harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
e_{int}\leq \frac{1}{n}\sum_{\mu \in \Xi}\sum_{l=1}^{k}\left( \int_{{\omega}}(u(\mu,{y}) - \mathcal{I}_{L}[u](\mu,{y}) ) \kappa_{l}({y}) \,d{y}\right )^{2}\leq \frac{1}{n}\sum_{\mu \in \Xi}k \|u(\mu,\cdot) - \mathcal{I}_{L}[u](\mu,\cdot) \|_{L^{2}({\omega})}^{2}.
lim_{\Delta x\rightarrow0}\frac{(x+\Delta x)^{4}-x^{4}}{\Delta x}=lim_{\Delta x\rightarrow0}\frac{x^{4}+4x^{3}(\Delta x)+6x^{2}(\Delta x)^{2}+4x(\Delta x)^{3}+(\Delta x)^{4}-x^{4}}{\Delta x}
\frac{uv_y-vu_y}{v^2}-\frac{2uvu_x-v^2v_x-u^2v_x}{v^2}&=\frac{\varepsilon uv_{xx}-\varepsilon vu_{xx}}{v^2}-\frac{2\varepsilon uv_x^2-2\varepsilon vu_xv_x}{v^3}+\tilde{\Gamma}^1_{22}\frac{1}{v}+2\tilde{\Gamma}^1_{12}\frac{u}{v}+\tilde{\Gamma}^1_{11}\frac{u^2-v^2}{v}.
R^{\Lambda,\Lambda'}_{x,y}= \sum_{i=0}^{D-1}\frac{S_{\Lambda,\Phi_1(i)}}{S_{0,\Phi_1(i)}}\cdot\frac{S_{\Lambda',\Phi_2(i)}}{S_{0,\Phi_2(i)}}\cdot\xi^i_x (\xi^i_y)^*
\begin{array}{llll}E \cdot u^*(u) = K(u^*(S(E) \cdot u)) + E(u^*(S(1) \cdot u)) = K(u^*(-K^{-1}E \cdot u)) + E(u^*(1 \cdot u)) = 0 E \cdot u^*(v) = K(u^*(S(E) \cdot v)) + E(u^*(S(1) \cdot v)) = K(u^*(-K^{-1}E \cdot v)) + E(u^*(1 \cdot v)) = -q^{-1}. \end{array}
\begin{cases}\left(w_i,e^{i\theta_\infty}\phi_i-\nabla\cdot(\sigma_{i\infty}\nabla\phi_i)\right)_{L^2(\mathbb{R}^d)}-(w_e,e^{i\theta_\infty}\phi_i)_{L^2(\mathbb{R}^d)}=0, (w_i,e^{i\theta_\infty}\phi_e)_{L^2(\mathbb{R}^d)}-\left(w_e,e^{i\theta_\infty}\phi_e-\nabla\cdot(\sigma_{e\infty}\nabla\phi_e)\right)_{L^2(\mathbb{R}^d)}=0, \end{cases}
\sum_{j=1}^{\infty}j |B(x_0, 2^jr)|^\frac{1}{q'}\Big( \int_{2^{j}r\leq |x_{0}-y|\leq 2^{j+1}r}|W(x,y)-W(x_{0},y)|^{q}dy \Big)^{1\over q}\leq C.
\bigl[ [x_1, x_2] x_3, x_4, x_5 \bigr] = [x_1, x_2] [x_3, x_4, x_5] + [x_1, x_2, x_4] [x_3, x_5] + [x_1, x_2, x_5] [x_3, x_4] + [x_1, x_2, x_4, x_5] x_3 .
\big|R_j(b\psi)(g)\big| =\left|\int_{\mathcal G}K_j(g,g')b(g')\psi(g')dg'\right| =\int_{F}\left|K_j(g,g')\right| \left|b(g')\right|dg' \geq A_4\int_{F}\rho(g')^{-Q}\left|b(g')\right|dg' \geq{A_4A_1\over 2A_2}\int_{F}M\rho(g')^{-Q+1}dg' \geq{A_4A_1\over 2A_2}M^{1\over Q}|F| \geq A_5 M^{1+{1\over Q}}.
X_mX_{m+n}= q^{\lfloor\frac{n}{2}\rfloor}X_{\lfloor m+\frac{n}{2}\rfloor}X_{\lceil m+\frac{n}{2}\rceil}+\sum\limits_{k=1}^{n-1}(\sum\limits_{l=1}^{(k,n-k)}q^{-\frac{1}{2}+l})hX_{m+n-k}+\sum\limits_{l=1}^{n-1}q^{-\frac{n-1-l}{2}}c_lF_{n-1-l}(X_\delta),
E(f) = E(v_1\cdots v_{m+1}) = E(v_1) \cup \bigcup_{\mathclap{r=2}}^m E(x_iv_r) \cup E(x_iv_{m+1}) &= E(v_1) \cup \bigcup_{\mathclap{r\in R}}E(x_iv_r) \cup E(x_iv_{m+1}) = E(g).
\sum_{n=0}^{\infty}p_{-6}\left(5^{j}(5n+3)+\dfrac{3\times5^{j}+1}{4}\right)q^{5n+3}=\dfrac{1}{q^{7}E_{25}^{6}}\sum_{l=1}^{\infty}b(j,l)T^{l}H\left(\zeta^{-(6l+6)}\right) &=\dfrac{1}{q^{7}E_{25}^{6}}\sum_{l=1}^{\infty}\sum_{k=1}^{\infty}b(j,l)m(6l+6,k)T^{l-k}.
Re(e^{i(\theta - \frac{\pi}{2})}A) = \frac{e^{i(\theta - \frac{\pi}{2})}A+e^{-i(\theta - \frac{\pi}{2})}A^{\ast}}{2}&= \frac{-ie^{i\theta}A+ie^{-i\theta}A^{\ast}}{2}&= \frac{e^{i\theta}A-e^{-i\theta}A^{\ast}}{2i}&= Im(e^{i\theta}A).
