harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
D(\lambda, T) := \frac{1}{N(T)}\sum_{\substack{0<\gamma \le T \gamma^+ -\gamma \le \frac{2 \pi \lambda}{\log T}}}\!\!\!\!\! 1 D_d(\lambda, T) := \frac{1}{N(T)}\sum_{\substack{0<\gamma_d \le T \gamma_d^+ -\gamma_d \le \frac{2 \pi \lambda}{\log T}}}\!\!\!\!\! 1.
{\mathbb E}\!\left\{{{{\left\|{{{\bf{g}}_n}}\right\|}^4}}\right\}={{\bf{V}}\!{\rm{ar}}}{\left\{{{{\left\|{\bf{g}}_n \right\|}^2}}\right\}}+{\left({\mathbb E}\!\left\{{{{\left\|{{{\bf{g}}_n}}\right\|}^2}}\right\}\right)}^2.
\begin{array}{l}(\overline{\mathcal{L}}_{\xi}g)_{11}=(\overline{\mathcal{L}}_{\xi}g)_{22}=(\overline{\mathcal{L}}_{\xi}g)_{33}=(\overline{\mathcal{L}}_\xi g)_{44}=-2\alpha, (\overline{\mathcal{L}}_\xi g)_{13}=(\overline{\mathcal{L}}_\xi g)_{24}=2(1-\beta) \end{array}
\phantom{=}\left(a (t_\lambda^\flat)^{p-1}-\frac{b(t_\lambda^\flat)^{2\cdot p^\flat-1}}{p^\flat}\right) \int_{\mathbb{R}^N}|u^\flat|^{p}dx &\geq (t_\lambda^\flat)^{p-1}\int_{\mathbb{R}^N}\tilde{a}(\lambda x)|u^\flat|^{p}dx-\frac{b(t_\lambda^\flat)^{2\cdot p^\flat-1}}{p^\flat}\int_{\mathbb{R}^N}|u^\flat|^{p}dx>0
4^4\frac{\partial}{\partial B'}\frac{\partial}{\partial b_3}\frac{\partial}{\partial b_4}\frac{\partial}{\partial b_5}\hat\vartheta(\mu)=\left(\psi_0\frac{\partial^4}{\partial \psi_0^4}\right) \frac{1}{\psi_0}\hat\vartheta(\mu) \ .
u_{n}(t) = e^{\lambda_{n}t}\langle u^{0}, \varphi_{n}\rangle_{X_{-1}}+ \int_{0}^{t}e^{\lambda_{n}(t - s)}\langle f(s) - \lambda_{n}D \gamma(s), \phi_{n}\rangle_{X_{-1}}\mathrm{d}s t \in [0, T].
\forall i \in \{1,\dots,n\},~ f(P_{\sigma(i)}) = f(P_i) \Leftrightarrow \forall i \in \{1,\dots,n\},~ f \circ \sigma(P_i) = f(P_i) & \Leftrightarrow \forall i \in \{1,\dots,n\},~ (f \circ \sigma - f)(P_i) = 0.
u_t(x,t+t_{0n};t_{0n},u_{0n}) =\Delta u(x,t+t_{0n};t_{0n},u_{0n}) -\chi\nabla\cdot (u(x,t+t_{0n};t_{0n},u_{0n}) \nabla v(x,t+t_{0n};t_{0n},u_{0n})) & + (a(x,t+t_{0n})-b(x,t+t_{0n})u(x,t+t_{0n};t_{0n},u_{0n}))u(x,t+t_{0n};t_{0n},u_{0n}).
t - t_{0}= \int \left( 9 V^{4 / 3}+ 3 \Lambda V^{2}+ 3 k_{4}^{2}\rho_{0}V^{2 - \gamma}+ \frac 1 4 k_{5}^{4}\rho_{0}^{2}V^{2 - 2 \gamma}+ C \right)^{- 1 / 2}d V ,
\varphi(x)=d(x,F),\,\,\mbox{if}\,\, F\ne\emptyset, \varphi(x)=1,\,\,\,\mbox{if}\,\, F=\emptyset, g(x)=\exp(-\sum\limits_{n=1}^\infty \|x_n\|_n), f(x)=\varphi(x)\cdot g(x).
\Omega_X= zdp\wedge dq\ \big(=p^{-\frac{2}{3}}q^{-\frac{2}{3}}(1-p)^{-\frac{2}{3}}(1-q)^{-\frac{2}{3}}(1-x_1 p-x_2q)^{-\frac{2}{3}}dp\wedge dq\big)
\sum_{m=1}^{\infty}[(\frac{m\pi}{a})^{4}Y_{m}sin\frac{m\pi x}{a}-2(\frac{m\pi}{a})^{2}\frac{d^{2}Y_{m}}{dy^{2}}sin\frac{m\pi x}{a}+\frac{d^{4}Y_{m}}{dy^{4}}sin\frac{m\pi x}{a}]=0
f^{( a , b )}\in L^{N - 1}\Big ( \mathrm{I R}^{D ( n_{a}+ n_{b})}, \prod_{i = 1}^{n_{a}}d^{D}z_{i}^{( a )}\prod_{j = 1}^{n_{b}}d^{D}z_{j}^{( b )}\Big ) ,
\begin{aligned}\mathcal{G}(T)= \{(u,T(u)): u \in I\}\subseteq \{(u, T(u)+F(u)-F(u)): u \in I\}\subseteq \{(u,T(u)+F(u)): u \in I\}+ \{(0, -F(u)):u \in I\}= \mathcal{G}(F+T))+ \{(0, -F(u)):u \in I\}. \end{aligned}
E_{0}=\frac{\left|\mu_{eff}^{2}\right|^{1/2}}{16\pi^{2}r}\sum_{n=1}^{\infty}\frac{1}{n^{3}}\left\{\eta^{3}\left[\frac{H_{1}\left(\eta \right)}{2\eta}-\frac{H_{2}\left(\eta \right)}{\eta^{2}}-\frac{H_{3}\left(\eta \right)}{2\eta}+\frac{\eta}{60\pi}\right]-\eta N_{2}\left(\eta \right)+\eta^{2}N_{3}\left(\eta \right)\right\}_{\lambda =1}.
v_{i}( p^{+}) = \frac{i N_{c}g^{2}}{4 \pi^{2}}\int d k^{+}d k^{-}\frac{\cal P}{( p^{+}- k^{+})^{2}}\frac{1}{k^{--}\frac{m_{i}^{2}}{k^{+}}- v_{i}( k^{+}) + i \epsilon \mathrm{s g n}( k^{+})}.
