harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
J=2\Gamma(2-\alpha)\lambda\tau^\alpha\left[\begin{matrix}0 ~ p_1 ~\cdots p_{n-2}p_{n-1}0 ~ 0 ~\cdots ~p_{n-3}~p_{n-2}\vdots ~ \vdots ~ \ddots ~\vdots ~\vdots 0 ~0 ~\cdots ~ 0 ~p_1 0 ~0 ~\cdots ~0 ~0 \end{matrix}\right]_{n\times n}.
\begin{aligned}\nabla_{\boldsymbol{y}}\mathcal{L}(\boldsymbol{y},\boldsymbol{u},\boldsymbol{p}) = 0, \nabla_{\boldsymbol{u}}\mathcal{L}(\boldsymbol{y},\boldsymbol{u},\boldsymbol{p}) = 0, \nabla_{\boldsymbol{p}}\mathcal{L}(\boldsymbol{y},\boldsymbol{u},\boldsymbol{p}) = 0,\end{aligned}
S_k(c;\xi\,(\ell))={\sum_{a\,\hbox{\tiny mod}\, k}}^{\!\!\!*}e_{k}(-a h^2)S_{k_1}\big(a\bar k_2, \bar k_2\bar \ell_2 c;\xi\,(\ell_1)\big)\,S_{k_2}\big(a\bar k_1, \bar k_1\bar \ell_1 c;\xi\,(\ell_2)\big).
H = \left.\left(\frac{\partial_rL}{\partial\dot{q}}\dot{q}-L\right)\right|_{\frac{\partial_rL}{\partial\dot{q}}=P},\,\,\,\,\,\, q = (x,e,\chi,\psi^n),\; P=(p,P_e,P_\chi,P_n)\;.
\begin{array}{l}\mathbf{B}(\mathbf{x},t) =\displaystyle\frac{1}{4\pi}\int\rho_m(\mathbf{x}',t-r)\frac{\mathbf{r}}{r^3}\,d^3x' ~ -\displaystyle\frac{1}{4\pi}\int\mathbf{J}_e(\mathbf{x}',t-r)\,{\bf \times}\,\frac{\mathbf{r}}{r^3}\,d^3x'+\displaystyle\frac{1}{4\pi}\int\frac{1}{r}\frac{\partial\mathbf{J}_e(\mathbf{x}',t-r)}{\partial t}\,d^3x' .\end{array}
\begin{aligned}\left| \iint (g_{[<\mu]}- g_{[<\mu]}^{x_0}) \left(M(v_\mu^{x_0}) E(v_\lambda)- P(v_\lambda) P(v_\mu^{x_0})\right) dx dt \right|\lesssim \ \epsilon^2 C^2 \lambda^2 |x_0| d_{\lambda}^2 d_{\mu}^2.\end{aligned}
M := \alpha_1 L_1^*L_1 + \alpha_2 (L_1^\perp)^*L_1^\perp + \alpha_3 L_3^*L_3 = \frac{1}{1-\rho^2}\left( \begin{array}{ccccc}I_n - \rho I_n - \rho I_n I_n \end{array}\right)
\frac{\sum_{k=0}^n V_k}{n^{2/\delta}}= \left( \frac{m_n^{2/\delta}}{n^{2/\delta}}\cdot \frac{\sum_{i=1}^{m_n}Q_i}{m_n^{2/\delta}}\right) + \frac{\sum_{k=0}^n V_k - \sum_{i=1}^{m_n}Q_i}{n^{2/\delta}}\equiv (I) + (II).
1 + \sum_{d=1}^\infty \frac{z^{2d}}{d!d!}N^{2d}\sum_{\substack{\lambda \vdash d \ell(\lambda) \leq N}}s_\lambda(a_1,\dots,a_N) s_\lambda(b_1,\dots,b_N) \frac{(\dim \mathsf{V}^\lambda)^2}{(\dim \mathsf{W}_M^\lambda) (\dim \mathsf{W}_N^\lambda)},
V(\psi_{\uparrow},\psi_{\downarrow})=\frac{p}{M}\left(\bar{\psi}_{\uparrow}\psi_{\uparrow}+\bar{\psi}_{\downarrow}\psi_{\downarrow}\right)^2+\frac{q}{M}\left(\bar{\psi}_{\uparrow}\psi_{\uparrow}-\bar{\psi}_{\downarrow}\psi_{\downarrow}\right)^2+W(\psi_{\uparrow},\psi_{\downarrow}),
b_{j}^{\left( \nu ,1\right)}=4\pi\left[\frac{\left( \frac{1}{4}+\nu^{2}\right)^{j}}{j !}+\sum\limits_{i=1}^{j}\frac{(-1)^i\left( \frac{1}{4}+\nu^{2}\right)^{j-i}}{\left( j-i\right) !i!}B_{2i}(\nu+1/2)\right]
\left( \begin{matrix}{{x^{\prime \prime}}}&{{y^{\prime \prime}}}\\ \end{matrix}\right) = \left( \begin{matrix}{{x}}&{{y}}\\ \end{matrix}\right) \left( \begin{matrix}{{a^{\prime}}}&{{b^{\prime}}}\\{{c^{\prime}}}&{{d^{\prime}}}\\ \end{matrix}\right)
