harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\Phi_{K_{p,q}}(B_0) = B_0 + b_1 + b_2 + a_1 + a_2, \Phi_{K_{p,q}}(B_1) = B_1 + b_1 + b_2 + a_2 + \varepsilon_p c_2 + \varepsilon_q d_1, \Phi_{K_{p,q}}(B_2) = B_2 + b_2 + a_1 + a_2 + \varepsilon_p c_2, \Phi_{K_{p,q}}(B_3) = B_3 + b_2 + \varepsilon_p c_2, \Phi_{K_{p,q}}(B_4) = B_4 + a_2 + \varepsilon_p c_2 + \varepsilon_q d_1
a_{ij}^{(2k)}=\begin{cases}(-1)^{k+i}\dfrac{m^{1\dots k}_{1\dots i-1i+1\dots kj}}{m_{k}}\medskip \mbox{if}1\leq i\leq k, k+1\leq j\leq n \dfrac{m^{1\dots ki}_{1\dots kj}}{m_{k}}\mbox{if}k+1\leq i\leq m,k+1\leq j\leq n\end{cases}.
\int_{- \infty}^{\infty}\! \! \mathrm{d}x \, \frac{- \nu_{2}+ \alpha \left[ \nu_{1 +}^{2}- ( u^{( 0 )}( x ) )^{2}\right] - \mu_{2}\left[ u^{( 0 )}( x ) - v^{( 0 )}( x ) \right]}{\cosh^{2}\left[ \nu_{1 +}\, m \, ( x - z_{u}) \right]}= 0
\frac{1}{M_p^2 p^2}\left( 1 + \langle T_{CFT}\left( p\right) T_{CFT}\left( -p\right)\rangle \frac{1}{M_p^2p^2}\right) + \frac{1}{M_w^8}\langle T_{CFT}\left( p\right) T_{CFT}\left( -p\right)\rangle ,
\Gamma_{A^{\mu}A^{\nu}}( p , - p ) \equiv \Gamma_{\mu \nu}( p , - p ) = ( \eta_{\mu \nu}- \frac{p_{\mu}p_{\nu}}{p^{2}}) \Gamma^{T}( p^{2}) + \frac{p_{\mu}p_{\nu}}{p^{2}}\Gamma^{L}( p^{2})
g(\zeta) = \tan^{d-3}\bigg(\frac{\zeta}{2}\bigg)\, \frac{d}{d\zeta}\int_\zeta^\pi \frac{Q(\theta)\, \cos^{d-3}(\theta/2) \sin\theta \, d\theta}{\sqrt{\cos \zeta-\cos\theta}}, \alpha \leq \zeta \leq \pi.
f_{X_n}\!(x) = \frac{1+q_n^{2}}{2q_n\theta_n}e^{-\frac{\left( 1+q_n^{2}\right)^{2}}{4q_n^{2}\theta_n}x}I_{0}\!\left( \frac{1-q_n^{4}}{4q_n^{2}\theta_n}x\right),x\geq 0
A H_{0}= c_{H}\, K_{H}\, \int{\frac{d^{2}a_{1}d^{2}a_{2}d^{2}a_{3}\, | a_{4 5}\, a_{4 6}\, a_{5 6}|^{2}}{T^{5}\prod_{i < j}^{6}| a_{i j}|^{2}}}\prod_{l = 1}^{4}{\frac{d^{2}z_{l}( r - z_{l})}{y ( z_{l})}}I ( r ) \bar{F}( \bar{a}, \bar{z}) ,
\frac{d\sigma}{d\Omega}=(\frac{\alpha}{4E})[csc^{4}\frac{\chi}{2}+sec^{4}\frac{\chi}{2}+\frac{Acos(\frac{\alpha}{\hbar\nu}lntan^{2}\frac{\chi}{2})}{sin^{2}\frac{\chi}{2}cos\frac{\chi}{2}}]^{2}
\lim_{x \uparrow 0}\frac{\partial}{\partial x}\hat{m}(s,x,s+\epsilon,\phi'(s+\epsilon)) =\int_{0}^{\infty}e^{\phi'(s+\epsilon)y}\lim_{x \uparrow 0}\frac{\partial}{\partial x}m(s,x,s+\epsilon,y)dy
\frac{d^{v}}{dt^{v}}\left\{\mathcal{F}\left( t,k;\lambda \right) \right\}=\frac{\left( -1\right)^{v}\left( k\right)^{\left( v\right)}\lambda{^{2v}}}{2^{v}}\mathcal{F}\left( t,k+v;\lambda \right)
F_{n+(k-1)}^{(k)}= \sum_{j=0}^{\lfloor n/(k+1)\rfloor}(-1)^j\,\frac{(n-jk)+j+\delta_{n,0}}{2(n-jk)+\delta_{n,0}}\, \binom{n-jk}{j}\, 2^{n-j(k+1)}\,.
U_t+(A U)_x+A^T U_x =0 A = \begin{bmatrix}a_{11}a_{12}begin{align*}0.05cm] a_{21}a_{22}begin{align*}0.05cm] \end{bmatrix}= \begin{bmatrix}\alpha \frac{U_2}{\sqrt{U_1}}(1-3 \alpha) \sqrt{U_1}begin{align*}0.05cm] 2 \alpha \sqrt{U_1}\frac{1}{2}\frac{U_2}{\sqrt{U_1}}begin{align*}0.05cm] \end{bmatrix}
2\left\{3{\rm Cl}_2(\theta_7)-3{\rm Cl}_2(2\theta_7)+{\rm Cl}_2(3\theta_7)\right\}=Z_7=7\left\{{\rm Cl}_2\left(\frac{2\pi}{7}\right)+{\rm Cl}_2\left(\frac{4\pi}{7}\right)-{\rm Cl}_2\left(\frac{6\pi}{7}\right)\right\}
\int (t)\,dt\lesssim \left( \|\nabla \xi\|_\infty + \|\nabla^2 \xi\|_\infty\right)\times \left( \iint \big|G_{h/2}\ast \partial_t^h u^h\big|^2 dx \,dt\right)^{\frac12}\left( \iint \Big(h + \int G_h(z) \big|u^{h}(x-z)-u^h(x)\big|^2 dz \Big)dx\,dt \right)^{\frac12}.