\sum_{n=T}^{[\tau]}N_A\left(\frac{n+\xi}{T}\right) = \frac{T}{\phi(Q)}\frac{e^{\frac{\xi}{T}}e^{\frac{[\tau]+1}{T}}}{(e^{\frac{1}{T}}-1)([\tau]+1)}+ O\Bigg(\frac{T^2}{\phi(Q)}\sum_{n=T}^{[\tau]}\frac{e^{\frac{n+1}{T}}}{(n+1)^2}+ \frac{1}{\phi(Q)}\Bigg).
n^{*}(x)=\sum_{\begin{matrix}n\le x\\n\in\mathbb{N}^{*}\end{matrix}}1;n_{0}^{*}(x)=\sum_{\begin{matrix}n\le x\\n\in\mathbb{N}_{0}^{*}\end{matrix}}1;n_{1}^{*}(x)=\sum_{\begin{matrix}n\le x\\n\in\mathbb{N}_{1}^{*}\end{matrix}}1.
\frac{d}{dt}z^{(k,\ell)}_j =\mathcal{W}^{(k,\ell)}_j,\ \frac{d}{dt}\bar{z}^{(k,\ell)}_j-m \bar{z}^{(k,\ell)}_j =\overline{\mathcal{W}}^{(k,\ell)}_j.
\begin{aligned}\pi^{O(4,3)}_{1,\ell}|_{O(4)\times O(3)}= \sum_{\substack{d-\ell-3 \ge e \ge 0 begin{align*}.2ex] e\equiv d-\ell-3 \pmod{2}}}\pi^{O(4)}_d \otimes\pi^{O(3)}_e \pi^{O(3,4)}_{2,\ell}|_{O(3)\times O(4)}= \sum_{\substack{d' -\ell -4 \ge e' \ge 0 begin{align*}.2ex] e'\equiv d'-\ell-4\pmod{2}}}\pi^{O(3)}_{d'}\otimes\pi^{O(3)}_{e'}.\end{aligned}
e_{9a}=e_9|_{b=0}=2 s^3 p^2-18 p^2 s^2+30 s p^2+99 s^2 p-12 p^2-192ps-3s^2+93p+3 &=2p^2+(1-s)[6(1-p)+(1-s)(99p+2p^2s-14p^2-3)]\ge 0
a^s_3(A_\varepsilon)=-\frac{1}{i\pi}\log(\varepsilon)\t_{12}+\left(\tau^s\right)^{1,2}+\left(\tau^s\right)^{12,3}+\mathcal O(\varepsilon|\log\varepsilon|^\bullet),a^s_3(B_\varepsilon)=-\frac{1}{i\pi}\log(\varepsilon)\t_{23}+\left(\tau^s\right)^{2,3}+\left(\tau^s\right)^{1,23}+\mathcal O(\varepsilon|\log\varepsilon|^\bullet).
\delta L_{T G}= \varepsilon^{\mu \nu \rho}( R_{\beta \mu \nu}^{\alpha}\delta \Gamma_{\alpha \rho}^{\beta}- \bar{R}_{\beta \mu \nu}^{\alpha}\delta \bar{\Gamma}_{\alpha \rho}^{\beta}) - \partial_{\mu}[ \varepsilon^{\mu \nu \rho}N_{\beta \nu}^{\alpha}( \delta \Gamma_{\alpha \rho}^{\beta}+ \delta \bar{\Gamma}_{\alpha \rho}^{\beta}) ]
V_{\theta}^{D}\equiv{\frac{\mu^{\epsilon}}{2{\cal A}}}\mathrm{T r}{\ln}\left({\frac{P_{\theta}( D )}{\mu^{2}}}\right) = -{\frac{\mu^{\epsilon}}{2{\cal A}}}\lim_{s \to 0}\partial_{s}\zeta_{\theta}( s , D ) ,
\|V^{k+1}(t)\|_{H_{s,\delta}}^2 \leq e^{C_1t}\left(\|U^{k+1}_0-U_0^0\|_{H_{s,\delta}}^2 +\int_0^t\|F^{k}(t,\cdot)\|_{H_{s,\delta}}^2 d\tau\right) \leq e^{C_1t}\left( 2^{-k}+4C_0^2\|U_0\|_{H_{s,\delta}}^2 +2\int_0^t \|F(\tau,\cdot)\|_{H_{s,\delta}}^2d\tau+t2^{-k+1}\right).
\mathbb{E}\left[\left(X_{t}^{x}\right)^{\kappa}\right]=\int_{\mathbb{R}_{\geqslant0}}\int_{\mathbb{R}_{\geqslant0}}(y+z)^{\kappa}\mu_{Y_{t}^{x}}(\mathrm{d}y)\mu_{Z_{t}}(\mathrm{d}z)<\infty.