A_{p}\! = \! e^{\frac{\phi}{2}}[ z_{1}\, \gamma_{p}\otimes 1 \! \! 1 + z_{2}\Pi^{( 1 ) j}\, h_{j p}\otimes \sigma_{3}+ z_{3}\Pi^{( 2 ) j}\, h_{j p}+ z_{4}\Pi_{p}^{( 3 )}\otimes \sigma_{1}+ z_{5}{\tilde{\gamma}}^{j}h_{j p}\otimes \sigma_{2}]
\pi ( u ) = \cosh \left( \frac{\sigma}{2}\right) \left[ ( \sinh^{2}\left( \frac{\sigma}{2}\right) \right]^{- \frac{1}{2}- k}( 2 )^{- 2 k}F \left( k , k + \frac{1}{2}; 2 k ; - c o s e c h^{2}\left( \frac{\sigma}{2}\right) \right)
\mathcal{H}= \sum_{i=1}^L \sum_{\alpha = \uparrow,\downarrow}(c^\dag_{i,\alpha}c^{}_{i+1,\alpha}+ c^{}_{i,\alpha}c^\dag_{i+1,\alpha}) +U \sum_{i=1}^L n_{i,\uparrow}n_{i,\downarrow},
a_{k\lambda}(t)=a_{k\lambda}(0)e^{-i\omega_{k}t}+ie\sqrt{\frac{2\pi}{\hbar\omega_{k}V}}\int_{0}^{t}dt^{\prime}e_{k\lambda}\cdot\dot{x}(t^{\prime})e^{i\omega_{k}(t^{\prime}-t)}
\sum_{k \geq 1}\partial_{\alpha}\left(\int \hat \Psi_{\alpha}\cdot \hat \Psi_{\alpha}\circ F_{\alpha}^k \cdot h_{\alpha}\: dm\right)= \sum_{k \geq 1}\int \partial_{\alpha}\left(P_{\alpha}^k(\hat \Psi_{\alpha}h_{\alpha}) \hat \Psi_{\alpha}\: dm \right).
P_{\pi}=[\begin{matrix}e_{\pi(1)}\\e_{\pi(2)}\\e_{\pi(3)}\\e_{\pi(4)}\\e_{\pi(5)}\end{matrix}]=[\begin{matrix}e_{1}\\e_{4}\\e_{2}\\e_{5}\\e_{3}\end{matrix}]=[\begin{matrix}1&0&0&0&0\\ 0&0&0&1&0\\ 0&1&0&0&0\\ 0&0&0&0&1\\ 0&0&1&0&0\end{matrix}].
\delta{\cal L}= \mu \left( \psi_R^{(i)\dagger}\psi_R^{(i)}+\psi_L^{(i)\dagger}\psi_L^{(i)}\right)+ \mu_5 \left( \psi_R^{(i)\dagger}\psi_R^{(i)}- \psi_L^{(i)\dagger}\psi_L^{(i)}\right)
\rho_1\rho_2\xrightarrow{\rho_1(U_2)}\rho_1\hat\rho_2\xrightarrow{U_1}\hat\rho_1\hat\rho_2=\hat\rho_2\hat\rho_1\xrightarrow{U_2^{-1}}\rho_2\hat\rho_1\xrightarrow{\rho_2(U_1)^{-1}}\rho_2\rho_1.
f_{4, 1, 0}(x_1, x_2, x_3; q, 2) = -(q - 1)\frac{(x_2 + x_3)(x_1 - qx_3)(x_1 - qx_2)}{(qx_2 - x_3)(qx_1 - x_3)(qx_1 - x_2)}, f_{2, 1, 1}(x_1, x_2, x_3; q, 2) = -(q-1)^2\frac{(q+1)(x_2 - x_3)\big(x_1^2 + x_2x_3 + 2x_1x_2 + 2x_1x_3\big)}{(qx_1 - x_2)(qx_1 - x_3)(qx_2 - x_3)}.
\chi^{i\alpha}\rightarrow \chi^{i\alpha}+ \epsilon g^{i\bar j}(\Omega_2)_{\bar j \bar k}\bar\chi^{\bar k}_{\alpha}\qquad\qquad\bar \chi^{\bar i}_\alpha \rightarrow \bar \chi^{\bar i}_\alpha + \epsilon g^{\bar i j}(\Omega_2)_{j k}\chi^{k\alpha}
\alpha\left(u_1''+\frac{\mu_1}{\alpha^2}\left(\alpha - v^+\right)v^+ - k \omega u_1 u_2\right) = v''+d u_2''+\frac{\mu_1}{\alpha}\left(\alpha - v\right)v^+ - \alpha k \omega u_1 u_2 = \frac{\mu_2}{d^2}\left(d + v \right) v^- +d u_2'' -\alpha k \omega u_1 u_2 = d u_2''+\frac{\mu_2}{d^2}\left(d - v^- \right) v^- -\alpha k \omega u_1 u_2
{\widetilde{\cal K}}_a={(\sqrt{t})^{-2d}\over\eta^{2d}({\mbox{e}}^{-2\pi t})}\left((2\sqrt{t})^{d(J_a)}V(J_a)\right) \sum_{{\widetilde p}\in 2\Lambda(J_a)}\exp(-\pi t{\widetilde p}^2) \left({(\sqrt{t})^{d(RJ_a)}\over V(RJ_a)}\right) \sum_{{\widetilde w}\in{\widetilde\Lambda}(RJ_a)}\exp(-\pi t{\widetilde w}^2)~.
