harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
E^2=-\frac{d}{pq}A\cdot E=\frac{1}{q},B\cdot E=\frac{1}{p},(A\cdot A)_{S_{P,\omega}}=(A\cdot A)_S-\frac{p}{d q},(B\cdot B)_{S_{P,\omega}}=(B\cdot B)_S-\frac{q}{d p}.
\|D^*x_0\|_1+\rho\geq\|D^*\hat{x}\|_1 =\|D^*x_0+D^*h\|_1 &=\|D^*_{T_0}x_0+D^*_{T_0}h+D^*_{T_0^c}x_0+D^*_{T_0^c}h\|_1 &\geq\| D^*_{T_0}x_0\|_1-\|D^*_{T_0}h\|_1-\|D^*_{T_0^c}x_0\|_1+\|D^*_{T_0^c}h\|_1,
-(t*x)ygh_{t}-(t*x)yth_{g}-gty\beta_{t*x}-gt(t*x)\beta_{y}&+x(g*y)gh_{t}+x(g*y)th_{g}+gtx\beta_{g*y}+gt(g*y)\beta_{x}= -(t*x)ygh_{t}-(t*x)yth_{g}+x(g*y)gh_{t}+x(g*y)th_{g}&-y\beta_{t*x}gt-y((gt)*(t*x))gt-(t*x)\beta_ygt-(t*x)((gt)*y)gt &+x\beta_{g*y}gt+x((gt)*(g*y))gt+(g*y)\beta_xgt+(g*y)((gt)*x)gt.
[u,w]+\lambda J[u,w] + J[u,Jw]-\lambda[u,Jw] = v-w+\lambda Jv-\lambda Jw+J[u,Bv+Cw]-\lambda[u,Bv+Cw] = v-w+\lambda Xu + \lambda Yv+\lambda Zw -\lambda Au-\lambda Bv-\lambda Cw- B(Xu+Yv+Zw) + &+C(Xu+Yv+Zw)-C(Au+Bv+Cw)+\lambda Bv-\lambda Cv+\lambda Cw
\|(Y(u)-Y(s))\, (X(t)-X(w))\|_p\leqslant \|Y(u)-Y(s)\|_q\,\|X(t)-X(w)\|_r &\leqslant \psi^{-1}(\psi(\|Y(u)-Y(s)\|_q))\,\phi^{-1}(\phi(\|X(t)-X(w)\|_r)) &\leqslant \psi^{-1}(V_{\psi,r}( Y;[s,t]))\,\phi^{-1}(V_{\phi,q}( X;[s,t]) ) &\leqslant \psi^{-1}(\varphi(t)-\varphi(s))\,\phi^{-1}(\varphi(t)-\varphi(s))=:f(s,t),
\| (P_j f_j)_{l_j^q}\|_p \lesssim \| (\mathcal M(|f_j|^r) )^{\frac 1r}_{l_j^{\frac qr}}\|_p= \| (\mathcal M (|f_j|^r) )_{l_j^{\frac qr}}\|^{\frac 1r}_{\frac pr}& \lesssim \| (|f_j|^r)_{l_j^{\frac qr}}\|_{\frac pr}^{\frac 1r}= \| (f_j)_{l_j^q}\|_p.
\delta(\Delta{f}) = \frac{d}{dt}(g^{ij}f_{ij}) - \frac{d}{dt}(g^{ij}\Gamma_{ij}^k f_k) &= 2u \langle h, f \rangle + \Delta\dot f + 2u \langle \nabla H, \nabla f \rangle + 2 \, h (\nabla{u},\nabla{f}) - 2H \langle \nabla u, \nabla f \rangle,
\left\lvert \sum_{n=1}^{N}e^{2\pi ih f(n+c_0-1)/q}\right\rvert \ll \left(\frac{hf(N)}{qN}+1\right)\left(\frac{N\sqrt{q}}{\sqrt{h f(N)}}+1\right) &\ll \frac{hf(N)}{qN}+\frac{N\sqrt{q}}{\sqrt{hf(N)}}+\sqrt{\frac{hf(N)}{q}}+1\,,
||v_{n\,m+6}^1||^2= -\frac{\overline{d_{n\,m+6}}}{a_{n-1\,m+3}}||v_{n-1\,m+3}^1||^2= \frac{\overline{d_{n\,m+6}}}{a_{n-1\,m+3}}\cdot \frac{\overline{b_{n-1\,m+3}}}{c_{nm}}||v_{nm}^1||^2.
Y^{n,j}_t=\xi^j+\int_t^T f^j(r,Y^n_r)\,dr+\int_t^T\,dV^j_r+\int_t^Tn(Y^{n,j}_r-H^j(r,Y^n_r))^-\,dr-\int_t^T\,dM^{n,j}_r
\sigma^{2}(\tau)\approx\int_{0}^{\infty}S[g*\dot{x}_{f}](\omega)d\omega=\int_{0}^{\infty}S[g](\omega)\cdot S[\dot{x}_{f}](\omega)d\omega=\int_{0}^{\infty}S[h](\tau\omega)\cdot S[\dot{x}_{f}](\omega)d\omega
\frac{t^{\frac{1}{2}+\frac{Nr}{2}}}{t(t)^{\frac{N}{2}}}=\begin{cases}t^{-\frac{1}{2}-\frac{N}{2}(\frac{1}{p}-\frac{1}{q})}\ \ \q>p t^{-\frac{1}{2}}\ \q=p\end{cases}
\Psi^\star_L(\mu, \nabla \tilde G_L) \le \Psi_L(\mu,j^G) \le \frac 12\sum_{i\in\mathbb T_L^d}\sum_{k=1}^d \hat \chi^0_{i,i+e_k}(\mu) \bigl[2L \sinh\bigl(\tfrac 12 \nabla^{i,i+e_k}G(\cdot/L)\bigr)\bigr]^2.