\gamma^{-1}S_{\gamma_N x}\gamma = \lbrace \gamma^{-1}\gamma' \gamma, \gamma' \in S_{\gamma_N x}\rbrace &= \lbrace \gamma^{-1}\gamma' \gamma, \gamma'_N \gamma_N \cdot x \in \Bar{P}^0 \gamma_N x \rbrace &= \lbrace \gamma^{-1}\gamma' \gamma, \gamma_N^{-1}\gamma'_N \gamma_N \cdot x \in \Bar{P}^0 x \rbrace &\subset S_x,
m(L'')=\pi^{-4}\cdot 2\cdot m_2(L'')\cdot 2\cdot \frac{3^8}{2^4}\cdot (1-2^{-2})^2\cdot (1-3^{-2})^2\cdot \zeta(2)^2=m_2(L'')\cdot \frac{3^4}{2^4}.
\vec{\vec{\Delta}}\hat{W}= \vec{\nabla}\otimes \vec{\nabla}\hat{W}= \sum_{a = 1}^{2}\sum_{b = 1}^{2}\vec{e}_{a}\otimes \vec{e}_{b}\frac{\partial^{2}\hat{W}}{\partial \phi^{a}\partial \phi^{b}}\quad , \Delta \hat{W}= \vec{\nabla}\vec{\nabla}\hat{W}= \sum_{a = 1}^{2}\frac{\partial^{2}\hat{W}}{\partial \phi^{a}\partial \phi^{a}}
\mathrm{dim}(h \times f)(\mathcal{W}_0) = \mathrm{dim}\mathcal{W}_0 - \mathrm{dim}M + \mathrm{dim}(h \times f)(M) &= \mathrm{dim}\mathcal{W}_g - \mathrm{dim}M + \mathrm{dim}(h \times f)(M),
\begin{aligned}\left(\frac{3 \boldsymbol{\xi}_{u,h}^n-3\widetilde{\boldsymbol{\xi}}_{u,h}^n}{2 \Delta t}+\nabla (\xi_{p,h}^n-\xi_{p,h}^{n-1}), \nabla \phi_h \right)=0. \end{aligned}
= \sum_{j=0}^{n-1}\mathbb{E}\bigg(\bigg\langle W_{E_{t_{j+1}\wedge t}}-W_{E_{t_{j}\wedge t}}, \phi^*(h) \bigg\rangle_K \bigg| \mathcal{G}_s\bigg) = \sum_{j=0}^{n-1}\bigg\langle W_{E_{t_{j+1}\wedge s}}-W_{E_{t_{j}\wedge s}}, \phi^*(h) \bigg\rangle_K
\sup\left\{\mathrm{e}^{\alpha t}\left\Vert \frac{\partial y_{1}(t,\eta)}{\partial\eta}a-\frac{\partial y_{2}(t,\eta)}{\partial\eta}a\right\Vert :(t,\eta,a)\in\mathbb{R}_{+}\times\mathbb{L}_{\xi}^{-}\times\mathbb{L}_{\xi}^{-},\left\Vert \eta\right\Vert \le r,\left\Vert a\right\Vert =r\right\}.
s_2^{!}(A) =c_1(L)\cap A\overset{\eqref{lem:eq:c1}}{=}\sum_{V\in J}{\rm ord}_V\cdot (V\cdot_{{\rm id}_L}A) &\overset{\eqref{eq:refinedGysing01}}{=}\sum_{V\in J}{\rm ord}_V\cdot i_V^!A\overset{\eqref{eq:fibration100:101}}{=}\sum_{V\in J}{\rm ord}_V\cdot i_V^! g^!(Z) &=\sum_{V\in J}{\rm ord}_V\cdot g^!(i_V^!(Z))
e^{-{1\over 2}a\phi-(p+1)A}=1+{{\lambda_1}\over{a\sqrt{2}}}{1\over{r^{p+1}}},\ \ \ \e^{+{1\over 2}a\phi-(p+1)A}=1+{{\lambda_2}\over{a\sqrt{2}}}{1\over{r^{p+1}}},
\int \mathrm{Y}\mathrm{Y}^T d\hat{\nu}= \mathbf{A}^{-1}\left( \int (\mathrm{X}- \mathrm{c}) (\mathrm{X}- \mathrm{c})^T d\hat{\nu}\right) \mathbf{A}^{-T}&= 2 \mathbf{I}
\frac{1}{c_1}=-\frac{G_{t-4}(\theta)}{F_{t-2}(\theta)}= -\frac{G_{t-2}(\theta)}{F_{t-2}(\theta)}+1=-\frac{\lambda G_{t-3}(\theta)-(k-1)G_{t-4}(\theta)}{F_{t-2}(\theta)}+1=-\frac{\lambda G_{t-3}(\theta)}{F_{t-2}(\theta)}-\frac{k-1}{c_1}+1,
\begin{array}{l l}{U = - 2 \pi \int_{0}^{r_{0}}\tilde{T}_{_{( 0 )}t}^{t}r d r ;}\\{T = - 2 \pi \int_{0}^{r_{0}}\tilde{T}_{_{( 0 )}z}^{z}r d r .}\\ \end{array}
x_{1}^{4}+ x_{2}^{4}+ x_{3}^{2}- 2 \sqrt{2}\omega x_{1}x_{2}x_{3}= ( x_{3}- \sqrt{2}\omega x_{1}x_{2})^{2}+ ( x_{1}^{2}- \omega^{2}x_{2}^{2})^{2}= 0 \ .