f^{bcd}f^{d}{}_{[e a}\Lambda_{a_1 a_2 ... a_{2m-1}]}{}^{b}= 2f^{cd}{}_{[e}f^{db}{}_{a}\Lambda_{a_1...a_{2m-1}]}{}^{b}= -2 f^{cd}{}_{[e}\Omega^{d}{}_{a a_1...a_{2m-1}]}= 0 \ .
{\bf v}_h(s,y):=\begin{cases}{\bf v}(s,y), \; \mbox{for}t \leq s <t+ h, \enskip y \in \mathbb{R}^n{\bf \tilde v}(s,y), \; \mbox{for}t+ h \leq s \leq T,\enskip y \in \mathbb{R}^n. \end{cases}
\nu(\hat{x}) =\sum_{x\in X}\pi(x,\hat{x})=\pi(\hat{x},\hat{x})+\sum_{\substack{x\in X x\neq\hat{x}}}\pi(x,\hat{x}) &\leq\sum_{y\in X}\pi(\hat{x},y)+\sum_{\substack{x\in X x\neq\hat{x}}}\pi(x,\hat{x})=\mu(\hat{x})+\sum_{\substack{x\in X x\neq\hat{x}}}\pi(x,\hat{x}).
b_{1}={\frac{a_{1}}{2}}[ 1 +{\frac{({\bar{u}}{\bar{z}}^{\prime}{\bar{u}})}{( u x^{\prime}{\bar{u}})}}] , \; \;{\bar{b}_{1}}={\frac{a_{1}}{2}}[ 1 +{\frac{( u{z}^{\prime}u )}{( u x^{\prime}\bar{u})}}] .
P_{T-t}u_t(x)= P_T u_0(x) + \int_0^t P_{T-s}\big(\beta\,u_s(1-u_s)\big)(x)\,ds &+ \int_{[0,t]\times \Gamma}p(T-s,x,y)\,\ell(y)\sqrt{\gamma(y)\,u_s(y)\big(1-u_s(y)\big)}\,dW(s,y) &+\frac{1}{2}\int_0^t\sum_{v\in V}p(T-s,x,v)\,\ell(v)\,\hat{\beta}(v)\,u_s(v)(1-u_s(v))\,ds
\tilde C = \left[\begin{array}{ccccc}C_{1,1}C_{1,t}C_{1,t-1}\hdots C_{1,2}C_{s,1}C_{s,t}C_{s,t-1}\hdots C_{s,2}C_{s-1,1}C_{s-1,t}C_{s-1,t-1}\hdots C_{s-1,2}\vdots \vdots \vdots \ddots \vdots C_{2,1}C_{2,t}C_{2,t-1}\hdots C_{2,2}\end{array}\right].
\Delta_1(t) = i \int_{T \wedge \mu}^{t \wedge \mu}\mathbf S(t-s) \left[ \mathbf N(\mathbf U(s)) - \mathbf N(\mathbf V(s)) \right] \, ds, \Delta_2(t) = i \int_T^t \mathbf S(t-s) \left[ \mathbf M(\mathbf U(s)) - \mathbf M(\mathbf V(s)) \right] \, dW(s).
\int_0^{1}\Bigg|\sum_{\substack{n\geq 1 P^{+}(n)\leq y}}\frac{\psi_p(n) \cos(2\pi n\alpha)}{n}- U(\alpha)\Bigg|^2d\alpha = \sum_{\substack{n\geq 1 P^+(n)\leq y}}\frac{(\psi_p(n)- h(n))^2}{n^2}+ \sum_{\substack{n\geq 1 P^+(n)> y}}\frac{h(n)^2}{n^2}\ll \sum_{\substack{n\geq 1 P^{+}(n)>H}}\frac{1}{n^2}\ll \frac{\log y}{\sqrt{y}},
\begin{array}{l}Z_0^{r,s}(\bar{D},V):=\{\omega \in \Lambda_0^{r,s}(\bar{D},V): \bar{\partial}\omega=0 \}\qquad \medskip B_0^{r,s}(\bar{D},V):=\bar{\partial}\{\omega \in \Lambda^{r,s-1}_0(\bar{D},V): \bar{\partial}\omega \in \Lambda_0^{r,s}(\bar{D},V)\}.\end{array}
\int_X |Du_\epsilon|^2 \,d\mu = -\int_X u_\epsilon A u_\epsilon \,d\mu=-\int_X u A \beta_\epsilon u_\epsilon \,d\mu \leq -2 \int_X (K|Df|^2+Df\cdot DAf) \beta_\epsilon u_\epsilon \,d\mu.