\delta{\cal F}= - \frac{i}{2 4}{\bar{D}}^{4}\left( D^{\alpha{\mathbf j}}L_{{\mathbf i}{\mathbf j}}D_{\alpha}^{\mathbf i}{\cal F}\right) - 3 \sigma{\cal A}{\cal A}_{D}- \frac{i}{4 8}{\bar{D}}^{4}\left[ ( H -{\bar{H}}){\bar{\cal A}}{\cal A}_{D}\right] ,
\langle Q ~ Q ~ Q \rangle_{*}- \frac{3}{\hat{\beta}}\langle Q ~ \frac{\Delta Q}{\Delta \phi}\rangle_{*}= \mu^{n - 6}\left( \frac{n - 6}{\hat{\beta}}\right)^{3}L_{Y}\times \ \mathrm{l o c a l \s t r u c t u r e}+ ( \mathrm{f i n i t e}) .
\frac{c_{1}^{\prime}(s)c_{2}^{\prime\prime}(s)-c_{2}^{\prime}(s)c_{1}^{\prime\prime}(s)}{(c_{1}^{\prime}(s)^{2}+c_{2}^{\prime}(s)^{2})^{3/2}}and\frac{c_{2}^{\prime}(s)}{c_{1}(s)\sqrt{c_{1}^{\prime}(s)^{2}+c_{2}^{\prime}(s)^{2}}}
-\frac kna_{nm}d_{n+1\;m+3}+ \frac{n+1-k}nb_{nm}c_{n+1\,m-3}-\frac{n-k}{n-1}c_{nm}b_{n-1\,m+3}+ \frac{k-1}{n-1}d_{nm}a_{n-1\,m-3}= \frac{m-n-1+2k}2.
V ( z_{i}, \bar{z}_{i}) = \epsilon_{\mu \lambda}{D^{\lambda}}_{\nu}: \partial X^{\mu}( z_{i}) \exp ( i k_{i}. X ( z_{i}) ) : : \bar{\partial}X^{\nu}( \bar{z}_{i}) \exp ( i k_{i}. D . X ( \bar{z}_{i}) ) : .
\dot{M}+ \frac{1}{4 M^{2}}\frac{a [ k^{2}; Q ]}{d ( M )}+ \frac{1}{4 M^{4}}\frac{b [ k^{2}; Q ]}{d ( M )}+ \frac{c [ k^{2}; Q ]}{d ( M ) M^{6}}= 0 ,
t_\mathrm{s,1}= \Delta t_{\mathrm{s}_{1,1}}+ \Delta t_{\mathrm{s}_{1,2}}+ \Delta t_{\mathrm{s}_{1,3}}&=g_\mathrm{DL}^{-1}l_\mathrm{r}+ t_\mathrm{w}+ g_\mathrm{UL}^{-1}l_\mathrm{r}+ g_\mathrm{DL}^{-1}l_\mathrm{b}.
\mathcal{L}_{n+1}\left[\begin{array}{c}u_{\alpha}u_{\beta}\end{array}\right]=\left[\begin{array}{cc}I_{N_n}\mathcal{L}_{\alpha,\beta}\mathcal{L}_{\beta,\alpha}\mathcal{L}_{\beta,\beta}\end{array}\right]\left[\begin{array}{c}u_{\alpha}u_{\beta}\end{array}\right]=\lambda_i(n+1)\left[\begin{array}{c}u_{\alpha}u_{\beta}\end{array}\right],
\tilde{G}_{\sigma}^{( n )}( \zeta ) = ( \sqrt{- 1})^{- n}\sum_{k_{0}, k_{1}, \cdots k_{2 n}}t_{k_{1}}^{k_{0}}( u_{1}- u_{0}) \otimes \cdots \otimes t_{k_{2 n}}^{k_{2 n - 1}}( u_{2 n}- u_{0}) \tilde{F}_{\sigma}^{( n )}( \zeta )^{k_{0}k_{1}\cdots k_{2 n}}.
\sum_{\vec{z}+q\vec{y}\in P\mathcal{B}}e(\vec{\beta}\cdot\vec{f}(\vec{z}+q\vec{y}))= I(\mathcal{B},P^d\vec{\beta})\frac{P^n}{q^n}+O\left(C\widetilde{C}^{r-1}\frac{P^{n+\eta-1}}{q^n}\right).
l_2(y) ={\begin{cases}3 (y_1, y_2)=(1,1), (1,2), 4 (y_1, y_2)=(2,1), (2,3), (3,1), (3,3), \end{cases}}k_2(y) ={\begin{cases}2 (y_1, y_2)=(1,1), (2,1), (3,1), 3 (y_1, y_2)=(1,2), (2,3), (3,3), \end{cases}}
\mathcal{K}^{p,q}(\mathcal{U},U_{0}):=\left\lbrace T\in\mathcal{K}^{p,q}(\mathcal{U})\,\mid\, T_0=0\right\rbrace=\mathcal{K}^{p,q}(U_{1})\oplus\mathcal{K}^{p,q-1}(U_{01}).
