harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
{\cal G}_{\alpha_{1}, \ldots , \alpha_{n}}^{( A )}( x_{1}, \ldots , x_{n}) = \left({\frac{A}{\pi}}\right)^{( \sum \alpha_{i}- Q ) / b}{\frac{F_{\alpha_{1}, \ldots , \alpha_{n}}( x_{1}, \ldots , x_{n})}{\Gamma ( ( Q - \sum \alpha_{i}) / b )}}\, .
\theta^{\rm N}_n\big((x, y, z)\big) = \theta^{\rm LB}_n \Big(\big( (x, y), (x, \langle x,y,z \rangle )\big) \Big) &= \theta^{\rm LB}_n \Big(\big( (x, y), (x, -x+y+z )\big) \Big) &=2^{-2}(x-y) \frac{\big(x-y+2(-x+y+z)\big)^n +\big( x+y\big)^n- 2\big( x+(-x+y+z)\big)^n}{n}&=2^{-2}(x-y) \frac{(-x+y+2z)^n +( x+y)^n- 2( y+z)^n}{n},
R_1 = - e^{-\varphi}\frac{\sinh \Theta}{\cosh \Theta}v F_k \varphi^k, R_2 = e^{-\varphi}\tilde{F}\frac{\sinh \Theta}{\Theta \cosh \Theta}v^{-1}|D \varphi|^4 \sinh^{-3}u\{u^3 \cosh u- u^2 \sinh u\}\geq 0.
\Gamma(\pi)\ge\frac{1}{N}\sum_{\ell=1}^N\left(\frac{\rho_\ell\beta_\ell^\pi}{2T_{\ell,\pi}^{av}}+\frac{\rho_\ell T_{\ell,\pi}^{av}}{2}+\rho_\ell\gamma_\ell+\frac{c_\ell}{T_{\ell,\pi}^{av}}\right),
a(r) = \frac{1}{(1-2r^2)^2}2\sqrt{1-2r^2}(l^2-|k|^2) &= \frac{1}{(1-2r^2)^{3/2}}\left( 1+2r^2(1-4Q_0^2)-4\sqrt{2}Q_0r^3\cos(3\alpha)\right),
\frac{\partial \widehat{w}^+(\vec{u})}{\partial u_k}=\sum_{i, j}x_i y_j \frac{\partial w_{ij}^{(1)}(\vec{u})}{\partial u_k}+\sum_{i, i', j, j'}x_i x_{i'}y_j y_{j'}\frac{\partial w_{ii'jj'}^{(2)}(\vec{u})}{\partial u_k}+ \cdots =0
\dfrac{1}{-z-\sum_{i=0}^{\infty}[\mathcal{D}'_i]}=\dfrac{-1}{z}+\sum_{s\geq 1}\sum_{r=1}^{\infty}(-z)^{-s-1}[\mathcal{D}'_{r}](\sum_{i=1}^{\infty}[\mathcal{D}'_i])^{s-1}
M_{i,I}(X_l^\natural) =CB_l({\ell^2(I)}_c,CB_{l}(X,M_{i,1}(B))_{B}))\simeq CB_l({\ell^2(I)}_c\otimes_h X,M_{i,1}(B))_{B}) &\subset CB_l({\ell^2(I)}_c\otimes_{eh}X,M_{i,1}(B))_{B})=CB_l(M_{I,1}( X),M_{i,1}(B))_{B}).
\zeta \left( s ; \not \! \! D^{\dag}\not \! \! D \right) ={\mu}^{4}\Omega \frac{a b}{4{\pi}^{2}}\left[ 2 \sum_{n_{a}= 1}^{\infty}\left( 2 n_{a}a + c \right)^{- s}+ 2 \sum_{n_{b}= 1}^{\infty}\left( 2 n_{b}b + c \right)^{- s}+ \right.
(I)\quad\phi^*_t \left(\sum u_{i}\sigma_i\right) = e^{\lambda t}u_1\sigma_1+e^tu_2\sigma_2 (II)\quad\phi^*_t \left(\sum u_{i}\sigma_i\right) = e^{\lambda t}u_1\sigma_1+(te^tu_1+e^tu_2)\sigma_2 (III)\quad\phi^*_t \left(\sum u_{i}\sigma_i\right) = e^{\lambda t}\left( (\cos(t)u_1-\sin(t)u_2)\sigma_1+(\sin(t)u_1+\cos(t)u_2)\sigma_2\right)
{\lambda_{\ell}^{2}A_{0,\ell}\; q(\mathbf{0}) -(\mathbb{E}\left[{\cal N}^c(f_{\ell}) \right])^{2}= \lambda_{\ell}^{2}\left[ \frac{2}{\ell}+\frac{2 \cdot 3^2}{\pi^2}\frac{\log \ell}{\ell^2}+O(\ell^{-2}) \right] q(\mathbf{0}) =\left[ 2 \ell^3 +\frac{2 \cdot 3^2}{\pi^2}\ell^2 \log \ell \right] q(\mathbf{0})+O(\ell^2).}
(L^2, \ell^2)=L^2, &(L^{2}, \ell^{q}) \hookrightarrow L^{q}\;\; \;\; 1 < q< 2, &(L^2,\ell^1) \hookrightarrow h^1 \hookrightarrow L^1, &L^{q}\hookrightarrow (L^{2}, \ell^{q}) \;\; \;\; 2 < q< \infty, &L^{\infty}\hookrightarrow bmo \hookrightarrow (L^2,\ell^{\infty}),
\chi(S;a,b) \cdot{V}^x = \binom{|S \backslash \{x\}|}{a-|S \cap \{x\}|}\binom{|\bar{S}\backslash \{x\}|}{b}= \binom{|S|-S(x)}{a - S(x)}\binom{r-|S| - 1 + S(S)}{b}.
p^{\beta_2}_{k-1}(2,t)=(-\lambda)^{k-1}\underset{\Omega^{k-1}_{2}}{\sum}\frac{t^{k_0\beta_0+k_1\beta_1+k_2\beta_2}}{\Gamma\left(k_0\beta_0+k_1\beta_1+k_2\beta_2+1\right)}.