\begin{cases}(-\Delta)^{\frac{n}{2}}u(x)=u^p(x), x\in\Omega, u(x)=-\Delta u(x)=\cdots =(-\Delta)^{\frac{n-1}{2}}u(x)=0, x\in\mathbb{R}^n \backslash \Omega,\,\, \mbox{if}\,\, n\,\, \mbox{is odd}; u(x)=-\Delta u(x)=\cdots =(-\Delta)^{\frac{n}{2}-1}u(x)=0, x\in\partial \Omega,\,\, \mbox{if}\,\, n\,\, \mbox{is even}. \end{cases}
\left| \big( A t^{-2\ell -4C}\cup A t^{2\ell+4C}\big) \cap B_S(n + D') \right| \ge \frac{1}{\ell+1}\binom{\ell +1}{2}\, | R_{q,\ell}^\flat(n)| > \frac{q(n)}{2}\, |R_{q,\ell}^\flat(n)|.
i \langle \phi ( k , \omega ) \; \phi ( - k , - \omega ) \rangle = 1 / \left( - \frac{2 \pi}{\log ( | k | / \Lambda )}+ \frac{e^{2}}{\pi}\frac{v_{F}k^{2}}{v_{F}^{2}k^{2}- \omega_{k}^{2}}\right) \; . \;
q_{F}(x^{*}_{k_{i}}-x^{*}_{k_{j}}) \leq q_{F}(x^{*}_{k_{i}}-z^{*}_{k_{i}})+q_{F}(z^{*}_{k_{i}}-z^{*}_{k_{j}})+q_{F}(z^{*}_{k_{j}}-x^{*}_{k_{j}}) &\leq 2\epsilon+q_{F}(z^{*}_{k_{i}}-z^{*}_{k_{j}}),i,j=1,2,...
\left( f_{1}^{\prime}f_{2}\right)^{2}f_{1}f_{2}^{\prime \prime}-2\left(f_{1}^{\prime}f_{2}^{\prime}\right)^{2}f_{1}f_{2}+\left(f_{1}f_{2}^{\prime}\right)^{2}f_{2}f_{1}^{\prime \prime}+f_{1}f_{2}^{\prime \prime}+2af_{1}^{\prime}f_{2}^{\prime}+a^{2}f_{1}^{\prime \prime}f_{2}=0.
\sum_{i=1}^{k}\frac{1}{a_i}\leq \sum_{i=1}^{n}\frac{1}{a_i}< \sum_{i=1}^{n-1}\frac{1}{a_i}+ \frac{1}{a_n -1}\leq \sum_{i=1}^{k-1}\frac{1}{a_i}+ \frac{1}{a_k-1}
X : = \frac{\delta F(y)}{\delta L}&= \sum_{k=1}^n D^{-k}\xi^{k-1}\delta(x-y) + \frac{\delta(x-y)}{\xi-w(x)}+ \frac{\delta(x-y)}{(\xi-w(x))^2}(D-w(x)) &= \sum_{k=-1}^n D^{-k}a_k(x) - \epsilon a_{-1}'(x),
\bigcup_{i\in I}\bigcap_{j\in J}(L_{i}\backslash R_{j})=\bigcap_{j\in J}\bigcup_{i\in I}(L_{i}\backslash R_{j})and\bigcup_{j\in J}\bigcap_{i\in I}(L_{i}\backslash R_{j})=\bigcap_{i\in I}\bigcup_{j\in J}(L_{i}\backslash R_{j})
[ \hat{x}_{1}^{\prime}, \hat{x}_{2}^{\prime}] = i \alpha^{2}, [ \hat{p}_{1}^{\prime}, \hat{p}_{2}^{\prime}] = i \alpha^{- 2}, [ \hat{x}_{i}^{\prime}, \hat{p}_{j}^{\prime}] = i \delta_{i j}.
\begin{aligned}T_{f}(A^{{''E}^{-}_{0}}_{0}, h^{{''E}^{-}_{0}})=T_{f}(A^{{'E}^{-}_{0}}_{0}, h^{{'E}^{-}_{0}})+T_{f}(A^{{'E}^{-}_{1}}_{1}, h^{{'E}^{-}_{1}}),\end{aligned}
n_w \leq \max\biggl\{\sum_{j\in K}n_{e_j}: K \subseteq [|E|], |K| \leq (|E|-1)\biggr\}\leq \max\biggl\{\sum_{j\in K}n_{e_j}: |K| = (|E|-1)\biggr\}= \max\biggl\{\sum_{e\in E}n_e - n_{e'}: e' \in E\biggr\}=\sum_{e\in E}n_e - s.
F_{k,t}(r)=2\pi i^k\int_0^\infty e^{-2\pi t\rho}J_{-k}(2\pi r\rho)\,d\rho =2\pi (-1)^k i^k \int_0^\infty e^{-2\pi t\rho}J_k(2\pi r\rho)\,d\rho\,.
\begin{bmatrix}-1.7287 -1.1761 -1.0597 -0.3236 -0.2340 0.4706 0.4712 0.5435 0.6309 0.7533 0.8020 0.9237 1.1394 1.4373 1.5351 1.6941 8.6501 10.7254 12.7694 13.0349. \end{bmatrix}
l(\mathfrak{s}(t_{\lambda})\varphi_{K,\chi})=Q^{-1}c_{w_0}(\chi)\delta^{1/2}_{B_n}\ \chi^{-1}(\mathfrak{s}(t_{\lambda}))=Q^{-1}c_{w_0}(\chi)\delta^{1/4}_{B_n}(t_{\lambda}).