(q_n +1) (q_n +2) \tilde b_{q+ \overline{n}, \kappa}+ \sum_{i \neq n}(q_i + 1) (q_i + 2) \tilde b_{q+ 2 \overline{i}- \overline n, \kappa}- (\kappa +1) \tilde b_{q- \overline{n}, \kappa+1}+ \tilde c^{m,\ell}_{q,\kappa}\tilde b_{m,\ell}= 0,
1+ a = \dfrac{r(r-p-q)_p+(r-p)i-q j}{r(r-p-q)_p}\ge \dfrac{r(r-p-q)_p - q(r-p-q)_q}{r(r-p-q)_p}= \dfrac{r(r-p-q)_p - p (r-p-q)_p}{r(r-p-q)_p}= \dfrac{r-p}{r}> 0, 1- a = \dfrac{r(r-p-q)_p-(r-p)i+q j}{r(r-p-q)_p}\ge \dfrac{r(r-p-q)_p - (r-p)(r-p-q)_p}{r(r-p-q)_p}= \dfrac{p}{r}> 0,
\lvert m_{n+1}\rvert_S-\lvert m_{n}\rvert_S= (k+\sum_{j\in A^+}(m_j+n+1)-\sum_{j\in A^-}(m_j-n-1)+\sum_{j\in A^0}\lvert m_j\rvert) &-(k+\sum_{j\in A^+}(m_j+n)-\sum_{j\in A^-}(m_j-n)+\sum_{j\in A^0}\lvert m_j\rvert) = \sum_{j\in A^+}1 + \sum_{j\in A^-}1 = \lvert A^+\rvert+\lvert A^-\rvert .
\frac{dx_i}{dt}= \underbrace{\left(x_{i+1}- x_{i-2}\right)x_{i-1}}_{}-\underbrace{\mathrel{\phantom{\left(x_i\right)}}x_{i}\mathrel{\phantom{\left(x_i\right)}}}_{damping}+ \underbrace{\mathrel{\phantom{\left(x_i\right)}}F\mathrel{\phantom{\left(x_i\right)}}}_{}
z^{\mu}I_{\nu-\mu}^{(3)}(2z;q)=\sum_{m=0}^{\infty}\frac{\left(q^{\mu};q\right)_{m}}{\left(q;q\right)_{m}}I_{\nu+m}^{(3)}(2zq^{m/2};q)q^{\binom{m+1}{2}}\left(-\frac{q^{\nu-\mu}}{z}\right)^{m},
\int f(x) h(x/q) \Big( \frac{1}{q}D_{q^{-1}}u(x)+\frac{1}{q}u(x) u(x/q)+ A(x) u(x/q)+r(x)\Big) y(x) d_qx &= f(x/q) h(x/q) \Big( y(x/q)u(x/q) - D_{q^{-1}}y(x)\Big),
M'_{k, k}= \lambda_{q+1}^f \prod_{i=1}^q \left(\sum_{s =0}^{m+1}\binom{m+1}{s}(-\lambda_i)^s\lambda_{q+1}^{m+1-s}\right) = \lambda_{q+1}^f\prod_{i=1}^q(\lambda_{q+1}- \lambda_i)^{m+1}.
x^\alpha\succeq_b x^\beta :\iff x_1^{\alpha_1}x_2^{\alpha_2}\succ_{grlex}x_1^{\beta_1}x_2^{\beta_2}x_1^{\alpha_1}x_2^{\alpha_2}=x_1^{\beta_1}x_2^{\beta_2}\frac{x^\alpha}{x_1^{\alpha_1}x_2^{\alpha_2}}\succeq_{grlex}\frac{x^\beta}{x_1^{\beta_1}x_2^{\beta_2}}.
\alpha_V\circ (g\otimes 1)\circ \alpha_V^{-1}\circ (g^{-1}\otimes 1)(b\otimes v)=\alpha_V\circ (g\otimes 1)\circ \alpha_V^{-1}(g^{-1}(b\otimes \rho(g)(v))=b\otimes \rho(g)(v).
{\cal A}_{b}= \frac{1}{4 \pi b^{2}}\int d^{2}x \left[ \frac 1{2}( \partial_{\mu}\tilde{\varphi})^{2}+ ( m k ( G ) )^{2}\sum_{i = 0}^{r}e^{e_{i}\cdot \tilde{\varphi}}\right] + O ( 1 ) .
\mathtt{L}_u(xy) = \mathtt{B}_u(x)\mathtt{L}_{\overline{u}}(y), \mathtt{R}_u(xy) = \mathtt{R}_{\overline{u}}(x)\mathtt{B}_{u}(y), \mathtt{B}_u(xy) = \mathtt{L}_u(x)\mathtt{R}_u(y).
\textbf{i}^2=\textbf{j}^2=\textbf{k}^2=\textbf{i}\textbf{j}\textbf{k}=-1 &\textbf{i}\textbf{j}= -\textbf{j}\textbf{i}= \textbf{k}&\textbf{j}\textbf{k}= -\textbf{k}\textbf{j}= \textbf{i}&\textbf{k}\textbf{i}= -\textbf{i}\textbf{k}= \textbf{j}~.
