0&0\\ 1&0&1&0&0&0&0&0\\ 1&1&1&1&0&0&0&0\\ 1&0&0&0&1&0&0&0\\ 1&1&0&0&1&1&0&0\\ 1&0&1&0&1&0&1&0\\ 1&1&1&1&1&1&1&1\end{matrix})
\stackrel{\wedge}{\Sigma}_{( 2 )}( p , m , \lambda , \Lambda_{0}) = \frac{i \lambda^{2}}{3 8 4 \pi^{3}}\int_{0}^{1}d x \int_{0}^{1}d y \frac y{\left[ y C \right]^{3 / 2}}F ( p^{2}, m^{2}, x , y , \Lambda_{0})
\ddot{\Phi}- \frac{2}{a}\dot{\Phi}_{, r}+ \frac{1}{a^{2}}\Phi_{, r r}+ H \dot{\Phi}+ \frac{2}{a^{2}}\left( \frac{1}{r}- \dot{a}\right) \Phi_{, r}= 8 \pi G \dot{\varphi}\left( \delta \dot{\varphi}- \frac{1}{a}\delta \varphi_{, r}\right) \ .
\left( L^{2}(\O_{N-m_{0}+1}; H^{1}_{c,x^{(0)}}) \right)^{*}= L^{2}(\O_{N-m_{0}+1}; H^{-1}_{c,x^{(0)}}) & \left( L^{2}(\O_{N-m_{0}+1}; H^{-1}_{c,x^{(0)}}) \right)^{*}= L^{2}(\O_{N-m_{0}+1}; H^{1}_{c,x^{(0)}}) .
\langle [\alpha_1, \ldots, \alpha_n], X \rangle = \sum_{k=1}^n (-1)^{n-k}\langle \mathcal{L}_{\Pi^{\sharp}(\alpha_1 \wedge \cdots \wedge \widehat{\alpha}_k \wedge \cdots \wedge \alpha_n)}\alpha_k, X \rangle- (n-1) \langle d (\Pi(\alpha_1,\ldots, \alpha_n)), X \rangle.
\left\{i \omega \gamma^{0}- i \vec{\gamma}\left( \vec{\partial}_{\vec{r}}- i e \vec{A}( \vec{r}) \right) - m + \gamma^{0}\mu \right\}\widetilde{G}_{\omega}( \vec{r}, \vec{r}^{\prime}) = \delta ( \vec{r}- \vec{r}^{\prime}) ,
f^{-1}(G)\cap D_1(n) \subseteq \{G'\in C_r(n): \Delta(G_1,G')\subseteq E_1\}, f^{-1}(G)\cap D_2(n) \subseteq \bigcup_{i=2}^9 \{G'\in C_r(n): \Delta(G_i,G')\subseteq E_i\}, f^{-1}(G)\cap D_3(n) \subseteq \{G'\in C_r(n): \Delta(G_{10},G')\subseteq E_{10}\}.
a(v) =\tfrac{1}{2}(v + v_1)^{2}+\tfrac{1}{2}(v + v_2)^{2}- v^{2}+ \sum_{u\in N(v)\setminus \{v_1,v_2\}}\tfrac{1}{2}(v+u)^{2}= \tfrac{1}{2}(v_1+v_2)^{2}+ \tfrac{1}{2}v_1^{2}+ \tfrac{1}{2}v_2^{2}+ \sum_{u\in N(v)\setminus \{v_1,v_2\}}\tfrac{1}{2}(v+u)^{2}\, .