d_{\mathcal A}c^k(X_0, X_1, \ldots , X_k) = \sum_{i= 0}^k (-1)^i\rho(X_i)(c^k(X_0, \ldots, \widehat{X}_i, \ldots, X_k)) & + \sum_{0\leq i<j \leq k}(-1)^{i-1}c^k(X_0, \ldots, \widehat{X}_i, \ldots, X_{j-1}, [X_i, X_j], X_{j+1}, \ldots X_k),
[\begin{matrix}A&B\\C&D\end{matrix}]^{-1}=[\begin{matrix}A^{-1}+A^{-1}B(D-CA^{-1}B)^{-1}CA^{-1}&-A^{-1}B(D-CA^{-1}B)^{-1}\\ -(D-CA^{-1}B)^{-1}CA^{-1}&(D-CA^{-1}B)^{-1}\end{matrix}],
\pi_l^N(n):M_{N,n}^F \ni\begin{pmatrix}z^1_1, \ldots, z^1_n \vdots \vdots z^N_1 \ldots, z^N_n\end{pmatrix}\mapsto\begin{pmatrix}z^1_1, \ldots, z^1_n \vdots \vdots z^l_1 \ldots, z^l_n\end{pmatrix}\in M_{l,n}^F.
\partial_x^m \big( \tilde{v}_\epsilon (\partial_y\tilde{w}_\epsilon + u^s_{yy}) \big) = \tilde{v}_\epsilon \partial_x^m \partial_y \tilde{w}_\epsilon + (\partial^m_x\tilde{v}_\epsilon) (\partial_y \tilde{w}_\epsilon+u^s_{yy}) &+\sum_{1\le j\le m-1}C^j_m\, \partial^j_x\tilde{v}_\epsilon\partial^{m-j}_x \partial_y\tilde{w}_\epsilon.
\mathcal{D}[H_N^\mathrm{HC}]:=\; \Big\{\psi_N \in L^2_\mathrm{sym}(\Lambda^N)\;|\; \psi_N(x_1,\dots,x_N)=0 \exists\, i\ne j &\qquad\qquad\qquad |x_i-x_j| \le \frac{\mathfrak{a}}{N}\Big\}\cap H^1(\Lambda^N).
\begin{cases}\partial_t \psi_{n+1}=\partial^2_s \psi_{n+1}+h(\psi_n)~ \mbox{for}~(s,t)\in D_{t_0}, \psi_{n+1}(s,0)=\varphi_0(s) \mbox{for}~ s\in\mathbb R.\end{cases}
R^{\sigma(Y)}_{s,t}= \int_0^1\int_0^1 r \sigma''(ur Y_t + u(1-r)Y_s + (1-u)Y_s )( Y_s'B^H_{s,t}+ R^Y_{s,t})(Y_s'B^H_{s,t})drdu + \int_0^1 \sigma'(rY_t + (1-r)Y_s)R^Y_{s,t}dr.
\widetilde{F}_{0}^{(\rho)}(z)=\begin{cases}1+C_{1}^{(\rho)}z^{-1}+O(z^{-2}), \mbox{if}\ \rho\not\equiv p \mod (p+1) z+C_{0}^{(\rho)}+O(z^{-1}), \mbox{if}\ \rho\equiv p \mod (p+1)\end{cases}
\cfrac{\ell_{X_1}(\beta)}{\ell_{X_2}(\beta)}\leq K_3 \cdot \max \{1, cfrac{\ell_{X_1}(\alpha)}{\ell_{X_2}(\alpha)}, cfrac{\ell_{X_2}(\gamma)}{\ell_{X_1}(\gamma)}, cfrac{\ell_{X_2}(\gamma')}{\ell_{X_1}(\gamma')}\}\ .
S_{N-1,0,0}= \bigl[1+(-1)^{N-1}\bigr]\frac{N}{N+1}, S_{N-1,1,\ell-1}= (-1)^{\ell}\biggl[\binom{N+\ell}{\ell}^{-1}+N\sum_{j=0}^{\ell-1}(-1)^{N-j}\binom{N-1}{j}\frac{1}{N+\ell-j}\biggr] = (-1)^{\ell}N\sum_{i=0}^{N-1-\ell}(-1)^i\binom{N-1}{i}\frac{1}{i+\ell+1}.
\sum_{|\alpha|\le{m+2}}\left|\int_{\mathbb{R}}(\partial^\alpha\partial_y w^{n}\partial^\alpha\partial_y w^{n})\big|_{y=0}dx\right| &\le \frac 1{16}\|\partial_y w^n\|^2_{H^{m+2}_{k+\ell}(\mathbb{R}^2_+)}+C\|w^{n-1}\|^{m+2}_{H^{m+2}_{k+\ell}(\mathbb{R}^2_+)}\| w^n\|^{m+2}_{H^{m+2}_{k+\ell}(\mathbb{R}^2_+)}.
