harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
C_{i_1,\ldots,i_k}(1) = \sum_{r=0}^{i_1-1}C_{r,i_1-1-r,i_2,\ldots,i_k}(1) + \sum_{r=0}^{i_1-2}A_{r,i_1-2-r}C_{i_2,\ldots,i_k}+ 2\sum_{r=2}^k i_rC_{i_1+i_r-1,i_2,\ldots,\widehat{i_r},\ldots,i_k}
Z = \frac{1}{\mathrm{V O L ~ G}}\sum_{n}\int [ d g_{n}] e^{- i \theta n}~ \int \prod_{x}d A^{a}( x ) ~ < A \vert e^{- H / T}\vert g_{n}^{\dagger}A g_{n}+ i g_{n}^{\dagger}\nabla g_{n}>
\Phi(R,t):=\Phi_{\Psi}(R,t):=\sup_{r\in(0,\infty)}\Bigl\{\frac{R}{r}-\frac{t}{\Psi(r)}\Bigr\}=\sup_{\lambda\in(0,\infty)}\Bigl\{\frac{R}{\Psi^{-1}(\lambda^{-1})}-\lambda t\Bigr\}.
d s^{2}= - \frac{2 Q^{2}}{\gamma^{4}}\left( 1 - \frac{4 M}{r}\right) d t^{2}+ \frac{2 Q^{2}}{r^{2}}\left( 1 - \frac{4 M}{r}\right)^{- 1}d r^{2}+ 2 Q^{2}d \Omega^{2}
D_{p}(\delta) \leq C^{N^2}(D_{p}(\delta^{1 - \frac{1}{2^{N + 1}}}) + \delta^{-\frac{1}{2^{N + 1}}(1 + \frac{2}{p}\sum_{j = 0}^{N}(\frac{2}{p - 2})^{j})}\prod_{j= 0}^{N}D_{p}(\delta^{1 - \frac{1}{2^{j + 1}}})^{\frac{p - 4}{p - 2}(\frac{2}{p - 2})^{N - j}}).
B_{2.1}(x) = \sum_{n \geq 0}\frac{\nu^{n}}{n!}\frac{\Gamma(\chi_{2}+1)\Gamma( \frac{n \kappa}{k}+ \frac{1}{k}+\nu_{2}+1 )}{\Gamma( \frac{n \kappa}{k}+\frac{1}{k}+\nu_{2}+ \chi_{2}+ 2)}x^{\frac{n \kappa}{k}+ \frac{1}{k}+\nu_{2}+ \chi_{2}+ 1}
V_{+}= \bigoplus_{j \in J_+}V_{\lambda_{j+1}}^{\lambda_j}(s_{j}), \V_{-}= \bigoplus_{j \in J_-}V_{\lambda_j}^{\lambda_{j+1}}(s_{j}), J_+ = \{\j \ | \ 0 \leq j \leq \ell, \lambda_{j}\geq \lambda_{j+1}\ \}, \J_- = \{\j \ | \ 0 \leq j \leq \ell, \lambda_{j}< \lambda_{j+1}\ \}.
\nabla^0_k=d-\sum_{1\leq i<j\leq k}\theta^0_{ij}\t_{ij}+\cdots,\ \theta^0_{ij}=\frac{1}{2\pi i}d\log(z_i-z_j)=\frac{1}{2\pi i}\frac{dz_i-dz_j}{z_i-z_j},
\begin{gathered}(1+X^1+X^{1+1}+\ldots)(1+X^2+X^{2+2}+\ldots)\ldots(1+X^k+X^{k+k}+\ldots) =\frac{1}{1-X}\frac{1}{1-X^2}\ldots\frac{1}{1-X^k}=\frac{1}{X^k!}.\end{gathered}
T_{ik}=(1-2\zeta )\partial_{i}\varphi \partial_{k}\varphi+(2\zeta -1/2)g_{ik}\partial^{l}\varphi \partial_{l}\varphi-2\zeta \varphi \nabla_{i}\nabla_{k}\varphi +(1/2-2\zeta)m^{2}g_{ik}\varphi^{2}.
nR_i \leq I(W_i;\textbf{Y}_i^n,\Lambda^n|\Omega^n,W_{i+1},\ldots,W_K) &=I(W_i;\textbf{Y}_i^n|\Omega^n,\Lambda^n,W_{i+1},\ldots,W_K) &\ \ \ +\underbrace{I(W_i;\Lambda^n|\Omega^n,W_{i+1},\ldots,W_K)}_{=0}
\langle \theta_2,\theta_1 |T_{\mu}^{\mu}| \theta_1,\theta_2\rangle_{\rm sym}= 8\pi m^2 \left(\frac{2 \sin\pi\alpha \cosh\theta_{12}}{\sinh^2\theta_{12}+ \sin^2\pi\alpha}\right)\cosh^2\frac{\theta_{12}}{2}\,\,\, .
A=[\begin{matrix}0&0&0&1\\ 0&0&1&0\\ 0&1&0&0\\ 1&0&0&0\\ 0&0&0&0\end{matrix}][\begin{matrix}4&0&0&0\\ 0&3&0&0\\ 0&0&2&0\\ 0&0&0&1\end{matrix}][\begin{matrix}0&0&0&1\\ 0&0&1&0\\ 0&1&0&0\\ 1&0&0&0\end{matrix}],
q_{\sigma}\left( x,t\right) =q\left( x,t\right) & +2\left[ \int\psi_{\sigma}\left( x,t;\mathrm{i}s\right) \psi\left(x,t;\mathrm{i}s\right) \mathrm{d}\sigma_{t}\left( s\right) \right]^{2}+4\int\psi_{\sigma}\left( x,t;\mathrm{i}s\right) \psi^{\prime}\left(x,t;\mathrm{i}s\right) \mathrm{d}\sigma_{t}\left( s\right) ,
\mu_{1}(\varphi)-\mu_{1}(\frac{\pi}{2}) =\frac{\cos\varphi}{1-\cos^{2}\varphi}\Big(-2R(x, qx, q^{2}x, x) &+2(\cos\varphi) R(x, qx, qx, q^{2}x) &-(\cos\varphi) R(qx, q^{2}x, q^{3}x, x) &-2R(qx, q^{2}x, q^{2}x, x)\Big),
A_{L}^{\mu \nu}= - g^{\mu \nu}[{\frac{7 \zeta ( 3 )}{4 \pi^{4}}}\Lambda^{2}+ k^{2}\sum_{n = 0}{\frac{( 2^{2 n - 1}- 1 ) n ! B_{2 n}\pi^{2 n - 2}{\sqrt \pi}}{2^{2 n + 4}( 2 n ) ! \Gamma ( n + \frac{5}{2})}}\left({\frac{k^{2}}{\Lambda^{2}}}\right)^{n}] .