\frac{1}{2}(S,S)^a+\int dx\varepsilon^{ab}\Phi_{Ab}^* \frac{\delta}{\delta{\bar\Phi}_{A}}S=0,\quad\varepsilon^{ab}=-\varepsilon^{ba},\quad\varepsilon_{ab}\varepsilon^{bc}=\delta^c_a,\quad\varepsilon_{12}=\varepsilon^{21}=1,
[x_{i, r+1}^{+}, x_{i+1, s}^{+}] - [x_{i, r}^{+}, x_{i+1, s+1}^{+}] = - \varepsilon_2 x_{i, r}^{+}x_{i+1, s}^{+}- \varepsilon_1 x_{i+1, s}^{+}x_{i, r}^{+},
\int_{\Omega}\big( \nu \, \textbf{curl}\, \boldsymbol{y}_0^c \cdot \textbf{curl}\, \boldsymbol{v}_0^c \big)\,d\boldsymbol{x}= \int_{\Omega}\boldsymbol{u}^c_0 \cdot \boldsymbol{v}_0^c\,d\boldsymbol{x}
\int_{W}g_{u_j}^p \,d\mu \le \int_{U}g_{u_j}^p \eta^p \,d\mu= \lim_{k \to \infty}\int_{V_k}g_{u_j}^p \eta_k^p \,d\mu & \le \lim_{k \to \infty}\int_{V_k}(2^p\eta^p_k g_f^p + (4pM)^p g_{\eta_k}^p) \,d\mu\le \int_{U}(2^p g_f^p + (4pM)^p g_{\eta}^p) \,d\mu < \infty,
\mathrm{d}s_{K}^{2}= \sum_{m , n \neq k}^{\infty}{\frac{\partial^{2}K ( z_{( k )},{\bar{z}}_{( k )})}{\partial z_{( k )}^{m}\partial{\bar{z}}_{( k )}^{n}}}\mathrm{d}z_{( k )}^{m}\mathrm{d}{\bar{z}}_{( k )}^{n}.
\int_{0\leq z_p\leq\cdots\leq z_1\leq Z}\frac{dz_1}{z_1}\cdots\frac{dz_{p-1}}{z_{p-1}}\frac{dz_p}{1-z_p}(z_p)^A = \sum_{n=1}^{+\infty}\frac{Z^{n+A}}{(n+A)^p}
(a) \; \prod_{1\leq i<j\leq n}\dfrac{(qt^{j-i})_{\lambda_i-\lambda_j}}{(qt^{j-i-1})_{\lambda_i-\lambda_j}}= \prod_{s\in \lambda}\left( \dfrac{1 - q^{a'_\lambda(s)+1}t^{-l'_\lambda(s)+n-1}}{1-q^{a_\lambda(s)+1}t^{l_\lambda(s)}}\right)
\sum_{j=1}^{\infty}t^{j-1}\|(V^{(k,q-1)}_{I_{k;q}})^{diag}_j \|=\frac{d}{dt}\,\Big(\sum_{j=1}^{\infty}\frac{t^{j}}{j}\|(V^{(k,q-1)}_{I_{k;q}})^{diag}_j \|\Big)
j^1s_t(m)=T\Phi_{\tau_t(p)\xi_0(p)^{-1}}\left(\operatorname{Hor}_p^{\Gamma_0}+\left(TR_{\tau_t(p)^{-1}}T_p\tau_t\!\cdot\!\operatorname{Hor}_p^{\Gamma_0}\right)_P(p)\right),
W_X = (\eta_A - \hbox{$\frac{1}{2}$}m^2 (P_+)^B_A \bar{\phi}_B) \phi^A -\phi^*_{A a}\pi^{A a}-\bar{\phi}_A (\lambda^A - \hbox{$\frac{1}{2}$}m^2 (P_-)^A_B \phi^B).
A_{j}^{i_1,...,i_{k-2j}}(\beta,\tilde\gamma,B) = 2\beta^{-1}\sum_{\alpha=1}^{k-2(j-1)-1}\sum_{\delta>\alpha}A_{j-1}^{l_1...l_{\alpha-1}\eta l_{\alpha}.... l_{\delta-2}\xi l_{\delta-1}...l_{k-2j}}(\beta,\tilde\gamma,B)\frac{1}{2}(\tilde\gamma_{\eta\xi}+\tilde\gamma_{\xi\eta})G_{i_1...i_{k-2j}}^{l_1...l_{k-2j}}(-\tilde\gamma)
\int_{(B_r \backslash B_{\rho/2}) \cap \{0<d_U < \kappa \}}\frac{V(d_U(y))}{d_U(y)}\frac{dy}{|x-y|^{n-2}\varphi(|x-y|)}\le \sum_{k=1}^M \frac{1}{(2^{-k}r)^{n-2}\varphi(2^{-k}r)}\int_{(C_{k-1}\backslash C_k) \cap \{0< d_U < 6 \cdot 2^{-k}r \}}\frac{V(d_U(y))}{d_U(y)}|\nabla d_U(y)|dy.