\lambda_y = (\lambda_{y,\gamma})_{\gamma=1,2}, \lambda_{y,\gamma}:= \lambda_{y,\gamma}(0) = \frac{\chi_{\Lambda}(k)}{2\pi |k|^{1/2}}\left( \begin{array}{c}\mathbf{e}^{(1)}_{\gamma}(k)2 \cos(k_1 y) -\mathbf{e}^{(2)}_{\gamma}(k) 2i\sin (k_1 y) -\mathbf{e}^{(3)}_{\gamma}(k) 2i\sin (k_1 y)\end{array}\right) .
f_B\left(\mathbf{Q}\right): = -\frac{A}{2}\textrm{tr}\mathbf{Q}^2 - \frac{B}{3}\textrm{tr}\mathbf{Q}^3 + \frac{C}{4}\left(\textrm{tr}\mathbf{Q}^2\right)^2 +M\left(A, B, C \right)
B^{(a)}_{\underline{z}}=\begin{bmatrix}(\partial_{z_1 \overline{z}_1}g_a)(\underline{z}) (\partial_{z_2 \overline{z}_1}g_a)(\underline{z}) (\partial_{z_1 \overline{z}_2}g_a)(\underline{z}) (\partial_{z_2 \overline{z}_2}g_a)(\underline{z})\end{bmatrix},
\sum_{A_1,\ldots,A_{k-2}\in F_1}\deg_G(A_1,\ldots,A_{k-2})\ge \frac{\binom k2}{k}\cdot \frac{(k-1)^{k-1}}{k^{k-1}}\cdot \frac{(m/2)^k}{t^{k-1}}=\frac{(k-1)^{k}}{2k^{k-1}}\cdot \frac{(m/2)^k}{t^{k-1}}.
\left\lbrace\begin{array}{ll}u_r'' -\Delta u_r + \sum_{s=1}^p\alpha_{rs}u_s =0 \hbox{in}(0, +\infty)\times\Omega,\cr u_r =0 \hbox{on}(0, +\infty)\times\Gamma_0,\cr \partial_\nu u_r + \sum_{s=1}^p\beta_{rs}u_s =0 \hbox{on}(0, +\infty)\times \Gamma_1\end{array}\right.
\phi(g \cdot a_{i}) = \phi(\det(g) a_{i}\circ g) = \det(g) \phi(a_{i}\circ g) = \det(g) c_{i}(b_{i}\circ g). g \cdot \phi(a_{i}) = g \cdot c_{i}b_{i}= \det(g) c_{i}(b_{i}\circ g).
\prod_{l=1}^N w^{(\rm cJ)}(\theta_l) \prod_{1 \le j < k \le N}| e^{i \theta_k}- e^{i \theta_j}|^\beta, w^{(\rm cJ)}(\theta) = e^{q \theta}|1 + e^{i \theta}|^{\beta p}\: \: (-\pi < \theta \le \pi),
\left|\sum_{j=0}^{\kappa_1q_1+\cdots+\kappa_{N-1}q_{N-1}+sq_{N}-1}\log J^u(f^jy)\right| \leq \kappa_{N-1}q_{N-1}\left(\lambda^u(\mu_{N-1})+\frac{2}{N-1}\right)+s q_{N}\left(\lambda^u(\mu_{N})+\frac{1}{N}\right).
\begin{aligned}\left(\frac{3 \boldsymbol{w}_t^n-3\widetilde{\boldsymbol{w}}_t^n}{2 \Delta t}+\nabla (r_t^{n}-r_t^{n-1}), \boldsymbol{v}\right) =0 (\nabla \cdot \boldsymbol{w}_t^n, q) =0 \boldsymbol{w}_t^n|_{\partial\Omega}=0 \end{aligned}
y(t)\geq -\frac{\left(L_{1}y\right)^{\frac{1}{\alpha}}(t)\cdot\pi_{1}(t)}{1-\frac{\lambda}{\alpha}}=-\frac{r_{1}^{\frac{1}{\alpha}}(t)y'(t)\pi_{1}(t)}{1-\frac{\lambda}{\alpha}}.
\mathcal{Q}_{t}(x_{1:t-1})=\rho_t( \mathfrak{Q}_{t}(x_{1:t-1}, \xi_t) ) = \displaystyle \sup_{p \in \mathcal{P}_t}\;\displaystyle \sum_{j=1}^{M}p_j \Phi_{t, j}\mathfrak{Q}_{t}(x_{1:t-1}, \xi_{t, j})
L(x,\dot{x}, \{q_{n},\dot{q}_{n}\},t)=\frac{M}{2}\dot{x}^{2}- V(x,t) + \sum_{n}\left\{\frac{1}{2}m \dot{q}_{n}^{2}- \frac{1}{2}m \omega_{n}^{2}\left[ q_{n}- Q_{n}(x,t)\right]^{2}\right\}.
\left|\mathfrak{F}(x)\right|\leq\exp\!\left(\sum_{k=1}^{\infty}|x_{k}x_{k+1}|\right)\!,\ \ \left|\mathfrak{F}(x)-1\right|\leq\exp\!\left(\sum_{k=1}^{\infty}|x_{k}x_{k+1}|\right)-1,
S_1\le C \sup_{t \in I}\Bigg[\sum_{\alpha=1}^\infty \lambda_\alpha^{\beta-H+\frac12}\bigg(\int_0^t \phi_\alpha^2(t-s) ds\bigg) \Bigg]^\frac p2\le C \Bigg(\sum_{\alpha=1}^\infty \lambda_\alpha^{\beta-H-\frac{1}{2}}\Bigg)^\frac p2,
H^2(A,A) \,\cong \,H^2({\cal X},{\bigwedge}{\cal T}) \, =\, H^0 ({\cal X},{\bigwedge}{}\!^2{\cal T}) \bigoplus H^1({\cal X},{\bigwedge}{}\!^1{\cal T}) \bigoplus H^2 ({\cal X},{\bigwedge}{}\!^0{\cal T}).
