harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\begin{array}{l c r}{M^{+{\frac{n ( n + 1 )}{2}}}= ( \eta^{+})^{{\frac{n ( n + 1 )}{2}}}= N^{+{\frac{n ( n + 1 )}{2}}}}\\{S^{+ ( n + 1 )}= ( \eta^{+})^{n + 1}.}\\ \end{array}
\Big( \begin{array}{cc}{\rm e}^V \end{array}\Big) =\frac1{\sqrt{(\bar{z}_{+}\cdot z_{+})(\bar{z}_{-}\cdot z_{-}) - (\bar{z}_{+}\cdot z_{-})( \bar{z}_{-}\cdot z_{+})}}\left( \begin{array}{cc}\bar{z}_{-}\cdot z_{-}-\bar{z}_{-}\cdot z_{+}-\bar{z}_{+}\cdot z_{-}\bar{z}_{+}\cdot z_{+}\end{array}\right)\ ,
X_n =S_{\Delta t}^{n-\ell}X_{\ell}+\Delta tS_{\Delta t}^{n-\ell}G(X_\ell)+S_{\Delta t}^{n-\ell}e^{\tau A}\sigma(X_\ell)\Delta W_\ell \&~+\Delta t\sum_{m=\ell+1}^{n-1}S_{\Delta t}^{n-m}G(X_m)+\sum_{m=\ell+1}^{n-1}S_{\Delta t}^{n-m}e^{\tau A}\sigma(X_m)\Delta W_m,
{\bf M}= \left( \begin{array}{c c}{{\left({\frac{\gamma^{\prime}}{\bar{\gamma}^{\prime}}}\right)^{1 / 4}}}&{{0}}\\{{0}}&{{\left({\frac{\gamma^{\prime}}{\bar{\gamma}^{\prime}}}\right)^{- 1 / 4}}}\\ \end{array}\right) .
\partial_t f(t,m,x) \ = \ - \frac{1}{\sqrt{T-t}}\varphi\Big(\frac{m-x}{\sqrt{T-t}}\Big), \partial_{xx}^2 f(t,m,x) \ = \ \frac{2}{\sqrt{T-t}}\varphi\Big(\frac{m-x}{\sqrt{T-t}}\Big),
\det \mathbf{T}_{n,m,0,0}(x;q)=\prod_{i=1}^n\prod_{j=1}^m\frac{(1-q^{x+i-j})(1-q^{x+2i+j+a-b-2})}{(1-q^{x+2i-j})(1-q^{i+j-1})}.
\int_t^T\int_{\mathcal{O}}(u-k_m)^+\,\mu(dxds) \leq \int_t^T\int_{\mathcal{O}}(u-\xi)^+\,\mu(dxds)+\int_t^T\int_{\mathcal{O}}(\xi-\hat{\xi}^+)^+\,\mu(dxds)=0.
F_T(x)-1-G_1(x) =x(F_T(x)-1)+\frac{x^2}{1-x}(K-1) &+\sum_{m\geq2}J_m(x) -\frac{x}{1-x}J_1(x)+\sum_{m\geq2}x^m\sum_{j=1}^{m-1}\left(\frac{1}{(1-x)^j-1}\right)(K-1)\,.
x([2]R) =-\frac{3}{4}(t^2-6t+1)(t^2+6t+1) x([3]R) =-\frac{8}{9}(t^2+1)^4(t^2-18t+1)(t^2+18t+1)(t^2-6t+1)^{-2}(t^2+6t+1)^{-2}x([4]R) =\frac{15}{16}(t^2+1)^{-2}(t^2-18t+1)^{-2}(t^2+18t+1)^{-2}(t^2-6t+1)(t^2+6t+1) &\mbox{}\times (t^4 - 36t^3 - 106t^2 - 36t + 1)(t^4 + \frac{118}{5}t^2 + 1)(t^4 + 36t^3 - 106t^2 + 36t + 1).
(Q^{1}_{r}f)(y,m_{1},m_{2})=(2\pi iyr)f(y,m_{1},m_{2}) &(Q^{2}_{(s,t)}f)(y,m_{1},m_{2})=2\pi i(m_{1}s+m_{2}t)f(y,m_{1},m_{2}) &(Q^{3}_{w}f)(y,m_{1},m_{2})=w\frac{\partial f}{\partial y}(y,m_{1},m_{2}).
\begin{bmatrix}-\frac{\delta}{q_0}\frac{\delta}{q_1}0 0 \cdots 0 -\frac{\delta}{q_0}0 \frac{\delta}{q_2}0 \cdots 0 -\frac{\delta}{q_0}0 0 \frac{\delta}{q_3}\cdots 0 \vdots \vdots \vdots \ddots \ddots 0 -\frac{\delta}{q_0}0 0 0 \cdots \frac{\delta}{q_n}\end{bmatrix}.
x=\frac 1 s \int_0^s\varphi^{-1/(q-1)},\y=\varphi^{-1/(q-1)}(s),\c=M^{1/(q-1)},\ \z = \left(\frac 1 s \int_0^s \varphi\right)^{-\frac 1{q-1}}.
A_{\boldsymbol{\eta}, \boldsymbol{\nu}}^{(c)}=(A^{(c)}\boldsymbol{u}_{\boldsymbol{\eta}}, \boldsymbol{u}_{\boldsymbol{\nu}})=\sum_{l=1}^{c}q^{c-l}(\boldsymbol{u}_{\boldsymbol{\eta}}, S_{c-1}\cdots S_{l}\boldsymbol{u}_{\boldsymbol{\nu}}).