A^{\mu}= \overline{{{g}}}^{\mu \nu}A_{\nu}; \F_{\mu \nu}= \partial_{\mu}A_{\nu}- \partial_{\nu}A_{\mu}; \F^{\mu \nu}= \overline{{{g}}}^{\mu \alpha}\overline{{{g}}}^{\nu \beta}F_{\alpha \beta}
\|V_{t}\|(\phi(\cdot,t))\Big|_{t=t_1}^{t_2}\leq \int_{t_1}^{t_2}{\mathcal B}(V_t,u(\cdot,t),\phi(\cdot,t))\, dt+\int_{t_1}^{t_2}\int_U \frac{\partial\phi}{\partial t}(\cdot,t)\, d\|V_t\|dt.
\widehat{\rho}_{1}^{\left( 2\right)}=\frac{6S_{n}^{\left( 2\right)}-4+\sqrt{3S_{n}^{\left( 2\right)}-2}}{4S_{n}^{\left( 2\right)}-3},2/3<S_{n}^{\left( 2\right)}<3/4.
\psi_{n,p}= \Big[{\frac{n(2j^{(1)}+1-n)(2j^{(2)}+1-n)(2j^{(1)}+2j^{(2)}+2-n)(N-n-p+2j^{(3)}+2)(N-n-p+1)}{(2j^{(1)}+2j^{(2)}+2-2n)^2(2j^{(1)}+2j^{(2)}+1-2n)(2j^{(1)}+2j^{(2)}+3-2n)}}\qquad\times{(p-n-N+2j^{(1)}+2j^{(2)}+1)(n-p-N +2j^{(0)}+2j^{(4)})}\Big]^{\frac12}\,,
v_s v_t\xi(x,r) =u(s,s^{-1}r)(x)u(t,(st)^{-1}r)(\varphi_{s^{-1}}(x))\xi(\varphi_{(st)^{-1}}(x),(st)^{-1}r) &\stackrel{\eqref{eq:cocycle_identity}}{=}u(s,t)(x) u(st,(st)^{-1}r)(\varphi_{(st)^{-1}}(x))\xi(\varphi_{(st)^{-1}}(x),(st)^{-1}r) =\overline{\pi}(u(s,t))v_{st}\xi(x,r).
D(n,k,r,b,p)= \!\!\prod_{(i,j) \in{\cal{A}}_{n,k,r}}\!\!\!{i+b+jp \over i}\!\!\prod_{(i,j) \in{\cal{B}}_{n,k,r}}\!\!\!{(k-i+1)(p+1)+(i+j-1)r+r(b+np) \over k-i+1+(i+j-1)r},
\tilde{{\bf E}}{}={}\frac{1}{2}\, \left({\bf u}\, \mathrm{e}^{\mathrm{i}\Theta}{}+{}{\bf u}^{*}\, \mathrm{e}^{- \mathrm{i}\Theta}\right) \quad , \qquad \tilde{{\bf B}}{}={}\frac{1}{2}\, \bar{\bf k}{}\times{}\left({\bf u}\, \mathrm{e}^{\mathrm{i}\Theta}{}+{}{\bf u}^{*}\, \mathrm{e}^{- \mathrm{i}\Theta}\right) \quad ,
\Pi_{\alpha \beta}( x - y ) = \frac{1}{\beta L}\frac{e^{2}}{\pi}\sum_{n , k = - \infty}^{+ \infty}e^{i \omega \cdot ( x - y )}\left[ \delta_{\alpha \beta}- \frac{\omega_{\alpha}\omega_{\beta}}{\omega^{2}}\right]
\begin{aligned}\sum_{j \in \mathbb{Z}}\overrightarrow{W}^\ell_j ( \overline{W}_{j - 1}- \overrightarrow{W}^\ell_j ) \varphi_j = \sum_{j \in \mathbb{Z}}\overrightarrow{W}^\ell_j \sum_{i = 0}^{\ell - 1}( \overline{W}_{j + i - 1}- \overline{W}_{j + i}) \psi_i \varphi_j \end{aligned}
h_{\mu,f}\bigl(f^0,g^0\bigr)=\frac{|A_N(e^{i\lambda})(1-e^{i\lambda\mu})^{n}\lambda^{2n}g^0(\lambda)+\lambda^{2n}C^{\mu,0}_{N}(e^{i\lambda})|}{|\lambda|^{n}|1-e^{i\lambda\mu}|^{n}p^0(\lambda)},
{\cal B}^{(\varphi)}_{WZWC}+{\cal B}^{(\varphi)}_{WZW}=\frac \ell{16 \pi G}\int_{\cal V}\partial_\varphi\left [ 2 \, Q_2^{-1}\partial_r Q_2 Q^{-1}\partial_t Q - Q^{-1}\partial_r Q Q^{-1}\partial_t Q \right ] \, dr \, d\varphi \, dt \, .
I_{3,\max}(G_\omega)= (X_3^2)S\bigcap(X_1^2,X_4^2)S\bigcap(X_1^2,X_5^2)S\bigcap(X_2^2,X_4^2)S\bigcap(X_2^2,X_5^2)S &\bigcap(X_3^3,X_4^2)S\bigcap(X_4^2,X_6^3)S\bigcap(X_2,X_3^3)S\bigcap(X_2,X_6^3)S.