\mathcal{R}(v,\gamma,\gamma_t,\sigma_t) = -\gamma_t v -\frac{1}{N}\sigma_t v + \beta\mathcal{J}\left[\frac{\gamma_{xx}}{\left(1-\gamma_x\right)^2}v - \frac{\gamma_x^2}{1-\gamma_x}\phi'\right] , \mathcal{P}(v,\gamma) = -\beta\mathcal{J}\left[\gamma_x + \frac{\gamma_x}{1-\gamma_x}\right] v,
V_{r a d}= V_{\|}\: \left[ \bf r \cdot \left( v_{\mathrm{1}}+ v_{\mathrm{2}}\right) \right]^{\mathrm{2}}/ 4 \mathrm{r^{2}\; + \; V_{L}\: \left[ \bf r \cdot \left( v_{\mathrm{1}}- v_{\mathrm{2}}\right) \right]^{\mathrm{2}}/ 4 \mathrm{r^{2}.}}
S^{env}_{s,t}= \int_{s}^t 2\hat b_+^j(r,q_r) (\tilde \Sigma^{-1})_{jk}(r,q_r)\circ dq^k_r &-\int_{s}^t 2\hat b_+^j(r,q_r) (\tilde \Sigma^{-1})_{jk}(r,q_r)b^k_-(r,q_r)dr -\int_s^t\nabla\cdot b_-(r,q_r)dr,
ln(\frac{P_{subl}}{1bar})=8.514-\frac{458.4K}{T}-\frac{19870K^{2}}{T^{2}}+\frac{4.800\cdot10^{5}K^{3}}{T^{3}}-\frac{4.524\cdot10^{6}K^{4}}{T^{4}}(\beta)
\|Y\|_2 \le \delta^2 m + \delta^2 s^{2}_{k+1}( M) + \sqrt{C \log (\ell) \max\{1, \delta^2m \ell, \delta^3 m \ell^2 \}}\cr = o(\delta^2 s^2_{k}(M)) + \sqrt{C \log (\ell) \max\{1, \delta^2m \ell \}},
\hat{v}_{\mathrm{H , \pm}}( p ) = \frac{\hbar}{\hbar}{L}\frac{1}{p^{2}}\{p \frac{d}{d{\alpha}_{\pm p}}\mp ( 1 - \mathrm{N}^{2}) ( \frac{e_{+}\mathrm{L}}{4{\pi}{\hbar}})^{2}{\alpha}_{\mp p}\}.
L_1 \cong 2^{-1}A(0,\,0)\bot\cdots\bot 2^{-1}A(0,\,0)\bot 2^{-1}A(0,\,0)\,, L_1 \cong 2^{-1}A(0,\,0)\bot\cdots\bot 2^{-1}A(0,\,0)\bot 2^{-1}A(2,\,2\rho)\,.
h(x) \le \Bigl(g_{6+}(x)+\frac2{6^7}(3^7x-911)(54-2^7x)\Bigr)\Bigr|_{x=\frac{2627}{6305}}+ \frac1{10\cdot 6^7}- \eta &=-\frac{296152864180283626343198670531396721}{357807656810258627806425237458320688544000}<0,
|d_{Z^\alpha}(z,z_0)-d_{Z^\alpha}(z',z_0)| = |d_{Z}(z,z_0)^\alpha-d_{Z}(z',z_0)^\alpha| &\leq \alpha d_{Z}(z,z_0)^{\alpha-1}|d_{Z}(z,z_0)-d_{Z}(z',z_0)| &\leq d_{Z}(z,z_0)^{\alpha-1}d_{Z}(z,z') &= d_{Z^\alpha}(z,z_0)^{\frac{\alpha-1}{\alpha}}d_{Z^\alpha}(z,z')^{1/\alpha}.
g_{1}(\nabla^{^{M_1}}_{U}V,X) =g_{1}(V,\mathcal{T}_{U}\phi \mathcal{B}X+\hat{\nabla}_{U}\phi\mathcal{B}X)+g_{1}(V,\mathcal{T}_{U}\omega\mathcal{B}X+\mathcal{H}\nabla^{^{M_1}}_{U}\omega\mathcal{B}X) &-g_{2}(\pi_{\ast}(\mathcal{C}X),\pi_{\ast}(\mathcal{H}\nabla^{^{M_1}}_{U}\varphi V))+g_{1}(V,\varphi U)\eta(X).
s_k = O(e^{-\lambda_1k}e^{(\max\{\lambda_{i},\lambda_1 a\}+5\tau(1+\lambda_1))r(0,k)}) &= O(e^{-\lambda_1(1-\tau)r(0,k+1)}e^{(\max\{\lambda_{i},\lambda_1 a\}+5\tau(1+\lambda_1))r(0,k)}) &= O(e^{(\max\{\lambda_{i},\lambda_1 a\}-\lambda_1+6\tau(1+\lambda_1))r(0,k+1)}).
\begin{cases}\frac{dx}{dt}= \frac{dx}{dz}\frac{dz}{dt}= -2\pi i x(1 + e^{4\pi it}xy) \frac{dy}{dt}= \frac{dy}{dz}\frac{dz}{dt}= -2\pi i y(1 + e^{4\pi it}xy)\end{cases}\, .
\log\left(e^Xe^Y\right)= X+Y+\frac{1}{\mathrm{ad}_{Y}}\left(1-\frac{\exp\left(\mathrm{ad}_{X}\right)-1}{\mathrm{ad}_{X}}\frac{\mathrm{ad}_{X}+\mathrm{ad}_{Y}}{\exp\left(\mathrm{ad}_{X}+\mathrm{ad}_{Y}\right)-1}\right)[X,Y].
\Gamma_{a , s}= \tilde{\Gamma}_{s}\Gamma_{a , s}^{( o )}( l_{1 s}) \tilde{\Gamma}_{s}^{- 1}, \qquad \Gamma_{b , s}= \tilde{\Gamma}_{s}\Gamma_{b , s}^{( o )}( l_{2 s}) \tilde{\Gamma}_{s}^{- 1}= \Gamma_{a , s}({1 / 2}- l_{2 s}) \Gamma_{s}\qquad
|\partial_{t}^{n}D_{x}^{m}q(t,x)| &\leq N|x|^{-d-m}t^{-n+\alpha-1}(R^2+R^2 \ln R \cdot 1_{d=2}) & + N|x|^{-d}t^{-n+\alpha-1}\left[ R^{1/2}\cdot 1_{d=1}+ R \cdot 1_{d=2}+R^2 \ln R \cdot 1_{d=4}\right] 1_{m = 0},
\begin{array}{r c l}{}&{}&{\delta A_{m}= \sigma_{m n}A_{n}+ m \lambda_{m},}\\{}&{}&{}\\{}&{}&{\delta V_{m n}={\cal D}\sigma_{m n},}\\{}&{}&{}\\{}&{}&{\delta B_{m}= 2 \partial \lambda_{m}+ 2 \partial \sigma_{m n}A_{n}+ m \sigma_{m n}B_{n},}\\ \end{array}
\ln \mathrm{d e t}= \frac{1}{2}\int_{0}^{\infty}\frac{d t}{t}\left\{\mathrm{T r}\left( \! \e^{- P^{2}t}- \exp [ - ( D^{2}+ \frac{1}{2}\sigma F ) t \! \ ] \! \ \right) + \frac{\| F \|^{2}}{2 4 \pi^{2}}\right\}e^{- t m^{2}}.