X'(q_1,\ldots,q_j)\equiv\sum\limits_{1\leq k<l\leq j}(q_k(q_l+1)+q_kq_l )+\sum_{k=1}^{j-1}k\equiv\sum\limits_{k=1}^{j-1}(q_k(k+j)+k)\equiv\sum\limits_{k=1}^{j-1}(q_k+1)(k+j),
|F_\epsilon^k|^2 = \alpha^2 \frac{x_k^2}{(|x|^2+\epsilon^2)^2}+ \sum_{i=1}^{k-1}\frac{1}{((x_k-x_i)+\epsilon^2)^2}+ \sum_{i=k+1}^{d}\frac{1}{((x_i-x_k)+\epsilon^2)^2}-2\frac{\alpha}{|x|^2+\epsilon^2}\sum_{i=1}^{k-1}\frac{x_k}{((x_k-x_i)+\epsilon^2)}+ 2\frac{\alpha}{|x|^2+\epsilon^2}\sum_{i=k+1}^d \frac{x_k}{((x_i-x_k)+\epsilon^2)}+ ET(k),
(e^{tB}- e^{tA})f_n(1) = \frac{1}{3n^2}(1-e^{-t}) -\frac{e^{-t/2}}{2n}(1-e^{-t}) &= \left(\frac{1}{3n^2}-\frac{e^{-t/2}}{2n}\right)(1-e^{-t})
\frac{4n\pi y}{e^{2n\pi y}-1}-\frac{4\,n^2\,\pi^2\,y^2\,e^{2n\pi y}}{(e^{2n\pi y}-1)^2}=\frac{4\pi yn}{(e^{2n\pi y}-1)^2}\left(e^{2n\pi y}-1-yn\pi\,e^{2n\pi y}\right).
\varepsilon_t \geq \frac{1}{r+1}\left ( \frac{2}{r^2-2}\right )^{t-1}> \frac{1}{r+1}\left ( \frac{1}{\tau}\frac{2 \tau}{r^2-2}\right )^{\frac{\log\log n}{3\log \tau}}= \frac{1}{r+1}(\log n)^{-1/3}\left (\frac{r^2-r-2}{r^2-2}\right )^{\frac{\log\log n}{3\log \tau}}.
f^\lambda = \left(f_1 e_a^\lambda e^{b \nu}+ f_2 e^{b \lambda}e^\nu_a+ f_3 g^{\nu \lambda}\delta^a_b \right)\bar{\Omega}^a_{~b \nu}+ N^{\mu \alpha \beta ~ b \nu}_{~~~~b}\nabla_\alpha \nabla_\beta \bar{\Omega}^a_{~b \nu}
\begin{array}{ll}A_4(\hat{S}_3) A_1(\hat{S}_3)=i[[\hat{S}_3]]~,~~~~~~~ A_2(\hat{S}_3)=2i (A_1(\hat{S}_3)-A_1(\hat{S}_3-1)) begin{align*}3mm]A_3(\hat{S}_3)=2i (A_1(\hat{S}_3)-A_1(\hat{S}_3+1))~, A_5=4(2A_1(\hat{S}_3)-A_1(\hat{S}_3-1)-A_1(\hat{S}_3+1))~.\end{array}
\begin{array}[c]{c}\frac{\partial}{\partial t}\varphi=\log\frac{(\widehat{\omega}_{t,N}+\sqrt{-1}\partial_{B}\overline{\partial}_{B}\varphi)^{2}\wedge\eta_{N}}{\Omega_{N}\wedge\eta_{N}},\varphi(T_{0})=\phi_{T_{0}},t\in(T_{0},T^{\prime}]\end{array}
V_{5}^{\lambda \mu \nu}( x ) = S y m m{\lim_{\epsilon \rightarrow 0}}\{\bar{\psi}( x + \epsilon / 2 ) \frac{1}{2}[ \gamma^{\lambda}, \sigma^{\mu \nu}] \gamma_{5}U_{P}( x + \epsilon / 2 , x - \epsilon / 2 ) \psi ( x - \epsilon / 2 ) \}.
A \vert 0 \rangle =A_{\mathcal{A}(W)}\vert0 \rangle, A\in\mathcal{A}_{\rm in}(W), A_{\mathcal{A}(W)}\in\mathcal{A}(W), S(A)_{\mathcal{A}(W)}\vert 0 \rangle =(A_{\mathcal{A}( W)})^{\ast}\vert0 \rangle =S_{\rm scat}A^{\ast}S_{\rm scat}^{-1}\vert 0 \rangle, S=S_{\rm scat}S_{\rm in}, S_{\rm scat}A^{\ast}S_{\rm scat}^{-1}\in\mathcal{A}_{\rm out}(W).
E\left[P\left[E_v\in C\cap f^{-1}(D)\mid W_v\right]P\left[F_v\in B\mid W_v\right]; W_v\in A\right] \stackrel{\rm (EX)}{=}E\left[P\left[F_v\in B, E_v\in C\cap f^{-1}(D)\mid W_v\right]; W_v\in A\right] = P[ W_v\in A, F_v\in B,E_v\in C\cap f^{-1}(D)],
\begin{array}{lll}\overline y_i = (y_{s+\overline m}(1)+\beta)^{\overline d\tau_i}y_i(1)\mbox{for}1\le i\le s\mbox{and}\overline y_{s+\overline m}= (y_{s+\overline m}(1)+\beta)^{\overline d}-\overline \beta\end{array}
\lambda_a\lambda_b(c) =a^{-1}(a\circ\lambda_b(c))=a^{-1}(a\circ (b^{-1}(b\circ c))) &=a^{-1}(a\circ b^{-1})a^{-1}(a\circ b\circ c)=(a\circ b)^{-1}(a\circ b\circ c)=\lambda_{a\circ b}(c).
v_p\left( \frac{f(k)^p - f(k_{-})^p}{k - k_{-}}\right) = v_p\left( \frac{f(k)-f(k_{-})}{k-k_{-}}\cdot \sum_{i=0}^{p-1}f(k)^{i}f(k_{-})^{p-1-i}\right) &\geqslant -\left\lfloor \frac{\log n}{\log p}\right\rfloor + 1,
G_1=\begin{pmatrix}{y}_{2}^{2}{y}_{1}{y}_{2}{y}_{1}^{2}{-{x}_{1}}0 0{x}_{2}0 0 0{-{x}_{1}}0 0{x}_{2}0 0 0{-{x}_{1}}0 0{x}_{2}0 0 0 0 0 0 \end{pmatrix},C_1 =\begin{pmatrix}0 0 1 0 1 0 1 0 0 \end{pmatrix}.