\left( \begin{array}{l}c_{r}^{a}\overline{c}_{r}^{a}d_{r}^{a}\overline{d}_{r}^{a}\end{array}\right) =\frac{1}{2}\left( \begin{array}{cccc}i 1 i 1 -i 1 -i 1 -1 -i 1 i 1 -i 1 i\end{array}\right) \left( \begin{array}{l}b_{r}^{a}\overline{b}_{r}^{a}b_{-r}^{a}\overline{b}_{-r}^{a}\end{array}\right)
x^{s}q^{r}\overset{\mathrm{Weyl}}{\longrightarrow}\frac{1}{2^{s}}\sum_{\ell=0}^{s}\begin{pmatrix}s \ell\end{pmatrix}\widehat{p}^{s-\ell}\widehat{x}^{r}\widehat{p}^{\ell}
L_x^{}= \sum_{1\leq j\leq n}\frac{\partial^2}{\partial x_j^2}- \sum_{1\leq j<n}a_j e^{-x_j+x_{j+1}}-a_{n-1}e^{-x_{n-1}-x_n}& - \frac{\frac{1}{4}g_S(g_S+2g_L-1)}{\sinh^2 \textstyle{\frac{1}{2}}(x_n)}- \frac{g_L(g_L-1)}{\sinh^2 (x_n)}
\frac{dT}{ds}=\kappa N=\frac{-a}{a^{2}+b^{2}}cos\frac{s}{\sqrt{a^{2}+b^{2}}}i+\frac{-a}{a^{2}+b^{2}}sin\frac{s}{\sqrt{a^{2}+b^{2}}}j+0k
\!\!\!\!\!\!\!\!\!\!\! \mathcal{L}_{\mathcal{I}^{({\rm CEU})}_{\rm u \rightarrow d}}(s) =\exp\left( - \frac{2}{\eta}\pi^{2}\lambda \csc \left(\frac{2 \pi}{\eta}\right) s^{\frac{2}{\eta}}\mathbb{E}\left[ P_{\rm{u}}^{\frac{2}{\eta}}\right] \right),
e(G_c) =\frac{1}{2}\sum_{i=1}^{c}f_i=\frac{1}{2}(\sum_{v\in A'}d(v)+\sum_{v\in A\backslash A'}d(v)+\sum_{v\in B'}d(v)+\sum_{v\in B\backslash B'}d(v)) &\leq \frac{1}{2}\left((i-1)i+(|A|-i+1)\alpha+|B'|(c-i)+(|B|-|B'|)(c-1)\right) &\leq i^2-i(|B|+\alpha+2)/2+\alpha(c+1)/2+|B|(c-\alpha)/2,
\frac{dX}{dt}=-XZ+L_{1}(1-Z-\mu(1-Y)) \frac{dY}{dt}=-L_{3}YZ+\frac{L_{4}}{L_{2}}(1-Y) \frac{dZ}{dt}=\frac{1}{L_{2}}((-XZ+1-Z-\mu(1-Y))+\mu)-L_{3}YZ+\frac{L_{4}}{L_{2}}(1-Y)))
V_{p^{+}, p^{-}}^{r , s}\left( z \right) = e^{i X^{-}p^{+}}e^{- \frac 1 2 X_{2}\left( 1 - 2 i s \right)}F_{r , s}\left( e^{- X_{2}}p^{-}p^{+}\right) e^{i X^{+}p^{-}}
d s^{2}= d t^{2}+ A_{1}( t )^{2}\sigma_{1}^{2}+ A_{2}( t )^{2}\sigma_{2}^{2}+ B ( t )^{2}\sigma_{3}^{2}+ C ( t )_{1}^{2}( \sigma_{4}^{2}+ \sigma_{5}^{2}) + C ( t )_{2}^{2}( \sigma_{6}^{2}+ \sigma_{7}^{2}) .
\lambda = \frac{1}{2\log_2 t}\sum_{Z < p \leq t}\frac{|g(p)|}{p}=\frac{\log 2}{2\log_2 t}\left(\sum_{0 \leq l \leq L-1}\xi_l +\frac{1}{\log 2}S_{\kappa}(t)+ O_B\left(\frac{1}{\log Z}\right)\right).
6Cm \geq \|S_L \widetilde{\Phi}h + S_L R^T v\|_2 &\geq \sigma_L(S)\sqrt L\|\frac{1}{\sqrt L}(\widetilde{\Phi}_Lh + (R^T)_{{}_L}v)\|_2 &= \sigma_L(BD^{-r})\sqrt L\|\frac{1}{\sqrt L}(\widetilde{\Phi}_Lh + (R^T)_{{}_L}v)\|_2 \geq \sqrt m \left(\frac{m}{L}\right)^{r/2-1/4}\sqrt L \|\frac{1}{\sqrt L}(\widetilde{\Phi}_Lh + (R^T)_{{}_L}v\|_2.
F^{ij}\nabla_i\nabla_j H= -F^{ij, rs}\nabla_m h_{ij}\nabla_m h_{rs}+F^{ij}h_{ij}|A|^2-F^{ij}h_{im}h_{jm}H &+ qu^{q-1}\Delta u +q(q-1)u^{q-2}|\nabla u|^2.
\lambda_{\mathrm{rd}}(x_{0}, \omega) = \lim_{n \to\infty}\frac{1}{n}\log\left\| \varphi_{\mathrm{rd}}(n; x_{0}, \omega)\right\| ,\displaybreak[0] \lambda_{\mathrm{rc}}(x_{0}, \omega) = \lim_{t \to\infty}\frac{1}{t}\log\left\| \varphi_{\mathrm{rc}}(t; x_{0}, \omega)\right\| ;
\sup_{0<s<C\epsilon_{j-1}^{-\frac{1}{16}}}\lVert \widetilde{M}_s(\alpha,\theta,E,0)\rVert \leq C \epsilon_{j-1}^{-\frac{1}{96}}+C \epsilon_{j-1}^{-\frac{1}{96}}\sum_{i=0}^{s-1}C'' i|g(n+i)|\leq C''' \epsilon_{j-1}^{-\frac{1}{96}},
\begin{aligned}T_{f}(A', h^{E}_{1})-T_{f}(A', h^{E}_{0})=\widetilde{f}(\nabla^{E}, h^{E}_{0}, h^{E}_{1})-\widetilde{f}(\nabla^{H(E)}, h^{H(E)}_{0}, h^{H(E)}_{1}).\end{aligned}
\tilde{q}^{( a b )}( r ) = \frac{1}{8 \pi^{2}b^{2}r^{2}}\sum_{l = 1}^{\infty}( 2 l + 1 ) \sum_{\lambda = 0 , 1}\int_{0}^{\infty}{z \tilde{\Omega}_{\lambda l}( z , \sigma ) F_{\lambda l}^{( q )}( z , z y ) d z},
\sum_{j\in I}\binom{n}{j}^r\sim 2^{nr}\, \sqrt{\frac{n}{4}}\, \left(\frac{\pi n}{2}\right)^{-r/2}\int_{-a+\frac{2}{\sqrt{n}}}^{-a+\frac{2}{\sqrt{n}}}e^{-rt^2/2}d\, t,
x_0: =\frac{(1+4n)-\sqrt{(1+4n)^{2}-12[(n-2)b_n+2(n+1)+u_n]}}{6}&=\frac{2[(n-2)b_n+2(n+1)+u_n]}{1+4n+\sqrt{(1+4n)^{2}-12[(n-2)b_n+2(n+1)+u_n]}}.