{}\frac{D}{4}=\frac{1}{x^2(1+x^p)^2}\left\{-px^pw^2-\frac{2w}{p}\left(1+px^p-\frac{x^p\log(x^p)}{1+x^p}\right)-\frac{x^p\log^2(x)}{p(1+x^p)^2}\right\}.
\begin{array}{ll}g(U(t), U(t)) g(\dot{\alpha}(t), (\nabla f)_{\alpha(t)}) + \frac{\lambda^2}{2}g\left(\nabla_\nu \frac{1}{\lambda^2}, U(t) \right) g(X(t), X(t)) + g((T_U U)(t), X(t)) = 0,\end{array}
A_{U ( 1 ) , ~ ( U ( 1 ) , S O ( 8 ) , S U ( 1 2 ) )}= \frac{1}{3}2 ( - 3 \sqrt{3}) 1 2 \frac{\sqrt{3}}{2}( 2 4 ( \frac{1}{2}) , - 1 , \frac{1}{2}) = ( - 4 3 2 , 3 6 , - 1 8 ) ,
u_{\vec{\xi}}(\vec{x}) = \sum_{\sigma \in S_N}A_{\sigma}\prod_{j=1}^{N}\xi_{\sigma (j)}^{x_j}= \sum_{\sigma \in S_N}A_{\sigma \tau^2}\prod_{j=1}^{N}\xi_{\sigma \tau^2(j)}^{x_j}=\sum_{\sigma' \in S_N}A_{\sigma' \tau}\prod_{j=1}^{N}\xi_{\sigma' \tau(j)}^{x_{\tau(j)}}= \sum_{\sigma' \in S_N}- A_{\sigma'}\prod_{j=1}^{N}\xi_{\sigma'(j)}^{x_j}= - u_{\vec{z}}(\vec{x}).
\sum_{1\leq i\leq n}(|Dw|^2)_i\gamma^i= \sum_{1\leq i\leq n}(|D'w|^2)_i\gamma^i = 2\sum_{1\leq i,k,l\leq n}c^{kl}w_{ki}w_l\gamma^i = 2\sum_{1\leq i,k,l\leq n}c^{kl}u_{ki}u_l\gamma^i-2\sum_{1\leq k,l\leq n}c^{kl}u_lD_k\psi,
\times\exp\Biggl\{\int d^4x\Biggl[-\frac14{\bf F}_{\mu\nu}^2-\eta^2\sum\limits_{a=1}^{3}\left(\partial_\mu\theta_a-g_m{\bf e}_a{\bf B}_\mu\right)^2+ik\sum\limits_{a=1}^{3}\theta_a^{\rm reg.}\Biggr]\Biggr\}.
\langle 0| a(f_{1}) \cdots a(f_k) \gamma_{N,\ell}^{(k)}a^\dagger(g_1) \cdots a^\dagger(g_k) |0 \rangle &={N\choose k}^{-1}\langle \Psi_{N,\ell}| a^\dagger(g_k) \cdots a^\dagger(g_1) a(f_1)\cdots a(f_k) | \Psi_{N,\ell}\rangle
g_2 = \frac{\sinh \xi}{4 \cosh^5 \xi}- \frac{5 \sinh^3 \xi}{4\cosh^7 \xi}+ C_2 \left( \frac{\sinh^2 \xi}{2 \cosh^4 \xi}-\frac{3 \sinh^4 \xi}{2 \cosh^6 \xi}\right) + C_3 \frac{\sinh^3\xi}{\cosh^5 \xi}.
-2a=\left( 1+a^{2}\right) \left( \frac{f_{1}}{f_{1}^{\prime}}\right) \left( \frac{f_{2}^{\prime \prime}}{f_{2}^{\prime}}\right) +\left( \frac{f_{1}^{\prime \prime}}{f_{1}^{\prime}}\right) \left( \frac{f_{2}}{f_{2}^{\prime}}\right) ,
[e_1, e_1]=\alpha_1e_3+ \alpha_2e_4+ \alpha_3e_5, [e_1, e_2]=\alpha_4e_3+\alpha_5e_4+ \alpha_6e_5, [e_2, e_1]=\beta_1e_3+ \beta_2e_4+ \beta_3e_5, [e_2, e_2]=\beta_4e_3+ \beta_5e_4+ \beta_6e_5, [e_1, e_3]=\gamma_1e_5, [e_2, e_3]=\gamma_2e_5, [e_1, e_4]=\gamma_3e_5, [e_2, e_4]=\gamma_4e_5.
\tilde{e}_{i_1}^{}\left( b \otimes b^\prime \right) = \tilde{e}_{i_1}^{}\left( \tilde{f}_{i_1}^{a_1}\cdots \tilde{f}_{i_l}^{a_l}(b_\lambda) \otimes b^\prime \right) &= \tilde{f}_{i_2}^{a_2}\cdots \tilde{f}_{i_l}^{a_l}(b_\lambda) \otimes \tilde{e}_{i_1}^{r_1}(b^\prime)
\alpha_{k,l}^{(n,p)}= \pi^n\frac{\Gamma\left(\frac{p+k-l}{2}+1\right)\Gamma\left(\frac{p-k+l}{2}+1\right)}{\Gamma\left(\frac{p+k+l}{2}+n\right)\Gamma\left(\frac{p-k-l}{2}+1\right)}.
\left({\mathcal Z}^{W_\alpha}_M(x)\right)_{x\ge 0}:=\left(L^{W_\alpha}_{{\mathcal T}_{M+1}}(x)-L^{W_\alpha}_{{\mathcal T}_M}(x)\right)_{x\ge 0}\Longrightarrow \left(Z^{(\alpha,0)}(x)\right)_{x\ge 0}, \ \Z^{(\alpha,0)}(0)=1.
\begin{array}{l l}ch(L_1)=(u+\frac{1}{2}u^2)\cdot(v_{2n-4}+\frac{1}{2}v_{2n-2}) ch(L_2)=(u+\frac{1}{2}u^2)v_{2n-2}begin{align*}15pt]ch(L_3)=u^2(v_{2n-4}+\frac{1}{2}v_{2n-2}) ch(L_4)=u^2v_{2n-2}.\end{array}
\int_0^\infty \dots \int_0^\infty f_n(z_1,\dots,z_n)dz_1\dots dz_n = n\int_0^\infty g_n(z_1)dz_1 \int_0^{z_1}\dots \int_0^{z_1}dz_2\dots dz_n = n \int_0^\infty g_n(t)t^{n-1}dt =1.