W_{\nu}^{(\nu)}(x;\lambda_{1})=-\left[ \log \left( \frac{\lambda_{1}+\int_{x_{0}}^{x}dt\psi_{\nu}^{2}(t)}{\psi_{\nu}(x)}\right) \right]^{\prime}=-w_{\nu}(x)-\frac{\psi_{\nu}^{2}(x)}{\lambda_{1}+\int_{x_{0}}^{x}dt\psi_{\nu}^{2}(t)}.
\sum_{i=1}^{r_1}\left(L_i^\prime\right)^d = \sum_{i=1}^{r_1}\big(\alpha_{i,0}(x_0-a'_1 y_1 +\ldots - a'_m y_m)+\alpha_{i,1}x_1\ldots+\alpha_{i,n}x_n\big)^d = (x_0-a'_1 y_1 +\ldots - a'_m y_m)x_1^{a_1}\cdots x_n^{a_n}.
\delta u_{\underline{{m}}}^{( 0 )}= u_{\underline{{m}}}^{i}i_{\delta}f^{i}, \qquad \delta u_{\underline{{m}}}^{i}= - u_{\underline{{m}}}^{j}i_{\delta}A^{j i}+ u_{\underline{{m}}}^{( 0 )}i_{\delta}f^{i},
F[f](x) - c_{2}^{\prime}\delta ((x^{0})^{2}- |{\bf x}|^{2}) =\sum_{k = 0}^{N}a_{k}\left( \sum_{\nu = 0}^{3}\eta^{\nu \nu}\left( \frac{\partial}{\partial x^{\nu}}\right)^{2}\right)^{k}\delta (x).
P^{(D_n)}_{(r)}(x|q,t)={(q;q)_r\over (t;q)_r}\sum_{0\leq 2j\leq r}G_{r-2j}(x;q,t){( t^{-n}q^{-r};q)_{j}(1/t;q)_{j}\over (t^{-n+1}q^{-r+1};q)_{j}(q;q)_{j}}{1-t^{-n}q^{-r+2j}\over 1- t^{-n}q^{-r}}(t^2/q)^{j}.
(\frac{\partial}{\partial t}-c\cdot\nabla)(\frac{\partial}{\partial t}+c\cdot\nabla)u=(\frac{\partial^{2}}{\partial t^{2}}+(c\cdot\nabla)c\cdot\nabla)u=(\frac{\partial^{2}}{\partial t^{2}}+(c\cdot\nabla)^{2})u=0.
\underset{y\rightarrow\overline{y}}{\lim}\frac{\triangle_{2}+\triangle_{3}}{y-\overline{y}}=\int_{\left\{x:\,w(x,\overline{y})=t\right\}}\frac{\partial w}{\partial y}(x,\overline{y})\,\frac{f(x,\overline{y})}{\left\vert \nabla_{x}w\right\vert}\varphi(x)\,d\mathcal{H}^{n-1}(x).
2i\int_{D_r\setminus D_\varepsilon}\frac{f_{\overline{z}}(z)}{z-z_0}\, dx\, dy = \int_{\partial D_r}\frac{f(z)}{z-z_0}\, dz - \int_{\partial D_\varepsilon}\frac{f(z)}{z-z_0}\, dz\, .
\tilde{A}_{1}= C_{1},\ \tilde{A}_{2}= C_{2}+ z (c_{0})^{2}e^{-t_{2}}E & \tilde{B}= (-t_{1}+ c +\alpha z) C_{1}+ C_{2}-\frac{z}{2}D + z(c_{0})^{2}e^{-t_{2}}E.
\frac{\partial w^{\varepsilon}(x, t)}{\partial t}=\big( - \frac{b}{\varepsilon}\big) \cdot \nabla u (x^\varepsilon, t) + \frac{\partial u}{\partial t}(x^\varepsilon, t) + \varepsilon \varkappa_1 \big(\frac{x}{\varepsilon}\big)\otimes \big( - \frac{b}{\varepsilon}\big) \cdot \nabla \nabla u (x^\varepsilon, t) + \phi_\varepsilon^{(0)}(x,t),
\partial_{U}\left( U^{8}\Big ( 1 -{\frac{U_{0}^{7}}{U^{7}}}\Big ) \partial_{U}\phi \right) + \left({\frac{c g_{Y M}^{2}N \omega^{2}U}{1 - U_{0}^{7}/ U^{7}}}- \ell ( \ell + 7 ) U^{6}\right) \phi = 0 \, .
P_\ell(x) := \mathrm{det}(\mathrm{Id}- \rho_{\overline{f}}(1)(\mathrm{Fr}_\ell) x : T_{\overline{f}}(1)) & = 1 - \overline{a_\ell(f)}\ell^{-1}x + \overline{\psi}(\ell)\ell^{-1}x^2
\beta_{0 0}( t ) = \frac{\pi g \mathrm{e}^{- \pi g t / 4}}{1 6 \tilde{\omega}\tilde{\tilde{\omega}}}\left[ \left( 2 \pi g - i \tilde{\tilde{\omega}}\right) \mathrm{e}^{i \tilde{\tilde{\omega}}t}- \left( 2 \pi g - i \tilde{\tilde{\omega}}\right) \mathrm{e}^{- i \tilde{\tilde{\omega}}t}\right] .