\hat Q_{m}- \eta \, k^2 \, a^{5-h}\, \sum_{j=1 \atop j \neq m}^\aleph{\bf{P}}_{0}\cdot \Upsilon_k(z_{m}, z_j) \cdot \hat{Q}_{j}= i \, k \, a^{5-h}\,{\bf{P}}_{0}\cdot \hat H^{Inc}(z_{m}).
\left. \left[ \prod_{k \neq i,j}\, K(X_{k},sD) \right] \frac{\partial}{\partial D_{b}}K(X_{i},sD) \, \frac{\partial}{\partial D_{b'}}K(X_{j},sD) \leq \exp{\left[ - M \, \sum_{k}|X_{k}| \right]}\ . \right.
{\bf e}_{N + 1}\equiv{\bf p}_{N}^{\bot}/ c_{N}, \quad{\bf e}_{2 N + 1 - a}= -{\frac{1}{\sqrt{\Delta_{a}\Delta_{a + 1}}}}\left| \begin{array}{c c c}{{{\Phi}_{N . N}}}&{{\cdots}}&{{{\Phi}_{N . a}}}\\{{\vdots}}&{{}}&{{\vdots}}\\{{{\Phi}_{a + 1 . N}}}&{{\cdots}}&{{{\Phi}_{a + 1 . a}}}\\{{{{\bf p}^{\bot}}_{N}}}&{{\cdots}}&{{{{\bf p}^{\bot}}_{a},}}\\ \end{array}\right| ,
a \left( \partial^{\alpha}A^{\beta}- \partial^{\beta}A^{\alpha}\right) + \frac{i b}{2}\left( \partial_{\rho}A_{\sigma}- \partial_{\sigma}A_{\rho}\right) \epsilon^{\rho \sigma \alpha \beta}= \frac{1}{2}\epsilon^{\alpha \beta \rho \sigma}\left( \partial_{\rho}B_{\sigma}- \partial_{\sigma}B_{\rho}\right) .
\int_{[\Lambda, \infty)}\lambda^{-2 \gamma}d\mu(\lambda)=\int_{[\Lambda, \infty)}\lambda^{-2 \gamma}d I(\lambda)=-\Lambda^{-2 \gamma}I(\Lambda) +2 \gamma\int_{[\Lambda, \infty)}\lambda^{-2 \gamma - 1}I(\lambda)d \lambda.
(m+p)(\partial_t - X^k_j\cdot \nabla_Y)\zeta^k_j(X^k_j,Y^k_j,t^k_j) \geq (p-2)\Delta^N_{\infty,X}\zeta^k_j(X^k_j,Y^k_j,t^k_j) + \Delta_X \zeta^k_j(X^k_j,Y^k_j,t^k_j) \geq (p-2) \lambda( \nabla^2_X \zeta^k_j(X^k_j,Y^k_j,t^k_j) ) + \Delta_X \zeta^k_j (X^k_j,Y^k_j,t^k_j) = (p-2)\lambda (-H_k^X) - \textrm{tr}(H_k^X).
\sum_{i=m}^{n}f(i)=\int_{m}^{n}f(x)dx+\frac{f(n)+f(m)}{2}+\sum_{k=1}^{\lfloor\frac{p}{2}\rfloor}\frac{B_{2k}}{(2k)!}(f^{(2k-1)}(n)-f^{(2k-1)}(m))+R_{p},
H_{,i}^{k^{*}}=\sum_{j}h_{ij}^{k^{*}}\langle x, e_{j}\rangle, 1\leq i,k\leq 2, &H_{,ij}^{k^{*}}=\sum_{m}h_{im,j}^{k^{*}}\langle x, e_{m}\rangle+h_{ij}^{k^{*}}-\sum_{m,p}(H-\xi)^{p^{*}}h_{im}^{k^{*}}h_{mj}^{p^{*}},1\leq i,j,k\leq 2.
\Delta_n\mu(u,v) =-\int_0^ve^{\int_{v'}^vL_n(u,v'')dv''}\left(K_n\Delta_n\nu-\Xi_n\right)(u,v')dv', \Delta_n\nu(u,v) =\int_0^ue^{-\int_{u'}^uK_n(u'',v)du''}\left(L_n\Delta_n\mu+\Xi_n\right)(u',v)du'.