\overline{{{I}}}_{2}={\frac{( 2 i e \pi )^{2}}{2}}\int{\frac{d \overline{{{k}}}}{( 2 \pi )^{2}}}\, \epsilon ( \overline{{{k}}}) \epsilon ( \overline{{{k}}}+ \overline{{{p}}}) \left[ n ( | \overline{{{k}}}| ) + n ( | \overline{{{k}}}+ \overline{{{p}}}| ) - 2 n ( | \overline{{{k}}}| ) n ( | \overline{{{k}}}+ \overline{{{p}}}| ) \right]
a(g)( f_1\otimes_s\cdots\otimes_s f_n ) =\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\langle g,f_i \rangle f_1\otimes_s\cdots\otimes_s \hat{f}_i\otimes_s\cdots\otimes_s f_n a^\dagger(g)( f_1\otimes_s\cdots\otimes_s f_n ) =\sqrt{n+1}g\otimes_s f_1\otimes_s\cdots\otimes_s f_n
xy= \sum_{(x)_2, (y)_2}\tau(x_{(0)},y_{(0)})\tau(x_{(2)},Sy_{(2)})y_{(1)}x_{(1)}, yx= \sum_{(x)_2, (y)_2}\tau(Sx_{(0)},y_{(0)})\tau(x_{(2)},y_{(2)})x_{(1)}y_{(1)}
\mathcal{D}_A =\left(\begin{array}{ccc}0 d_A^* d_A d_A^+ 0 0 d_A^* 0 0 \end{array}\right):\begin{pmatrix}a b c \end{pmatrix}\mapsto\begin{pmatrix}d_A^* b +d_A c d_A^+ a d_A^* a \end{pmatrix}
\mathbb{P}[w_nx_0 \in S_{x_0}(u_n^{-1}x_0,R_n) ] = \sum_{g\in G}\mathbb{P}[w_nx_0 \in S_{x_0}(u_n^{-1}x_0,R_n) \ | \u_n^{-1}=g ] \cdot \check{\mu}_n(g) &= \sum_{g\in G}\mu_n(S_{x_0}(gx_0,R_n)) \check{\mu}_n(g) & \le f(l(n)),
{\bf R}_i= \sum\limits_{l=1}^{3}\sum\limits_{m=1}^{2}p_{lm}{\bf G}_{il}{\bf v}_{lm}{\bf v}^{T}_{lm}{\bf G}^{T}_{il}+\sigma^{2}{\bf I}_2,\quad\forall i\in\mathcal{I},
\Bigl({\rm two\summations\in\ \eqref{b1}}\Bigr)=\sum_{k=1}^N \det\left((\lambda_i(t) I_N-H_{\alpha}(t)(\lambda_j(t) I_N-H_{\alpha}(t))_{k|k}\right) -2\sum_{\l-k>1}\det\left((\lambda_i(t) I_N-H_{\alpha}(t))_{k|\l}\right)\det\left((\lambda_j(t) I_N-H_{\alpha}(t))_{\l|k}\right).
g(\theta t + (1-\theta)s) = |\theta \nabla u + \theta t \nabla h + (1-\theta) \nabla u + (1-\theta) s \nabla h | &\le |\theta \nabla u + \theta t \nabla h |+ |(1-\theta) \nabla u + (1-\theta) s \nabla h | = \theta g(t) + (1-\theta) g(s),
\eqref{0-3term2}= \frac{1}{\sqrt{N}}\int dx_1dx_2 dy_1 dy_2 dy_3 \Big\{&\quad\ \bar u(y_1, x_1)v_N(x_1-x_2)\bar\phi(x_2)\bar u(y_2, x_2) \delta(x_1-y_3) &\quad\ \bar u(y_1, x_1)v_N(x_1-x_2)\bar\phi(x_2)\bar u(y_2, x_2) \bar p(x_1, y_3) \Big\}a_{y_1}a_{y_2}a_{y_3}.
\int_{0}^{\infty}\frac{x^{\alpha - 1}\cos{b x}}{x \sinh{c x}}d x = \frac{\Gamma ( \alpha )}{( 2 c )^{\alpha}}\left\{\zeta \left[ \alpha , \frac{1}{2}\left( 1 + \frac{i b}{c}\right) \right] + \zeta \left[ \alpha , \frac{1}{2}\left( 1 - \frac{i b}{c}\right) \right] \right\},
\sup_{\alpha\in \mathcal A}\inf_{\beta\in \mathcal B}\big\{-Tr\big(\sigma_{\alpha\beta}(x)\sigma_{\alpha\beta}^T(x)D^2u(x)\big)-I_{\alpha\beta}[x,u]+b_{\alpha\beta}(x)\cdot Du(x)+c_{\alpha\beta}(x)u(x)+f_{\alpha\beta}(x)\big\}=0,\,\, \,\,\mathbb R^n
{\cal Z}_{GU}={\displaystyle \int}{\cal D}(n^{a}, \mu^{\;\prime}) ~({\sf det}|-n^an_a|)^{1/2}~{\exp{\left(i{\displaystyle \int}dxdt \left [ - \frac{\displaystyle (\partial_1{\widetilde n}^a_{_{GU}}) (\partial_1{\widetilde n}_{a_{(GU)}})}{\displaystyle 4}+ |n|^2(\mu^{\prime\prime})^2\right]\right)}},
d\hat{H}_{p_1}(x)=\sum^{K_1}_{i=1}\hat{w}_i\delta_{t_i}(x)+\sum^{K_2}_{i=K_1+1}\hat{w}_i\,dM^{(1)}_i(x)+\sum^{K}_{i=K_2+1}\hat{w}_i\,dM^{(2)}_i(x),
T_{F,G,N}^L(1,\dots,1) \ll_{c,C,\varepsilon}\frac{1}{N^{d-m}}\sum\limits_{\substack{n_{m+1},\dots,n_d \in \mathbb{Z}\vert n_{m+1}\vert,\dots,\vert n_{d}\vert \leqslant N}}\widetilde{G}(\sum_{j=m+1}^d \mathbf{v_j}n_j),
\mathbb{H}(\eta,\overline{\eta}^{\diamond}) = [LH(\, \cdot \,,\overline{\eta}^{\diamond})](\eta) - [\overline{L}^{\diamond}H(\eta,\, \cdot\,)](\overline{\eta}^{\diamond}) - V^{\diamond}(\overline{\eta}^{\diamond})H(\eta,\overline{\eta}^{\diamond})
E^{\nu}[\frac{d\mu}{d\nu}(X_{0})|\mathcal{F}^{\mathcal{Y}}_{0,\infty}]=E^{\nu}[\frac{d\mu}{d\nu}(X_{0})|\bigcap_{n \geq 0}\mathcal{F}^{\mathcal{X}}_{n,\infty}\vee \mathcal{F}^{\mathcal{Y}}_{0,\infty}]~~~P^{\mu}~a.s.