{\cal H}_{p h y s}^{'}( x ) ={\frac{1}{2}}[{\vec{\pi}}^{2}( x ) +{\vec{B}}^{2}( x ) ] +{\frac{1}{2}}M^{2}{\vec{A}}^{2}( x ) +{\frac{1}{2}}\vec{\partial}\cdot \vec{\pi}( x ) \,{\frac{1}{\triangle + M^{2}}}\, \vec{\partial}\cdot \vec{\pi}( x ) .
\int_{\tau_{j+1}}^{\tau_{j+\ell+1}}G_{j+\ell}(v) \: dv = \sum_{m=1}^{\ell}\int_{\tau_{j+m}}^{\tau_{j+m+1}}G_{j+\ell}(v) \: dv &= \sum_{m=1}^{\ell}\bigg( \int_{\tau_{j+m}}^{\tau_{j+m+1}}G_{j+m}(v) \: dv + s(\ell - m)(\tau_{j+m+1}- \tau_{j+m}) \bigg).
\Pi_{\mu}^{\lambda}\Theta_{\lambda}^{\nu}=0\;,\qquad \Pi_{\mu}^{\nu}+\Theta_{\mu}^{\nu}=\delta_{\mu}^{\nu}\left(T_{1}+T_{2}+T_{3}+T_{4}\right) +T_{5}+T_{6}+T_{7}+T_{8}
\left|z+\dfrac{a_{n-1}}{a_n}-(t_1-t_2)\right|\leq \,\,\dfrac{2t_2|a_{k+1}|+2|a_k|}{|a_n|t_1^{n-k-1}}\cos\alpha+2\dfrac{\sin\alpha}{|a_n|}\sum\limits_{\nu=0}^{n-2}\dfrac{|a_{\nu}|}{t_1^{n-\nu-1}}&+\left(t_2+\left|\dfrac{a_{n-1}}{a_n}\right|\right)(\sin\alpha-\cos\alpha).
\sup_{t\in[0,1]}(|G_{1}(t)|+|G_{1}'(t)|+|G_2(t)|+|G_2'(t)|) \ll \Omega_G\Bigl(\int_{t\ge 0}G(t)dt\Bigr)^{j+1}\Bigl(\prod_{i=j+2}^r G(u_i)\Bigr) &=\Omega_G\int_{t\ge 0}G_2(t)dt.
\left|\sum_{n=1}^N y_n\right|^2 \leq \frac{N+H}{H+1}\sum_{n=1}^N |y_n|^2 + \frac{2(N+H)}{H+1}\sum_{h=1}^H \left(1 - \frac{h}{H+1}\right)\left|\sum_{n=1}^{N-h}y_{n+h}\overline{y_n}\right|.
\frac{\langle\theta|\bar{\psi}(x)\psi(x)|\theta\rangle}{\langle\theta|\theta\rangle}=\frac{1}{\langle\theta|\theta\rangle}\Bigl[\langle\theta|\psi_{1}^{+}(x)\psi_{2}(x)|\theta\rangle+\langle\theta|\psi_{2}^{+}(x)\psi_{1}(x)|\theta\rangle \Bigr] \ .
f(x)-f(x^\star) =\int_0^1 \tfrac{1}{\lambda}\langle \nabla f(\lambda x+(1-\lambda)x^\star), \lambda x+(1-\lambda)x^\star-x^\star\rangle d\lambda &\geq \int_0^1 \tfrac{\gamma}{\lambda}\left( f(\lambda x+(1-\lambda)x^\star)-f(x^\star)+\tfrac{\mu_s\lambda^2}{2}\|x-x^\star\|^2\right)d\lambda & \geq \tfrac{\gamma\mu_s}{4}\|x-x^\star\|^2,
G^{( \psi )}( z_{1}, z_{2}) = \frac{1}{1 6}\left[ \begin{array}{l l}{{\bar{D}_{1}^{2}\bar{D}_{2}^{2}G_{+ +}( z_{1}, z_{2})}}&{{\bar{D}_{1}^{2}D_{2}^{2}G_{+ -}( z_{1}, z_{2})}}\\{{D_{1}^{2}\bar{D}_{2}^{2}G_{- +}( z_{1}, z_{2})}}&{{D_{1}^{2}D_{2}^{2}G_{--}( z_{1}, z_{2})}}\\ \end{array}\right]
\widehat{\partial}^{n+1}(\alpha_{i,j}^{n}) =\alpha_{i,j}^{n}\partial^{n+1}=\beta_{i,j+1}^{n+1}+\gamma_{i,j}^{n+1}+(-1)^{n+1}(\beta_{i-1,j+1}^{n+1}+\gamma_{i-1,j}^{n+1}).
\begin{aligned}N_\lambda^{l,2}v = \L(P_{\lambda}g(u_{<\lambda}), \partial^2 v_{<\lambda})+ L(P_\lambda g(u_{<\lambda}), \partial^2 v_{\lambda}) +\sum_{\mu \gg \lambda}L( P_\mu g(u_{<\lambda}),\partial^2 v_\mu) := \N_\lambda^{l,2,lo}v + N_\lambda^{l,2,med}v + N_\lambda^{l,2,hi}v,\end{aligned}
\iota_0 : G \times G \longrightarrow \mathrm{Sp}_{2n}: \left( \begin{pmatrix}a_1 b_1 c_1 d_1 \end{pmatrix}, \begin{pmatrix}a_2 b_2 c_2 d_2 \end{pmatrix}\right) \mapsto \begin{pmatrix}a_1 b_1 a_2 b_2 c_1 d_1 c_2 d_2 \end{pmatrix}
m! \, N_0 ((Rf^{i_1 \dots i_k})_{p_1 q_1 \dots p_{m-k}q_{m-k}}) &= \sigma(i_1 \dots i_k) \,\sum\limits_{r=0}^k (-1)^r \binom{k}{r}\frac{\partial^{r}}{\partial x_{i_1}\dots \partial x_{i_r}}(R^k(G_{m-r}))_{p_1 q_1 \dots p_{m-k}q_{m-k}i_{r+1}\dots i_k},
S_{n}=\sum_{k=1}^{n}(X_{n,k}-\mathbb{E}_{k-1}X_{n,k})+\sum_{k=1}^{n}\mathbb{E}_{k-1}X_{n,k}& =\sum_{k=1}^{n}D_{n,k}+\sum_{k=1}^{n}\mathbb{E}_{k-1}X_{n,k}.