e_k = \phi_k\left(\langle 2\phi_{k, x}+ \varphi_{k, x}, \varphi_{k, x}\rangle- \langle 2\phi_{k, t}+ \varphi_{k, t}, \varphi_{k, t}\rangle\right) \;\;\;\;\;\; - \varphi_k \left(|\tilde \phi_{k+1, t}|^2- |\tilde \phi_{k+1, x}|^2 \right)+ \mathbf{1}_{\omega}\left(h_k^{p^{\perp}}+ f_{k-1}^{\phi_k^{\perp}}- f_{k}^{\tilde \phi^{\perp}_{k+1}}\right).
\begin{cases}A_{1}A_{2}A_{3}B_{1}B_{2}B_{3}=A_{1}'A_{2}'A_{3}'B_{1}'B_{2}'B_{3}'=-1, A_{2}B_{3}A_{2}'B_{3}'=A_{3}B_{1}A_{3}'B_{1}'=1, A_{3}B_{2}\overline{A_{3}'B_{2}'}=A_{1}B_{3}\overline{A_{1}'B_{3}'}=1,\end{cases}
{\bf E}\Bigl[\exp\bigl(q\log\beta_{M, N}(a,b)\bigr)\Bigr] =\exp\Bigl(\int\limits_0^\infty (e^{-tq}-1) e^{-b_0t}\frac{\prod\limits_{j=1}^N (1-e^{-b_j t})}{\prod\limits_{i=1}^M (1-e^{-a_i t})}\frac{dt}{t}\Bigr).
|w(t,z)|\leq 2e^{t{\rm Re}\widehat{a}(z)}\leq 2t|\widehat{a}(z)|e^{t{\rm Re}\widehat{a}(z)}\leq Ct|\widehat{a}(z)|e^{t\sup{\rm Re}\widehat{a}(z)}, z=k+i\tau,
(v_{S},v_{S}) =(s_i v_T-\rho v_T,s_i v_T-\rho v_T) &=(v_T,v_T)+\rho^2(v_T,v_T)-\rho(s_iv_T,v_T)-\rho(v_T,s_iv_T) &=(v_T,v_T)+\rho^2(v_T,v_T)-\rho(\rho v_T+v_S,v_T)-\rho(v_T,\rho v_T+v_S) &=(1-\rho^2)(v_T,v_T).
\mbox{Problem 3:}~~\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 &\mathbb{E}\left[P_i\right] \le P_i^{av},~\forall i &\sum_{i=1}^K g_iP_i\le I.
\delta S^{C T P}\left[ \phi^{+}, \phi^{-}\right] = - i \ln \int D \Phi^{+}D \Phi^{-}\; e^{i \left\{S_{H}\left( \Phi^{+}\right) - S_{H}\left( \Phi^{+}\right) + S_{l H}\left( \phi^{+}, \Phi^{+}\right) - S_{l H}\left( \phi^{-}, \Phi^{-}\right) \right\}}
-i \eta \mathrm{P}_0 \xi_1 \left( \left(\mathrm{P}_0 e\right) + \left(\mathrm{P}_1 e\right)\right)=\sigma (\mathrm{P}_0 e), &-i \eta \mathrm{P}_1 \xi_1 (\mathrm{P}_0 e) -i\eta \mathrm{P}_1\xi_1 (\mathrm{P}_1 e) + L (\mathrm{P}_1 e) =\sigma (\mathrm{P}_1 e).