e_{\Gamma'}(S) \geq e_{\Gamma}(S) - \gamma e(\Gamma) \geq d\binom{|S|}{2}- \frac{\gamma}{2}n^2 = d\binom{|S|}{2}- \frac{\gamma}{\rho^2}\frac{|S|^2}{2}\geq d' \, \binom{|S|}{2},
\begin{array}{c}{{\chi ( a , b ) = \chi ( a ) + \chi ( b ) + n_{2}( a ) n_{1}( b )}}\\{{\chi ( c , d ) = \chi ( c ) + \chi ( d ) + n_{2}( c ) n_{1}( d )}}\\{{\chi ( a , d ) = \chi ( a ) + \chi ( d ) + n_{2}( a ) n_{1}( d )}}\\{{\chi ( b ) = \chi ( c )}}\\{{( \epsilon_{( b , c )})_{n}= ( - )^{n_{2}( b )}q^{\frac{1}{2}( n_{1}( b ) - n_{2}( b ) )}}}\\ \end{array}
f = \phi_1( p|_{P( d(r)P(s) )}, v) = P(p|_{P( d(r)P(s) )}v) + P(d(p|_{P( d(r)P(s) )}) P(v) ) - p|_{P( d(r)P(s) )}P(v) + \lambda P(d(p|_{P( d(r)P(s) )}) v ) g = \phi_1(r,s) = P(d(r) P(s)) - rP(s) + P(rs) - \lambda P(d(r) s),
\left\lVert T_\epsilon(0)\right\rVert_{C^{0,\alpha}_{\epsilon,\gamma+2,\delta}(M_G)}=\lVert 1-e^{f_\epsilon}\rVert_{C^{0,\alpha}_{\epsilon,\gamma+2,\delta}(M_G)}\leq C\epsilon^{(4+\gamma)\frac n{n+1}}.
\frac{|G|}{|C|}I_j = \sum_{\chi}\overline{\chi}(g)J_j(\chi) = k_j(1)-k_j(0) \sum_{\chi}\overline{\chi}(g)\left\{a(\chi)-\delta(\chi)\right\}-\sum_{\chi}\overline{\chi}(g)\left(\sum_{\rho_{\chi}\in Z(\chi)}k_j(\rho_{\chi})\right) -\sum_{\chi}\overline{\chi}(g)V_j(\chi)
k=-2.5log_{10}[(\frac{\nu_{c}}{\nu_{d}})^{2}(\frac{\Delta_{c}}{\Delta_{d}})(\frac{\nu_{b}}{\nu_{a}})^{2\frac{\nu_{c}-\nu_{d}}{\nu_{a}-\nu_{b}}}(\frac{\Delta_{b}}{\Delta_{a}})^{\frac{\nu_{c}-\nu_{d}}{\nu_{a}-\nu_{b}}}]
\begin{aligned}\hat\lambda_j^h = \frac{168\sin^2(\frac{t_j}{2})}{h^2}\frac{33 - 1,200 \eta + 2 (13 + 900 \eta) \cos(t_j) + (1- 720 \eta) \cos(2t_j) + 120 \eta \cos(3t_j)}{1,208 + 1,191\cos(t_j) + 120\cos(2t_j) + \cos(3t_j)}, \hat{U}_{j}= c_j \big(\sin(k t_j)\big)_{k \in\{1,\ldots, N-1\}}\end{aligned}
\widetilde{\eta}^{i}\equiv U^{i}{}_{j}\eta^{j}, \qquad \widetilde{\eta}_{i}\equiv \eta_{j}( U^{- 1})^{j}{}_{i}\ , \qquad \widetilde{\theta}^{i}\equiv \tilde{U}^{i}{}_{j}\theta^{j}, \qquad \widetilde{\theta}_{i}\equiv \theta_{j}( \tilde{U}^{- 1})^{j}{}_{i}\ ,
Z(k_1,\dots,k_n)=\det\Big[\mathbb{I}_{n!|\mathcal{E}|^n}-E(k_n) Y_{n-1}(k_{n}-k_{n-1})\dots Y_1(k_{n}-k_{{1}}) \left(\mathbb{I}_{n!}\otimes S_v(k_{n})\otimes\mathbb{I}_{|\mathcal{E}|^{n-1}}\right) Y_1(k_{1}+k_{n})\dots Y_{n-1}(k_{n-1}+k_{n})\Big],
\theta \circ D_{*}(t_{e}) = \theta(\delta(\bar{e})t_{e}\delta(e)^{-1}) = \delta(\bar{e}) t_{e^{\prime}}t_{e^{\prime \prime}}\delta(e)^{-1}& = D_{*}^{\prime}(t_{e^{\prime}})D_{*}^{\prime}(t_{e^{\prime \prime}}) = D^{\prime}_{*}(t_{e^{\prime}}t_{e^{\prime \prime}}) & = D^{\prime}_{*}\circ \theta(t_{e}).
\frac{\sharp \{\{q^n \nu\}\in I \mid n\leq N \}}{N}\leq \frac{\sharp \{\{q^n \nu\}\in [\frac{l}{q^k}, \frac{m}{q^k}) \mid n\leq N \}}{N}& \ll \frac{m-n}{q^k}+ O\left((q^k)^{1/3}\frac{1}{N^{1/6}}\right) &< \vert I \vert + \delta + O\left((q^k)^{1/3}\frac{1}{N^{1/6}}\right) &= \vert I \vert + \delta + O_\delta \left( \frac{1}{N^{1/6}}\right).