V_k(\lambda)^{-1}=\left[ \begin{array}{ccccc|c}-1 -\lambda^\ell -\lambda^{2\ell}\cdots -\lambda^{(k-1)\ell}\lambda^{k\ell}& -1 -\lambda^\ell \ddots \vdots \lambda^{(k-1)\ell}& -1 \ddots -\lambda^{2\ell}\vdots & \ddots -\lambda^{\ell}\lambda^{2\ell}& -1 \lambda^\ell & 1\end{array}\right].
=\frac{p(s+t,X(t,s,x))\partial_t L_X(t,s,x)}{1+p(s+t,X(t,s,x))L_X(t,s,x)}-\frac{L_X(t,s,x) p(s+t,X(t,s,x))u(s+t,X(t,s,x))}{1+p(s+t,X(t,s,x))L_X(t,s,x)}
0\rightarrow \mathrm{Hom}_A(T\oplus A, \ \mathrm{rad}(P_i))\xrightarrow{f_i}\mathrm{Hom}_A(T\oplus A,P_i)\rightarrow \mathrm{Hom}_A(T\oplus A,S_i)\rightarrow \rightarrow\mathrm{Ext}^1_A(T\oplus A,\mathrm{rad}(P_i))\rightarrow \mathrm{Ext}^1_A(T\oplus A,P_i)=0 .
- \frac{i}{2}g^{2}\left[p_{1}\cdot \varepsilon (k_{1}) p_{2}\cdot \varepsilon(k_{2}) C_{a_{1}b_{1}f}C_{a_{2}b_{2}f}+ p_{1}\cdot \varepsilon (k_{2}) p_{2}\cdot \varepsilon(k_{1}) C_{a_{2}b_{1}f}C_{a_{1}b_{2}f}\right]
\Pi_{\beta_i}=\{\delta_1-\delta_2,\ldots,\delta_{k'-2}-\epsilon_1, \epsilon_1-\epsilon_2,\ldots,\epsilon_{l-1}-\epsilon_l\cr \epsilon_l-\delta_{k'-1}(=-\beta_{i}),\cr \delta_{k'-1}-\epsilon_{n}(=\beta_{i+1}),\cr\epsilon_n-\delta_{k'},\delta_{k'}-\delta_{k'+1},\ldots, \delta_{m-1}-\delta_m\}l+1=n.
\mu_{+t}= \frac{2}{3}\frac{u_t}{u}\mu_+ -\frac{1}{3}\nu, \mu_{-t}= -\frac{2}{3}\frac{u_t}{u}\mu_- +\frac{1}{3}\nu, \nu_{t}= \frac{2}{3}u^2\mu_- +\frac{2}{3}\frac{\omega}{u^2}\mu_+
\Xi'_{\imath}:=\begin{cases}\xi(n-1)+\xi(n) \ \flat=> \ \imath=n, 2\Xi_{\imath},\end{cases}\Pi_{\imath}:=\begin{cases}(\imath, \Xi'_{\imath}) \ 1\leq \imath\leq n, (2n-\imath, \Xi'_{\imath}) \n+1\leq \imath \leq 2n-1.\end{cases}
\mathfrak{bc}=\mathfrak{cs}=\sum_{i=0}^{(s-1)/2}\binom{(d-1)/2+i}{\lfloor i/2 \rfloor}\left( \binom{(s+d-2)/2}{(d-1)/2+i}+ \binom{(s+d-4)/2}{(d-1)/2+i}\right).
\int e^f d\nu - 1 =\int \Phi(f)\, d\nu - \Phi \left( \int fd\nu \right) & = \inf_{t \in \mathbb{R}}\int \left(\Phi(f)- \Phi(t) - \Phi'(t)(f-t) \right) d\nu & \leq b\, \inf_{t \in \mathbb{R}}\int \left(\Phi(f)-\Phi(t) - \Phi'(t)(f-t)\right) d\mu \, = \,b\, \left( \int e^f d\mu - \exp \left\{\int f d\mu \right\}\right) .
\int_{\{\delta(x)\leq t\}}G_{|\Omega}^s(x,y)\,\delta(x)^{2-2s}dx\leq\left\lbrace\begin{aligned}C\,t^{2-2s'}\,\delta(y)^{2s'}\delta(y)\geq t & C\,t\,\delta(y) \delta(y)<t\end{aligned}\right.
\frac{P'_{k}(x_0)}{P_k(x_0)}-\frac{P'_{k-1}(x_0)}{P_{k-1}(x_0)}= \frac{1}{x_0-r_1}+ \sum_{j=1}^{k-1}\frac{r_{j+1}-s_j}{(x_0-r_{j+1})(x_0-s_j)}\, ,
m_{11}:=e^{(a+d)/2}\Big[\Delta \cosh{\frac{1}{2}\Delta}+(a-d)\sinh{\frac{1}{2}\Delta}\Big], m_{12}:=2be^{(a+d)/2}\sinh{\frac{1}{2}\Delta}, m_{21}:=2 c e^{(a+d)/2}\sinh{\frac{1}{2}\Delta}, m_{22}:=e^{(a+d)/2}\Big[\Delta \cosh{\frac{1}{2}\Delta}+(d-a)\sinh{\frac{1}{2}\Delta}\Big].