\xi_1^{y,s,\varepsilon}(x,t) \le g(y,s)-2\varepsilon-M_1\rho(x)-M_2|t-s| &\le g(x,t)+\omega(|x-y|)+\omega(|t-s|)-2\varepsilon-M_1\rho(x)-M_2|t-s| &\le g(x,t).
\left\{K ( \Lambda ) , K ( \bar{\Lambda}) \right\}= \sum_{I = 1}^{n_{1}}k_{I}^{1}\int_{\partial D_{1}}\Lambda_{I}^{1}d \bar{\Lambda}_{I}^{1}+ \sum_{J = 1}^{n_{2}}k_{J}^{2}\int_{\partial D_{2}}\Lambda_{J}^{2}d \bar{\Lambda}_{J}^{2},
\langle\hat{N}\rangle\equiv\langle\hat{B}_{1}^{\dagger}\cdot\cdot\cdot\hat{B}_{K}^{\dagger}\hat{a}_{1}^{\dagger}\cdot\cdot\cdot\hat{a}_{N_{\hat{a}}}^{\dagger}\hat{a}_{N_{\hat{a}}}\cdot\cdot\cdot\hat{a}_{1}\hat{B}_{J}\cdot\cdot\cdot\hat{B}_{1}\rangle
(G^{(2)}_{kp_2j_2})_{ab}= (G^{(2),p_1j_1}_{kp_2j_2})_{ab}-c_{p_1j_1}W_{p_1j_1}N^{-1/2}(G^{(2),p_1j_1}_{kp_2j_2})_{ap_1}(G^{(2)}_{kp_2j_2})_{j_1b}&-c_{p_1j_1}W_{j_1p_1}N^{-1/2}(G^{(2),p_1j_1}_{kp_2j_2})_{aj_1}(G^{(2)}_{kp_2j_2})_{p_1b}
qxy\sum_{k=-1}^{n-1}(-1)^{k+1}\frac{M_k^{N}+M_{k+1}^{N}}{M_kM_{k+1}}&=qxy\left(\sum_{k=0}^{n-1}(-1)^ka_{k+1}\sum_{i=0}^{N-2}M_{k-1}^{N-2-i}M_{k+1}^i\right)+M_0^{N-1}x+M_{n-1}^{N-1}y.
Y_1=\begin{bmatrix}b a-b 0 a\end{bmatrix}, Y_2 = \begin{bmatrix}\frac{1}{b}\frac{1}{a}-\frac{2}{b}0 \frac{1}{a}\end{bmatrix}, Y_3=\begin{bmatrix}1 1 0 1\end{bmatrix}.
|u_e(x)-u_e(\hat{x})| = |(u-L)_e(x)-u_e(\hat{x}) + L_e(x)| = |(u-L)_e(x)|\leq & \leq \frac{C||(u-L)||_{L^{\infty}(B_{|x-\hat{x}|}(x))}}{|x-\hat{x}|}\leq \frac{C|x-\hat{x}|\sigma(|x-\hat{x}|)}{|x-\hat{x}|}= C\sigma(|x-\hat{x}|)
{\cal C}\Gamma_{a}= \left( \begin{matrix}{{c \, \tilde{\gamma}_{a}}}&{{0}}\\{{0}}&{{c^{T}\gamma_{a}}}\\ \end{matrix}\right) \otimes \Omega \ ; \qquad{\cal C}\Gamma_{a^{\prime}}= \left( \begin{matrix}{{0}}&{{- c}}\\{{c^{T}}}&{{0}}\\ \end{matrix}\right) \otimes \Omega \gamma_{a^{\prime}}\ .