\chi=I_\psi \int_{0}^{2\pi}d\xi_1\int_0^{2\pi}d\xi_2\int_0^{\frac{\pi}{2}}\sin\varphi\cos\varphi d\varphi =\frac{1}{\pi^2}\int_{0}^{2\pi}d\xi_1\int_0^{2\pi}d\xi_2\int_0^{\frac{\pi}{2}}\sin\varphi\cos\varphi d\varphi=\frac{1}{\pi^2}\cdot 4\pi^2\cdot\frac{1}{2}=2,
\frac{\hat{u}_\alpha^{2^*(s)-1}}{d_{\hat{g}_\alpha}(X,X_{0,\alpha})^s}=\mu_\alpha^{({\frac{n}{2}-1})(2^*(s)-1)+s}\frac{u_\alpha^{2^*(s)-1}(x)}{d_g(x,x_0)^s}.
\frac{3}{2}d(d+1) = \begin{pmatrix}2d+2 2\end{pmatrix}- \begin{pmatrix}d+2 2\end{pmatrix}&= \begin{pmatrix}2d+2 2\end{pmatrix}- \begin{pmatrix}2d-k+2 2\end{pmatrix}+ \alpha(d-k) + (d-k)(k+3)-1 &\geq \begin{pmatrix}2d+2 2\end{pmatrix}- \begin{pmatrix}2d-k+2 2\end{pmatrix}+ \alpha(d-k).
F(a,b,c;z) = \sum_{i = 0}^\infty \frac{(a)_i (b)_i}{(c)_i}\frac{z^i}{i !}&= 1 + \frac{ab}{c}\frac{z}{1!}+ \frac{a (a + 1) b (b + 1)}{c (c + 1)}\frac{z^2}{2 !}+ \cdots.
k S_k= (k+1) S_{k+1}- \eta_{T-k}q_{T-k}= k S_{k+1}+ S_{k+1}- \eta_{T-k}q_{T-k}\le k S_{k+1}+ \frac{1}{k+1}\sum_{t=T-k}^T \eta_t (q_t - q_{T-k}) \: .
\tau_{H}=-m_{1}\cot \left( \frac{\vartheta}{2}\right) \alpha +m_{1}\tan \left( \frac{\vartheta}{2}\right) \alpha+n\tan \vartheta (2\alpha)-m_{2}\cot \vartheta (2\alpha) &=2\{-(m_{1}+m_{2}) \cot \vartheta+n\tan \vartheta\}\alpha.
\|\psi(f\circ u) -(\psi(f))\circ u\|_{\ell^p}= \|M_{\psi(f\circ u)}-M{(\psi(f))\circ u}\|_{S^p}=\|\Psi(M_{f\circ u})-U\Phi(M_f)U^*\|_{S_p}&= \|U\Psi(M_{f})U^*-U\Phi(M_f)U^*\|_{S_p}\leq C[\Psi]\|M_f\|_{S^p}=C[\Psi]\|f\|_{\ell^p}
\frac{\partial^2B_j^{(1)}}{\partial r^2}+\frac{2}{r}\frac{\partial B_j^{(1)}}{\partial r}+\frac{1}{r^2\sin \theta}\frac{\partial}{\partial \theta}\left (\sin \theta\frac{\partial B_j^{(1)}}{\partial \theta}\right )=0,\r>1, & B_j^{(1)}\sim b_jr\cos \theta \mbox{as}r \to \infty;B_j^{(1)}=0 \mbox{on}r=\ell_j.
\int\limits_{E_{2}}|\sigma_{i}^{2}(x,\xi)-\sigma_{i}^{2}(y,\xi)|^{1+\tau}m(d\xi)\leq\max\limits_{j\in\{1,\dots,d\}}\int\limits_{|z|>1}|z|^{1+\tau}\mu_{j}(dz)\sum\limits_{k=1}^{d}|x_{k}-y_{k}|,
{\cal F}_{a}^{\mu \nu}= \frac{1}{a}\phi_{a}{\cal F}^{\mu \nu}= \frac{1}{a}\phi_{a}\, \frac{1}{a^{3}e}\phi \cdot ( \partial^{\mu}\phi \wedge \partial^{\nu}\phi ) + \partial^{\mu}A^{\nu}- \partial^{\nu}A^{\mu}.
\begin{bmatrix}n_{11}n_{12}n_{13}n_{21}n_{22}n_{23}n_{31}n_{32}n_{33}\end{bmatrix}\begin{bmatrix}1 1 1 1 1 1 1 1 1\end{bmatrix}= \begin{bmatrix}22 22 22 22 22 22 22 22 22\end{bmatrix}
0\geq \sum\limits_{i,j=1}^{n}\left[\frac{(1-K)}{A^2}(u_1)_{x_i x_j}(y)+\frac{K}{B^2}(u_2)_{x_i x_j}(z) - \frac{1}{C^2}u_{x_i x_j}(x)\, \right]\eta_i \eta_j
x = \frac{\ell^{-a_1(x)}}{\displaystyle 1+\frac{(\ell-1)\ell^{-a_2(x)}}{\displaystyle 1 + \frac{(\ell-1)\ell^{-a_3(x)}}{\displaystyle 1 + \ddots}}}=:[a_1(x), a_2(x), a_3(x), \ldots]_{\ell},
u \left( x \right) = \left( \begin{array}{c}{e^{- i \varphi}u_{1}\left( r , t \right)}\\{u_{- 1}( r , t )}\\ \end{array}\right) \exp i \left[ \left( l - l_{0}\right) \varphi - \frac{m^{2}}{2}\int \frac{d u^{0}}{\lambda + g \left( u^{0}\right)}- \frac{\lambda u^{3}}{2}\right] ,
\Gamma_{4 V}^{[ r , 0 ]}= - \big ( C_{1}^{[ r ]}\big )_{2 V , 2 V}- \big ( C_{2}^{[ r ]}\big )_{2 V , 2 V}- \big ( C_{3}^{[ r ]}\big )_{2 V , 2 V}+ V_{4 V}^{[ r , 0 ]},
\int_{\mathbb{R}_+}f(t,x,y)e^{uy}dy= \big(1-\frac{\sigma^2}{2a}u(1-e^{-at})\big)^{-\frac{2a\theta}{\sigma^2}}\cdot \exp \Big (\frac{xue^{-at}}{1-\frac{\sigma^2}{2a}u(1-e^{-at})}\Big) .