(\bar{S}\bar{R})_{oo^{\prime}}={\frac{16\,i^{o+o^{\prime}+2}}{\pi^{2}o^{\prime}}}\sum_{e=2}^{\infty}{\frac{e^{3}}{(e^2-(o^{\prime})^{2})(e^2-o^{2})}}=\pm \infty \,.
\sum_{i=1}^{n-1}(v-v^{-1})^2\left(v^{2(n+1-i)}F_iK_{\varpi_{i+1}}E_iK_{\varpi_i}+ v^{-2(n-i)}F_iK_{-\varpi_i}E_iK_{-\varpi_{i+1}}\right)+ (v^2-v^{-2})^2v^2F_nK_{-\varpi_n}E_nK_{\varpi_n}.
\int_{\Omega}uv(x,t)dx+{\displaystyle\iint_{Q_{t}}}\left[ -uv_{t}+a(u,\nabla u)\nabla v+H(\nabla u)v+G(u)v\right]dxd\tau & =\int_{\Omega}u_{0}v(x,0)dx+{\displaystyle\iint_{Q_{t}}}\left( f_{0}v+F\nabla v\right) dxd\tau,
2 V^{- 1}< H > = \frac{1}{2}\int \frac{d^{2}p}{( 2 \pi )^{2}}\left[ G^{- 1}( p ) + \kappa^{2}G ( p ) + ( p^{2}+ 4 \kappa^{2}) [ G^{- 1}( p ) + \kappa^{2}G ( p ) ]^{- 1}\right]
\frac{1}{2\pi}\int_{-\infty}^\infty \Big(\log N -\sum_{n=0}^N \frac{2(2n+a+\frac12)}{(2n+a+\frac12)^2+(t+b)^2}\Big)\hat{g}(\frac12+it)dt &+ O\big(\int_{-\infty}^\infty\frac{1+|t+b|}{N}\Big|\hat{g}(\frac12+it)\Big|dt\big).
D_{\mu \nu}^{( 0 )}( p ) = \frac{1}{p^{2}+ \kappa^{2}}\left[ ( \delta_{\mu \nu}- \frac{p_{\mu}p_{\nu}}{p^{2}}) - \kappa \epsilon_{\mu \nu \lambda}\frac{p_{\lambda}}{p^{2}}\right] + \xi \frac{p_{\mu}p_{\nu}}{( p^{2})^{2}}
\epsilon^{i j k}\partial_{j}[ v e^{\phi}( \delta_{k l}( 1 + e^{- \phi}B_{i}^{1}B_{i}^{1}) - e^{- \phi}B_{k}^{1}B_{l}^{1}) B_{l}^{2}- E_{k}^{1}- v e^{\phi}\psi B_{k}^{1}] = 0 .
\mathbb G^b_{n,j}(\theta)-\mathfrak G^b_{n,j}(\theta)=\mathfrak G^b_{n,j}(\theta)\left(\frac{\sigma_{P,j}(\theta)}{\hat\sigma_{n,j}(\theta)}-1\right)=\mathfrak G^b_{n,j}(\theta)\eta_{n,j}(\theta),~\theta\in\Theta.
\langle\Psi,\Psi_c\rangle =-\|Qv^*\mathbf{n}\|^{-2}&\quad\times\bigl\langle G_{0,0}vT\Phi,G_{0,0}v(I-q_0^\dagger v^*G_{0,2}v)Qv^*\mathbf n\bigr\rangle &=0.
{\partial\widetilde{A}_b^{Mi}({\bf x})\over\partial A_c^{Mi}({\bf x})}=\delta_{bc}+{\partial\,\delta A_b^{Mi}({\bf x})\over \partial A_c^{Mi}({\bf x})}=\delta_{bc}-g f_{bd}^c\delta\Lambda_d^M({\bf x}).
= B^{\alpha}{\cal T}_{\alpha}+ \frac{1}{2}B^{\beta}B^{\alpha}U_{\alpha \beta}^{\gamma}\Pi_{\gamma}( - 1 )^{\varepsilon_{\beta}+ \varepsilon_{\gamma}}+ \Pi_{*}^{\alpha}\bar{\cal P}_{\alpha}( - 1 )^{\varepsilon_{\alpha}}- \Pi_{*}^{\beta}B^{\alpha}U_{\alpha \beta}^{\gamma}B_{\gamma}^{*}( - 1 )^{\varepsilon_{\beta}},
\frac{\lambda_3}{q^2}a_{1,0,0}=a_{0,0,0}+(q^3+q^2+q)a_{2,1,1}&\frac{\lambda_1}{q}a_{2,1,0}=(q^3+q^2)a_{2,1,1}+qa_{2,2,0}+a_{3,1,0}.