\begin{array}{rcl}V(x,\psi) := \frac{1}{2 \nu \omega_N}\int_\Gamma ( \frac{1}{(n-2)|x-y|^{N-2}}+ \frac{(x-y)(x-y)^T}{|x-y|^N}) \psi(y) \; d\sigma(y); Q(x,\psi) := \frac{1}{\omega_N}\int_\Gamma ( \frac{(x-y)}{|x-y|^N}) \psi(y) \; d\sigma(y). \end{array}
\mathbb{E}[(\mathbf{S}_1-\hat{\mathbf{S}}_1(\gamma))(\mathbf{S}_1-\hat{\mathbf{S}}_1(\gamma))^T] &=\mathbb{E}[(\mathbf{S}_1-\mathbf{S}_1(\gamma))(\mathbf{S}_1-\mathbf{S}_1(\gamma))^T] &=\mathbf{\Theta}_{1}(\gamma) &\preceq\mathbf{D}_1,
\rho_{\nu}( x ) = \rho_{\nu^{\prime}}( x ) \frac{q^{C ( \nu ) / 2}+ \epsilon ( \nu ) \epsilon ( \nu^{\prime}) x q^{C ( \nu^{\prime}) / 2}}{x q^{C ( \nu ) / 2}+ \epsilon ( \nu ) \epsilon ( \nu^{\prime}) q^{C ( \nu^{\prime}) / 2}}, \forall \nu \neq \nu^{\prime}.
Z[J]=\int D\phi e^{i\int d^{4}x(\frac{1}{2}\partial^{\mu}\phi\partial_{\mu}\phi-\frac{m^{2}}{2}\phi^{2}-\frac{\lambda}{4!}\phi^{4}+J\phi)}=Z[0]\sum_{n=0}^{\infty}\frac{1}{n!}\langle\Omega|T\{\phi(x_{1})\cdot\cdot\cdot\phi(x_{n})\}|\Omega\rangle.
A_{i,k}:= \left[(1 + 2 i) a - 2 i, (1 +2i) a - 2 i + (6i+2) \right]_2, B_{i,k}:= \left[(2 + 2 i) a - 2 (i+1), (2 + 2 i) a - 2 (i+1) + (6i+6) \right]_2,
C_{\mu \nu}^{- 1}= - \frac{1}{m^{2}}\left( \begin{array}{c c c c}{{0}}&{{\partial_{1}^{x}}}&{{\partial_{2}^{x}}}&{{- 1}}\\{{\partial_{1}^{x}}}&{{0}}&{{- m}}&{{0}}\\{{\partial_{2}^{x}}}&{{m}}&{{0}}&{{0}}\\{{1}}&{{0}}&{{0}}&{{0}}\\ \end{array}\right) \delta^{2}( x - y ) .
\begin{aligned}a_1 = u\{2fgu+(3f^2-g^2)v\}\{(3f^2+g^2)u^2+(12f^2-4g^2)uv & -(27f^2+24fg+9g^2)v^2\}, b_1 = v\{(3f^2-g^2)u-6fgv\}\{(9f^2+8fg+3g^2)u^2 & +(12f^2-4g^2)uv-(9f^2+3g^2)v^2\}, b_2 = (fu+gv)(gu+3fv)\{(9f^2-4fg-g^2)u^2 & \quad+(12f^2-24fg+4g^2)uv-(9f^2+12fg-9g^2)v^2\}.\end{aligned}
\Phi^{\pm\infty}(x,t,k)=\Delta_{R,L}(k)e^{-i(X_{R,L}(k)x+\Omega_{R,L}(k)t)\sigma_3}, \Phi^{\pm\infty}:\mathbb{R}\times[0,+\infty)\times(\mathbb{C}\backslash[-C,C])\rightarrow\mathbb{C},
p_{v_0}- \sum_{s(e) =v_0, r(e) \notin H_\delta}s_es_e^* = p_{v_0}- \left( \sum_{s(e)= v_0, r(e) \notin H}s_es_e^* + \sum_{s(e) = v_0, r(e) \notin H_\delta \setminus H}s_es_e^* \right) & = \left(p_{v_0}- \sum_{s(e)v_0, r(e) \notin H_\delta}s_es_e^* \right) - \sum_{s(e) = v_0, r(e) \notin H_\delta \setminus H}s_es_e^* \notin h(\delta)
D_P(A) =\begin{pmatrix}0 D_0(A) -D^*_0(A) 0 \end{pmatrix}\oplus \begin{pmatrix}0 -D_1(A) D^*_1(A) 0 \end{pmatrix}, \;\epsilon_P =\begin{pmatrix}1 0 0 -1\end{pmatrix}\oplus \begin{pmatrix}-1 0 0 1\end{pmatrix}\colon \Gamma(E_0(P) \oplus E_1(P)) \to \Gamma(E_0(P) \oplus E_1(P)).
\beta_{0,2}:=\lim_{r\to0}\lim_{k\to+\infty}\frac{1}{2\pi}\int_{B_{r}}\left|x\right|^{2\alpha_{k,2}}U_{k}e^{u_k(x)}=\frac{\beta_0(4(\alpha_1+1)-a\beta_0)}{4((\alpha_1+1)-a(\alpha_2+1))}
\sup_{|\alpha|=k}\big\|D^{\alpha}\dot{\Delta}_j f\big\|_{L^{\infty}\left(\mathbf{R}^{n}\right)}\leq C 2^{j(k+\frac{n}{p})}\|\dot{\Delta}_jf\|_{L^{p, \infty}\left(\mathbf{R}^{n}\right)},
L_1(t) := \{(x_1, x_2) \in \mathbb{Z}^2 \mid (x_2 \le t) \wedge (3/5 x_2 \le x_1 \le 5/8 x_2)\}, L_2(t) := \{x_2 \in \mathbb{Z}\mid 0 \le x_2 \le t\}, L_3(t) := \{x_2 \in L_2(t) \mid \# \{x_1 \in \mathbb{Z}\mid (x_1, x_2) \in L_1(t)\}< m\}.