\mu^a\big(B_{r_\alpha}(x_i)\big)\leq \sum \mu^a\big(B_{\bar r_j}(y_j)\big)\leq 4\omega_k\sum \bar r_j^k = 4\cdot 10^k\mu'\big(B_{r_\alpha}(x_i)\big)\leq 2^{5n}D(n) r_\alpha^k\, ,
\lambda_1:=\nu+\sqrt{\left(\sum_{i=1}^{\nu}\overline{\jmath}_i^2\right)\left(\sum_{i=1}^{\nu}\overline{\jmath}_i^{-2}\right)}, \lambda_2:=\nu-\sqrt{\left(\sum_{i=1}^{\nu}\overline{\jmath}_i^2\right)\left(\sum_{i=1}^{\nu}\overline{\jmath}_i^{-2}\right)}.
2 \dfrac{\phi_1(\epsilon)- \phi_1(0)}{\phi_2(\epsilon)- \phi_2(0)}=2 \dfrac{\phi_1'(\gamma)}{\phi_2'(\gamma)}= \dfrac{\langle\nabla f(x_0+\gamma h),h\rangle - \langle\nabla f(x_0),h\rangle}{\gamma}= \dfrac{\langle\nabla f(x_0+\gamma h)- \nabla f(x_0) ,h\rangle}{\gamma}\leq \dfrac{|\nabla f(x_0+\gamma h)- \nabla f(x_0)|}{\gamma}\leq L
\begin{cases}\,\,\, \,\, c_{n,n}=1 n\geq1 \sum_{j=1}^na_{i,j}\,c_{n,j}\,=0 1\leq i\leq n-1 \sum_{j=1}^na_{n,j}\,c_{n,j}=\frac{b_n}{b_{n-1}}\,\, n\geq1.\end{cases}
{\cal W}_{{\cal C}}^{U ( N )}= 1 - \frac{1}{2}~ N g^{2}{\cal A}_{{\cal C}}+ \frac{1}{8}( \frac{2}{3}N^{2}+ \frac{1}{3}) ~ g^{4}{\cal A}_{{\cal C}}^{2}- \frac{1}{4 8}( \frac{5}{1 5}N^{3}+ \frac{1 0}{1 5}N ) ~ g^{6}{\cal A}_{{\cal C}}^{3}+{\cal O}( g^{8}) ,
S_t = \{\epsilon_t, \epsilon_i + \epsilon_j \, | \, 1 \leq i < j \leq n\}, t = 1, 2, \ldots, n S^*_t = \{\epsilon_t, \epsilon_i + \epsilon_j, \epsilon_{i^\prime}- \epsilon_n \, | \, 1 \leq i < j < n, 1 \leq i^\prime <n\}, t = 1, 2, \ldots, n
\alpha_{k}\triangleq \max_i \frac{\rho^+_{k}(i)}{Mp(i)}, \rho^+_{k}\triangleq \max_i \rho^+_{k}(i), \quad\overline{\rho}^+_k \triangleq \frac{1}{M}\sum_{i=1}^M \rho^+_k(i).
[k*]_n(1,t) = t^{n-k}+ \sum_{j=k}^{n-2}\left( t^3 (t+1)^{n-j-2}- t^{n-j+1}+ t^{n-j}\right) &= t^{n-k}+ t^3 \left( \frac{(t+1)^{n-k-1}-1}{(t+1)-1}\right) - t^{n-k+1}+ t^2 &= t^2 (t+1)^{n-k-1}- t^{n-k+1}+ t^{n-k}.
Z ( \kappa ) = \int \! [ d \phi ( x ) ] \, \exp \Biggl [ - \int \! d^{2}x \, \Bigl ({\frac{1}{2}}\, \partial_{a}\phi \partial_{a}\phi +{\frac{\Lambda^{2}}{2 \kappa}}: \! \phi^{2}\! : + \Lambda^{2}\sqrt{\frac{2 \pi}{\kappa}}\, : \! \phi^{3}\! : + \pi \Lambda^{2}: \! \phi^{4}\! : \Bigr ) \Biggr ] \, .
\boldsymbol{A}_{\rm sph}=\left[\begin{matrix}y^2+z^2 -xy -zx -xy x^2+z^2 -yz -zx -yz x^2+y^2\end{matrix}\right ] , \boldsymbol{B}_{\rm sph}=\left[\begin{matrix}0 yp_x-xp_y zp_x-xp_z -(yp_x-xp_y) 0 zp_y-yp_z -(zp_x-xp_z) -(zp_y-yp_z) 0\end{matrix}\right ] .
\Phi_{+}^{[ 0 ]}( n , p ) = \left( \begin{matrix}{| \eta_{+}( n ) >}\\{0}\\ \end{matrix}\right) \otimes | p > , \Phi_{-}^{[ 0 ]}( n , p ) = \left( \begin{matrix}{0}\\{| \eta_{-}( n ) >}\\ \end{matrix}\right) \otimes | p > ,
\beta_{i}=\alpha_{i}+\widetilde{M}_{i}-\widetilde{N}_{i}-\frac{1}{2}(M_{i}-N_{i}+1)+q-p, \gamma_{i}=\alpha_{i}-\widetilde{M}_{i}+\widetilde{N}_{i}-\frac{1}{2}(N_{i}-M_{i}+1)+p-q.