\cos \angle P A_{1}C_{1}= \frac{1 3 + 8 + 1 3}{2 \times \sqrt{1 3}\times 2 \sqrt{2}}= \frac{8}{4 \sqrt{2 6}}= \frac{2 \sqrt{2 6}}{2 6}= \frac{2 \sqrt{2 6}}{1 3}\Rightarrow \sin \angle P A_{1}C_{1}= \frac{\sqrt{6 5}}{1 3}
P_{2k+1}(m_r+1,n_1,m_1,\dots,n_k,m_k) =m_kP_{2k}(m_r+1,n_1,m_1,\dots,m_{k-1},n_k) &+P_{2k-1}(m_r+1,n_1,m_1,\dots,n_{k-1},m_{k-1}) &=1+m_r+\sum\limits_{i=1}^k m_im_{1,N_i}.
CB.g({h})= \frac{1}{2}(x-y)\big((\mu-h^{2}+2h)g({h}-2)-(-\mu+h^{2}+2h)g({h}+2)\big) & \equiv\frac{1}{2}\big((\mu-h^{2}+6h-8)g(h-4)-(-\mu+h^{2}-2h)+g(h)x &-(-\mu-h^{2}-2h)g(h)y-(-\mu+h^{2}+6h+8)g(h+4)y\big).
\stackrel{( 0 )}{\omega}_{1}= - W_{a b}\left( H_{0}^{a}\eta^{( 2 ) b}+ C_{i}^{( 2 ) a}A^{b i}\right) - M_{b c}^{a}\left( B_{a}^{0 i}\eta^{( 2 ) b}A_{i}^{c}+ \frac{1}{2}\eta_{a}^{( 2 ) i j}A_{i}^{b}A_{j}^{c}\right) .
\int_{B_{\rho}}\left\vert w-(w)_{\rho}\right\vert^{p_0}\leq\int_{B_{\rho}}\left\vert w-w(0)\right\vert^{p_0}& \leq\int_{0}^{\rho}\int_{\mathbb{S}^{n-1}}\tau^{p_0}\left\vert Dw\right\vert^{p_0}\tau^{n-1}d\tau d\phi & =C_{8}(n,p_0,\lambda)\rho^{n+p_0}\left\| Dw\right\|_{L^{\infty}(B_{1/4})}^{p_0}.
y_{t_{n+1}}=y_{n}+\tilde{\phi}(t_{n},y_{n};h_{n})+\frac{h_{n}}{2}(k_{1}+k_{2})+g(t_{n})\Delta w_{n}+\frac{(g(t_{n+1})-g(t_{n}))}{h_{n}}J_{(0,1)}(7.5)
(H_n^{(0)})^{-1}\partial_t H_n^{(0)}-(H_{n-1}^{(1)})^{-1}\partial_t H_{n-1}^{(1)}=(H_n^{(0)})^{-1}H_n^{(1)}-(H_{n-1}^{(0)})^{-1}H_{n-1}^{(1)}, &(H_n^{(1)})^{-1}\partial_t H_n^{(1)}-(H_n^{(0)})^{-1}\partial_t H_n^{(0)}=(H_n^{(1)})^{-1}H_{n+1}^{(0)}-(H_{n-1}^{(1)})^{-1}H_n^{(0)}.
(d_{X,0}F)(a_1, \ldots, a_{p+2}) = [a_1 , F(a_2, \ldots, a_{p+2})]_X, (d_{X,i}F)(a_1, \ldots, a_{p+2}) = F(a_1, \ldots, [a_i , a_{i+1}]_X, \ldots, a_{p+2}), (d_{X,n}F)(a_1, \ldots, a_{n+1}) = [F(a_1, \ldots, a_{p+1}) , a_{p+2}]_X
\Gamma=\left\{(s, i)=\pi_{t_n}^{\varrho(n)}\circ\cdots\circ\pi_{t_2}^{+}\circ\pi_{t_1}^{-}( s^+_*, i^+_*):0\leq t_1,t_2,\cdots,t_n; \; n\in \mathbb N \right\}.
( \bar{\sigma}_{\nu \mu})_{u}^{~ v}= \left( \begin{matrix}{{0_{[ N - 2 ] \times [ N - 2 ]}}}&{{0_{[ N - 2 ] \times [ 2 ]}}}\\{{0_{[ 2 ] \times [ N - 2 ]}}}&{{( \bar{\sigma}_{\nu \mu})_{\dot{\alpha}}^{~ \dot{\beta}}}}\\ \end{matrix}\right) ~ ~ ,
L_{L 1}={\frac{K \, v^{2}}{2 \, R}}~ \left[ 1 + \sum_{L = 1}^{\infty}\sum_{q = 0}^{L - 1}\, d_{L L}( 0 , q ) ~ K^{q}~ N^{- q}\left({\frac{N \, v^{2}}{R^{2}M_{p}^{9}V_{T}~ b^{7 - p}}}\right)^{L}~ ( 1 + S_{L 1}) \right]
\begin{array}{c}\Bigl(a^{+}(P^{0}-P_{2})+a^{-}\left( iP_{1}-P_{3}\right)\Bigr) |\psi\rangle = 0, begin{align*}-2mm] begin{align*}-2mm] \Bigl(a^{-}(P^{0}+P_{2})-a^{+}\left( iP_{1}+P_{3}\right) \Bigr) |\psi\rangle = 0.\end{array}
\frac{1}{\alpha k_{0}^{2}-|\mathbf{k}|^{2}-m^{2}+i\varepsilon}+\frac{1}{\alpha k_{0}^{2}-|\mathbf{k}|^{2}-m^{2}+i\varepsilon}\bullet \frac{-\delta \alpha k^{2}_{0}}{\alpha k_{0}^{2}-|\mathbf{k}|^{2}-m^{2}+i\varepsilon}+......