\overset{0}{R}ic (Y,Z) = \overset{g}{R}ic (Y,Z)- \frac{1}{2}{D}(AZ,Y) - \frac{1}{4}(\overset{2}{D}(AY,Z) + \overset{3}{D}(AY,Z)) + \frac{n-1}{4}\pi (Y) \pi (Z).
\limsup_{n \to \infty}\sup_{V \in \mathbb{V}_{n,k}}\mathbb{E}\Big\lVert\sum_{j=1}^n n^{1/p}\frac{Z_j}{(\lVert Z^{(n)}\rVert_p^p + W_n)^{1/p}}V_{\bullet,j}\Big\rVert_2 < \infty,
-\mathcal{A}[u_{0}]u_{1}= u_{0T}+ f'(u_{0})u_{0X}+ \mathcal{K}*_{\theta}u_{0X}- \mathcal{K}^{(1)}*_{\theta}(k_{X}u_{0\theta}+ ku_{0X\theta}) + \frac{1}{2}kk_{X}\mathcal{K}^{(2)}u_{0\theta\theta}.
R \left( \frac{s L_{1}^{2}}{4}; \mu_{s}^{2}L_{1}^{2}, \mu_{u}^{2}L_{1}^{2}, \mu_{t}^{2}L_{1}^{2}; m_{0}L_{1}, w \right) \equiv 1 6 \pi^{2}\frac{\sigma^{( \infty )}( s ) - \sigma^{( 0 )}( s )}{g \sigma^{( 0 )}( s )}.
G(z) = \sum_{k=1}^{M \wedge N}\frac{1}{\lambda_k-z}\left({\begin{array}{*{20}c}{\xi_k \xi^*_k}z^{-1/2}\sqrt{\lambda}_k \xi_k \zeta_k^* z^{-1/2}\sqrt{\lambda}_k \zeta_k \xi_k^* \zeta_{k}\zeta_{k}^* \end{array}}\right).
ds^2 = (H_1 H_5)^{-\frac{1}{2}}\left [ -dt^2 + dx^2 + H_1 \sum_{i=6}^{9}dx_i^2 \right ] + (H_1 H_5)^{\frac{1}{2}}[ dr^2 + r^2 d\Omega_3^2 ],
\displaystyle\sum\limits_{n=0}^{n_{12}+j}Q_n^{\mu_1}(x) Q_{n_{12}+j-n}^{\mu_2}(y) \langle n, n_{12}+j-n |n_{12},j\rangle = Q_{n_{12}}^{\mu_{12}}(z) \displaystyle\sum\limits_{n=0}^{j}Q_n^{\mu_1}(x) Q_{j-n}^{\mu_2}(y) \langle n, j-n | 0,j \rangle.
\begin{aligned}\lambda_m^{1,1}[g(\theta)] ={\rm tr}({\rm sym}^m[g(\theta)]) = \frac{\sin[(m+1)\theta]}{\sin \theta}, \log D(x,{\rm sym}^m[g(\theta)]) ={\rm tr}({\rm sym}^m[g(\theta)])x+O(x^2) (|x|<1).\end{aligned}
G ({\bf{x}}_{b},{\bf{x}}_{a}; E ) = \frac 1{\sqrt{r_{b}r_{a}}}\sum_{l_{D}}G ( r_{b}, r_{a}; E_{D}, l_{D}) \sum_{k = - l_{D}}^{l_{D}}Y_{l_{D}, k , q}( \theta_{b}, \varphi_{b}) Y_{l_{D}, k , q}^{\star}( \theta_{a}, \varphi_{a}) ,
\frac{1}{\hat{\Pi /}- M + i 0}= \int_{0}^{\infty}\! \! \! d T \! \int \! d \chi \, \exp \left[ \, - \frac{i}{2 \kappa_{0}}( M^{2}T + M \chi ) \, \right] \, \exp \left[ \, \frac{i}{2 \kappa_{0}}( \hat{\Pi /}^{2}T + \hat{\Pi /}\, \chi ) \, \right]
\begin{array}{rcl}L[n,y(n,m,\nu)] = \sum\limits_{k=-\infty}^{+\infty}G(n,k+1)f(k,L[k,y(k,m,\nu)]) & -\sum\limits_{k=-\infty}^{+\infty}G(n,k+1)g(k,y(k,m,\nu))+y(n,m,\nu).\end{array}
{\cal L}_{\mathrm{g f}}+{\cal L}_{\mathrm{g h o s t}}= - \frac{1}{2 \xi}( \partial_{\mu}A^{\mu}) \star ( \partial_{\nu}A^{\nu}) + \partial^{\mu}\bar{c}\star ( \partial_{\mu}c - i e \left[ A_{\mu}, c \right]_{\mathrm{M B}})
e^{-(q+b_0) t}\prod\limits_{j=1}^N (1-e^{-b_j t}) = e^{-(q+b_0) t}\sum\limits_{p=0}^N (-1)^p\sum\limits_{k_1<\cdots<k_p=1}^N\exp\bigl(-(b_{k_1}+\cdots+b_{k_p})t\bigr), = \bigl(\mathcal{S}_{N}e^{-xt}\bigr)(q|b),
\xi=\begin{pmatrix}-1 0 0 -1 \end{pmatrix}, \alpha=\begin{pmatrix}i 0 0 -i \end{pmatrix}, \beta=\begin{pmatrix}0 i i 0 \end{pmatrix}, \tau=\dfrac{1}{\sqrt{2}}\begin{pmatrix}1 i i 1\end{pmatrix}.
\begin{cases}v_i(x,t)=S_{d,i}(t,0)v_{i,0}(x) \forall\:i\in[1,p], v_i(x,t)=S_{d,i}(t,t_{i,ex}(x,t))S_{c,i}(t_{i,ex},t_{i,en}(x,t))S_{d,i}(t_{i,en}(x,t),0)v_{i,0}(x) \forall\:i\in[p+1,n]; \end{cases}
\left(\kern-.1em\left(\kern-.1em\nabla\widetilde{\psi}_{h},\left( \Pi^{\operatorname*{curl},\ast}-\Pi_{h}^{\operatorname*{curl},\ast}\right) \mathbf{v}_{h}\kern-.1em\right)\kern-.1em\right) =0\qquad\forall\widetilde{\psi}_{h}\in S_{h}.