\||x|^{\gamma_1}|x'|^{\alpha}u\|^s_{L^s(R_k\cap K_{\epsilon_1})}\le C\||x|^{\gamma_1}|x'|^{\alpha}u\|^s_{L^s(R_k\cap K_{\epsilon_2}\setminus K_{\epsilon_1})}+C\||x|^{\gamma_2}|x'|^{\mu}\nabla u\|_{L^p(R_k\cap K_{\epsilon_2})}^{as}\quad\cdot\||x|^{\gamma_3}|x'|^{\beta}u\|_{L^q(R_k\cap K_{\epsilon_2})}^{(1-a)s}.
\setlength{\abovedisplayskip}{4 pt}\setlength{\belowdisplayskip}{4 pt}E_{\tau,i}\!\left( \theta_{i,n_{i\tau},\tau}^j\right)\!=\! - \mathop \frac{1}{{\theta_{i,n_{i\tau},\tau}^j \tau}}\log_2 \mathbb{E}_{d_i,h_i}\!\!\!\left[{{e^{- \theta_{i,n_{i\tau},\tau}^j C_{\tau,i}}}}\right],
\left( L_{0}, L_{0}\, | \, L_{+}, A , L_{-}\right) = 2 \hbar^{3}\left[ L_{0}, A \right] + \hbar \left[ L_{-}, \left[ L_{+}, \left[ A , L_{0}\right] \right] \right] + \hbar \left[ L_{+}, \left[ L_{-}, \left[ A , L_{0}\right] \right] \right] \; ,
\mathcal{H}_{C_{n}^{(1)}}^{\mathit{Toda}}={\frac{1}{2}}P^{2}+m^{2}e^{-2Q_{1}}+m^{2}\sum_{j=1}^{n-1}e^{Q_{j}-Q_{j+1}}+m^{2}e^{2Q_{n}},
\sqrt{x_i}\left(u_{x_i}(P_2)-u_{x_i}(P_1)\right) = \sqrt{x_i}\int_0^{x^2_j-x^1_j}u_{x_ix_j}(P_1+te_j)\, dt &= \sqrt{x_i}\int_0^{x^2_j-x^1_j}\sqrt{x^1_j+t}\, u_{x_ix_j}(P_1+te_j)\frac{1}{\sqrt{x^1_j+t}}\, dt,
J_{x}=[\begin{matrix}0&0&0&0\\ 0&0&0&0\\ 0&0&0&-1\\ 0&0&1&0\\ \end{matrix}],J_{y}=[\begin{matrix}0&0&0&0\\ 0&0&0&1\\ 0&0&0&0\\ 0&-1&0&0\end{matrix}],J_{z}=[\begin{matrix}0&0&0&0\\ 0&0&-1&0\\ 0&1&0&0\\ 0&0&0&0\end{matrix}]
\lim_{|b|\to\infty}\frac{\mathcal{H}(b,u,w)}{|b|\log(|b|+1)}=\frac{1}{2}\qquad\mathrm{and}\lim_{|b|\to 0}\frac{\mathcal{H}(b,u,w)}{b^2}=\frac{1}{4(1+uw)}.
p(h_{j}^{(1t)}=1|v^{t},h^{(2t)})=g(\sum_{k=1}^{K}W_{kl}^{(1t)}v_{k}^{(t)}+\sum_{l=1}^{F_{2}^{t}}W_{jl}^{(2t)}h_{l}^{(2t)}+Mb_{j}^{(1t)})
H^{2}(t)=\left( \int_{0}^{R}fvdr\right)^{2}=\left( \int_{0}^{R}\frac{f}{{f}^{\prime}}vdf\right)^{2}\leq\left( \int_{0}^{R}\frac{f^{2}}{{f}^{\prime2}}df\right) \left( \int_{0}^{R}v^{2}df\right)
F(x) =\left(f(x),f\circ S^{-1}(x),f\circ S^{-2}(x),\dotsc,f\circ S^{-n+1}(x) \right) &=\left(f(x_1,x_2\ldots,x_n),f(x_2,\ldots,x_{n},x_1),\ldots,f(x_n,x_1,\ldots,x_{n-1})\right).