D_n : = \frac{\sum_{i = 1}^n b_{i,n}(1)b_{i,n}(2)}{\sqrt{\sum_{i = 1}^n a_{i,n}^2(1)}\sqrt{\sum_{i = 1}^n a^2_{i,n}(2)}}- \frac{\sum_{i = 1}^n b_{i,n}(1) b_{i,n}(2)}{\sqrt{\sum_{i = 1}^n b_{i,n}^2(1)}\sqrt{\sum_{i = 1}^n b^2_{i,n}(2)}}.
D(P|_{\mathcal{F}\vee \mathcal{G}}\|Q|_{\mathcal{F}\vee \mathcal{G}}) =D(P|_{\mathcal{F}}\|Q|_{\mathcal{F}})+D(P|_{\mathcal{G}}|\mathcal{F}\|Q|_{\mathcal{G}}|\mathcal{F})
Y_s^{t,\eta}\ = \Y_{T_0}^{t,\eta}+ \int_s^{T_0}F(r,\mathbb W_r^{t,\eta},Y_r^{t,\eta},Z_r^{t,\eta}) dr - \int_s^{T_0}Z_r^{t,\eta}dW_r, t \leq s \leq T_0.
M'_s (t) = \sigma(s, X(s)) + \int_0^t U(s, r) dr + \int_0^t V(s, r) dW(r) &+\int_0^t \nabla_x b(r, X(r)) M'_s(r) dr + \int_0^t \nabla_x \sigma (r, X(r)) M'_s(r) dW(r),
(T\varphi_t)^* \tilde \Omega_L =(T\varphi_t)^*(\pi_{TQ})^* \Omega=(\pi_{TQ}\circ T\varphi_t)^* \Omega=(\varphi_t \circ \pi_{TQ})^* \Omega=(\pi_{TQ})^*\varphi_t^* \Omega = \tilde \Omega_L,
\Delta R_{k}= \mathbb{E}_{\mathbf{H}, \mathbb{W}}\{R_{P,k}- R_{Q,k}\}& \leq \Delta R = \log_2\left( 1+ c P ~ 2^{-\frac{B}{n_T-1}}\right) + \frac{\log_2(e)}{n_T-1}\sum_{i=1}^{n_T-1}\beta\left( n,\frac{i}{n_T-1}\right),
\tilde{L}^{(m,n)}_1(t+\delta) =2 \lambda \sum_{j=0}^{n-1}\tilde{\mu}^{(m,n)}_1(t + \delta + j \delta) =2 \lambda \sum_{j=0}^{n-1}\tilde{\mu}^{(m,n)}_1(t + j \delta) =\tilde{L}^{(m,n)}_1(t).
\sum\limits_{l = 1}^N |{\bf{w}}_r^T{{\bf{g}}_l}{\lambda_l}{{\bf{h}}_l^T{{\bf{w}}_{\bf{t}}}}|^2\leq \lambda_n^2\sum\limits_{l = 1}^N\beta_l^2\alpha_l^2\leq \lambda_n^2\sum\limits_{l = 1}^N\beta_l^2\sum\limits_{l = 1}^N\alpha_l^2=\lambda_n^2,
D_{ii}u_\epsilon(x) = \int_{\mathbf{R}^n \cap \{y ; x-y \not\in E_u\}}D_{ii}u_0(x-y)\rho_\epsilon(y) \dy \quad\quad+\int_{\{y; x-y \in E_u\}}\rho_\epsilon(y) \d\mu^{ii}, \geq \int_{\mathbf{R}^n \cap \{y ; x-y \not\in E_u\}}D_{ii}u_0(x-y)\rho_\epsilon(y) \dy
\begin{aligned}\left\vert\sum\limits_{j}\mu_{j}M_{\Phi}(I^{loc}_{\alpha}(b_{j}))(x)\right\vert &\leq\sum\limits_{j}\mu_{j}\vert M_{\Phi}(I^{loc}_{\alpha}(b_{j}))(x)\vert\chi_{P_{j}^{*}}(x)+\sum\limits_{j}\mu_{j}\vert M_{\Phi}(I^{loc}_{\alpha}(b_{j}))(x)\vert\chi_{(P_{j}^{*})^{c}}(x) &=:\uppercase\expandafter{\romannumeral1}+\uppercase\expandafter{\romannumeral2}.\end{aligned}
K \cdot \kappa(r_2) = K E^{3t}K^{-1}= q^{6t}E^{3t}= E^{3t}= \kappa(r_2) F \cdot \kappa(r_3) = 0 = -\alpha q^2 K^{3s_0}+ q^2 \alpha K^{3s_0}= \kappa(r_5) + q^2 \kappa(r_4).
[\zeta_{1,\varepsilon}, \mathcal{T}_{a^+,a^-}](c^k) =\zeta_{1,\varepsilon}((c^k)^{aps})-(\zeta_{1,\varepsilon}c^k)^{aps}&=\sum_{n\in \mathbb{Z}}\Big[(\sum_{|j|\leq N_1}\varsigma_j\mbox{sign}(n-j)p^k_{n-j})-(\sum_{|j|\leq N_1}\mbox{sign}(n)\varsigma_jp_{n-j})\Big]e^{int}&=\sum_{|n|\leq N_1}\pm 2(\sum_{|j|\leq N_1}\varsigma_j p_{n-j})e^{int}.