S_p\big((x,y,t),(x',y',t')\big) = \int_{\Sigma_p}\frac{1}{C^1_{\eta,\tau}}e^{2\pi\eta((x+x')+i(y-y'))}e^{-2\pi\tau\cdot(p(x)+p(x')-i(t-t'))}\, d\eta\,d\tau.
(J_j f)(\lambda ) := \begin{cases}-\sum_{k=1}^{a_j(\lambda)}f(\lambda-k\alpha_j) \a_j(\lambda)> 0 , 0 \a_j(\lambda)=0 , \sum_{k=0}^{-a_j(\lambda)-1}f(\lambda+k\alpha_j) \a_j(\lambda) <0,\end{cases}
\frac{i}{2 \pi}\delta^{4}( p + q ) \left[ \frac{f_{+}({\bf p})}{\omega ({\bf p})}\frac{p_{0}}{( p_{0}+ i \epsilon )^{2}-{\bf p}^{2}- m^{2}}+ f_{-}({\bf p}) \frac{1}{( p_{0}+ i \epsilon )^{2}-{\bf p}^{2}- m^{2}}\right] \ ,
\P_\eta \left(G_\infty \cap \Lambda_\infty \right) = \P_\eta(\tau = \infty, \Lambda_\infty) + \P_\eta (\tau < \infty, G_\infty \cap \Lambda_\infty) & \ge p_\bullet \hat{p}p_{**}^{|\eta|-1}\left\{\P_\eta(\tau = \infty, \Lambda_\infty) + \P_\eta(\tau < \infty, \Lambda_\infty)\right\}= p_{**}^{|\eta|},
S^{(n+1)}(0)=-K^{(n+1)}(0)-5(n+1)K^{(n)}(0)-(n+1)!\dbinom{n+5}{n+1}K(0) =-3(n+3)!n\left( n+5\right) +5(n+3)!(n+2)(n+1)+12(n-1)(n+1)!\dbinom{n+5}{n+1}
X_{r s}= \left( \Gamma_{\mu}C \right)_{r s}X^{\mu}+ \left( \Gamma_{[ \mu \nu ]}C \right)_{r s}X^{[ \mu \nu ]}+ \left( \Gamma_{[ \mu_{5}- \mu_{s}]}C \right)_{r s}X^{[ \mu_{1}\ldots \mu_{5}]},
\int_0^{t/\lambda}|u_{\lambda}(s,x)|^{h-1}u_{\lambda}(s,x) db_{s,\lambda}^n = \int_0^{t/\lambda}\lambda^{\frac{1}{2(h-1)}}|u(\lambda s,\lambda^{1/2}x)|^{h-1}u(\lambda s,\lambda^{1/2}x) dw_{\lambda s}^n = \lambda^{\frac{1}{2(h-1)}}\int_0^{t}|u(s,\lambda^{1/2}x)|^{h-1}u(s,\lambda^{1/2}x) dw_{s}^n,
\frac{1}{T}\int_{T}^{2T}\bigg (R_Y(\sigma+it)\bigg)^{k}\cdot \bigg (\overline{R_Y(\sigma+it)}\bigg )^{\ell}dt = \mathbb{E}\bigg( \bigg(R_Y(\sigma,X)\bigg)^{k}\cdot \bigg (\overline{R_Y(\sigma,X)}\bigg )^{\ell}\bigg ) + O \bigg ( \frac{Y^{k + \ell}}{\sqrt{T}}\bigg ).
( 2 \pi )^{d}\; \delta^{( d )}\! \left( \sum_{i = 1}^{M}p_{i}\right) A_{P}( p_{1}\dots p_{M}) = \frac{N}{( 2 \alpha^{\prime})^{d / 2}}\frac{\langle B_{0}| D | B_{M}\rangle}{\pi}~ .
\varphi_{m+n+k}(\mathbf{f})=\frac{l_{k}^{z}}{l_{1}^{z}}\frac{\sum\limits_{i=1}^m \sum\limits_{j=1}^n{e_{ij1}x_{i}y_{j}}}{\sum\limits_{i=1}^m \sum\limits_{j=1}^n{e_{ijk}x_{i}y_{j}}},\thinspace k=1,\thinspace \mathellipsis ,p.
\partial^2_t Q(\overline{\partial}\eta) = Q( \overline{\partial}\eta) (\partial_t \overline{\partial}\eta)^2 + Q( \overline{\partial}\eta) \partial^2_t \overline{\partial}\eta=Q( \overline{\partial}\eta) (\overline{\partial}v)^2 + Q( \overline{\partial}\eta) \partial_t \overline{\partial}v,
k\omega_{p}(1)=\int_{\Gamma_p}\left( \frac{\partial^+ w_{p}}{\partial \nu_p^+}+ \frac{\partial^- w_{p}}{\partial \nu_p^-}\right) \psi \,ds- \int_{\Gamma_p}\frac{\partial^+\psi}{\partial\nu_p^+}(\gamma_p^+(w_{p})-\gamma_p^-(w_{p}))\,ds.
w_{n}^{\ast}(\zeta) \leq w_{n}(\zeta)\leq w_{n}^{\ast}(\zeta)+n-1, \qquad\widehat{w}_{n}^{\ast}(\zeta)\leq \widehat{w}_{n}(\zeta)\leq \widehat{w}_{n}^{\ast}(\zeta)+n-1, w_{=n}^{\ast}(\zeta) \leq w_{=n}(\zeta)\leq w_{=n}^{\ast}(\zeta)+n-1, \widehat{w}_{=n}^{\ast}(\zeta)\leq \widehat{w}_{=n}(\zeta)\leq \widehat{w}_{=n}^{\ast}(\zeta)+n-1.