\lbrack F_{_{( 1 )}}]^{- 1}= \left( \begin{array}{c c c}{{f^{- 1}}}&{{- f^{- 1}M_{_{( 1 )}}R_{_{( 1 )}}^{- 1}}}&{{0}}\\{{- R_{_{( 1 )}}^{- 1}M_{_{( 1 )}}^{^T}f^{- 1}}}&{{R_{_{( 1 )}}^{- 1}\omega_{_{( 1 )}}R_{_{( 1 )}}^{- 1}}}&{{- R_{_{( 1 )}}}}\\{{0}}&{{R_{_{( 1 )}}}}&{{0}}\\ \end{array}\right) \; ,
\|w^{1:N}\|_{1}^{q_{*}- p_{*}}\ge N^{(q_{*}- p_{*})/q_{*}}\|w^{1:N}\|_{q}^{q_{*}-p_{*}}= N^{1 - p_{*}/q_{*}}\|w^{1:N}\|_{q}^{q_{*}-p_{*}}&= N^{-p_{*}(1/p - 1/q)}\|w^{1:N}\|_{q}^{q_{*}-p_{*}},
0=\mathcal A_i=\div_{\mathbb G}\left[ \frac{(-\gamma'_i(x),x)}{\sqrt{(\gamma '_i(x))^2 + x^2}}\right]= - \partial_x \left[ \frac{\gamma'_i(x)}{\sqrt{\left(\gamma '_i(x) \right)^2 + x^2}}\right].
a_{23}^{3214}=a_{23}^{3241}=a_{23}^{3124}=a_{23}^{3142}=a_{23}^{3421}=a_{23}^{3412}= a_{23}^{1324}=a_{23}^{1342}=a_{23}^{1432}=a_{23}^{4321}=a_{23}^{4312}=a_{23}^{4132}
f\left(y_{T+1}\right)-f\left(x^{*}\right) \le\frac{2R_{1}^{2}\alpha_{T}}{\eta_{T}}=6R_{1}^{2}c\gamma^{2}L\alpha_{T}^{2}=6\max\left\{10^{4}L\gamma^{2}R_{1}^{2}(T+1)^{-2};6R_{1}\left(T+1\right)^{-1}\left(26T\right)^{\frac{1}{p}}\gamma^{\frac{p-1}{p}}\sigma\right\}.
{\frac{\Lambda^{2}}{P^{+}}}> \mid{\frac{P_{\perp}^{2}}{P^{+}}}-{\frac{k_{1 \perp}^{2}}{k_{1}^{+}}}-{\frac{k_{2 \perp}^{2}}{k_{2}^{+}}}-{\frac{k_{3 \perp}^{2}}{k_{3}^{+}}}\mid \; .
\quad~\#\{1 \leq i \leq \epsilon_{k-l-1}2^{k-l-1}:\lambda_i=0\}&= \#\{\epsilon_{k-l}2^{k-l}+1 \leq i \leq \epsilon_{k-l}2^{k-l}+ \epsilon_{k-l-1}2^{k-l-1}:\lambda_i=0\}& =\cdots & = \#\Big\{\sum_{j=k-l}^{k}\epsilon_j 2^j+1 \leq i \leq \sum_{j=k-l-1}^{k}\epsilon_j 2^j:\lambda_i=0\Big\}
\psi(k,v) \leq c_1\exp(\phi_v(k))\leq c_1\exp(\phi_v(\varepsilon_0v)) \leq c_1\exp\left(-\frac{c_2(\varepsilon v-\theta_\lambda v)^2}{v}\right)=c_1\exp\left(-c_2(\varepsilon_0-\theta_\lambda)^2v\right)\leq c_3k^{-\frac{1}{2}}.
\left. \begin{array}{c}\left( u_{t}-L_{0}u\right)^{2}\varphi_{\lambda ,\nu}\psi^{-\nu +1}=\left[\left( z_{1}+z_{3}\right) +z_{2}+z_{4}\right]^{2}\psi^{-\nu +1}\geq \geq \left( z_{1}+z_{3}\right)^{2}\psi^{-\nu +1}+2z_{1}z_{2}\psi^{-\nu+1}+2z_{1}z_{4}\psi^{-\nu +1}+2z_{2}z_{3}\psi^{-\nu +1}+2z_{3}z_{4}\psi^{-\nu +1}.\end{array}\right.
\vert u_{l}( r ) \vert \leq r^{l}( 1 + r^{2})^{{\frac{1 - l}{2}}}{\frac{\Gamma ( 2 ) \Gamma ( l +{\frac{3}{2}})}{\Gamma ({\frac{l}{2}}+ 2 ) \Gamma ({\frac{l + 3}{2}})}}.
\lefteqn{\psi_{U}\Big ( \sum_{{i},{j}}\,{f}_{{i}}{g}_{{j}}\otimes f_{{i}}^{\prime}g_{{j}}^{\prime}\Big ) \: = \: \sum_{{i},{j}}\,{f}_{{i}}{g}_{{j}}\otimes f_{{i}}^{\prime}g_{{j}}^{\prime}\otimes 1}.
(\lambda_{i+1}\lambda_{i+2}\dotsm\lambda_{n-j})a_ja_i =(\lambda_{i+2}\dotsm\lambda_{n-j})(a_{n-i}a_{i+1})(a_ja_i) & =(\lambda_{i+2}\dotsm\lambda_{n-j})a_ja_{n-i}a_{i+1}a_i & =(\lambda_{i+2}\dotsm\lambda_{n-j})a_ja_{n-i}a_ia_{j+1}& =(\lambda_{i+2}\dotsm\lambda_{n-j})(a_{n-i}a_i)(a_ja_{i+1}) & =(a_{n-i}a_i)(a_{n-i-1}a_{i+1})\dotsm(a_ja_{n-j})
\widehat{Y}(qx) = H \widehat{Y}(x), \widehat{Y}(x) = \begin{pmatrix}\widehat{y}_0(x) \widehat{y}_1(x) \widehat{y}_2(x) \widehat{y}_3(x) \end{pmatrix}\!, &H = F_{\infty}+ \frac{F_1}{1-\alpha_1 x}+ \frac{F_2}{1-\alpha_2 x}+ \frac{F_3}{1-\alpha_3 x}.