L(f,2) L(f,\chi,1) = \frac{N \pi^2 \tau(\chi)}{4 \phi(N)}\int_{\gamma_f}\eta(u_{\overline{\chi}},u_{\psi\chi}) = \frac{N \pi^2 \tau(\chi)}{4 \phi(N)}\int_{\gamma_f^-}\eta(u_{\overline{\chi}},u_{\psi\chi}).
\sigma^{\mathrm{r e g}}( \xi ) = -{\frac{| \xi |}{2}}-{\frac{| \xi |}{\pi}}\sum_{n = 1}^{\infty}{\frac{K_{1}( 2 n \pi | \xi | )}{n}}= -{\frac{| \xi |}{2}}-{\frac{| \xi |}{\pi}}G ( 2 \pi | \xi | ) \ .
~\frac{1}{(r+1)(\mu+1)}\sum^\mu_{j=0}\frac{1}{(\mu-j+\beta)(\mu-j+\beta+1)}\|\phi\|^2 = ~\frac{1}{(r+1)(\mu+1)}\bigg(\frac{1}{\beta}-\frac{1}{\beta+\mu+1}\bigg)\|\phi\|^2 = ~\frac{1}{(r+1)(\beta+\mu+1)\beta}\|\phi\|^2.
\Sigma_3 =\sum\limits_{D\leq d<2D}\sum\limits_{n\in \mathcal{N}(d)}\Bigg(-\sum\limits_{1\leq|m|\leq M}\frac{e\left(m\left(\frac{\sqrt{X}-n}{d}\right)\right)}{2\pi i m}+\mathcal{O}\left(f_{M}\left(\frac{\sqrt{X}-n}{d}\right)\right)\Bigg) &=\Sigma_4+\Sigma_5\,,
\hat{\mathbf{h}}^{(g)}=\boldsymbol{\Upsilon}_{U}^{(g)}\hat{\mathbf{c}}^{(g)}=\boldsymbol{\Upsilon}_{U}^{(g)}\left(\mathbf{W}_{mmse,D}^{(g)}\right)^H\mathbf{y}^{(g)}=\underbrace{\boldsymbol{\Upsilon}_{U}^{(g)}\left(\mathbf{W}_{mmse,D}^{(g)}\right)^H{\left(\boldsymbol{\Upsilon}_S^{(g)}\right)^H}}_{\textrm{Reduced Rank Wiener Filter}}\mathbf{y}
{\hat{G}}({\hat{\xi}}{\hat{\eta}};{\hat{\xi}}^{\prime}{\hat{\eta}}^{\prime}) = \sum_{n}{\phi}_{n}({\hat{\xi}}{\hat{\eta}}; P ){\phi}_{n}^{*}({\hat{\xi}}^{\prime}{\hat{\eta}}^{\prime}; P ) / ( P^{2}+ M^{2})
K ( t ,{\bf x}, s \vert t ,{\bf x}, 0 ) = \big \langle t ,{\bf x}\big \vert \exp - i \left[ \left({\partial_{t}}^{2}-{\nabla}^{2}+ 2 i q \phi ( \partial_{t}+ \partial_{z}) \right) s \right] \big \vert t ,{\bf x}\big \rangle .
e^TB_{SS}e\geq e^T\prod_{t=1}^kA_{SS}(t)e\geq \min_{\substack{\forall t:X(t)\geq \gamma I, X(t)e\leq e\cr e^T\sum_{t=1}^kX(t)e\geq k|S|-\Lambda}}e^T\prod_{t=1}^kX(t)e\geq ??
\lambda_\alpha \lambda_\beta -\lambda_\beta \lambda_\alpha = 0, \lambda_\alpha \theta_\beta^i-\theta_\beta^i \lambda_\alpha = 0, \theta_\alpha^i \theta_\beta^j+\theta_\beta^j \theta_\alpha^i = -2\delta_{\alpha,\beta}\delta_{i,j}\lambda_\alpha,
|v|_{V_{\pi}}\coloneqq\inf_{H\in H_{1}(\Omega)^{3}}\{|\gamma H|_{H_{1/2}(\Gamma)^{3}}\,|\,\pi_{\tau}H=v\}, |v|_{V_{\gamma}}\coloneqq\inf_{H\in H_{1}(\Omega)^{3}}\{|\gamma H|_{H_{1/2}(\Gamma)^{3}}\,|\,\gamma_{\tau}H=v\}.
\sup_{0 \leq t \leq T}\|{\cal I}_{a, m}^{-1}\frac{\partial}{\partial t}P_t^{m}f \|_{\infty}=\sup_{0 \leq t \leq T}\|{\cal I}_{a, m}^{-1}G_m P_t^{m}f \|_{\infty}=\sup_{0 \leq t \leq T}\|{\cal I}_{a, m}^{-1}P_t^{m}G_m f \|_{\infty}< \infty.
\begin{array}{lll}[R/(I,L)]_3 \cong [K[x_1,x_2,x_3]/(x_1^3,x_2^3,x_3^3,x_1^2x_2,x_1+x_2+x_3)]_3 \cong[K[x_1,x_2]/(x_1^3,x_2^3,x_1^3+3x_1^2x_2+3x_1x_2^2+x_2^3, x_1^2x_2)]_3 \cong [k[x_1,x_2]/(x_1^3,x_2^3,x_1^2x_2,x_1x_2^2)]_3=0 \end{array}
S=\begin{bmatrix}\frac{\sqrt{p_1p_2}}{2}\Delta_1-\varepsilon_1\Delta_1A_{1,1}\eta\Delta_1A_{1,2}-\frac{1}{2}\Delta_1 J_{1,2}\Delta_2 \eta\Delta_2A_{2,1}-\frac{1}{2}\Delta_2 J_{2,1}\Delta_1 \frac{\sqrt{p_1p_2}}{2}\Delta_2 -\varepsilon_2\Delta_2A_{2,2}\end{bmatrix},
w : = \left( \begin{array}{c}{{\tilde{x}_{1}}}\\{{.}}\\{{.}}\\{{.}}\\{{\tilde{x}_{n}}}\\{{\hat{y}_{1}}}\\{{.}}\\{{.}}\\{{.}}\\{{\hat{y}_{n}}}\\ \end{array}\right) \; , \; z : = \left( \begin{array}{c}{{\tilde{y}_{1}}}\\{{.}}\\{{.}}\\{{.}}\\{{\tilde{y}_{n}}}\\{{\hat{x}_{1}}}\\{{.}}\\{{.}}\\{{.}}\\{{\hat{x}_{n}}}\\ \end{array}\right) \; \; .