\begin{gathered}\min\limits_{u\in\Gamma}J(u)=\int\limits_{0}^{t_f}\left( w_1 x_3^2(t)+ w_2 u^2(t)\right) dt, \dot{x}(t)=A(t) x(t)+B(x(t))u(t)+f(x(t)),x(0)=(S_0,E_0,I_0,Q_0,R_0,D_0,P_0,W_0),\end{gathered}
\sum_{m=0}^n S_1\left(n,m,\frac{z}{\rho}\right)\rho^{-m}B_{m,\rho,q}^{(k)}(z) =\frac{n!}{\rho^n[n+1]_q^k}\,, \sum_{m=0}^n S_2\left(n,m,\frac{z}{\rho}\right)\rho^{-m}c_{m,\rho,q}^{(k)}(z) =\frac{1}{\rho^n[n+1]_q^k}\,, \sum_{m=0}^n S_2\left(n,m,-\frac{z}{\rho}\right)\rho^{-m}\widehat c_{m,\rho,q}^{(k)}(z) =\frac{(-1)^n}{\rho^n[n+1]_q^k}\,.
\|v(\cdot,t)\|_{L^{2}(\mathbf{T})}^{2}=2\pi\sum_{n\in\mathbf{Z}}|v_{n}(t)|^{2}=2\pi\sum_{n\in\mathbf{Z}}\exp\biggl\{-f(t)\left(n+\frac{\mu w_{t}}{f(t)}\right)^{\!2}+\frac{\mu^{2}|w_{t}|^{2}}{f(t)}\biggr\}\ge2\pi\exp\biggl\{-f(t)+\frac{\mu^{2}|w_{t}|^{2}}{f(t)}\biggr\}.
g^1(N,C,D^2,\tau) =\displaystyle\frac{\tau}{2}+a_1N+c_1C+d_1D^2+e_1N^2+f_1NC+g_1N\tau+\cdots, g^2(N,C,D^2,\tau) =b_2\tau+a_2N+c_2C+d_2D^2+e_2N^2+f_2NC+g_2N\tau+\cdots
\Delta_{\Omega^{({D})}}= \sum_{j = 1}^{{D}- 1}\left[ \left( \prod_{k = 1}^{j - 1}\sin^{2}\theta_{k}\right) \, \sin^{{D}- j - 1}\theta_{j}\right]^{- 1}\frac{\partial}{\partial \theta_{j}}\left( \sin^{{D}- j - 1}\theta_{j}\, \frac{\partial}{\partial \theta_{j}}\right) \; ,
\bigl(\mathcal{S}_N\,B^{(f)}_n\bigr)(q\,|\,b) =0,\; n=0\cdots N-1, \bigl(\mathcal{S}_N\,B^{(f)}_N\bigr)(q\,|\,b) =f(0) \,N!\prod\limits_{j=1}^N b_j, \bigl(\mathcal{S}_N \,x^n\bigr)(q\,|\,b) =0,\;n=0\cdots N-1, \bigl(\mathcal{S}_N \,x^N\bigr)(q\,|\,b) =(-1)^N N!\prod\limits_{j=1}^Nb_j.
E_{n\ell}=\frac{\hbar^2}{24\mu b^2}\ell(\ell+1)-\frac{\hbar^2}{2\mu b^2}\left[\frac{\tilde{A}+\alpha(\alpha-1)-\left[n+\frac{1}{2}+\frac{1}{2}\sqrt{(1-2\alpha)^2+4\ell(\ell+1)}\right]^2}{2\left[n+\frac{1}{2}+\frac{1}{2}\sqrt{(1-2\alpha)^2+4\ell(\ell+1)}\right]}\right]^2,
\lambda^2\Vert (\bar{R}_{j_1}\, \cdot + \bar{c}_{j_1}) - (\bar{R}_{j_2}\, \cdot + \bar{c}_{j_2})\Vert^2_{L^\infty(Q_1 \cap Q_2)}& \le C(m^{-\frac{1}{2}})^2 m^{-1}\Vert (\bar{R}_{j_1}\, \cdot + \bar{c}_{j_1}) - (\bar{R}_{j_2}\, \cdot + \bar{c}_{j_2})\Vert^2_{L^2(\tilde{Q}_1 \cap \tilde{Q}_2 \cap U_Q)}.
\frac{1}{t_n}\sum_{i=1}^{t_n}\mathbf{1}_{[0, b)}(t_i^n)-b= &=\frac{1}{t_n}\frac{(\beta-1)}{(1+\beta^2)}(\beta^2(1-(-\beta)^n)-\beta+(-\beta)^n\beta^n(1+\beta)-(-\beta)^n)\ .
S^{00}(h_I^\epsilon h_I^{\epsilon'})=\sum_{J: J\supset I}a_J|J|^{-1/2}h_J\left(\int_I h_I^\epsilon h_I^{\epsilon'}\right)+\sum_{J: J\subsetneq I}a_J|J|^{1/2}h_J(h_I^\epsilon h_I^{\epsilon'}).
u_4=\frac{x_{13}'x_{24}'}{\omega x_{14}'x_{23}'}=\frac{x_{13}''x_{24}''}{\omega x_{14}''x_{23}''},~~u_5=\frac{x_4'}{x_3'}=\frac{x_{13}'''x_{24}'''}{\omega x_{14}'''x_{23}'''},~~~~u_6=\frac{x_4''}{x_3''}=\frac{x_4'''}{x_3'''}.~
\displaystyle \frac{1}{\mu - \hat{x}}= \displaystyle \frac{1}{\nu}= \displaystyle \frac{( - 1 )^{\delta}e^{i Z_{N}( \hat{x}+ \nu )}i Z_{N}^{\prime}( \hat{x}+ \nu )}{1 + ( - 1 )^{\delta}e^{i Z_{N}( \hat{x}+ \nu )}}+ . . .