\Psi_\alpha(t): =\sum_{i=0}^{m-1}\int_{I_i}\bigg[\int_{I_i}\bigg( \phi_\alpha(t-s)- \phi_\alpha(t-\tau) \bigg) d\tau \bigg]^2ds, \Upsilon_\alpha(t): =\sum_{i=0}^{m-1}\int_{I_i}\int_{I_i}\phi_\alpha(t-s)\bigg( \phi_\alpha(t-s)- \phi_\alpha(t-\tau) \bigg) d\tau ds.
\operatorname{supp}\mathcal{F}(f)\subset\lambda\mathcal{B}\ \ \Rightarrow\ \ \sup_{|\alpha|=k}\|\partial^{\alpha}f\|_{L^{r}}\leq C^{k+1}\lambda^{k+n(\frac{1}{p}-\frac{1}{r})}\|f\|_{L^{p}},
\frac{r^{f+1}+1}{p}=r\cdot \frac{r^f+1}{p}+\frac{1}{p}(1-r) &\ge r\cdot (r^{2f/p}-1)+\frac{1}{p}(1-r) &=r^{2(f+1)/p}-1+r^{2f/p}(r-r^{2/p})+(1-r)(1+1/p).
g^{- 1}= \left( \begin{array}{l l}{{- 2 \mu_{2}- 2 \mu_{2}^{2}+ \frac 2 3 \mu_{3},}}&{{- \mu_{3}( 6 + \frac{1 6}{3}\mu_{2})}}\\{{- \mu_{3}( 6 + \frac{1 6}{3}\mu_{2}) ,}}&{{\frac 8 3 \mu_{2}^{2}\mu_{3}- 8 \mu_{3}^{2}}}\\ \end{array}\right)
\widetilde{\cal H}_{\scriptscriptstyle P}=\{Q,\widehat{X}_{(1)}\cdots\widehat{X}_{(p)}\}=\sum_{i=1}^p(-1)^{i+1}\widehat{X}_{(1)}\cdots{\widehat{\widehat X}}_{(i)}\cdots\widehat{X}_{(p)}\widetilde{\cal H}_{\scriptscriptstyle X_{(i)}}
\Pi_{i}^{a}\left({\bf x}\right) \left| \Omega \right\rangle = \overline{{\psi}}\left({\bf y}\right) U ({\bf y},{\bf 0}) \psi \left({\bf 0}\right) \Pi_{i}^{a}\left({\bf x}\right) \left| 0 \right\rangle + g T^{a}\int_{\bf 0}^{\bf y}{d z_{i}}\delta \left({{\bf x}-{\bf z}}\right) \left| \Omega \right\rangle .
t^{8}\sim c_{\circ}^{\prime}( t^{\circ}+ t^{2}+ \cdots ) + c_{8}^{\prime}( t^{1}+ t^{3}+ \cdots ) + c_{1 0}^{\prime}( t^{2}+ t^{4}+ \cdots ) + c_{\overline{{{1 0}}}}^{\prime}( t^{2}+ t^{4}+ \cdots )
S(v,v) = \int_\Sigma \nabla v\nabla v-|A|^2 v^2-\int_{\partial \Sigma}v^2 &= \int_\Sigma -v(\Delta+|A|^2)v+\int_{\partial \Sigma}(D_\eta v-v)v &=\int_\Sigma -|A|^2 v^2+\int_{\partial \Sigma}(X_a+c\xi u-X_a-cu-d)v &=\int_\Sigma -|A|^2 v^2+\int_{\partial \Sigma}\Big((\xi-1)cu-d\Big)(X_a+cu+d) &=\int_\Sigma -|A|^2 v^2+\int_{\partial \Sigma}(\xi-1)c^2u^2-d^2<0.
\Delta_{2^{p_0}+\cdots+2^{p_s}}=\sum_{j=1}^s 2^{j-p_j+p_0}(2^{p_j-p_{j-1}}-2)\left[(3\ 2^{j-1}-1)\Delta_{2^{p_j}+\cdots+2^{p_s}}\right. &\left.+(2^{j-2}(2^{p_j-p_{j-1}}-2^2)+1)(1+2^{p_{j+1}-p_j}+\cdots+2^{p_s-p_j})\right]\ge 0
\Delta_N G(x/N) = N^2 \sum\limits_{j=1}^d \Big[ G\Big(\frac{x+e_j}{N}\Big) + G\Big(\frac{x-e_j}{N}\Big) - 2 G\Big(\frac{x}{N}\Big) \Big] \ \ \mbox{and}\ \ \Delta G (u)= \sum\limits_{j=1}^d \partial_{e_j}^2 G(u) .
\|u^h_t\|^2_{H_l}=\|\psi\|^2_{H^l}+ \int_0^t \left[\left( (L^h_s +I^h) u^h_s, u^h_s\right)_{H^l}+ (f_s, u^h_s)_{H^l}\right] \ds \leq \|\psi\|^2_{H^l}+ N\int_0^t \|u^h_s \|^2_{H^l}\ds+\int_0^T\|f_s\|^2_{H^l}\ds< \infty,
Q_{\nu,\ell}(R,M) \lesssim\begin{cases}R^{2\nu-n+3}M^{-n}\|b_{\nu,\ell}(\rho)\chi(\frac{\rho}M)\rho^{\frac{n-1}2}\|^2_{L^2},~R\lesssim 1; R^{-(n-2)}M^{-n}\|b_{\nu,\ell}(\rho)\chi(\frac{\rho}M)\rho^{\frac{n-1}2}\|^2_{L^2},~R\gg1.\end{cases}
\sum_{m=1}^{\infty}\lambda_f (m)\, e\left(-\frac{(a+bq)m}{p_1q}\right)\, m^{-i(t+\nu)}\,e\left(\frac{-mu}{p_1qQ}\right) V_2\left( \frac{m}{N}\right) = \, \frac{N^{3/4-i(t+\nu)}}{(p_1q)^{1/2}\,p_2^{1/4}}\sum_{\pm}\sum_{m \sim (p_1Qt)^2p_2/N}\frac{\lambda_g(m)}{m^{1/4}}e\left(-m \frac{\overline{(a+bq) p_2}}{q}\right) \mathcal{I}^{\pm}(...),
d_\alpha(\textit{\textbf{p}}_\lambda,\textit{\textbf{p}}_\mu)=s_0^{2(1-\alpha)}|\theta_\lambda-\theta_\mu|^2+s_0^{2\alpha}|x_\lambda-x_\mu|^2 + \frac{s^2_0}{1+s_0^{2(1-\alpha)}|\theta_\lambda-\theta_\mu|^2}|\langle e_\lambda, x_\lambda-x_\mu \rangle|^2.