\partial e^{t \Delta}f =\sum_{j=1}^{+\infty}\langle e^{t\Delta}f , \phi_j \rangle_\mathcal{H}\partial \phi_j =\sum_{j=1}^{+\infty}e^{-\lambda_j t}\langle f , \phi_j \rangle_\mathcal{H}\partial \phi_j =- \sum_{j=1}^{+\infty}\frac{1}{\lambda_j}e^{-\lambda_j t}\langle \partial f , \partial \phi_j \rangle_\mathcal{H}\partial \phi_j
\lim_{r\to \infty}\frac{u(r)}{u_0(r)}= \lim_{r\to \infty}\frac{v(r)}{v_0(r)}=\lim_{r\to \infty}\frac{u'(r)}{u_0'(r)}= \lim_{r\to \infty}\frac{v'(r)}{v_0'(r)}= 1.
\lVert T_g^\varphi \rVert_{e,\mathcal{X}\to H_v^\infty}\leq \lVert T_g^\varphi - I \circ K \rVert_{\mathcal{X}\to H_v^\infty}= \lVert I \circ D \circ T_g^\varphi + \delta_0 \circ T_g^{\varphi}- I \circ K \rVert_{\mathcal{X}\to H_v^\infty}&\leq \Vert I \Vert_{H_w^\infty \to H_v^\infty}\lVert D \circ T_g^\varphi - K \rVert_{\mathcal{X}\to H_w^\infty},
\langle V_{\alpha_{1}}V_{\alpha_{2}}V_{\alpha_{3}}\rangle = \left( C^{e v e n}( \alpha_{1}, \alpha_{2}, \alpha_{3}) + \eta \bar{\eta}C^{o d d}( \alpha_{1}, \alpha_{2}, \alpha_{3}) \right) \prod_{i < j}| Z_{i}- Z_{j}|^{\delta_{i j}}
E_{0}= \frac{1}{2 L}\sum_{n = - \infty}^{\infty}\int \frac{d k_{2}}{2 \pi}\left\{\Big ( ( n + \mathrm{\frac{1}{2}}) \mathrm{\frac{\pi}{L}}\Big )^{2}+ k_{2}^{2}+ m^{2}\right\}^{1 / 2}
- \hbar g^2 \mu^\epsilon\int\frac{d^d k}{(2\pi)^d}[D_0]_{\sigma\tau}(k)\frac{(2p-k)_\sigma (2q+k)_\tau (2k+q-p)_\mu}{(k-p)^2 (k+q)^2}= (q-p)_\mu\cdot\hbar g^2 \frac{\alpha}{8 \pi^2\epsilon}+ \cdots .
\frac{\binom nl \alpha^l (1-\alpha)^{n-l}}{\binom n{l_0}\alpha^{l_0}(1-\alpha)^{n-l_0}}=\frac{l_0(l_0-1)\cdots (l_0-h+1)}{(n-l_0+1)(n-l_0+2)\cdots (n-l_0+h)}\alpha^{-h}(1-\alpha)^{h}.
\frac{1}{M_{6}^{4}}f_{0}( \rho ) = - \frac{5}{8 M_{6}^{4}}{\Lambda}+ \frac{3}{8 M_{6}^{4}}f_{\theta}( \rho ) - \frac{{\left( \sigma^{3 / 4}\gamma^{1 / 2}\right)}^{\prime \prime}}{\sigma^{3 / 4}\gamma^{1 / 2}}\ .
\varphi(t,r,\theta,z)=\sum_{m}\int \bar{q}{\rm d}\bar{q}{\rm d}k_{z}\left[a_{\bar{q}mk_{z}}u_{\bar{q}mk_{z}}(t,r,\theta,z)+a_{\bar{q}mk_{z}}^{\dag}u_{\bar{q}mk_{z}}^{*}(t,r,\theta,z)\right] .
\begin{cases}h_t+u\cdot \nabla h=\Delta h+(u(a(t), t)-u)\cdot\nabla\Theta_a, u_t+u\cdot\nabla u+\nabla P=\mu\Delta u-\nabla h\cdot\Delta h -\nabla\Theta_a(\Delta h-\Delta h(a(t), t)), \nabla\cdot u =0.\end{cases}
\sum_{n=0}^{\infty}\frac{q^{2n^2}(q;q^2)_n^2 (q;q^2)_{2n}}{(q^2;q^2)_{2n}(q^6;q^6)_n^2}[8n+1]=\frac{(q^3;q^2)_\infty (q^3;q^6)_\infty}{(q^2;q^2)_\infty (q^6;q^6)_\infty}.
\tilde{R}_n (T)= &\sum_{m=1}^{M}\tau_lT\log_2\left(1+\frac{\frac{\eta\rho(1-T) g_n}{T}}{\frac{\eta\rho(1-T)\sum_{j_{m,k}>j_{m,n}}g_{A(m,j_{m,k})}}{T}+1}\right).