\cdot \frac{( a + b - \gamma ) !}{( b - a + \gamma ) !}{_3 F}_{2}\left\{\begin{array}{l}{{- a + \alpha , c + \gamma + 1 , - c + \gamma}}\\{{\gamma - a - b , b - a + \gamma + 1}}\\ \end{array}\biggr | 1 \right\}.
\begin{aligned}x_i{}={}\frac{U}{L}\sum\limits_{j=i}^{N-1}h_j g_i{}={}U f_i{}={}\Delta - \frac{U^2}{2L}\sum\limits_{j=0}^{i-1}h_j \Big(2 - h_j \frac{-\kappa}{1-\kappa}\Big) \end{aligned}
a_n =\frac{1}{n+1}\sum_{\lambda}\binom{n+k}{k}\binom{k}{k_1,k_2,...,k_n}\prod_{j=1}^n b_{j+1}^{k_j}&=\sum_{\lambda}\frac{1}{n+1}\binom{n+k}{k}\binom{k}{k_{d_1-1},k_{d_2-1},...,k_{d_r-1}}\prod_{i=1}^r b_{d_r}^{k_{d_r-1}}&=\sum_{\lambda}a_{\lambda}^{\ast},
\gamma^{( n )}= \gamma^{a_{1}\ldots a_{n}}= \gamma^{[ a_{1}}\gamma^{a_{2}}\ldots \gamma^{a_{n}]}= \frac{1}{n !}\sum_{p e r m^{\prime}s}( - 1 )^{p}\gamma^{a_{1}}\ldots \gamma^{a_{n}}
Z({\bf p})=\frac{2mM\left[\sqrt{{\bf p}^2+m^2}+\sqrt{{\bf p}^2+M^2}\right]}{\sqrt{{\bf p}^2+m^2}\sqrt{{\bf p}^2+M^2}\left[\sqrt{{\bf p}^2+m^2}+\sqrt{{\bf p}^2+M^2}+E\right]}
\hat f(k) = \frac{2}{\sqrt2\pi}\frac{1}{k}\Gamma(b +1)(a^2+k^2)^{-\frac{1}{2}(b+1)}\sin[(b+1)\tan^{-1}(\frac{k}{a})] = \times\frac{1}{k}(a^2+k^2)^{-\frac{1}{2}(b+1)}\sin[(b+1)\tan^{-1}(\frac{k}{a})]
\log \frac{\pi(x)}{m_t^{(n)}(x)}= - e^{-\frac{\sigma^2}{2}t}\log \frac{m_0(x)}{\pi(x)}&- \int_0^t \frac{\sigma^2}{2}e^{-\frac{\sigma^2}{2}(t-s)}\left( - \frac{2}{\sigma^2}\frac{\delta F}{\delta m}(m_s^{(n-1)},x) + \operatorname{KL}(m_s^{(n-1)}|\pi) \right) ds \,.
\delta S_{5}= \frac{\sqrt{2}}{8 ( 2 \pi )^{3}\kappa^{2}T_{5}}\frac{1}{6 \sqrt{2}}\frac{\kappa^{2}}{\lambda^{2}}\int_{M_{D}^{1 1}}d ( \hat{I}_{2}^{1}\wedge X_{8}) = - \frac{1}{4 8 ( 2 \pi )^{3}\lambda^{2}T_{5}}\int_{M^{1 0}}\hat{I}_{2}^{1}\wedge X_{8}
\left( \sum_{j_1,\ldots,j_s=1}^{n}|T_s(e_{j_1},\ldots, e_{j_s})|^{\frac{1}{1-\left|\frac{1}{\mathbf{p}}\right|_{\leq s}}}\right)^{1-\left|\frac{1}{\mathbf{p}}\right|_{\leq s}}\leq 2^{(s-1)\left[1-\left|\frac{1}{\mathbf{p}}\right|_{\leq s}\right]}\Vert T_s\Vert
\theta'_j =\frac{\sum_{i=0}^{\ell-1}(a/b)^{2ip^{r+j}}-\ell}{p^r}\cdot\left(\frac{a}{b}\right)^{-(\ell-1)p^{r+j}}= \frac{\sum_{i=0}^{\ell-1}\left( (a/b)^{2ip^{r+j}}-1 \right)}{p^r}\cdot\left(\frac{a}{b}\right)^{-(\ell-1)p^{r+j}}.
\Vert f\Vert^{2}\leq \Vert T^{\dagger}_{W}\Vert^{2}.\Vert T_{w}f\Vert^{2}=\Vert T^{\dagger}_{W}\Vert^{2}.\sum_{i\in I}v_{i}^{2}\langle \pi_{W_{i}}C^{\prime}f,\pi_{W_{i}}Cf \rangle^{2}
P_t(\lambda) = \sum_{i=0}^{\lfloor \frac{t}{2}\rfloor}(-1)^i \cdot \Theta (d^i) \cdot \lambda^{t-2i}= \sum_{i=0}^{\lfloor \frac{t}{2}\rfloor}(-1)^i \cdot \Theta (d^i) \cdot \Theta(d^{\alpha(t-2i)}) = \sum_{i=0}^{\lfloor \frac{t}{2}\rfloor}(-1)^i \cdot \Theta (d^{t\alpha + i(1-2\alpha)}).