\widetilde{V}^{(0^{2})}(x;1,1,\lambda_{1}) = \frac{3}{4\sin^2 x}+ \frac{3}{4\cos^2 x}- 9 - 12 \frac{\sin^2 2x \cos 2x}{\lambda_1 - 2 \cos^4 x + \frac{4}{3}\cos^6 x}\hphantom{\widetilde{V}^{(0^{2})}(x;1,1,\lambda_{1}) =}{}+ 2 \frac{\sin^6 2x}{\big(\lambda_1 - 2 \cos^4 x + \frac{4}{3}\cos^6 x \big)^2}
(0,0)_{p^r+1}= p^r-2, (0,i)_{p^r+1}= (i,0)_{p^r+1}= (i,i)_{p^r+1}= 0 \textrm{for}i \ne 0, (i,j)_{p^r+1}= 1 \textrm{for}i \ne j \textrm{and}i,j \ne 0,
A_{T i}^{a}({\bf{r}}) = \sum_{{\bf{k}}; s = 1 , 2}\frac{\epsilon_{i}^{s}({\bf{k}})}{\sqrt{2 k}}\, [ a_{s}^{a}({\bf{k}}) e^{+ i{\bf{k}}\cdot{\bf{r}}}+ a_{s}^{a \dagger}({\bf{k}}) e^{- i{\bf{k}}\cdot{\bf{r}}}] \; ,
\bigg\|\sum_{i=1}^{n}\vartheta_iu'\left(\theta_i,\frac{1}{2\theta_i}\log\left(\frac{E[e^{2\theta_i|\xi^i|}|\mathcal{F}_t]}{1-2\theta_i\vartheta_ic}\right)\right)\bigg\|_{\infty}\leq\frac{1}{4}, \sum_{i=1}^{n}\left(16\vartheta_i^2ch(2\theta_i\sqrt{c})\right)<1.
(\widehat{R}_{\epsilon_m}(x+\alpha){Q})^{-1}A^{E_m^{+}+\epsilon_m}(x)\widehat{R}_{\epsilon_m}(x){Q}=e^{\sqrt{\Delta}\left(\left[\begin{array}{cc}0 -1 1 0\end{array}\right]+\epsilon_m^3 \mathfrak{S}(x)\right)},
Z_2=\sum_{n=1}^{d_1^2}[X_{n1},X_{n2}], \nu=\sqrt{\sum_{n,m=1}^{d_1^2}\langle X_{n1},X_{m1}\rangle_1\langle X_{n2},X_{m2}\rangle_1}=\sqrt{\sum_{n=1}^{d_1^2}\|X_{n1}\|_1^2\|X_{n2}\|_1^2}, \|X_{n1}\|_1=\|X_{n2}\|_1\qquad n=1,\dots,d_1^2.
\Lambda(\Psi_* u_0,k,N) = \frac{1}{2N+1}\left(\sum_{\mathrm{sum \over \blocks}}+ R \right) &= (1+\mathcal{O}(K/N))\frac{\sum_{\mathrm{sum \over \blocks}}}{\sum_i (K+L_i)n_i}+ \mathcal{O}(K/N).
\Vert T^k(y) \Vert \leq d^{-k}\Vert y \Vert \prod_{i=0}^{k-1}m_{\omega(x,i)}+ d^{-k}\sum_{j=0}^{k-1}M^{k-1-j}d^j R \leq (1 - \lambda_k/d^k) \Vert y \Vert + \frac{R(M^k-d^k)}{d^k(M-d)}< \Vert y \Vert.
\mathcal{H}(\boldsymbol{x},\boldsymbol{u},{\boldsymbol{\lambda}})= \frac{1}{2}\boldsymbol{x}^{T}(t) Q(t) \boldsymbol{x}(t)+ \frac{1}{2}\boldsymbol{u}^{T}(t) R(t) \boldsymbol{u}(t)+ \boldsymbol{\lambda}^{T}(t) [A(t) \boldsymbol{x}(t)+B(t)\boldsymbol{u}(t)]
g^{[k](\alpha)}_{-}=\sum_{a,b=0}^{1}g^{[k](\alpha)}_{ab}(z)E_{ab},g^{[k](\alpha)}_{ab}(z)=\big\langle Q_{b}^{-1}v_{0},\psi_{a}^{-}(z)T^{-k}g_{C}^{(\alpha)}v_{0}\big\rangle/\tau_{k}^{(\alpha)},
\begin{aligned}W_i^{(1)}:= \Big\{\beta^{u_i t_i}\alpha_i^{u_i + q s_i}:u_i=0,1,\dots,s_i-1; q=0,1,\dots,\frac{p_i-1}{s_i}-1 \Big\}, W_i^{(2)}:= \Big\{\sum_{u_i=0}^{s_i-1}\beta^{u_i t_i}\alpha_i^{p_i-1}\Big\}, W_i := W_i^{(1)}\cup W_i^{(2)}.\end{aligned}
\mathcal{I}(\delta_1, 0) = \mathbb{E}[Y_2] - \mathbb{E}[Y_1] + (V_{1}/V_{1, XX})^{1/2}\delta_1^{1/2}= \mathcal{I}_{\mathrm{D}}(\delta_1, 0 ), \mathcal{I}(0, \delta_2) = \mathbb{E}[Y_2] - \mathbb{E}[Y_1] + (V_{2}/V_{2, XX})^{1/2}\delta_2^{1/2}= \mathcal{I}_{\mathrm{D}}(0, \delta_2 ).
\mathcal{M}_{|\Delta_n|^2}(s) = \left(\frac{\alpha_n}{2sb_{0,n}}\right)^{\frac{\alpha_n}{2}}\exp\left(\frac{\alpha_n}{4sb_{0,n}}\right) & \times W_{-\frac{\alpha_n}{2}, \frac{\alpha_n-1}{2}}\left(\frac{\alpha_n}{2sb_{0,n}}\right).
\begin{aligned}\widetilde{\varepsilon}=\frac{||u||\sin\angle(\mathcal{V},f)}{||(I-P_{V})u||}\varepsilon=\frac{\sin\angle(\mathcal{V},f)}{||(I-P_{V})\frac{u}{||u||}||}\varepsilon=\frac{\sin\angle(\mathcal{V},f)}{\sin\angle(\mathcal{V},u)}\varepsilon .\end{aligned}
\begin{array}{c c c}{\alpha_{1}\to \alpha_{1}+ \omega_{1}\alpha_{2},}&{\}&{\beta_{1}\to c \beta_{1}+ \omega_{2}\beta_{2}}\\{\alpha_{2}\to c \alpha_{2}\phantom{\omega_{2}c \alpha_{2}},}&{\}&{\beta_{2}\to \beta_{2}\phantom{n n c + \omega_{2}( \beta_{2}+ 1 )}.}\\ \end{array}
\tilde{\xi}_n \to 0 \; L^1_{loc}(\Omega) \;\;\;\; T_{\vec{{\bf f}}}(\tilde{\xi}_n) = \varphi T_{\vec{{\bf f}}}(\xi_n)+ \nabla \varphi \xi_n := \vec{\tilde{{\bf g}}}_n \to \varphi \vec{{\bf g}}\; (L^1(\Omega))^N.