\left |E^{Q_{n_j}}\left ( F(X)| X < N_j\right) - E^{Q_{n_j}}\left ( F(X)| X < \tilde N_j\right)\right | \ =\ \frac{|N_j - \tilde N_j |}{N_j -G_{n_j+1}}O(1) + \epsilon \frac{N_j}{| N_j -G_{n_{j+1}}|}O(1) &\ =\ \epsilon \frac{N_j}{N_j - G_{n_j+1}}O(1)=\epsilon O(1),
\overline{\gamma}_\theta ={\frac{\overline{\gamma}}{\Omega}}\mathbb{E}\left \{{W_{\theta}}\right \}={\frac{\overline{\gamma}}{\Omega}}(V_1^2+V_2^2+2V_1V_2\cos{\theta}+2\sigma^2\mu) ={\frac{\overline{\gamma}}{\Omega}}2\sigma^2\mu(1+\kappa_\theta).
\begin{aligned}\min_{0{}\leq{}i{}\leq{}N}\|\nabla f(x_i)\|^2{}\leq{}\, 2 L \big(f(x_0)-f_*\big) \max \left\{(1-h)^{2N}\ , \ \frac{1}{q + N \frac{h(2-h)(2-\kappa h)}{2-h \left(1+\kappa \right)}}\right\}\end{aligned}
\sum_{j=0}^{n-1}w^{ij}\wp_j C(j) \beta_r^{(j+1)}(x)=\sum_{j=0}^{n-1}w^{ij}\gamma_r\wp_{j+1}\beta_r^{(j+1)}(x) =\frac{\gamma_r}{w^i}\sum_{j=0}^{n-1}w^{ij}\wp_j \beta_r^{(j)}(x),
\big(M\circ \rho_\lambda(g)\big) f (\xi) = \xi_n \kappa\big(g^{-1}, \xi\big)^\lambda f\big(g^{-1}(\xi)\big)= \kappa\big(g^{-1},\xi\big)^{\lambda-1}\big(g^{-1}(\xi)\big)_nf\big(g^{-1}(\xi)\big) \hphantom{\big(M\circ \rho_\lambda(g)\big) f (\xi)}{}= \rho_{\lambda-1}(g) (Mf)\big(g^{-1}(\xi)\big)
\sum_{j=1}^{m}- 2\beta_{j}\partial_{s_{k}}\log \Lambda_{j}= -2 \sum_{j=1}^{m}\beta_{j}\partial_{s_{k}}(\beta_{j}) \log (4x_{j}c_{-x_{j}}\sqrt{r}) -2\sum_{j=1}^{m}\beta_{j}\sum_{\substack{\ell = 1 \ell \neq j}}^{m}\partial_{s_{k}}(\beta_{\ell})\log(\widetilde{T}_{\ell,j}).
B_{t\!+\!1\!,k_{t+1}}\!=\! B_{t,k_t}\!+\! T_f(E_{k_t}\!-\! \alpha_t\tilde{p}_{t}\!-\! (1\!-\!\alpha_t) \sum_{i=1}^2 |w_{t,k_ti}|^2 p_{t,i}), & B_{t\!+\!1\!,\bar{k}_{t+1}}\!=\! B_{t,\bar{k}_t}\!+\! T_f(E_{\bar{k}_t}\!-\! (1\!-\!\alpha_t)\sum_{i=1}^2 |w_{t,\bar{k}_ti}|^2 p_{t,i}),
M_{\pi'}(Z)= M_{\pi'}(1;Z) = \prod_{T\in{\cal T}^{\pi'}}(1-Z^T)^{-1}= \prod_{i_1,i_2,\ldots,i_m\geq 0}\c^{\pi'}_{(i_1)(i_2)\cdots(i_m)}\ (1 - z_1^{i_1}z_2^{i_2}\cdots z_m^{i_m})\,.
\left( Q^* Q \right)_{ij}= \sum_{\mu=1}^{M_A}\sum_{\nu=1}^{M_B}\langle f_\nu^B \vert i \rangle \langle i \vert g_\mu^A \rangle \langle g_\mu^A \vert j \rangle \langle j \vert f_\nu^B \rangle = W_{ij}\ ,
\bar{\phi}_{\bar{x}}^{\bar{i}}( \bar{y}) : = \bar{x}^{\bar{i}}+ \bar{y}^{\bar{i}}+ \frac 1 2 \bar{\phi}_{\bar{x}, \bar{j}\bar{k}}^{\bar{i}}\, \bar{y}^{\bar{j}}\bar{y}^{\bar{k}}+ \cdots
\, - \, \frac{1}{3 !}\int d^{3}x_{1}d^{3}x_{2}d^{3}x_{3}{\mathcal W}_{\mu_{1}\mu_{2}\mu_{3}}^{( 3 )}( x_{1}, x_{2}, x_{3}) b_{\mu_{1}}( x_{1}) b_{\mu_{2}}( x_{2}) b_{\mu_{3}}( x_{3}) \, + \, \ldots
\frac{\partial^\alpha u(x_i,y_j,t)}{\partial t^\alpha}\!+\!\kappa_x\sum\limits_{m=0}^{M_x}{a_{im}^{(1)}u(x_m,y_j,t)}\!+\!\kappa_y\sum\limits_{m=0}^{M_y}{b_{jm}^{(1)}u(x_i,y_m,t)}&\!-\!\varepsilon_x\sum\limits_{m=0}^{M_x}{a_{im}^{(2)}u(x_m,y_j,t)}\!-\!\varepsilon_y\sum\limits_{m=0}^{M_y}{b_{jm}^{(2)}u(x_i,y_m,t)}\!=\!f(x_i,y_j,t),
\begin{aligned}R_n(h_w) = (-1)^{L(w)}T_{I(w)}+ (-p)^{L(w)}T_{J(w)}+ T_{t_{n+1}}T_{\sigma_{n}}T_{t_{n+1}}\ (\sum_{\begin{smallmatrix}L(y)=L(w)-2 l(y)<l(I(w))-3 \end{smallmatrix}}\beta_y T_y )+ \sum_{L(x)< L(I(w))}\alpha_x T_x.\end{aligned}