f( ( \mathbf{a}\cdot \mathcal{B})_{K}- (\mathbf{a}\cdot \mathcal{B})_{K^{*}}) = \sum_{j=0}^{|K^{*}|}\sum_{|J| = j}|a_{J}| f^{(j)}( (\mathbf{a}\cdot \mathcal{B})_{J^{*}})
\mathbb E_x\left[e^{-p\int_{0}^{e(q)}\rm{\bf{1}}_{\{X^1_s \leq -b\}}ds}\rm{\bf{1}}_{\{X^1_{e(q)}>-y\}}\right]-\mathbb P_x\left(X^1_{e(q)}> -y\right)=J_2(-b-x;b-y),
\Omega_{\mathbf{m}}^{\pm}(\lambda,b) =\frac{1-b^{2}}{2b}\Lambda_{1}(\lambda,b)+\frac{1}{2}\Big(\Omega_{\mathbf{m}}(\lambda)-\Omega_{\mathbf{m}}(\lambda b)\Big) &\quad\pm\frac{1}{2b}\sqrt{\Big(b\big[\Omega_{\mathbf{m}}(\lambda)+\Omega_{\mathbf{m}}(\lambda b)\big]-(1+b^{2})\Lambda_{1}(\lambda,b)\Big)^{2}-4b^{2}\Lambda_{\mathbf{m}}^{2}(\lambda,b)},
{\cal S}_{N C C S}= \frac{k}{4 \pi}\int d t \, \epsilon^{\mu \nu \xi}\, 2 \theta \, \mathrm{T r}\left( A_{\mu}[ \partial_{\nu}, A_{\xi}] + \frac{2}{3}A_{\mu}A_{\nu}A_{\xi}\right) ~ .
\lim_{r\rightarrow\infty}\int_{S^{n-1}_1}g(\nabla R_g,\nu)r^{n-1}d\omega_{n-1}=\lim_{r\rightarrow\infty}\int_{S^{n-1}_1}\partial_i\left(\partial_k\partial_jg_{kj}- \partial_j\partial_jg_{kk}\right)\nu^e_ir^{n-1}d\omega_{n-1}.
Rm(X_1, X_2, X_1, X_2) = \frac{1}{f^2(r)}- \Big(\frac{f'(r)}{f(r)}\Big)^2 Rm(X, Y, X, Y) = \frac{f^2}{h^4}- \frac{f'\cdot h'}{f\cdot h}Rm(Y_1, Y_2, Y_1, Y_2) = \frac{4}{h^2}- \frac{3f^2}{h^4}- \Big(\frac{h'}{h}\Big)^2 Rm(\partial r, X, \partial r, X) = -\frac{f''}{f}\ , Rm(\partial r, Y, \partial r, Y)= -\frac{h''}{h}
\mathcal F_\alpha^V(\rho)= \int_\Lambda \Bigl[f(\rho(u))-f(\bar\rho_{\alpha,V}(u))-f'(\bar\rho_{\alpha,V}(u))\bigl(\rho(u)-\bar\rho_{\alpha,V}(u)\bigr)\Bigr]\;\!\mathrm du,
\begin{gathered}W_{z, z', a, b|N}(x) = (x + \epsilon)\frac{\Gamma(x + 2\epsilon)\Gamma(x + a + 1 )}{\Gamma(x + b + 1)\Gamma(x + 1)}\times\frac{1}{\Gamma(z - x + N)\Gamma(z' - x + N)\Gamma(z + x + N + 2\epsilon)\Gamma(z' + x + N + 2\epsilon)},\end{gathered}
\Psi(z):=\sqrt{\frac{\operatorname{Im}m(z)}{N \eta}}+\frac{1}{N \eta}, \ \ \underline{\Sigma}_o:=\left({\begin{array}{*{20}c}{\Sigma}0 0{I}\end{array}}\right), \ \ \underline{\Sigma}:=\left({\begin{array}{*{20}c}{z^{-1/2}\Sigma^{1/2}}0 0{I}\end{array}}\right).
C_{f,g}:=\frac1{6\sqrt{10}}C_{f,g}^0,C_{f,g}^0:=\frac{n_{f,g,3,1}+2n_{f,g,2,2}+n_{f,g,1,3}}{(n_{f,g,4,2}+2n_{f,g,3,3}+n_{f,g,2,4})^{1/2}}
E^{a}_{z}= \frac{N(z,t)}{f'(z,t)}\left(\begin{array}{c}f 1 f^{2}\end{array}\right) \; , \;\;E^{a}_{u}= \frac{n(u,t)}{\dot{f}(u,t)}\left(\begin{array}{c}f 1 f^{2}\end{array}\right)
\Delta_{H^{\otimes 3}}(B_1(a_1)\otimes a_2\otimes B_2(a_3)) = (B_1(a_1)\otimes a_3 \otimes B_2(a_5))\otimes (B_1(a_2)\otimes a_4\otimes B_2(a_6)) = (B_1(a_1)\otimes a_2 \otimes B_2(a_3))\otimes (B_1(a_4)\otimes a_5\otimes B_2(a_6)).
\rho_{11}=\rho_{22}=\rho_{33}=\rho_{44}=\frac{3}{2}\lambda_{1}^{2}, \rho_{12}=\rho_{14}=\rho_{23}=\rho_{34}=\frac{1}{2}\varepsilon\lambda_{1}^{2}, \rho_{13}=\rho_{24}=-\frac{1}{2}\lambda_{1}^{2},
W_{\pi}(g_{-2,l,v}) = q^{-1}G(\varpi^{-l},\chi)\chi^{-1}(v) + \sum_{\substack{\mu\in \mathfrak{X}_l, a(\mu\chi)=1}}\epsilon(\frac{1}{2},\mu^{-1}\chi^{-1})^2G(\varpi^{-l},\mu^{-1})\mu(v).
LHS= \Bigg( \sum_{j=1}^{i}\int_{\Lambda_j}u \mathcal{L}_{x_{i+ 1/2}}(x_j) du- \sum_{j=1}^{i-1}\int_{\Lambda_j}u \mathcal{L}_{x_{i- 1/2}}(x_j) du \Bigg)+{\cal O}(\Delta x^3).