\int_{\mathbb{R}^{d-1}}\psi\tilde \psi\, dx = 1, \int_{\mathbb{R}^{d-1}}\beta(|g_j^Q|^\frac{1}{p}\psi)\tilde \psi\, dx =(-1)^{j+1}|g_j^Q|^\frac{1}{p}\int_{\mathbb{R}^{d-1}}\tilde \psi \, dx = 0.
\begin{array}{cl}{\cal{L}}_{F}(x,e) ={\cal{L}}^r_{F}(x,r) begin{align*}4mm] =i\overline{L}\gamma^{\mu}({\partial}_\mu+L_\mu)L+i\overline{R}\gamma^{\mu}({\partial}_{\mu}+R_\mu)R+\lambda (\overline{L}\phi{R}+\overline{R}{\phi}^{\dag}L).\end{array}
\Gamma_{\mathit{e f f}}^{\prime}[ \phi ] = i \int \frac{d^{4}k}{( 2 \pi )^{4}}\mathrm{t r}\langle k | \g \ \phi ( \hat{x}) \S ( \hat{p}) + \frac{1}{2}\g \, p h i ( \hat{x}) \S ( \hat{p}) \g \ \phi ( \hat{x}) \S ( \hat{p}) + . . . | k \rangle .
U^* \left(\!\!\begin{array}{cccc}\sigma_1 x_1 \sigma_2 x_2 \sigma_3 x_3 x_4 \end{array}\!\!\right) Q \left(\!\!\begin{array}{cccc}0 0 0 -i 0 0 -1 0 0 -1 0 0 i 0 0 0 \end{array}\!\!\right)Q \left(\!\!\begin{array}{c}\sigma_1 x_1 \sigma_2 x_2 \sigma_3 x_3 x_4 \end{array}\!\!\right) U = 0,
w_{ij}^\tau(\gamma) :=|\{\delta\in\mathcal{N}_\tau:p^n\mid i-j\gamma\delta\}|-|\{\delta\in\mathcal{N}_\tau:p^n\mid i+j'\gamma\delta\}| &\mathrel{\phantom{:=}}+|\{\delta\in\mathcal{N}_\tau:p^n\mid i'-j'\gamma\delta\}|-|\{\delta\in\mathcal{N}_\tau:p^n\mid i'+j\gamma\delta\}|
\frac{\partial}{\partial u^{(n)}_j}\, \exp \left({\sum_{j=1}^{m+1}\sum_{n=1}^{\infty}E^{(n)}_{jj}u^{(n)}_j}\right) g= \left( \frac{\partial}{\partial u^{(n)}_j}g_{-}(u) \right) g_{+}(u) + g_{-}(u) \frac{\partial}{\partial u^{(n)}_j}g_{+}(u)
\begin{array}{l}d_{\min}=\sqrt{\|\overrightarrow{P P_0}\|_2^2 + (\|\overrightarrow{S_0 S_1}\|_2 - \|\overrightarrow{S_0 P_0}\|_2)^2}\mbox{and}d_{\max}=\sqrt{\|\overrightarrow{P P_0}\|_2^2 + (\|\overrightarrow{S_0 S_1}\|_2 + \|\overrightarrow{S_0 P_0}\|_2 )^2},\end{array}
\widehat{D}_0 \pi_a + i\varsigma{{\partial E_a}\over{\partial\phi_i}}\tau_i-i \overline{p}'{{\partial\overline{J}^a}\over{\partial\overline{\phi}_i}}\overline{\tau}_i+iE_a \beta-i \overline{J}^a \overline{\gamma}+{\cal O}(1/e^2)=0
=-\,\delta_{\epsilon,\bar{\epsilon}}\langle X\rangle=\frac{1}{2\pi i}\oint_{C}d z \,\epsilon(z)\langle T(z)X\rangle-\frac{1}{2\pi i}\oint_{C}d \bar{z}\,\bar{\epsilon}(\bar{z})\langle \bar{T}(\bar{z}) X\rangle
\int_\mathbb{R}f\circ( \phi_1^* * \phi_2^* ) (x) dx - \int_\mathbb{R}f\circ( \phi_1*\phi_2 ) (x) dx = 4 (\lambda f(y_1)+(1-\lambda )f(y_2)- f(\lambda y_1+(1-\lambda )y_2) ).