\Sigma_6 = \sum_{\alpha_1, \alpha_2 = \pm 1}\sum_{D_1, D_2}\frac{S(\alpha_2 n_2, \alpha_1 n_1, m_1, m_2; D_1, D_2)}{D_1D_2}\Phi_{w_6}\left( - \frac{\alpha_2 m_1n_2D_2}{D_1^2}, - \frac{\alpha_1 m_2n_1D_1}{D_2^2}\right),
A|z|^2-B\Re(z^2)=\frac{(a^2+b^2)(x^2+y^2)}{2a^2b^2}-\frac{(a^2-b^2)(x^2-y^2)}{2a^2b^2}=\frac{x^2}{a^2}+\frac{y^2}{b^2}\ .
W_{g}W_{h}=V_{b_1}^{*}V_{a_1}V_{a_1}^{*}V_{c_1}=V_{b_1}^{*}E_{a_1}V_{b_{1}}V_{b_{1}}^{*}V_{c_1}(\textrm{~Since $E_{a_1}$ and $E_{b_1}$ commute}) =E_{g}W_{hg}
\eta^{i}( \sigma , t ) \equiv - \frac{1}{4 M}\sum_{n > 0}\frac{\epsilon}{\tau}W^{'}( n + \frac{\epsilon}{\tau}) \Biggl [ \frac{c_{n}^{i}}{\sqrt{n}}x_{n}^{+}+ \frac{\tilde{c}_{n}^{i}}{\sqrt{n}}x_{n}^{-}+ h . c . \Biggr ] ,
\frac{\partial\Lambda_i(x,x_i)}{\partial x}-\frac{\partial\Lambda_j(x,x_j)}{\partial x}=0, &\frac{\partial\Lambda_i(x_{-i},x)}{\partial x}+\frac{\partial\Lambda_j(x,x_j)}{\partial x}=0, &\frac{\partial\Lambda_i(x_{-i},x)}{\partial x}-\frac{\partial\Lambda_j(x_{-j},x)}{\partial x}= 0.
\sum_{l}\langle j_{a,b}(f_s)\ |\kappa_{0}(f^{\prime}_{l})\rangle j_{a,b}(e_{i})\langle \kappa_{0}(f^{\prime}_{l})\ | j_{a,b}(f_{j}))\rangle &=j_{a,b}\left( (f^{*}_{s}\otimes 1_{Z}) \circ 1_{X^{b-a}}\otimes c_{Z,W})\circ (e_{i}\otimes 1_{W})\circ f_{j}\right)
\frac{M^2_4}{\sqrt{r}}\cdot\alpha(16r)-\frac{1}{\sqrt{r}}\cdot\alpha(r)=\frac{4 M_4^2}{3}(1+k^2_{16r})-\frac{1}{3}(1+k^2_r)-\frac{(1+k'_r)^2}{4},
\int | \Phi_{0}( w )^{2}| d w = \mathrm{K}^{2}\int \frac{d w}{z^{2}}\equiv \mathrm{K}^{2}\int_{- \infty}^{+ \infty}\frac{{\cal H}^{2}}{a^{3}\,{\varphi^{\prime}}^{2}}= 1 ,
\| Q^+_{<j}( Q_{> j-30}\chi_{I}Q^+_{< j-2}\phi) \|_{S_{k, j}^{ang}}\lesssim \ 2^{\frac{j}2}\| Q_{> j-30}\chi_{I}Q^+_{< j-2}\phi\|_{L^2 L^{2}}\lesssim 2^{\frac{j}2}\| Q_{> j-30}\chi_{I}\|_{L^2}\|Q^+_{< j-2}\phi\|_{L^\infty L^2}\lesssim \ \|\phi\|_{L^\infty L^2}.
{\omega^\sigma}_{\mu\mu}(\mu,\sigma) = 0,{\omega^\sigma}_{\mu\sigma}(\mu,\sigma) = 0,{\omega^\sigma}_{\sigma\sigma}(\mu,\sigma) = \frac{-\sigma^{-2}+ 3\sigma^{-4}(\sigma + \delta \sigma)^2 - 2\sigma^{-2}}{\sigma^{-1}- \sigma^{-3}(\sigma + \delta \sigma)^2}= -\frac{3}{\sigma}.
\mathcal{L}^{( 0 )}= p_{i}\dot{q}_{i}+ \epsilon \hat{\bar{p}}_{i}\dot{\bar{q}\,}_{i}+ \pi \dot{\phi}+ \epsilon \hat{p}_{i}^{b}\dot{b}_{i}+ p_{i}^{c}\dot{c}_{i}- \mathcal{H}_{e x t}\, .