M(\eta) = |\eta^+| + |\eta^-| + \sum \limits_{x \in \eta^+}e^{-E_{\kappa^+}(x, \eta^+ \backslash x)}e^{-E_{\tau^+}(x,\eta^-)}+ \sum \limits_{x \in \eta^-}e^{-E_{\kappa^-}(x,\eta^- \backslash x)}e^{-E_{\tau^-}(x,\eta^+)}
\tilde{R}= \frac{\tilde{a}M^{2}( \gamma - 1 )^{2}e^{\phi_{0}}}{\rho^{4}}\left( 1 - \frac{\rho_{1}}{\rho}\right) \left( 1 - \frac{\rho_{0}}{\rho}\right)^{- \frac{1 + 3 k^{2}- 2 l}{1 + k^{2}}}.
h(\rho \left | \bar \rho \right.) \geq \begin{cases}C_1 |\rho-\bar\rho|^2, \hbox{for}\ 0\leq \rho\leq R_0,\ \bar\rho\in K, C_2 |\rho-\bar\rho|^\gamma, \hbox{for}\ \rho> R_0,\ \bar\rho\in K.\end{cases}
F=\frac{KjDL}{2}\int_{-\theta_{0}}^{\theta_{0}}(\frac{cos^{2}\theta}{cos\theta_{0}}-cos\theta)d\theta=\frac{KjDL}{2}[\frac{\theta+sin\theta cos\theta}{2cos\theta_{0}}-sin\theta]_{-\theta_{0}}^{\theta_{0}}
D(\Phi_\alpha dx^\alpha) =r\Big(\mathring{\nabla}_\alpha \Phi_\beta+\mathring{\nabla}_\beta \Phi_\alpha-\mathring{\nabla}^\gamma \Phi_\gamma \mathring{\sigma}_{\alpha\beta}\Big)dx^\alpha dx^\beta, D^\dagger (\Psi_{\alpha\beta}dx^\alpha dx^\beta ) =-\frac{2}{r}\mathring{\sigma}^{\gamma\beta}\mathring{\nabla}_{\gamma}\Psi_{\alpha\beta}dx^\alpha.
\frac{1}{1-a_pp^{s-k+1}+p^{2s-k+1}\epsilon(p)}\begin{pmatrix}-p^{-1-s}-p^{1-k+s}\epsilon(p)+a_pp^{-k}p^{-k}(1-p)\epsilon(p) 1-p^{-1}-p^{-1-s}-p^{s-k+1}\epsilon(p)+a_pp^{-k+1}\end{pmatrix}.
\left( \frac{\delta A}{\delta y ( s )}\right)^{2}+ \left( \frac{\delta A}{\delta \vec{x}( s )}\right)^{2}= \frac{1}{y^{4}( s )}\left\{\left( \frac{d y}{d s}\right)^{2}+ \left( \frac{d \vec{x}}{d s}\right)^{2}\right\}.
[\begin{matrix}P\\e_{1}\\e_{2}\\e_{3}\end{matrix}]=[\begin{matrix}1&0&0&0\\ 0&cos\theta&sin\theta&0\\ 0&-sin\theta&cos\theta&0\\ 0&0&0&1\end{matrix}][\begin{matrix}P\\T\\t\\u\end{matrix}]
\lambda_{R}\equiv{\frac{1}{3 !}}{\frac{d^{4}{\cal V}( \Phi )}{d \Phi^{4}}}\biggr |_{\Phi = 0}= \lambda{\frac{1 - 6 \lambda I_{( 2 )}[ m_{R}^{2}]}{1 + 3 \lambda I_{( 2 )}[ m_{R}^{2}]}},
\langle \Phi | E^{a}E^{a}| \Phi \rangle = \langle \Phi | \left( \frac{1}{L^{2}}e^{p \dagger}e^{p}+ 2 \mathrm{t r}g \rho \frac{1}{D_{1}^{\prime}}g \rho \right) | \Phi \rangle .
h^B(\tilde{r},\hat{\delta}^{*},\tilde{\delta}, \hat{f}^{*}, \tilde{f})+ \mathcal{W}- p_W^{*}= 0 & g_1^B(\tilde{r}) \leq \hat{r}^{*}& g_2^B(\tilde{\delta}, \tilde{f}) \leq \overline{f},
r(x,y,t)=(x,t),\; r(x,g,0)=(x,0) s(x,y,t)=(y,t),\; s(x,g,0)=(x,0) (x,y,t)\cdot (y,z,t)=(x,z,t)\; t\neq 0 (x,g,0)\cdot (x,h,0)=(x,gh,0)\; g,h\in G_x (x,y,t)^{-1}=(y,x,t),\; (x,g,0)^{-1}=(x,g^{-1},0)
|\mathcal{L}_{r}| \leq 2^{d+1}C^\prime_{d,\epsilon}\big(\frac{n^{2+\epsilon}}{r^{d+1}}+\frac{n}{r}\big) &\leq 2^{d+1}C^\prime_{d,\epsilon}\alpha \frac{n^{2+\epsilon}}{r^{d+1}},
\big[U^{(m)}{\tilde{\circ}}U^{(m)}\big]=k_p([\omega])\big(\big[H^{(2)}\big]-f_p([\omega])\big) \phantom{\big[U^{(m)}{\tilde{\circ}}U^{(m)}\big]=}{}+l_p([\omega]) \prod\limits_{i=1}^{p}([\omega]- h_{i,1})\prod\limits_{i=1}^{p-1}([\omega]- h_{mp+p+i,1})\prod\limits_{i=1}^{m^2p}\bigg([\omega]-h_{p+\tfrac{i}{m},1}\bigg),
\cos\vartheta =-\frac{d|\vec{j}|+\eta |\sin\theta_0\vec{c}|}{\sqrt{d^2|\vec{j}|^2+\eta^2|\vec{c}|^2+ 2\eta d|\vec{j}\times\vec{c}|}}\sin\vartheta =\textup{sgn}(\sin\theta_0) \frac{\eta |\vec{c}|\cos\theta_0}{\sqrt{d^2|\vec{j}|^2+\eta^2|\vec{c}|^2+ 2\eta d|\vec{j}\times\vec{c}|}}.