\frac{\|u\|_{\widetilde W^r_p(\Omega)}^p}{\Lambda(r-1,d,p)}= 2 \iint_{A}\frac{|D u(x)- D u(y)|^p}{|x-y|^{d+(r-1)p}}\, dy dx + 2 \iint_{B}\frac{|D u(x)- D u(y)|^p}{|x-y|^{d+(r-1)p}}\, dy dx.
\frac{f(\omega h_j,\xi)}{g(\omega h_j)}= e^{\frac{\omega h_j}{2}(\xi+1)}- \frac{(-1)^{p+1}(\omega h_j)^{p+1}}{2g(\omega h_j)}\left[ R^-_{p+1}(\xi) + \sum_{k=1}^\infty \left(\frac{\omega h_j}{2}\right)^k R^{-,(-k)}_{p+1}(\xi) \right]
Z_i(x,\xi) \,\,\,= \d T (J_i)(x,\xi) = J_i(T) (T^{-1}(x,\xi) ) \stackrel{\eqref{sec.two_1:def_mapT}}{=}\Big(J_i(\pi),J_i(a\mapsto a_{j_1}),\ldots,J_i(a\mapsto a_{j_p}) \Big) (T^{-1}(x,\xi) ).
\sum_{i=0}^{n-1}\frac{\widetilde{p}_i(x)^2}{\widetilde{h}_i}= \frac{\widetilde{p}_{n}(x)^{\prime}\widetilde{p}_{n-1}(x)-\widetilde{p}_{n-1}(x)^{\prime}\widetilde{p}_{n}(x)}{\widetilde{h}_{n-1}}.
R_N^{SS}(z) =\frac{\Gamma(2N-1)}{2 \pi i}\sum_{n=1}^{\infty}\frac{1}{n}\Bigl( e^{-2 \pi n z i}\, \Gamma(2-2N, -2\pi n z i) & - \;\; e^{2 \pi n z i}\, \Gamma(2-2N, 2 \pi n z i ) \Bigr) \; .
\begin{array}{c c}{{}}&{{}}\\{{\tan \left( k_{2}a / 2 \right) = \mu_{1}^{\left( - \right)}/ \mu_{2},}}&{{\phi}}\\{{t e x t r m{e v e n}( B = B^{^{\prime}}, \; C = + A^{^{\prime}})}}\\{{}}&{{}}\\{{- \cot \left( k_{2}a / 2 \right) = \mu_{1}^{\left( - \right)}/ \mu_{2},}}&{{\phi}}\\{{t e x t r m{o d d}( B = - B^{^{\prime}}, \; C = - A^{^{\prime}})}}\\ \end{array}
\begin{array}{c}\displaystyle(2\pi)^2I_0(\tau,p)=\int_{\partial\Omega}dS_y\varphi(y,\tau) \displaystyle\times\int_{\partial D}e^{-\tau\,l_p(x,y)}\left\{\frac{H(x,p,\tau)}{\vert x-y\vert}-\frac{H(x,y,\tau)}{\vert x-p\vert}+2H(x,y,\tau)F(x,p,\tau)\right\}dS_x.\end{array}
p_{k,0}= \phi_n\left(\sum_{i}u_i^k\right),p_{k,1}= \frac{1}{k}\sum_i \left( (x_i+c_i)^k - x_i^k \right) = \frac{1}{k}\phi_n\left( \sum_{i}v_i^k - \sum_{i}u_i^k \right),
D =1-\big(205y+4300y^2+34000y^3+120000y^4+160000y^5\big)x \quad-\big(215y+4475y^2+35000y^3+122000y^4+160000y^5\big)x^2 \quad-\big(85y+1750y^2+13525y^3+46500y^4+60000y^5\big)x^3 \quad-\big(15y+305y^2+2325y^3+7875y^4+10000y^5\big)x^4 \quad-\big(y+20y^2+150y^3+500y^4+625y^5\big)x^5.
\Delta_{\alpha,\beta}(\rho)=\rho^{-1}\left((-1)^{\alpha}\left(\rho\cos\rho-W_{\alpha,\beta}(a,\rho)\right) +\int\limits_{0}^{1}(U_{\alpha,\beta}(t)\sin \rho t+V_{\alpha,\beta}(t)\cos \rho t)\,dt\right),
0 \le \sum_{i,j=1}^n \overline{c_i}\> c_j \> \Phi_{\tilde\alpha}(i,j)\beta(\pi(x_i,x_j)) &= \sum_{i,j=1}^n \overline{c_i}\> c_j \> \int_D\int_{A_i}\tilde K_h(x,A_j)\> d\tilde \omega_X(x)\cdot \tilde\alpha(h)\>\beta(\pi(x_i,x_j))\> d\tilde\omega_D(h).
E_N(f)(t) =\sum_{0\le k\le N}\| \nabla^k_xf\|^{2}_{L^2_{x,{p,\mathcal{I}}}},\cr\mathcal{D}_N(f)(t) =\|\{I-P\}f\|^2_{L^2_{x,p,\mathcal{I}}}+\sum_{1\le k\le N}\|\nabla^k_x f\|^2_{L^2_{x,p,\mathcal{I}}}.