\zeta_n(x) = \sum_{j=0}^n \frac{\prod_{k=0}^{n-1}(j + \theta k +a )}{\prod_{k \neq j}(j-k)}x^{j+a-1}= \frac{1}{2\pi i}\oint_{\Sigma_n}\frac{\prod_{j=0}^{n-1}(z + \theta j +a )}{\prod_{j=0}^{n}(z-j)}x^{z+a-1}dz ,
( - \partial_{i}^{2}+ m^{2}) \chi_{R}- b m^{2}g^{2}( \{\sigma^{2}, \chi_{R}\}+ \sigma \chi_{R}\sigma ) + \frac 3 4 c m^{2}g^{4}( \{\sigma^{4}, \chi_{R}\}+ \sigma^{2}\chi_{R}\sigma^{2}+ \sigma \{\sigma , \chi_{R}\}\sigma ) = \Lambda_{a}^{2}\chi_{R}\ .
\langle \vec{\sigma}\cdot{\vec{a}}\,\vec{\sigma}\cdot\vec{b}\rangle=\,\int_{M_4}\,d^4x\,\sum^N_{k=1}\, \int_{\varphi^{-1}_{k*}(T_xM_4)}\,d^4z_k\,n^2_k(\varphi^{-1}(x),z_k) \,A(\vec{\sigma}\cdot{\vec{a}},(\varphi^{-1}(x),z_k))\times \break \times B(\vec{\sigma}\cdot\vec{b},(\varphi^{-1}(x),z_k)).
\kappa_\mathrm{h}= \mathrm{rank}\left\lbrace \mathrm{span}\left[ \mathbf{a}_\mathrm{h}\left(\phi,\theta \right) \right] \right\rbrace \approx \lceil M_\mathrm{h}\frac{d_\mathrm{h}}{\lambda}\left(\max\left[\cos\phi \sin \theta \right] - \min \left[\cos \phi \sin \theta \right] \right) \rceil,
(e^{1/\kappa |v|}-1 ) g(x,v) = \int_0^1 \frac{e^{y/\kappa |v|}}{\kappa |v|}\rho_g (x+y) \left( \alpha\mathcal{M}_{T(x+y)}(v) + (1-\alpha) \mathcal{M}_{\tau(x+y)}(v) \right) \mathrm{d}y,
g ( \underline{{x}}) = \bigl ( \prod_{s = 1}^{l}\frac{\partial}{\partial y_{j_{s}}^{0}}\bigr )^{K}\sum_{| \underline{{m}}| \leq K - 1 , m_{j_{1}}^{0}= . . . = m_{j_{l}}^{0}= 0}( \frac{\partial}{\partial \underline{{y}}})^{\underline{{m}}}\nu_{\underline{{m}}}( \underline{{y}}) ,
|\omega|_{\mathcal{O}}^{lip}=\sup_{\xi,\zeta\in \mathcal{O}\atop{\xi\neq\zeta}}\frac{|\Delta_{\xi\zeta}\omega|}{|\xi-\zeta|},|\Omega|_{-\delta,\mathcal{O}}^{lip}=\sup_{\xi,\zeta\in\mathcal{O}\atop{\xi\neq\zeta}}\sup_{j\in\mathbb{Z}_*}|j|^{-\delta}\frac{|\Delta_{\xi\zeta}\Omega_j|}{|\xi-\zeta|}
\begin{array}{lclcl}C_r^{(I)}= C_{- r}^{(I)}={\displaystyle \frac{k}{2}~\sqrt{-\tau_0}{}~J_{\nu}(-r\tau_0 )}& D_r^{(I)}= D_{- r}^{(I)}={\displaystyle \frac{k}{2}~\sqrt{-\tau_0}{}~J_{-\nu}(-r\tau_0 )}& C_n^{(I)}= D_n^{(I)}= 0~~{\rm for}~ n\neq \pm r\end{array}
(\Delta_Eh,h)_{L^2}=\left\|D_1h\right\|_{L^2}^2+2\mu \left\|h\right\|^2_{L^2}-4(\mathring{R}h,h)_{L^2}-2\left\|\delta h\right\|_{L^2}^2, (\Delta_Eh,h)_{L^2}=\left\|D_2h\right\|_{L^2}^2-\mu \left\|h\right\|^2_{L^2}-(\mathring{R}h,h)_{L^2}+\left\|\delta h\right\|_{L^2}^2,
32\pi^2 (\chi(M)-1) = \int_{K_R}\vert W\vert^2 + 4 Q_g +\frac{2}{3}\Delta_g R_g \, dv_g +O\left( \frac{1}{R^\tau}\right) = \int_{K_R}\vert W\vert^2 +4 Q_g \, dv_g +O\left( \frac{1}{R^\tau}\right)
D_{s\chi_0\eta_0}(s^2\mathfrak{e}'_{k+1})=\chi_0^{k+2}\hat{\Theta}_s^0\Big(s^2\frac{(\mathfrak{e}'_{k+1}-\mathfrak{e}'_{k})}{\chi_0^{k+2}}\Big)+s^2\delta\mathcal{B}^{k+1}+s^2\delta\mathcal{C}^{k+1}+s^2\mathcal{B}^k+s^2\mathcal{C}^k.