\left|\frac{u_{h_1+i}(x)}{u_{h_1}(x)}-\frac{u_{h_1+i}(x_0)}{u_{h_1}(x_0)}\right|\leq C r^{\alpha}\forall x\in \overline{B_{r}(x_0)\cap \Omega_1},\i=2,\ldots, l
H_R=\sum_i \sum_n \frac{1}{2}\left[e^{-3Ht}L^3\pi_{\eta_i}(k_n,t)\pi_{\eta_i}(-{\bf k}_n,t)+ \frac{e^{3Ht}}{L^3}(e^{-2Ht}{\bf k}_n^2+\mu_i^2)\eta_i({\bf k}_n,t)\eta_i(-{\bf k}_n,t) \right],
\delta R_{M N}( h ) +{\frac{1}{6}}\bar{F}_{\theta \phi}\bar{F}^{\theta \phi}( h_{M N}+ 2 \bar{g}_{M N}{\cal F}) - ( \bar{F}_{M \theta}\bar{F}_{N}^{\theta}+ \bar{F}_{M \phi}\bar{F}_{N}^{\phi}){\cal F}= 0 ,
\left\{b_{\alpha}^{\omega}, ~ b_{\alpha^{\prime}}^{\omega^{\prime}}{}^{\dagger}\right\}= \left\{d_{\alpha}^{\omega}, ~ d_{\alpha^{\prime}}^{\omega^{\prime}}{}^{\dagger}\right\}= \delta ( \omega - \omega^{\prime}) \delta ( \alpha - \alpha^{\prime})
\begin{aligned}\lim_{n\to\infty}P_{n,k}(W_{\mathcal{S}^+(n,k;M_n)})=\Big(k\sum_{l=1}^k\binom kl(\frac c{1-c})^l\Big)\Big(\sum_{s=1}^\infty(1-c)^{k(s+1)}\frac 1{ks}\Big), & \M_n\sim cn, \c\in(0,1),\end{aligned}
\Psi'(0)\equiv \frac{d}{dx}\bigg({}_2F_1\bigg[\begin{matrix}-a+x -b &p-c\end{matrix}\bigg|\,z\bigg]\bigg)\bigg|_{x=0}\equiv\frac{d\big(\omega(x)\phi(x)\big)}{dx}\bigg|_{x=0}= \omega'(0)\phi(0)+\omega(0)\phi'(0)\equiv\frac{d\big(\Omega(x)\Phi(x)\big)}{dx}\bigg|_{x=0}\pmod{p}.
\sum_{\substack{a\bmod{q}a\bar{a}\equiv 1\bmod{q}}}e(n\bar{a}/q)L_f\bigl(1-s-u,-\tfrac{\bar{a}}{q}\bigr)=\sum_{\substack{a\bmod{q}a\bar{a}\equiv 1\bmod{q}}}e(n\bar{a}/q)\sum_{m=1}^\infty\frac{f_me(-m\bar{a}/q)}{m^{1-s-u}}= \sum_{m=1}^\infty\frac{f_mr_q(n-m)}{m^{1-s-u}},
E\Big[ \sum_{\sigma}\prod_{i=1}^p |\beta_{s_i}-\tilde{\beta}_{t_{\sigma(i)}}|^{-2}\Big] \le C^p p! \prod_{i=1}^p (s_i-s_{i-1})^{-1/2}\prod_{j=1}^p (t_j-t_{j-1})^{-1/2}\,.
\min_{\theta}(\mu_0 + \Phi \theta)^\top \ell\,, &\mbox{s.t.}(\mu_0+\Phi\theta)^\top \mathbf 1 = 1,\, \mu_0+ \Phi \theta \geq \mathbf 0,\, (\mu_0+\Phi\theta)^\top (P - B) = \mathbf 0 \; .
\frac{\vec{v}\cdot\nabla\rho}{\rho}=\frac{\vec{v}\cdot\nabla p}{a^{2}\rho}=-\frac{1}{a^{2}}\vec{v}\cdot\nabla[\frac{\partial\Phi}{\partial t}+\frac{\nabla\Phi\cdot\nabla\Phi}{2}]=-\frac{1}{a^{2}}\nabla\Phi\cdot\nabla[\frac{\partial\Phi}{\partial t}+\frac{\nabla\Phi\cdot\nabla\Phi}{2}]
\sum\limits_{i=1}^k\sum\limits_{l=0}^n c^{il}_m\langle P\psi^l_{\mu_m^i,\xi_m^i}, P\psi^h_{\mu_m^j,\xi_m^j}\rangle = c^{jh}_m \Big(c_h(1+o(1))\Big)+ O(1) \sum\limits_{l=0, l\neq h}^nc^{jl}_m + o\bigg(\Big(\frac{\epsilon}{|\ln\epsilon|^2}\Big)^{\frac{n}{n-2}}\bigg) \sum\limits_{i=1, i\neq j}^k\sum\limits_{l=0}^n c^{il}_m.
[z^n] \frac{\partial^2}{\partial x_1^2}(P_1+P_2+P_3) |_{x_1=x_2=x_3 = 1}= \frac{1}{n-1}[t^n] 2(n-1)(2n-3)t^2 (1+t)^{4n-6}&= 2(2n-3) [t^{n-2}](1+t)^{4n-6}= 2(2n-3) \binom{4n-6}{n-2}.
J_\nu(z) = \frac{{\rm i}^{-\nu}}{\pi}\int_0^{\pi}e^{{\rm i}z\cos(\theta)}\cos( \nu\theta) d\theta = \frac{{\rm i}^{-\nu}}{2\pi}\int_0^{2\pi}e^{{\rm i}z\cos(\theta)}\cos( \nu\theta) d\theta,
G^{(p)}_r(t,u,v,z) = \frac{t^{r+1}z^ruv(v^p-1)}{v\delta_1-1}+\frac{t(z(v-1)-v)}{v\delta_1-1}G^{(p)}_r(t,u,1,z) +\frac{uv^{p+1}t}{v\delta_1-1}G^{(p)}_r(t,uv,1,z).