\mathcal I_{M, \lambda}(w)=\frac{1}{2}\widehat M(\|w\|^2) -\frac{\lambda}{q}\int_{\Omega\times \{0\}}f(z)|w(z, 0)|^q dz-\frac{1}{2^*_\alpha}\int_{\Omega\times\{0\}}|w(z,0)|^{2^*_\alpha}dz.
\begin{array}{c c c}{{\varepsilon^{i}= \left( \begin{array}{c c}{{\bar{\varepsilon}_{a}}}&{{0}}\\{{0}}&{{\varepsilon^{a}{}^{t}}}\\ \end{array}\right) ~}}&{{~ \rho^{i}= \left( \begin{array}{c c}{{0}}&{{\bar{\rho}^{a}{}^{t}}}\\{{\rho_{a}}}&{{0}}\\ \end{array}\right) ~}}&{{~ a = 1 , 2 , \cdots , N}}\\ \end{array}
\frac{f^{\prime}\left( \frac{x_{1}}{\sqrt{\gamma}}+ i \sqrt{\gamma}x_{2}\right)}{f \left( \frac{x_{1}}{\sqrt{\gamma}}+ i \sqrt{\gamma}x_{2}\right)}= \frac{\left( 1 - \frac{2 \gamma_{1}\hbar}{\gamma \theta}\right) x_{1}- i \gamma x_{2}+ z \left( \frac{\sqrt{\gamma}}{2}+ \frac{\gamma_{1}\hbar}{\theta \sqrt{\gamma}}\right)}{\frac{\theta \sqrt{\gamma}}{2}- \frac{\gamma_{1}\hbar}{\sqrt{\gamma}}};
G [ \chi_{2}, \chi_{1}; \psi_{2}, \psi_{1}] = \prod_{k}^{\infty}\biggl ( \int_{0}^{\infty}d N_{k}N_{k}^{-{\frac{1}{2}}}e^{-{\frac{{( \Delta \psi_{k})}^{2}}{2 N_{k}}}}{\cal G}[ \chi_{2 k}, \chi_{1 k}; N_{k}] \biggr ) ,
V^{2}( r ) \equiv{\frac{\Delta ( r )}{R^{2}( r )}}\left\{\frac{\lambda}{R^{2}( r )}+ \frac{1}{R ( r )}\frac{\mathrm{d}}{\mathrm{d}r}\left[ \frac{\Delta ( r )}{R^{2}( r )}\frac{\mathrm{d}R ( r )}{\mathrm{d}r}\right] - \frac{m^{2}a^{2}}{R^{6}( r )}\left( r^{2}+ a^{2}+ 2 \mu r \cosh 2 \beta \right) \right\}.
\left(\frac{d}{dx}\right)^nA(x)^{-1}=\sum_{l=1}^n\prod_{j=1}^l\Bigg(\sum_{m_j=1}^n\Bigg)1_{\sum_{j=1}^lm_j=n}c^{(n)}(l,m_1,m_2,\cdots,m_l) &\quad\cdot(-1)^l\prod_{k=1\atop order}^l\Bigg(A(x)^{-1}\left(\frac{d}{dx}\right)^{m_k}A(x)\Bigg)A(x)^{-1},
\sum_{i=0}^{q-1}(-1)^i( \varphi(h_0(z_0),\ldots,h_0(z_i),h_1(z_i),\ldots,h_1(z_{q-1}))+\varphi(h_1(z_0),\ldots,h_1(z_i),h_2(z_i),\ldots,h_2(z_{q-1})) -\varphi(h_0(z_0),\ldots,h_0(z_i),h_2(z_i),\ldots,h_2(z_{q-1})))
- \int_{0}^{\tau}\! \mathrm{d}t \left[ \sum \dot{\phi}_{j}^{2}+ H_{\mathrm{b o s}}( i \dot{\phi}( t ) , \phi ( t ) ) \right] = - \int \! \mathrm{d}^{4}x{\cal L}_{\mathrm{E u c l}}\, .
\widehat{E}=(I+HH^{\ast}E_{S_{A}})(I+E_{S_{A}}+HH^{\ast}E_{S_{A}})^{-1}&=I-E_{S_{A}}(I+E_{S_{A}}+HH^{\ast}E_{S_{A}})^{-1}&=I-(I+E_{S_{A}}+E_{S_{A}}HH^{\ast})^{-1}E_{S_{A}}.
\sigma(t) \approx \frac{2}{5}\frac{1}{\gamma t}\left(\frac{t}{t_i}\right)^{1/3}\left[8U (v_s\gamma)^2 + 2\alpha(\phi_0)^2 \left(U+T+I^2\right)- \left( U-T-I^2\right) - 2\ln\left(\frac{r}{r_0}\right)I^2 \right]
c[n+1] = \sum_{k=0}^n \left( T - y[k] \right) = c[n] + T - y[n] \eta[n+1] = \eta[n] + \bar{w}[n]\left[ T - y[n] + \frac{c[n]}{\tau}\right] \bar{w}[n] = \frac{1}{\vert \mathcal{I}\vert}\sum_{k\in \mathcal{I}}w[k]
a = \frac{(p+q)(p-1)(q-2)}{pq(p+q-3)}b =\frac{(p+q)(p-1)^2}{pq(p+q-1)(p+q-3)}c =\frac{(p+q)(p-1)}{p(p+q-1)}d =\frac{q-2}{p(p+q-3)}e =\frac{p-1}{p(p+q-1)(p+q-3)}f =\frac{q}{p(p+q-1)}.
\lambda_{y,\gamma}(x) =\frac{\chi_{\Lambda}(k)}{2\pi |k|^{1/2}}e^{i(k_2 x_2 + k_3 x_3)}\left( \begin{array}{c}\mathbf{e}^{(1)}_{\gamma}(k) 2\cos(k_1 (x_1 - y)) \mathbf{e}^{(2)}_{\gamma}(k)2i \sin (k_1 (x_1 - y)) \mathbf{e}^{(3)}_{\gamma}(k) 2i \sin (k_1 (x_1-y))\end{array}\right).