((\partial_{1,0}^{\gamma})^{2}\eta)(u_{0},\ldots,u_{p+1}) & =-\sum_{0\leq i<j\leq p+1}(-1)^{i+j}L_{\operatorname{Curv}^{\gamma}(u_{i},u_{j})}\eta(u_{0},u_{1},\ldots,\hat{u}_{i},\ldots,\hat{u}_{j},\ldots,u_{p+1})
\frac{dT_{E}}{dt}=[\mu(\gamma\frac{\Delta_{h}T}{L}+\alpha\frac{\Delta_{v}T}{H}+\frac{\overline{\omega}L\Delta_{v}T}{H^{3}g^{\prime}\rho_{0}}-r^{\prime})-\frac{\overline{\omega}}{H}]T_{E}+\frac{\overline{\omega}\Delta_{v}T}{H^{2}}h_{W}
2\sum_{\mu = 1}^\infty{\frac{{H_\mu}}{{\mu^{2s + 1}}}}= ( - 1)^{s - 1}\zeta (s + 1)^2 + 2\sum_{i = 0}^{s - 2}{( - 1)^i \zeta (i + 2)\zeta (2s - i)}\,.
\mbox{Problem 4:}~~\max_{\boldsymbol{P}}~~ \mathbb{E}\left[\sum_{j=1}^K \ln\left(1+\frac{h_{j}P_j}{\sigma^2+\sum_{i=1,i\neq j}^K h_{i}P_i}\right)\right], \mbox{s.t.}~~ P_i\ge 0, ~\forall i &P_i \le P_i^{av},~\forall i &\mathbb{E}\left[\sum_{i=1}^K g_iP_i\right] \le I.
\left ( h^{J,j}_{0,n}, h^{J,j}_{1,n}\right ) = \left ( h^{J',j}_{0,n}, h^{J',j}_{1,n}\right ) + \sum_{k = J+1}^{J'}\left ( \lambda_{j,n}^{(N-2)/2}U^k_{L,n}(t_{j,n}, \lambda_{j,n}\cdot + x_{j,n}), \lambda_{j,n}^{N/2}\partial_t U^k_{L,n}(t_{j,n}, \lambda_{j,n}\cdot + x_{j,n}) \right ) .
\frac{E_V}{A}=\frac{\mu}{2}\sum_{n=1}^{\infty}\frac{1}{\Gamma(\frac{s}{2})}\int_{0}^{\infty}dt \,t^{\frac{s}{2}-1}e^{-(\frac{n\pi}{a\mu})^2t}\int_{-\infty}^{\infty}\frac{dk_ydk_z}{(2\pi)^2}e^{-[(\frac{k_y}{\mu})^2+(\frac{k_z}{\mu})^2]t}\rfloor_{s=-1}\,.
\tilde{X}(u, u_x, \dots)=\sum_{l\geq 1}\epsilon^{l-1}X_l(u)\left(\frac{\partial F_0(u)}{\partial u}u_{(l)}+\sum_{k\geq 1}\epsilon^k a_k(u)u_{(k+l)}+\mathrm{NQ1}\right)+\mathrm{NQ2},
E_\lambda^{G(\omega)}\zeta_t(\textbf{0}) =e^{-t}\sum_{y\in Z^d}e^{tB_\omega}(0,y) &=e^{-t}\sum_{n=0}^{+\infty}\frac{t^n}{n!}\big[\sum_{y\in Z^d}B_\omega^n(0,y)\big].
g(A) =\frac{(m(2^n-1)-1)(m(2^n-1)+d-1)}{2}+\frac{m(m-1)(2^n-1)}{2}+ m(2^{n-1}n-1) &=m(2^{n-1}n-1)+\frac{1}{2}(m^2(2^n-1)^2+(m^2+dm-3m)(2^n-1)-d+1).
2 \kappa^{2}e^{- 1}{\cal L}= R +{\frac{1}{2}}m^{2}( T^{2}- 2 T_{i j}T^{i j}) - T r ( P_{\mu}P^{\mu}) -{\frac{1}{2}}( V_{I}{}^{i}V_{J}{}^{j}F_{\mu \nu}^{I J})^{2},
\partial_t(r e)- \partial_r(r m) =0, & \partial_{t}(rm) - \partial_r (re) = - \frac{1}{2}\psi_t^2 + \frac{1}{2}\psi_r^2 + \frac{F(\psi)}{r^2}- \frac{2f(\psi)}{r}\psi_r =: L,
\mathbf{I}_{m_\textbf{s}\rightarrow m_\textbf{i}}[n_\textbf{i}] =\overset{I_{\textbf{s}\rightarrow\textbf{i}(l)}}{\overbrace{\frac{\sigma_{\mathbf{d}_\mathbf{s}}^2}{T}\sum\limits_{\tau=-KT+\frac{T}{2}}^{\frac{T}{2}}\underset{I_{\textbf{s}\rightarrow\textbf{i}}(l,\tau)}{\underbrace{\left|\int\limits_0^Tg(t-\tau)e^{j2\pi \frac{l t}{T}}dt\right|^2}}}}
S_{{\underline{{0}}}{\underline{{0}}}}\, = \, - \frac{2 e^{4{\cal U}}}{( 8 \pi )^{2}r^{4}}\mathrm{I m}{\cal N}_{\Lambda \Sigma}( p^{\Lambda}p^{\Sigma}+ \ell ( r )^{\Lambda}\ell ( r )^{\Sigma})
\mathit{t e n s o r \_c o e f f}: = R \,{_{\mu 1}}\,{_{\mathit{i 1}}}\,{_{\mathit{i 2}}}\,{_{\nu 1}}\, R \,{_{\mu 2}}\,{_{\mathit{i 3}}}\,{_{\mathit{i 4}}}\,{_{\nu 2}}
\phi_{\Phi_k, \beta}(x, y)= \frac{1}{2\pi i}\int_{b-i U}^{b+i U}\left(\sum_{n\le x}\left(\frac{\Phi_k(n)}{n^\beta}\right)^{-z}\right)\frac{y^z}{z}\,dz+O\left( \frac{y^{b}}{U}\sum_{n \leq x}\Phi_k(n)^{-b}n^{b\beta}\right).