\eta_k(\xi) =\frac{1}{2}-\frac{\sigma_k}{2\xi}-\frac{\log(C_k)}{\xi^2}-\frac{k}{2\xi^2}+\frac{k\log(\sigma_k +\xi)}{\xi^2}+ \frac{\log\left(\frac{1}{2}\left(1+\frac{\xi}{\sigma_k}\right)\right)}{2\xi^2}.
\tau(T(\lambda_C - g)) = \sum^m_{k=1}a'_k(c'_k - b_k)(x'_{k}- x'_{k-1}) &= \sum^m_{q=1}\sum^q_{j=1}\left(a'_q - a'_{q+1}\right) (c'_j - b_j)(x'_{j}- x'_{j-1}).
S_{I I}= k^{2}\frac{\sigma_{0}}{\pi}\int_{\tau_{0}}^{\tau_{f}}d \tau \sum_{n \neq 0}^{}| F_{\nu}\left( \frac{2 \pi}{\sigma_{0}}| n | \tau , \frac{2 \pi}{\sigma_{0}}| n | \tau_{0}\right) |^{2}| \phi_{n}( \sigma_{0}) |^{2}
\frac{\theta_0(1/4; \tau)}{(\tau; \tau) (1/2; \tau)}= \frac{1}{1 + i}\frac{(1/4; \tau)(-1/4; \tau)}{(\tau; \tau)(1/2;\tau)}= \frac{1}{1 + i}\frac{(1/2; 2\tau)}{(\tau; \tau)(1/2;\tau)}= \frac{1}{1 + i}\frac{1}{(\tau; \tau)(\tau+1/2; 2\tau)}
\log M \stackrel{(a)}{\leq}\log \left( \frac{\sqrt{nA}+ r_0}{r_0}\right)^n - 0.599n & \leq n\log \left( \frac{\sqrt{nA}+ r_0}{r_0}\right) - 0.599n & \stackrel{(b)}{=}\frac{1}{2}n \log \left( \frac{A}{\epsilon_n'}+ 1 \right) - 0.599n \,,
\tilde{A}_{2n}\Big(M(i\mu);\mathcal{O}_{(p,q)}\Big) = \big(c\cdot i\mu+d\big)^{2}\tilde{a}_{2n}\big[M(p,q)\big] (i\mu) &\big(c\cdot i\mu+d\big)^{2}\rho(M)\tilde{A}_{2n}[(p,q)] (i\mu),
R ( u_{1}- u_{2}) t_{b}^{c}( u_{1}) \otimes t_{a}^{b}( u_{2}) = \sum_{d}W \left( \left. \begin{array}{c c}{{c}}&{{d}}\\{{b}}&{{a}}\\ \end{array}\right| u_{1}- u_{2}\right) t_{a}^{d}( u_{1}) \otimes t_{d}^{c}( u_{2}) .
\widetilde{\mathcal{S}}_{n}(z) =\mathcal{S}_{n}(z-1/2), \widetilde{\mathcal{C}}_{n}(z) =\mathcal{C}_{n}(z-1/2), \widetilde{\mathcal{S}}_{n}^{\prime}(z) =\mathcal{S}_{n}^{\prime}(z-1/2), \widetilde{\mathcal{C}}_{n}^{\prime}(z) =\mathcal{C}_{n}^{\prime}(z-1/2).
\sum_{n=0}^\infty\overline{\textrm{spt}}_\omega(n)q^n=\sum_{n=1}^\infty\dfrac{q^n(-q^{n+1};q)_n(-q^{2n+2};q^2)_\infty}{(1-q^n)^2(q^{n+1};q)_n(q^{2n+2};q^2)_\infty}.
|c|^{2}- r^{2}= |z|^{2}- |p_{0}-z|^{2}= |p_{0}| \cdot (|z|-|p_{0}-z|) = |p_{0}| \cdot (|p_{0}|-2|p_{0}-z|) = (1-d_{0})(1-d_{0}-2|p_{0}-z|)\,.