\psi_{+}(\xi, \tau) = \frac{1}{\sqrt{2\pi}}\left[K_{0}(c^{\star}) + \frac{\beta-i\tau}{c^{\star}}K_{1}(c^{\star}) \right] & + \frac{1}{\sqrt{2\pi}}\left[- K_{0}(c) + \frac{\beta+i\tau}{c}K_{1}(c) \right]
\gamma_-(sv^++v^-) =I'(v^-)[v^-]+s^2\lambda\int_{\mathbb R^3}\phi_{v^-}(v^+)^2dx \leq I'(v^-)[v^-]+\lambda\int_{\mathbb R^3}\phi_{v^-}(v^+)^2dx=\gamma_-(v)<0.
\begin{aligned}D_{j,\alpha}(e^{it(x_1^3 + x_1x_2^2)}\,f)(x_1, x_2)= e^{it(x_1^3 + x_1x_2^2)}\, D_{j,\alpha}f(x_1, x_2) &+ e^{it(x_1^3 + x_1x_2^2)}\,\Phi_{j,t,\alpha}(f)(x_1, x_2),\end{aligned}
(u-v) \left\{\Delta_{\beta_1,\gamma_1}(u), \Delta_{\beta_2,\gamma_2}(v) \right\}= \sum_a \Big( \Delta_{\beta_1, J_a \gamma_1}(u) \Delta_{\beta_2, J^a \gamma_2}(v) - \Delta_{J_a \beta_1, \gamma_1}(u) \Delta_{J^a \beta_2, \gamma_2}(v) \Big)
\big(\Phi^V\wedge\Phi^V\big)_{X,Y}:=\Phi_X^V\Phi_Y^V-\Phi_Y^V\Phi_X^V,\big[R,\Phi^V\big]_X:=\big[R,\Phi_X^V\big], \big({\rm d}^{\nabla^V}\Phi\big)_{X,Y}:=\nabla_X^V\big(\Phi_Y^V\big)-\nabla^V_{Y}\big(\Phi_X^V\big)-\Phi_{[X,Y]}^V.
\left\Vert e^{\gamma \left\vert x\right\vert^{p}}\tilde{u}_{k}\left(x,0\right) \right\Vert_{X}=\left\Vert e^{\gamma \left( \frac{\alpha}{\beta}\right)^{p/2}\left\vert x\right\vert^{p}}u_{k}\left( x,0\right)\right\Vert_{X}=b_{0},
\left|V_{\mbox{bos}}\right>=\exp\left\{\frac{1}{2}\sum_{u,v=1}^3\left(\sum_{m,n}a^{I\dagger}_{m(u)}{\bar N}^{uv}_{mn}a^{I\dagger}_{n(v)}\right)\right\}\left|0\right>_{123}\,.
\mathrm{E}\{\alpha_{m_1}[l_1]\alpha_{m_1}^*[p_1]\alpha_{m_2}^*[l_2]\alpha_{m_2}[p_2]\}= \sigma_{l_1}^2\sigma_{l_2}^2\delta[l_1-p_1]\delta[l_2-p_2]+\sigma_{l_1}^2\sigma_{p_1}^2\rho^2\left(\frac{m_1-m_2}{M-1}\cdot\frac{D}{\lambda}\right)\delta[l_1-l_2]\delta[p_1-p_2].
a^{3412}=a^{3421},a_{1}^{3412}=a_{1}^{3421},a_{2}^{3412}=a_{2}^{3421},a_{3}^{3412}=a_{3}^{3421},a_{4}^{3412}=a_{4}^{3421}, a_{12}^{3412}=a_{12}^{3421},a_{13}^{3412}=a_{13}^{3421},a_{14}^{3412}=a_{14}^{3421},a_{23}^{3412}=a_{23}^{3421},a_{24}^{3412}=a_{24}^{3421}, a_{123}^{3412}=a_{123}^{3421},a_{124}^{3412}=a_{124}^{3421}
e(G) \leq e(G^{*})\leq e(G^{*}[C])+\sum_{i}\left(\frac{(d_i-2)}{2}|V(R_i)|+p_i\cdot |V(R_i)|\right) &\leq \binom{c-s+1}{2}+(s-1)s+\frac{d+2p-2}{2}(n-c).
T^{\,b_{N+1}-b_{N}+k-b_{n}}\,z= e_k-\Bigl( \prod_{j=b_{N}+1}^{b_{N}+k-b_{n}}w_{j}\Bigr)\, v_{N}^{-1}\,\Bigl(\,\prod_{j=b_{N}+1}^{b_{N+1}-1}w_{j}\Bigr)^{-1}\Bigl(\prod_{j=b_{N}+1}^{k}w_{j}\Bigr)^{-1}e_{b_{N}+k-b_{n}};
H^{2m-i-j+2}(h+H)^{i+j-2}=\sum_{k=1}^{j}c^k_{i,j}\mathrm{c^{sm}}(\Omega_k) &=\sum_{k=1}^{j}c^k_{i,j}H^{m-k+1}(h+H)^{k-1}.
{1\over 2}F^{ab}_\Lambda+{L_\Lambda}^3 G^{-ab}+{L_\Lambda}^i G_i^{+ab}+{(L^{-1})_3}^\Pi J_{\Pi\Lambda}G^{+ab}-{(L^{-1})_i}^\Pi J_{\Pi\Lambda}G^{i-ab}=0,
\sum_{m=1}^M c_m = \sum_{m=1}^M \| \left(\mathbf{W}^{\mathrm{RF}}\right)^{\mathrm{H}}\mathbf{p}_m\|^2_2 = \|\left(\mathbf{W}^{\mathrm{RF}}\right)^{\mathrm{H}}\mathbf{P}\|^2_\mathrm{F}= \mathrm{tr}\left\lbrace \mathbf{I}_R \right\rbrace = R.