G_{L_{n-1}}(z) = \prod_{i=0}^{n-2}f_{n-1-i}^{K-1}(z), ~ f_k(z) = \begin{cases}q_k + p_k f_{k+1}^{K}(z), ~1 \le k \le n-1 z^{\frac{1}{K}}, ~k=n.\end{cases}
A^{(r)}(y) = F_0^{(r)}(y) A^{(r)}_{\underline{\chi(\O_S)}}(y) (-1)^{r}& = \frac{1}{[r]_y}\prod_{k\geq 1}\frac{1}{(1-q^{rk})^{10}(1-y^{-r}q^{rk}) (1 - y^r q^{rk})}& = \frac{y^\frac{1}{2}- y^{-\frac{1}{2}}}{\phi_{-2,1}(q^r,y^r)^\frac{1}{2}\, \tilde\Delta(q^r)^\frac{1}{2}}\,,
e^{kt}E_x[V(X_t)]-V(x) = E_x[g(X_t,t)]-E_x[g(X_0,0)] = E_x\Big[\int_0^t\big(e^{ks}\cdot\mathcal{A}V(X_s)+ke^{ks}\cdot V(X_s)\big)ds\Big] \le E_x\Big[\int_0^t\big(e^{ks}\cdot \big(-kV(X_s)+M\big)+ke^{ks}\cdot V(X_s)\big)ds\Big] = \mathbb{E}_x\Big[\int_0^tMe^{ks}ds\Big] = \frac{M}{k}e^{kt}- \frac{M}{k}\le \frac{M}{k}e^{kt}.
\|\overline{u}^{\alpha+1}\|_{L^{m^\star}(\Omega_{R_1})}\le C\left[\frac{1}{R_1-R_2}+\left(\frac{G(R_1,R_2)}{m\alpha+1-Cm\epsilon^{m'}}\right)^{1/m}\right]\|\overline{u}^{\alpha+1}\|_{L^{m}(\Omega_{R_2})}+C\left(\frac{\|f\|_q}{(m\alpha+1-Cm\epsilon^{m'})k^{m-1}}\right)^{N/m(mq-N)}\|\overline{u}^{\alpha+1}\|_{L^{m}(\Omega_{R_2})}.
\left( -\frac{d^{2}}{d\theta^{2}}+\frac{(C_{m,n_1,n_2}-\frac{1}{4})}{\sin^{2}\theta}+\frac{g}{6 \cos^2\theta}\right) \Theta_{j,m,n_1,n_2}(\theta )=D_{j,m,n_1,n_2}\Theta_{j,m,n_1,n_2}(\theta ),
(p_m-k_m) \,|\, s_m^\wedge = (p_m-k_m) \,|\, (s_m-i\pi k_m) = -i \pi \, (p_m-k_m) s_m^\vee \,|\, (p_m-k_m) = (s_m+i\pi k_m) \,|\, (p_m-k_m) = i \pi \, (p_m-k_m) s_m^\vee \,|\, s_m^\wedge = (s_m+i\pi k_m) \,|\,(s_m-i\pi k_m) = 2 \pi^2 \,k_m \:.
\frac{\partial\tilde{\eta}_{1}}{\partial\tilde{t}}+\frac{\partial\tilde{u}_{1}}{\partial\tilde{x}}+\tilde{\eta}_{0}\frac{\partial\tilde{u}_{0}}{\partial\tilde{x}}+\tilde{u}_{0}\frac{\partial\tilde{\eta}_{0}}{\partial\tilde{x}}=0
\rho^Lu_1^L-\rho^Ru_1^R =S (\rho^L-\rho^R), p_{11}^L+\rho^L(u_1^L)^2-p_{11}^R-\rho^R(u_1^R)^2 =S(\rho^Lu_1^L-\rho^Ru_1^R).
H(X,Y|U'')-H(Y|U'') & = H(X,Y|U',T')-H(Y|U',T') & \overset{(a)}{=}H(X,Y|U,T')-H(Y|U,T') & \overset{(b)}{=}H(X,Y|U,T)-H(Y|U,T) & \overset{(c)}{=}H(X,Y|U)-H(Y|U),
d h_{0,2}=s_{0,2}d k_0 &=s_{0,1}d k_0+\psi_{0,1}^*(s_{1,2}d k_1) &=d h_{0,1}+\psi_{0,1}^*d h_{1,2}&=d(h_{0,1}+\psi_{0,1}^*h_{1,2}).
x(t,y) = x_0(y) \exp \big( \int_0^t \{\pi(s,y) b_0(s,y) + \lambda \ln(1+\pi(s,y)\gamma_0(s,y)) - \lambda \pi(s,y) \gamma_0(s,y) \}ds &+ \int_0^t \ln(1+ \pi(s,y) \gamma_0(s,y)) d\tilde{N}(s) \big).
\mathcal{K}_{\mathcal{I}_l}=\widetilde{\mathcal{K}}^{j}\cdot\widetilde{\mathcal{K}}_{{\mathcal{I}_l},j}=\widetilde{\mathcal{K}}^{j,k}\cdot\widetilde{\mathcal{K}}_{{\mathcal{I}_l},j,k}.
[\begin{matrix}N_{11}\\N_{22}\\N_{12}\end{matrix}]=\frac{2hE}{(1-\nu^{2})}[\begin{matrix}1&\nu&0\\ \nu&1&0\\ 0&0&1-\nu\end{matrix}][\begin{matrix}u_{1,1}^{0}\\ 0\\ 0\end{matrix}]
V(t,\xi):=\inf_{u \in\mathcal{U}}J(t,\xi,u),\textnormal{where}J(t,\xi,u):=\mathbb{E}^{\gamma_0}\left[ \int_{0}^{t}\Vert \mathbf{Z}_{r}^{u,t,\xi}(\theta)\Vert^2\hat{q}(dr,d\theta) \right].
\mathcal{L}_{\lambda}q(t)=\frac{1}{\pi}\int_0^{+\infty}e^{\lambda(iz^2t-iz x)}z\, \widehat{q}(\lambda z^2)dz,\lambda=\pm 1, \intertext{and}\widehat{q}(\tau)=\mathcal{F}(\chi_{(0,+\infty)}f)(\tau)=\int_{0}^{+\infty}e^{-i\tau t}f(t)dt,
\stackrel{.}{a_{1}}= \Gamma^{\rho}\sum_{i , j}T_{i}X_{0 , \rho}T_{j}e^{\left( m_{i}^{2}- m_{j}^{2}\right) t}- \sum_{i , j}T_{i}X_{1}T_{j}e^{\left( m_{i}^{2}- m_{j}^{2}\right) t}