{\mit \Gamma}_{\mathrm{c u s p}}( \theta ; \alpha_{s}) = - \frac{\alpha_{s}N_{c}}{\pi}\frac{1}{\sqrt{1 + x}}\left\{\sqrt{1 + x}- \left( 1 + \frac{x}{2}\right) \ln \frac{\sqrt{1 + x}+ 1}{\sqrt{1 + x}- 1}\right\},
-\dfrac{(1+h^{2}_q) \tilde\Psi u_p}{h^{3}_p}+\dfrac{h_q \tilde\Psi u_q}{h^{2}_p}+\left(-\dfrac{\tilde\Psi_p}{H^{3}_p}+\dfrac{1}{F^2}\rho \tilde\Psi-\dfrac{h^{2}_q \tilde\Psi_p}{h^{3}_p}-\left(\dfrac{1}{h^{3}_p}-\dfrac{1}{H^{3}_p}\right) \tilde\Psi_p \right)u=0.
\Theta_M(x):=\begin{cases}\Theta(M) x>M, \Theta(x) -M\leq x\leq M, \Theta(-M) x<-M,\end{cases}K_M(x):=\begin{cases}K(M) x>M, K(x) -M\leq x\leq M, K(-M) x<-M.\end{cases}
\frac{1}{(2\pi i)^2}\int_{1/\log{X}-i X^4}^{1/\log{X}+i X^4}\Bigl(\frac{X-1/2}{n_1n_2p}\Bigr)^{s}\frac{d s}{s}=\begin{cases}1+O(X^{-2}), O(X^{-2}), \end{cases}
S=\frac{1}{\beta^2}\int d^4x~ d^2\theta~ d^2\bar{\theta}~tr\left[-\frac{1}{2}(e^{-V}D^\alpha e^V)\bar D^2(e^{-V}D_\alpha e^V)-e^{-V}\bar\Sigma e^V \Sigma\right] \, .
[ \psi_{\uparrow}( z_{1}) , \psi_{\downarrow}( z_{2}) ]_{+}= \sum_{n = 1}^{\infty}n ( z_{1}^{n - 1}z_{2}^{- n - 1}- z_{1}^{- n - 1}z_{2}^{n - 1}) = \d^{\prime}( z_{1 2}) \ ,
I \left( a \right) = \inf_{\gamma \in \mathcal{R}\left( t \right)}\left[ \ell \left( \gamma ; a \right) + \psi \left( \gamma \right) \right] = \inf_{\gamma \in \left\lbrace \psi < \infty \right\rbrace}\left[ \ell \left( \gamma ; a \right) + \psi \left( \gamma \right) \right].
S_2 \ll \left(\sum_{j=0}^{\lceil \log_{b}{N}\rceil-1}b^{j\left(r-\frac{1}{p}+1\right)q}\left( b^j\left(\frac{1}{N}\frac{1}{b^j}\right)^p\right)^{\frac{q}{p}}\right)^{\frac{1}{q}}=\frac{1}{N}\left(\sum_{j=0}^{\lceil \log_{b}{N}\rceil-1}b^{jqr}\right)^{\frac{1}{q}}.
\sum_{i=l}^{\infty}\bigl (|\hat\gamma_{k, i}| + |\tilde\gamma_{k, i}| + |\gamma_i|\bigr)\le3 \sum_{i=l}^\infty\sum_{j=0}^\infty \delta_{j, p}\delta_{j+i, p}\le3 \Theta_{0, p}\Theta_{l,p},
w\left( x,s\right) =w_{0}\left( x_{3},s\right) +\widehat{w}\left( x,s\right),\widehat{w}\left( x,s\right) \in C^{2+\alpha}\left( \mathbb{R}^{3}\right) ,\ \forall s\geq \underline{s}\left( d\right) .
\mathcal{J}_3 \! = \! \bigl\{\emptyset, \{1\}, \{2\}, \{1,1\}, \{1,2\}, \{2,1\}, \{2,2\}, &\{1,1,1\},\! \{1,1,2\},\!\{1,2,1\},\!\{1,2,2\},\!\{2,1,1\},\!\{2,1,2\},\!\{2,2,1\},\!\{2,2,2\}\bigr\}.
n^{2\theta}\mathbb{P}(H_n^c \cap \{t(f_i) < n^{\theta}\}) \leq n^{2\theta}\mathbb{P}(H_n^c) \leq n^{2\theta}\frac{C}{n^{1+2\epsilon}}= \frac{C}{n^{2\epsilon-2\theta+1}}
N_c^R: = \big\{y\in N: \exists (y_i)_{i=1, ..., n}\in N, y_1 = y, y_n=x, (y_i, y_{i+1})\in E\backslash c \big\}, E_c^R: = \big\{(y,z)\in E: y,z\in N_c^R \big\}, H_c^R:= \big\{h\cap N_c^R: h\in H \big\}\backslash \{\emptyset \}, L_c^R: N_c^R \to \{1, ..., d\}, L_c^R = L|_{N_c^R}
R_{X Y W}{}^{Z}= f_{W}^{i A}f_{i B}^{Z}{\cal R}_{X Y A}{}^{B}\, , \qquad{\cal R}_{X Y A}{}^{B}= \frac 1 2 f_{i A}^{W}f_{Z}^{i B}R_{X Y W}{}^{Z}\, .
\alpha_{(1)}^{(1)}= -\frac{\sqrt{2}}{6}\sigma d (K_{\gamma}(t)-\sqrt{2}b K_{\phi}(t) + f_{\gamma}(t) -\sqrt{2}f_{\phi}(t)) +P^{\prime}(0,t) -\frac{\kappa^2}{6}(\rho + U )d e^{\sqrt{2}b \hat{\phi}}.