- b_{,k}d_{\pi,\mathcal{P}_{i^{*}}}\leq \frac{1}{t}\sum_{\tau=0}^{t-1}\sum_{\omega \in \Omega}\left(\pi(\omega) -\mathcal{P}_{i^{*}}(\omega) \right) \Phi_k(\tau) \leq b_{,k}d_{\pi,\mathcal{P}_{i^{*}}}.
\Delta_G =\left\{\widehat{x}_i-\widehat{x}_{i+1}\,, 1\leq i\leq p+q-1\,,\; 2\,\widehat{x}_{p+q}\right\}\,, \intertext{and}\Delta_K =\left\{\begin{cases}\widehat{x}_i-\widehat{x}_{i+1}\,, 1\leq i\leq p-1\,, 2\,\widehat{x}_{p}\,, \widehat{x}_i-\widehat{x}_{i+1}\,, p+1\leq i\leq p+q-1\,,2\,\widehat{x}_{p+q}\end{cases}\right\}\,,
X=\sum_{i=1}^n \left(\frac{\partial h}{\partial z}x_i-\frac{\partial h}{\partial y_i}\right) \frac{\partial}{\partial x_i}+\sum_{i=1}^n \frac{\partial h}{\partial x_i}\frac{\partial}{\partial y_i}+\left(h-\sum_{i=1}^n \frac{\partial h}{\partial x_i}x_i\right) \frac{\partial}{\partial z}.
\frac{d}{dt}\int f(x) d\mu_t(x) = \int \frac{d}{dt}f(\exp_x(tv(x)))d\mu_0(x) = \int \langle \nabla f(\exp_x(t v (x))), \Pi_{t,v(x)}( v(x) )\rangle_{\exp_x(tv(x))}d\mu_0(x),
[e_i,e_j]={\rm ad}_{e_i}e_j=U_0^{n-i}Ae_j-a_{ki}U_0^{n-k}e_j=a_{kj}e_{k+i-n}-a_{ki}e_{k+j-n}\phantom{[e_i,e_j]}=(a_{p+n-i,j}-a_{p+n-j,i})e_p.
iU(\alpha z_1)^{\ast}HU(\alpha z_2)=\left[\begin{array}{cc}\eta(\alpha z_1,\alpha z_2) \ast \ast \ast\end{array}\right],iU(z_1)^{\ast}HU( z_2)=\left[\begin{array}{cc}\eta(z_1,z_2) \ast \ast \ast\end{array}\right].
\sigma_{( a_{i}^{\dag}a_{k})}^{+}= <{\bf A}_{+}\mid \frac{1}{2}( a_{i}^{\dag}a_{k}+ a_{k}a_{i}^{\dag}) \mid{\bf A_{+}}> = \alpha_{i}^{*}\alpha_{k}t a n h \mid{\bf A}\mid^{2}+ \frac{1}{2}\delta_{i k}, \nonumber \,
\log \mathbb P[A] =\sum_{\alpha\in I_2}\log\left[1-e^{-e^{2\gamma |\alpha|_\infty}}\right]\simeq -\sum_{\alpha\in I_2}e^{-e^{2\gamma |\alpha|_\infty}}\gtrsim -\sum_{m\geq C_*\delta L_*}n m^{n-1}e^{-e^{2\gamma m}}&- \sum_{m\geq C_*\delta L_*}e^{-e^{\gamma m}}\simeq - e^{-e^{\gamma C_*\delta L_*}}.
\Delta_i^{i}f =f^{[0]}(\mu_i)=f(\mu_i), \Delta_i^{i+1}f =f^{[1]}(\mu_i,\mu_{i+1})=\frac{f^{[0]}(\mu_{i+1})-f^{[0]}(\mu_i)}{\mu_{i+1}-\mu_i}, \Delta_i^{i+m}f =f^{[m]}(\mu_i,\dots,\mu_{i+m})=\frac{f^{[m-1]}(\mu_{i+1},\dots,\mu_{i+m})-f^{[m-1]}(\mu_{i},\dots,\mu_{i+m-1})}{\mu_{i+m}-\mu_i}.
A_1=\big(\sum^m_{i=0}E^*_i\big)A_1\big(\sum^m_{i=0}E^*_i\big)=\sum^{m-1}_{i=0}(E^*_{i+1}A_1E^*_i+E^*_iA_1E^*_{i+1})+E^*_mA_1E^*_m
\mathcal{S}_{p,s}:=\inf_{u\in D^{s,p}(\mathbb{R}^N)\setminus\{0\}}\frac{\displaystyle\int_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^p}{|x-y|^{N+s\,p}}\,dx\,dy}{\displaystyle\left(\int_{\mathbb{R}^N}|u|^\frac{N\,p}{N-s\,p}\,dx\right)^\frac{N-s\,p}{N}},
\sum_{m=1}^\infty\prod_{i=0}^m(1-r_{m-i}^i) =\sum_{m=1}^{m_0-1}\prod_{i=0}^m\underbrace{(1-r_{m-i}^i)}_{\leq 1}+\sum_{m=m_0}^{\infty}\prod_{i=0}^m(1-r_{m-i}^i) &\leq m_0-1+\sum_{m=m_0}^\infty\prod_{i=i_0}^m(1-r_{m-i}^i)
z I_{\nu}' (z) + \nu I_{\nu}(z) = z I_{\nu - 1}(z), 2^{n}I_{\nu}^{(n)}(z) = \sum_{r=0}^n C^{r}_{n}I_{\nu+n-2r}(z), \hskip 10 pt (-2)^{n}K_{\nu}^{(n)}(z) = \sum_{r=0}^n C^{r}_{n}K_{\nu+n-2r}(z),