\begin{array}{l}{{h^{I_{1}\ldots I_{s}}( z ) = z^{2}V^{I_{1}\ldots I_{s}}, \; \;}}\\{{h^{I_{1}\ldots I_{s}}( z ) = z^{2 s + d - 3}V^{I_{1}\ldots I_{s}}}}\\ \end{array}
\binom{n}{k}/\binom{n}{(k+r)/2}=\frac{(n-k+1)\cdots (n-(k+r)/2)}{(\frac{k+r}{2}+1) \cdots k}\ge \frac{n-(k+r)/2}{(k+r)/2 +1}&=\frac{3k+r}{k+r+2}\ge \frac{3k+(k-2)}{k+(k-2)+2}=\frac{2k-1}{k}.
\mathcal{L}=\frac{r^{2}\partial_{r}W\partial_{r}W^{*}+\partial_{\theta}W\partial_{\theta}W^{*}-r^{2}\dot\lambda^{2}\partial_{\lambda}W\partial_{\lambda}W^{*}}{4r^{2}(1+|W|^{2})^{2}}.
\|f\| \leq \sum_{I\subset[N]}\left\|f^I\right\| & \leq \sum_{I\subset[N]}\left\|\xi^I\right\|\left\|\eta^I\right\| &\leq \left(\sum_{I\subset[N]}\left\|\xi^I\right\|^2\right)^{\frac{1}{2}}\left(\sum_{I\subset[N]}\left\|\eta^I\right\|^2\right)^{\frac{1}{2}}& =\|\xi\|\|\eta\| & = \|f\|.
\begin{array}{}{{]{c | c c c c}\mathrm{F i e l d}}}&{{S U ( n_{c})_{0}}}&{{S U ( n_{c})_{N + 1}}}&{{U ( 1 )}}&{{U ( 1 )_{R}}}\\ \hline{{Q_{0}}}&{{n_{c}}}&{{1}}&{{1}}&{{- 1}}\\{{Q_{i}}}&{{1}}&{{1}}&{{( - 1 )^{i}}}&{{- 1}}\\{{Q_{N}}}&{{1}}&{{\bar{n}_{c}}}&{{( - 1 )^{N}}}&{{- 1}}\\{{\lambda_{j}}}&{{1}}&{{1}}&{{0}}&{{1}}\\ \end{array}
B_{MN}= (-)^{M (N + \underline{N})}\partial_N z^{\underline{N}}\partial_M z^{\underline{M}}B_{\underline{MN}}\;, \;\;\;E_{\pm}^{\underline{A}}= E_\pm^M\partial_M z^{\underline{N}}E_{\underline{N}}^{\underline{A}}\;,
i\partial_{t}\epsilon_{t}(x) = \left(-\frac{1}{2}\Delta +U*|\epsilon_{t}|^{2}(x) +U*2\Re\left(\epsilon_{t}^{*}\right)(x) \right)\epsilon_{t}(x) + \left( U*|\epsilon_{t}|^{2}(x) +U*2\Re\left(\epsilon_{t}^{*}\right)(x)\right).
\sum_{k=0}^{(n-1)/2}[4k+1]_{q^2}\frac{(aq^2;q^4)_k (q^2/a;q^4)_k (q^2;q^4)_k q^{(n-1)k}}{(aq^4;q^4)_k(q^4/a;q^4)_k (q^4;q^4)_k}\equiv 0\pmod{\Phi_n(q)}.
N_q(Z) = \theta_q(\frac{n-3}{2}) \Bigl( d q^{\frac{n-3}{2}}+ \theta_q(\frac{n-5}{2}) - \theta_q(\frac{n-3}{2})\Bigr) +\theta_q(\frac{n-3}{2}) = \theta_q(\frac{n-3}{2}) \Bigl( (d-1)q^{\frac{n-3}{2}}+1 \Bigr).
(i)\ \ \ \nabla(\gamma_0J+\sum\limits_{k=1}^m\gamma_k f_k)(x^*)=0 (ii)\ \ \gamma_kf_k(x^*)=0\ \k\in\{1,...,m\}(iii)\ \gamma_{0}A_J+\sum\limits_{k=1}^m\gamma_kA_k\succeq0
a\left( y^{n},s^{m},p^{k}\right) \cdot p^{k}\left( x,t,y^{n},s^{m}\right)\rightarrow a\left( y^{n},s^{m},\nabla u+\sum\limits_{j=1}^{n}\nabla_{y_{j}}u_{j}+\delta c\right) \cdot \left( \nabla u\left( x,t\right) +\sum\limits_{j=1}^{n}\nabla_{y_{j}}u_{j}\left( x,t,y^{j},s^{m-d_{j}}\right) +\delta c\left(x,t,y^{n},s^{m}\right) \right)
s_a^1(x,\xi) =\sum_{z\in\mathbb{Z}^d}f\left[z\right]e^{i(\varphi_a(x-z,\xi)-\varphi_a(x,\xi))}-h_0(\nabla_x\varphi_a(x,\xi)) , s_a^2(x,\xi) =h(x,\nabla_x\varphi_a(x,\xi))-h_0(\xi) .