\Bigg\|\sum_{i=1}^{n}a^j_i e_i + x' + \sum_{i=1}^{n}y_i\Bigg\| \geq \theta \Bigg(\|x'\| + \sum_{i=1}^{n}\|y_i\|\Bigg) > \theta \Big(\|x'\| + n(1-\varepsilon)\Big).
\begin{array}{ll}{\rm{maximize}}\sum\limits_{i=1}^m S_i^2{\rm{subject \to}}\sum\limits_{i=1}^{m}S_i = \m 2^k \mathbb{E}[Y] \mathbb{E}[Y] 2^k - d \ \leq S_i \leq \ \mathbb{E}[Y] 2^k + d, \ \forall i \in [m].\end{array}
B_{b}^{a i}={( \frac{\partial^{2}A}{\partial \phi^{i}\partial \phi^{j}})}^{- 1}( \frac{1}{4}\lambda_{b j}^{a}- \frac{1}{2}\frac{\partial^{3}A}{\partial \phi^{j}\partial \phi^{k}\partial \phi^{p}}( \bar{\psi}_{a}^{k}\psi^{b p}+ \bar{\psi}^{b k}\psi_{a}^{p}) ) ,
D_{jj^{\prime}}^{\alpha\beta}=\frac{\hbar^{2}}{2m_{0}}[\delta_{jj^{\prime}}\delta_{\alpha\beta}+\sum_{\gamma}^{B}\frac{p_{j\gamma}^{\alpha}p_{\gamma j^{\prime}}^{\beta}+p_{j\gamma}^{\beta}p_{\gamma j^{\prime}}^{\alpha}}{m_{0}(E_{0}-E_{\gamma})}].
sup_{y^{*}\in Y^{*}}-F^{*}(0,y^{*})=sup_{y^{*}\in Y^{*}}inf_{x\in X}L(x,y^{*})\le inf_{x\in X}sup_{y^{*}\in Y^{*}}L(x,y^{*})=inf_{x\in X}F(x,0).
\tau = \langle 0,\xi,0,\xi-1,0,\xi-2,0,\dots,2,0,1,\underline{-1},0,-2,0,\dots,-(\xi-1),0,-\xi,0,-(\xi+1), 0,-\xi,0,-(\xi-1),0,\dots,-2,0,-1,1,0,2,0,\dots,\xi-1,0,\xi,0 \rangle.
S_{n}^j(t)=\frac{1}{(n\sigma^2)^{1/2}}\left( \log \|M^j_{\lfloor tn \rfloor}\|-nt\Lambda + (nt-\lfloor nt \rfloor) (\log \|M^j_{\lfloor tn \rfloor +1}\| - \log\|M^j_{\lfloor tn \rfloor}\|) \right).
\begin{array}{l}\mathsf{p}_{(0,0)}(1)=3p^2,\mathsf{p}_{(0,0)}(x^r) = 4\cos^2\left(\frac{2\pi r}{p+1}\right)-1, \mathsf{p}_{(0,0)}(y^s) = 1-4\cos \left(\frac{2(p-1)\pi s}{p^2+1}\right) \cos\left( \frac{2(p+1)\pi s}{p^2+1}\right).\end{array}
\begin{aligned}p\big| \sigma_r \big| \big(1+\sigma_r \big) = p\big| \big( \sigma_{r}^{(2)}+\cdots + \sigma_{r}^{(r-1)}\big) \big|\sigma_r &=\sum_{r-1\ge i\ge j \ge2}p\big|\big(\sigma_{r}^{(j)}\sigma_{r}^{(i)}+\sigma_{r}^{(i)}\sigma_{r}^{(j-1)}\big)=0,\end{aligned}
(gf) \cdot\tau = \alpha(r,b,c+e)\gf\ \tau + \beta(r,b,c+e)\d(gf)\wedge i_R\tau = &= \alpha(r,b,c+e)\gf\ \tau + \beta(r,b,c+e)g\df\wedge i_R\tau + \beta(r,b,c+e)f\dg\wedge i_R\tau.
b(t)=\begin{cases}t t\leq 0 \frac{t}{1-t}0\leq t\leq \frac{1}{2}3-\frac{1}{t}\frac{1}{2}\leq t\leq 1 t+1 1\leq t \end{cases}c(t)=\begin{cases}\frac{2t}{1+t}0\leq t\leq 1 t \end{cases}
{\cal P}({\cal E}) = \sum_{\vert \Psi \rangle}\vert{\cal A}({\cal E}) \vert^{2}= \int_{- \infty}^{\infty}d \tau \, \int_{- \infty}^{\infty}d \tau^{\prime}\, e^{- i{\cal E}( \tau - \tau^{\prime})}\,{\tilde{G}}\left[{\tilde{x}}( \tau ) ,{\tilde{x}}( \tau^{\prime}) \right] ,
t_k(\eta_{s_k(x)}) = \eta_{s_{k-1}(x)}\star_{k-1}\cdots \star_1 \eta_{s_0(x)}\star_0 \iota\pi(s_k(x)) &= s_k(\eta_{s_{k-1}(x)}\star_{k-1}\cdots \star_1 \eta_{s_0(x)}\star_0 \iota(x)),
f^B_{1,3}=(x_1-x_2)(x_1-x_3)x_1, f^B_{2,5}=(x_1-x_3)(x_1+x_3)x_1^2+(x_2-x_3)(x_2+x_3)x_2^2, f^B_{3,4}=x_1^2+x_2^2+x_3^2.
\det \begin{pmatrix}(2g)^{l_1}(l_1-\frac{m}{2})(-1)^{l_1}(-1)^{l_1}(2g)^{l_2}(l_2-\frac{m}{2})(-1)^{l_2}(-1)^{l_2}(2g)^{l_3}(l_3-\frac{m}{2})(-1)^{l_3}(-1)^{l_3}\end{pmatrix}\neq 0.