\frac{\partial \zeta_k}{\partial z_l}=g^{\overline{j}k}(p)g_{l\overline{j}}(z,\overline{p}),\ \ \ \frac{\partial z_k}{\partial \zeta_r}=g_{r\overline{\lambda}}(p)g^{\overline{\lambda}k}(z,\overline{p}).
\left<A_{N}f,A_{N}f\right> \leq \frac{\|f\|^{2}_{L^{2}(X,\mu)}}{N}+ \frac{1}{N}\sum_{k=-N+1,\k\neq 0}^{N-1}C'\cdot \lvert k\rvert^{-\alpha}\|f\|^{2}_{L^{2}(X,\mu)}&\leq \frac{\|f\|^{2}_{L^{2}(X,\mu)}}{N}+ 2C'\cdot N^{-\alpha}\|f\|^{2}_{L^{2}(X,\mu)},
\Big\| 1_{Q_0}\Big( \mathop{\sum_{R \in \mathcal{D}_{\textup{good}}:\, R \subset Q^*}}_{2^{-\kappa}< \ell(R) < 2^{s-r}}1_{R}\int_{\ell(R)/2}^{\ell(R)}\Big| \mathop{\sum_{\ell=r+1}^{s-\textup{gen(R)}}}\theta_t \Delta_{R^{(\ell)}}f \Big|^2\, \frac{dt}{t}\Big)^{1/2}\Big\|_{L^q(\mu)}\lesssim \mu(Q^*)^{1/q}.
2^{-ia}=\sup_{x_0\in\mathbb{R}^n,\,r\in(0,\infty)}\left(\frac{|x_0|+r}{|x_0|+2^ir}\right)^a\lesssim\sup_{x_0\in\mathbb{R}^n,\,r\in(0,\infty)}I(B(x_0,r),i) &\lesssim\sup_{x_0\in\mathbb{R}^n,\,r\in(0,\infty)}\left(\frac{|x_0|+r}{|x_0|+2^ir}\right)^a(i+1)^{-b}=2^{-ia}(i+1)^{-b},
\frac{\partial f_j}{\partial \lambda_k}=-\frac{1}{m|g|}\sum_{h',i'=0}^{m-1}[\tilde{v}_{i'}\tilde{w}_{i'}]_{k,j}[\tilde{v}_{i'-h'-1}\tilde{w}_{i'-h'-1}]_{j,k}\frac{\partial}{\partial\lambda_j}\frac{\lambda_j^{m-h'-1}\lambda_k^{h'}}{\lambda_j^m-\lambda_k^m},
_\pm \langle \{\lambda \}, \pm q | \prod_{i = 1}^{N}\Psi^{\pm}( z_{i}) | \pm ( q - N \beta ) \rangle = \prod_{i < j}( z_{i}- z_{j})^{\beta}\prod_{i}^{N}( z_{i})^{q - N \beta}J_{\{\lambda \}}^{( \beta )}( z_{1}, . . . z_{N}) \ ( j_{\lambda}^{\beta})^{- 1 / 2}
\varphi_1(x)= \Theta\left(\cos\left(\frac{5}{\sqrt{21}}x\right)-3\cos\left(\frac{1}{\sqrt{21}}x\right)+2\cos\left(\frac{4}{\sqrt{21}}x\right)\right), \varphi_2(x)=\Theta\left(-\sin\left(\frac{5}{\sqrt{21}}x\right)-3\sin\left(\frac{1}{\sqrt{21}}x\right)+2\sin\left(\frac{4}{\sqrt{21}}x\right)\right),
\begin{array}{l c c l}{{}}&{{( 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ - 1 )}}\\{{\mid \mathrm{g n d}>}}&{{( 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ - 1 )}}&{{=}}&{{({\bf 1}, ~{\bf 1})}}\\{{}}&{{\Downarrow \beta_{8}^{-}}}\\{{\mid \mathrm{1 s t}>}}&{{( 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 1 ~ - 1 )}}&{{=}}&{{({\bf \overline{{8}}}, ~{\bf 1}) .}}\\ \end{array}
\frac{1}{n^2}\; (tr_{g}\tau_s)(tr_{\tau_s}\omega) \cdot \frac{g^n}{\omega^n}\geq \frac{1}{n^2}\; tr_{g}\omega \cdot \frac{g^n}{\omega^n}=\frac{1}{n}\; \frac{\omega \wedge g^{n-1}}{\omega^n}.