\left[\tilde{\mathbf{H}}_{k,q,p}^{\dagger}\mathbf{y}_{k,q,p}\right]_{q,1}= \sum^{P}_{n=1}\alpha_{k,q,n}s_{k,q,n}+\tilde{\mathbf{H}}_{k,q,p}^{\dagger}\mathbf{n}_{k,l},
\frac{1}{2}\partial_1 |\nabla H-h^S_{\nu\nu}HV|^2 = -h^S_{22}(\partial_2 H)^2 +(\partial_2 H)(\partial_1 \partial_2 H) = (h^S_{\nu \nu}-h^S_{22}) (\partial_2 H)^2 + H \partial_2 H \partial_2 h^S_{\nu \nu}.
{\cal A}^{\prime \prime}+ \ddot{{\cal A}}-{\cal A}_{w}^{\prime}- \dot{{\cal A}}_{\theta}+ 2{\cal H}({\cal A}^{\prime}-{\cal A}_{w}) = \alpha{\cal L}^{2}f^{2}{\cal A}- \alpha{\cal L}^{2}f \Delta_{2},
d s^{2}= \frac{1}{H_{1}\tilde{H}_{1}}d t^{2}- \frac{H_{1}}{\tilde{H}_{1}}d s_{1}^{2}- \frac{\tilde{H}_{1}}{H_{1}}d s_{1}^{2}- H_{1}\tilde{H}_{1}d s_{3}^{2}\quad , \quad e^{- 2 \phi}\sim 1 \ .
L = \frac{1}{2 k z}\left[ A_{\mu}( \partial^{2}\eta^{\mu \nu}- z \partial_{z}( \frac{1}{z}\partial_{z}) \eta^{\mu \nu}- \partial^{\mu}\partial^{\nu}) A_{\nu}+ 2 A_{5}\partial_{z}\partial^{\mu}A_{\mu}- A_{5}\partial^{2}A_{5}\right] .
W [ \phi , \pi ] = \mathcal{N}\exp \left( - \frac{1}{\hbar}\int d x \, \left( \left( \phi \left( x \right) \, \sqrt{m^{2}- \nabla_{x}^{2}}\, \phi \left( x \right) \right) + \left({\pi}\left( x \right) \, \left( \sqrt{m^{2}- \nabla_{x}^{2}}\right)^{- 1}{\pi}\left( x \right) \right) \right) \right) ,
\theta^1_p = dp^1, \theta^2_p= dp^2,\theta_{x}= dx, \theta_y^1 =dy^1 - p^1 dx, \theta_y^2= dy^2 - p^2 dx, \theta^1_z = dz^1 - p^2dy^1 - p^1dy^2 + p^1p^2dx, \theta^2_z = dz^2 -x dy^2, \theta^3_z = dz^3 -xdz^1 -2p^1dz^2 + (2xp^1 - y^1)dy^2 \theta^4_z = dz^4 + (xp^1 -y^1)dz^1 + (p^1)^2dz^2 - p^1dz^3 -p^1(xp^1 - y^1)dy^2.
\frac{1}{2}(d{^\vee}\vartheta (d\pi) + a \circ d{^\vee}\vartheta(d\pi)) - (d{^\vee}\vartheta (X) -a \circ d{^\vee}\vartheta (X)) &= d{^\vee}\vartheta \left( \frac{1}{2}( d\pi + a(d\pi)) - ( X -a ( X))\right) & \leftrightarrow{^\vee}\vartheta \circ \nu
\omega_{W P}= i \overline{\partial}{\frac{2 \sum_{i , j = 1}^{h}\omega_{i}^{\prime \prime}A_{i j}\overline{\omega}_{j}- 3 \left( \sum_{i , j = 1}^{h}\omega_{i}^{\prime}A_{i j}\overline{\omega}_{j}\right)^{2}}{2 \left( \sum_{i , j = 1}^{h}\omega_{i}A_{i j}\overline{\omega}_{j}\right)^{2}}}.
H = \frac{{\bf P}^2}{2m}- \frac{{\bf P}^2}{2m^2c^2}\left(\frac{{\mbox{\boldmath$\pi$}}^2}{2\mu}+ U(q)\right)- \frac{{\bf P}^4}{8m^3c^2}+ \frac{{\mbox{\boldmath$\pi$}}^2}{2\mu(q)}+ v(q)\,.
\eta_(A) = \# \{(i,j) \mid 1 \leq j \leq i \leq n-1,\, d_{i+1,j}\leq d_{i,j}= d_{i+1,j+1}\}&= \# \{(i,j) \mid 1 \leq j \leq i \leq n-1,\, b_{i+1,j}\leq b_{i,j}= b_{i+1,j+1}-1 \}=: \eta_(B).
\exp ( \phi ( z , \bar{z}) ) = a \tau_{2}\eta^{2}\bar{\eta}^{2}\tilde{\tau}_{2}\tilde{\eta}^{2}\bar{\tilde{\eta}}^{2}\left| \prod_{i = 1}^{1 2}( z - z_{i})^{- \frac{1}{1 2}}\right|^{2}\left| \prod_{j = 1}^{1 2}( z - y_{j})^{- \frac{1}{1 2}}\right|^{2}.
\Big[ M_1 \otimes_{R_1}M_2 \otimes_{R_2}\ldots \otimes_{R_{n-1}}M_n \otimes_{R_n}- \Big]_{}:=\,\big(M_1 \otimes_{\mathbb Z}M_2 \otimes_{\mathbb Z}\ldots \otimes_{\mathbb Z}M_n\big)\big/\sim\,