\left[{\frac{d^{2}}{d r^{2}}}+{\frac{2}{r}}{\frac{d}{d r}}-{\frac{( l ( l + 1 ) + 2 )}{r^{2}}}\right] b_{l}+{\frac{2 l ( l + 1 )}{r^{3}}}c_{l}= M^{2}b_{l},
A_{K}^{N S N S}={\frac{1}{2}}\int_{0}^{\infty}{\frac{d t}{2 t}}( 4 \pi^{2}t )^{- 5}{\frac{f_{4}( e^{- 2 \pi t})^{8}}{f_{1}( e^{- 2 \pi t})^{8}}},
\sum_{j=1}^\infty\|b_j\|_b^2 =\sum_{j=1}^\infty\|q_j\|_{H^2}^2+\sum_{j=1}^\infty \Big\|\Big ( q - \sum_{k = 1}^{L}q_k \widetilde{\phi_k}\Big)e_j \Big\|_{H^2}^2.
E_{0}=(\begin{matrix}1\\ 0\\ 0\\ 0\end{matrix}),E_{1}=(\begin{matrix}0\\ 1\\ 0\\ 0\end{matrix}),E_{2}=(\begin{matrix}0\\ 0\\ 1\\ 0\end{matrix}),E_{3}=(\begin{matrix}0\\ 0\\ 0\\ 1\end{matrix})
A \nabla_{\nu}B \geqslant A \sharp_{\nu}B+\sum_{k=0}^{\infty}r_{k}[ A\sharp_{\frac{m_k}{2^k}}B -2A\sharp_{\frac{2m_k+1}{2^{k+1}}}B +A\sharp_{{\frac{m_k+1}{2^k}}}B].
a_3^{56}=a_1^{57}=:x_2,a_2^{36}=z_2^{26}=z_3^{27}=a_4^{57}=:y_3,z_3^{16}=z_4^{17}, &a_1^{57}-z_4^{17}=c^{15}=-a_3^{56}+z_3^{16}, &a_1^{57}-a_4^{57}=z_3^{35}=-a_2^{36}+a_3^{56}.
\lambda ( \vec{k}) \, = \, i \frac{{\vec{\epsilon}}_{L}( 0 ) \cdot \vec{k}}{{\vec{k}}^{2}}\, + i \, \frac{{\vec{\epsilon}}_{L}( 0 ) \cdot \vec{k}}{( \omega_{\vec{k}}\, + \, | \theta | ) \, | \theta |}.
\frac{(n+\beta+1)}{(2n+\alpha+\beta+2)}Q_n(x) +\frac{(n+\alpha+1)}{(2n+\alpha+\beta+2)}Q_{n+1}(x)= \frac{(N-x)}{N}\hat Q_n(x) , \frac{(n+1)(n+\alpha+\beta+N+2)}{(2n+\alpha+\beta+3)}\hat Q_n(x) + \frac{(N-n-1)(n+\alpha+\beta+2)}{(2n+\alpha+\beta+3)}\hat Q_{n+1}(x)= N Q_{n+1}(x) .
\tilde{\Phi}_{1}= | \Phi | \cos \alpha \ , \ \ \, t i l d e{\Phi}_{2}= | \Phi | \sin \alpha \ , \ \ \| \Phi | = \sqrt{\tilde{\Phi}_{1}^{2}+ \tilde{\Phi}_{2}^{2}}= \sqrt{\Phi_{1}^{2}+ \Phi_{2}^{2}}\ .
f_{1}\sim \exp \left( - \mu k_{1}( \zeta_{W}( u + u_{\infty}) + \zeta_{W}( u - u_{\infty}) ) + C u \right) \prod_{\alpha}\frac{\sigma ( u - u_{\alpha}) \sigma ( u - u_{\alpha}^{\prime})}{\sigma ( u - u_{\infty}) \sigma ( u + u_{\infty})}.
\left[ e^{i\gamma \varphi}\right]_{n}(\underline{\theta}(\lambda))=\left[ e^{i\gamma \varphi}\right]_{m}(\underline{\theta}^{\prime})\,\left[ e^{i\gamma \varphi}\right]_{n-m}(\underline{\theta}^{\prime\prime})+O(e^{-\lambda})\,.
F_{\log,\Lambda}(x) :=\sum_{v\in\Lambda}\int_1^\infty e^{-|x+v|^2t}\frac{\mathrm{d}t}{t}+\frac{1}{|\Lambda|}\sum_{w\in\Lambda^*\setminus\{0\}}e^{2\pi i w\cdot x}\int_0^1 \frac{\pi^{\frac{d}{2}}}{t^{\frac{d}{2}}}e^{-\frac{\pi^2|w|^2}{t}}\frac{\mathrm{d}t}{t}.
n^2(x)_{\lambda_A\psi_A+\lambda_B\psi_B}=\,|\lambda_A|^2\,n^2_A(x)+\,|\lambda^2_B|n^2_B(x)+\,N\,2 \,\Re (\lambda_A\lambda^*_B\langle\psi_A|\psi^*_B\rangle).
\frac{h_{11t}+ h_{11}}{h_{t}+h}-\frac{( h_{1}+h_{1t})^{2}}{(h_{t}+ h)^{2}}&=\sum w^{ii}\nabla_{11}w_{ii}-\sum w^{ii}w^{kk}(\nabla_{1}w_{ik})^{2}+\nabla_{11}\chi.
\partial_t \left( \rho - \frac{1}{\rho}\right) + \nabla_x\cdot (J_1 - J_2) = 0 \,,\frac{1}{\rho}J_1 = -D_1\nabla_x\rho \,, \rho J_2 = -D_2\nabla_x \frac{1}{\rho}\,,
\|u\|_{\tilde{L}^1_T(B^{s+1}_{2,2})}\leq C_2(\nu)\|e^{\nu t\Delta}u_0\|_{\widetilde{L}^1_T(\dot{B}^{s+1}_{2,2})}+TM_0^{\frac{1}2}+C_2(\nu)\int^T_0\|b\|^2_{B^{s}_{2,2}}\,dt &+C_2(\nu)T^{\frac{r-2}{2r}}\|u\|^{\frac{r-2}{r}}_{\widetilde{L}^{\infty}_T(B^{s-1}_{2,2})}\|u\|^{1+\frac{2}{r}}_{\widetilde{L}^2_T(B^{s}_{2,2})}.