\frac{\partial^{2}}{\partial x_{2}^{2}}V ( x_{2}) = \phi_{0}( x_{2}) \phi_{0}^{\dagger}( x_{2}) \int \frac{d^{2}k}{2 \pi}\frac{- i k_{0}+ i e V_{0 , 0}}{( - i k_{0}+ i e \gamma_{0}V_{0 , 0})^{2}}.
|X|>|Y_{t^s}| =\frac{n-|Z|-|X\setminus X_{t^s}|-|Y\setminus Y_{t^s}|}{2}&=\frac{n-st(2k-1)-2t^sn^{\frac{1}{s+1}}}{2}&=\frac{n-s \alpha n^{\frac{1}{s+1}}(2k-1)-2(\alpha n^{\frac{1}{s+1}})^s n^{\frac{1}{s+1}}}{2}&> \frac{n}{2}-ks \alpha n^{\frac{1}{s+1}}-\alpha^s n.
G_{\pm}( \alpha_{\pm}) = e^{\alpha_{\pm}\bar{T}_{\pm}}= 1 + \alpha_{\pm}\bar{T}_{\pm}= \left( \begin{array}{c c c}{{1}}&{{\frac{\alpha_{\pm}}{\sqrt{2}}}}&{{\mp i \frac{\alpha_{\pm}}{\sqrt{2}}}}\\{{0}}&{{1}}&{{0}}\\{{0}}&{{0}}&{{1}}\\ \end{array}\right)
H_{2,1}(F_{f}/2) =\frac{1}{2304}\left(-3\tau^4_{1}+12(1-\tau^2_{1})\tau^2_{1}\tau_{2}-8(1-\tau^2_{1})(2+\tau^2_{1})\tau^2_{2}\right. &\left.+24\tau_{1}\tau_{3}(1-\tau^2_{1})(1-|\tau^2_{2}|)\right).
w^{S_0 \oplus S_1}_{(a^-, i)}= 1 + \beta \sum_{j \in N}p_{ij}g^{S_0 \oplus S_1}(1, j) - \beta g^{S_0 \oplus S_1}_{(0, i)}= 1 + \beta \sum_{j \in S_1}p_{ij}g^{S_1}_j - \beta g^{S_1}_i = w^{S_1}_i,
\frac{d_{+}}{d_{-}}= \frac{\Gamma ( - \frac{2 j + 1}{k + 2})}{\Gamma ( \frac{2 j + 1}{k + 2})}\sqrt{\frac{\Gamma ( \frac{2 j}{k + 2}) \Gamma ( \frac{2 j + 2}{k + 2})}{\Gamma ( - \frac{2 j}{k + 2}) \Gamma ( - \frac{2 j + 2}{k + 2})}}= \frac{\gamma_{-}}{\gamma_{+}},
\mathcal{K}_m' (\varphi_m) v (s) = \tilde{h}\sum_{j=1}^m \sum_{i=1}^\rho w_i \; \frac{\partial \kappa}{\partial u}(s, \zeta_i^j, \varphi_m (\zeta_i^j)) v (\zeta_i^j), \;\;\; s \in [a, b].
E_n^{C}[f] = \sum_{m \in \Delta(n)}a_m E_n^{C}(T_m) & \sim \sum_{m \in \Delta(n)}\left( \sum_{k = 0}^{\infty}\frac{\mu_k}{m^{d_k}}\right) E_n^{C}(T_m) & = \sum_{k=0}^{\infty}\mu_k \sum_{m \in \Delta(n)}\frac{1}{m^{d_k}}E_n^{C}(T_m).