\frac{1}{\pi}\int_{-T}^T\left|\frac{\varphi_Z(t)-\varphi(t)}{t}\right|dt \le 2\int_{0}^{n^\epsilon}\frac{1}{t}O\left(\frac{t^3e^{-\frac{t^2}{3}}}{n}+\frac{t}{\sqrt{n}}\right)dt+2\int_{n^{\epsilon}}^{\sqrt{n}}\frac{1}{t}O\left(\frac{1}{\sqrt{n}t^{1-\epsilon}}\right)dt &=O_\epsilon\left(\frac{1}{n^{\frac{1}{2}-\epsilon}}\right)
\kappa_1^2 = a^2\alpha^4 + b^2\beta^4, \kappa_1^2 \kappa_2^2 = a^2\alpha^6 + b^2\beta^6 -\kappa_1^4, \kappa_1^2 \kappa_2^2 \kappa^3_3 = a^2\alpha^8 + b^2\beta^8 -\kappa_1^2(\kappa_1^2 +\kappa_2^2)^2,
P(\ell,\Phi)=\left[\ell^3 - \frac{\ell^2\,\Phi}{\pi}+ \ell \left( \frac{{\Phi}^2}{2\,{\pi}^2}-1 \right) + \frac{2\,{\pi}^2\,\Phi -{\Phi}^3}{8\,{\pi}^3}\right],
b^\perp_N =1-2\alpha_N^\perp, a^\perp_N = -\frac{D^\perp_N(q)}{2\sqrt{N}}\bigg|_{q=e^{-2\pi/\sqrt{2N}}}, \intertext{and}X^\perp_N =4\alpha_N^{\perp}\left(1-\alpha_N^{\perp}\right),
\rho^{2}= 2 R \left( | \vec{x}_{1}- \vec{x}_{2}| + | \vec{x}_{2}- \vec{x}_{3}| + \cdots + | \vec{x}_{N - 1}- \vec{x}_{N}| + | \vec{x}_{N}- \vec{x}_{1}| \right) .
\int_{I}|f(e^{is})-f_{I}|ds \leq\int_{I}\frac{1}{|I|}\int_{I}|f(e^{is})-f(e^{it})|dtds \leq \frac{1}{|I|}\int_{I}\int_{I}|f(e^{is})-f(e^{it})|^{2}dtds &\lesssim |I|^{5-4\beta}\|f\|_{Q_{p}^{\beta}(\mathbb{T})}^{2}.
\mathbb E_{0}[ Z_{t}^{-1}\vert \mathcal F^{X}_{T}] \, =\, \rho_{t}({\bf 1}) \, =\, \exp \Big( \int^{t}_{0}\pi_{s, 2}(b){\mathrm d}X_{s}- \frac{\,1\,}{\,2\,}\int^{t}_{0}\lvert \pi_{s, 2}(b) \rvert^{2}{\mathrm d}s \Big) \, ; 0 \le t \le T\, .
\lambda_{\mathtt{U}_{j}}q_{\mathtt{U}_{j}}^{(n)}=0;\ \tilde{\lambda}_{\mathtt{U}_{j}}(q_{\mathtt{U}_{j}}^{(n)}-\overline{q}_{\mathtt{U}_{j}})=0,\ \forall j=1,\ldots,K_{\mathtt{U}},
2\varepsilon\int_{Q}\Bigl(\int_{S\setminus Q_{3}}J_{\varepsilon}(x-y)|\nabla u(x)|^{2}dx\Bigr)dy & =2\varepsilon\sum_{\alpha\in\mathbb{Z}^{n}\cap(S\setminus Q_{3})}\int_{Q}\Bigl(\int_{\alpha+Q}J_{\varepsilon}(x-y)|\nabla u(x)|^{2}dx\Bigr)dy~.
\prod_{\substack{(a,b,c)=1 a,b,c \geq 1}}\left( \frac{1}{1-x^a y^b z^c}\right)^{\frac{1}{a^s b^t c^u}}= \exp\left\{\left( \sum_{i=1}^{\infty}\frac{x^i}{i^s}\right) \left( \sum_{j=1}^{\infty}\frac{y^j}{j^t}\right) \left( \sum_{k=1}^{\infty}\frac{z^k}{k^u}\right)\right\}.
h_k =\begin{pmatrix}n+k n\end{pmatrix}-\begin{pmatrix}n+k-2 n\end{pmatrix}\leq 2\begin{pmatrix}n+k-1 n\end{pmatrix}\leq 2e^n\begin{pmatrix}{\frac{n+k-1}n}\end{pmatrix}^n \leq 2e^{n+k-1},
\eta \log x = \exp\left\{-\left[D_1^{1/(1+d_1)}(2\log\log x)^{(1+\delta)/(1+d_1)}\right]\right\}\cdot \log x & = \exp\left[\log \log x - D_1^{\frac{1}{1+d_1}}2^{\frac{1+\delta}{1+d_1}}(\log\log x)^{\frac{1+\delta}{1+d_1}}\right] &\succ \exp\left(\frac{1}{2}\log\log x\right) = \sqrt{\log x}\succ 2\log \log x + \log 2.
{\cal{S}}^{[ p_1 ,..., p_N ]}\;\; =\;\; - \left( \prod_{i=1}^{N}\frac{1}{p_i !}\right) \int d^D x \; A^{{\mu}^1_1 ...{\mu}^1_{p_1}...{\mu}^i_1 ...{\mu}^i_{p_i}...{\mu}^N_{p_N}}E_{{\mu}^1_1 ...{\mu}^1_{p_1}...{\mu}^i_1 ...{\mu}^i_{p_i}...{\mu}^N_{p_N}}
\phi_Y(t) = \mathcal{F}^\mu\left(g_Y\right)(t) = \sum_{\Sigma\subseteq [\![1,n]\!]}e_{_\Sigma}\sqrt{\frac{\lambda_{_\Sigma}}{\pi}}\int_\mathbb{R}e^{-\lambda_{_\Sigma}x^2}e^{\mu tx}dx = \sum_{\Sigma\subseteq [\![1,n]\!]}e_{_\Sigma}e^{-\frac{t^2}{4\lambda_{_\Sigma}}}.