X_{1}Y_{2}X_{3}=[\begin{matrix}c_{2}&s_{2}s_{3}&c_{3}s_{2}\\s_{1}s_{2}&c_{1}c_{3}-c_{2}s_{1}s_{3}&-c_{1}s_{3}-c_{2}c_{3}s_{1}\\ -c_{1}s_{2}&c_{3}s_{1}+c_{1}c_{2}s_{3}&c_{1}c_{2}c_{3}-s_{1}s_{3}\end{matrix}]
\frac{1}{\beta_L}\mathbb{E}\bigg(G^{\Lambda}(z_L;n,n)\bigg) \leq \frac{1}{Imz}~\pi\bigg(1+\frac{k}{2}\bigg)S_\mu\bigg(\frac{2~Imz_L}{k}\bigg),~~\Lambda=C_p,~\Lambda_L &\leq C ~\bigg(\frac{2~\beta_L^{-1}~Imz}{k}\bigg)^{\alpha}&\leq C ~(2L+1)^{-d},~~(\mathrm{since}~~ \beta_L=(2L+1)^{d/\alpha}).
I_{4}= \frac{1}{1 6 \pi G}\int d^{4}x \sqrt{- g}e^{- \eta}[ R_{g}+ g^{\mu \nu}\partial_{\mu}\eta \partial_{\nu}\eta - \frac{1}{1 2}g^{\mu \sigma}g^{\nu \lambda}g^{\rho \tau}H_{\mu \nu \rho}H_{\sigma \lambda \tau}
\Lambda\triangle e_\lambda =-\lambda^3 e_\lambda=-\lambda\sqrt[3]{\Theta^2}e_\lambda=\frac{\lambda\theta}{\sqrt[3]{2}}e_\lambda-\frac{\lambda P_1'}{\sqrt[3]{2}}e_\lambda=\frac{\lambda}{\sqrt[3]{2}}\triangle e_\lambda-\frac{\lambda P_1'}{\sqrt[3]{2}}e_\lambda,
g(fX,Y) =g(X,f^{\ast}Y), g(fX,f^{\ast}Y) =g(X,Y)-\widetilde{g}(hX,h^{\ast}Y), \widetilde{g}(hX,V) =g(X,t^{\ast}V), \widetilde{g}(h^{\ast}X,V) =g(X,tV), \widetilde{g}(sU,V) =g(U,s^{\ast}V), \widetilde{g}(sU,s^{\ast}V) =\widetilde{g}(U,V)-g(tU,t^{\ast}V).
deg_{LS}(I-(P+QN_{f}- \frac{B_{\varphi,b}P}{T}+HP) ,\Omega,0) & = deg_{B}\left(I-(P+QN_{f}- \frac{B_{\varphi,b}P}{T}+HP)\left|_{\overline{\Omega \cap \mathbb{R}^{2}}}\right. ,\Omega \cap \mathbb{R}^{2}, 0\right) &=deg_{B}(G,\Omega \cap \mathbb{R}^{2},0)\neq 0.
q(1,1) =d_1^{-1}d_{32-1}=d_1^{-1}d_{31}, q(1,3) =d_3^{-1}d_{32-3}=d_3^{-1}d_{29}, q(1,5) =d_5^{-1}d_{32-5}=d_5^{-1}d_{27}, q(1,7) =d_7^{-1}d_{32-7}=d_7^{-1}d_{25}, q(1,9) =d_9^{-1}d_{32-9}=d_9^{-1}d_{23}, q(1,11) =d_{11}^{-1}d_{32-11}=d_{11}^{-1}d_{21}, q(1,13) =d_{13}^{-1}d_{32-13}=d_{13}^{-1}d_{19}, q(1,15) =d_{15}^{-1}d_{32-15}=d_{15}^{-1}d_{17};
T_{[ 2 ]}= \frac{1}{4 \pi}\int d^{2}x \left\{2 i \epsilon_{i j}\partial_{i}( \Psi^{\dagger}\partial_{j}\Psi ) + \epsilon_{i j}\partial_{i}\left[ ( 2 w - \Psi^{\dagger}\sigma^{3}\Psi ) A_{j}\right] \right\}.
\begin{matrix}{{N_{A}}^{B}={M^{B}}_{A}( - 1 )^{\varepsilon_{A}( \varepsilon_{B}+ 1 )},}&{{( N^{- 1})_{A}}^{B}={( M^{- 1})^{B}}_{A}( - 1 )^{\varepsilon_{A}( \varepsilon_{B}+ 1 )}.}\\ \end{matrix}
{\frac{3 2 \pi^{3}}{\alpha^{\prime}\kappa^{2}}}\cdot{\frac{1}{2 \pi \alpha^{\prime}}}\cdot{\frac{\alpha^{\prime}}{2}}\cdot \kappa^{2}\tau_{3}= 4 \pi^{2}\tau_{3}\cdot{\frac{2}{\alpha^{\prime}}}.
\: g_{c}^{2}\sim \frac{4}{\kappa}\, [ 1 - \frac{\kappa^{2}\lambda^{2}}{M^{2}}] \, , \hspace{0 . 5 c m}y_{0}\sim \frac{1}{\kappa \sqrt{\kappa}\,}\, M \, , \hspace{0 . 5 c m}\alpha \sim - 2 \, \frac{\kappa \lambda}{M}\, , \:
\begin{bmatrix}Q_0-\Tilde{A}X row(Q_j^*)_{j\geq m+1}X^* (Q_{i-j}\otimes I_d)_{i=1,j=1}^m-S(m-1)\otimes I_d (Q_{i-j}\otimes I_d)_{i=1,j=m+1}^{m+1,\infty}col(Q_j)_{j\geq m+1}(Q_{i-j}\otimes I_d)_{i=m+1,j=1}^{\infty,m}T_Q\otimes I_d^{m+1}\end{bmatrix}