\mathcal{D}^*=\left\{\begin{aligned}(d_1,d_2)\in \mathbb{R}^2_+:\d_1\geq \tfrac{\tau_1}{\gamma(P_1)},\d_2\geq \tfrac{\tau_2}{\gamma(P_2)}, [\gamma(P_1+P_2)-\gamma(P_2)]d_1+\gamma(P_2)d_2\geq \tau_1+\tau_2 \end{aligned}\right\}
\frac{d\phi}{dr}= \dot{\phi}/ \dot{r}= \left(\frac{-r}{2}+ \frac{(n-1)2r}{r^2-s^2}\right)\cot \phi + \left(\frac{s}{2}+ \frac{(n-1)2s}{r^2-s^2}\right) + \lambda \csc \phi &= I \cot \phi + II + \lambda \csc \phi
1 = \lim_{k\to \infty}\frac{\gamma'(0) \cdot (y_k-x)}{|y_k-x|}= \lim_{k\to \infty}\frac{\gamma'(h_k) \cdot (y_k-x)}{|y_k-x|}&= \lim_{k\to \infty}\frac{\gamma'(h_k) \cdot (\gamma(h_k)-x)}{|y_k-x|}.
\int_{X}\psi(u - v_{j}^{k}) \theta^{n}_{v_{j}^{k}}\leq \sum_{l=j}^{k}\int_{X}\psi(u - u_{l}) \theta^{n}_{u_{l}}\leq \sum_{l = j}^{k}I_{\psi}(u,u_{l})\leq C^{-2j+2}.
i \ln W^{{\rm SR}}_{q\overline{q}}= \int_{t_{\rm i}}^{t_{\rm f}}dt \,\left\{- \frac{4}{3}\frac{\alpha_s}{r}+ \frac{1}{2}\frac{4}{3}\frac{\alpha_s}{r}(\delta^{hk}+ \hat{r}^h\hat{r}^k) \dot{z}_1^h \dot{z}_2^k \right\}\> .
|I| \le \frac{|Q|}{t}= \frac{|Q| - 2 +|I_u \cap I_v|}{t}+ \frac{2 - |I_u \cap I_v|}{t}< |I| - |I_u \cup I_v| + 2 + \frac{2 - |I_u \cap I_v|}{t}\,,
\Delta u(\phi) := -\int_{\mathbb{R}^2}\nabla u \cdot \nabla \phi d\mathcal{H}^2 \phi \in C^{\infty}_c(\mathbb{R}^2) \Delta u(U) := \sup \{\Delta u(\phi): \phi \in C^{\infty}_c(U), |\phi|_{\infty}\le 1\}\ge \int_{\partial_{red}\{u>0\}\cap U}|x_2|^{\gamma}d\sigma(x_1, x_2).
V\left( \varphi_{1}^{2}+\varphi_{2}^{2}\right) =\frac{1}{2}\mu^{2}\left(\varphi_{1}^{2}+\varphi_{2}^{2}\right) +\frac{1}{4}\left\vert \lambda\right\vert \left( \varphi_{1}^{2}+\varphi_{2}^{2}\right)^{2}~.
|\hat u^{(N)}-\bar u^* |^2 ={\mathbb E}|R^{-1}B^T(y^{(N)}-{\bf y})|^2 \le C{\mathbb E}|y^{(N)}-{\bf y}|^2 = CE|y^{(N)}-{\mathbb E}y^{(N)}+{\mathbb E}y^{(N)}-{\bf y}|^2 \le C (1/N) +C |{\bf y}_N-{\bf y}|^2 =O(1/N)+ O(|x^{(N)}(0)-m_0|^2).
\begin{tabular}[c]{l}$dy\left( t\right) +A\left( t,y\left( t\right) \right) dt=f\left(t\right) dt+B\left( t,y\left( t\right) \right) dW\left( t\right)+dM\left( t\right) $\medskip $y\left( 0\right) =y_{0},\;\;t\in\left[ 0,T\right] ,$\end{tabular}
B_{1}= \left( \begin{matrix}{{0}}&{{\,}}&{{0}}\\{{\,}}&{{\,}}&{{\,}}\\{{0}}&{{\,}}&{{v t}}\\ \end{matrix}\right) , \, \, \, B_{2}= \left( \begin{matrix}{{0}}&{{\,}}&{{0}}\\{{\,}}&{{\,}}&{{\,}}\\{{0}}&{{\,}}&{{b}}\\ \end{matrix}\right) , \, \, \, B_{3}= 0 .
Y_{\alpha}^N(\eta) = a^{-\frac{n}{2}}e^{\pm im_{n-1}\theta_1}\prod_{k=0}^{n-2}(sin{\theta_{n-k}})^{m_{k+1}}C_{m_k-m_{k+1}}^{m_{k+1}+\frac{1}{2}(n-k)}(cos{\theta_{n-k}})
\frac{1}{c_{n}-c_{n-1}}- \frac{1}{c_n}\ = \ \frac{c_{n-1}}{c_n^{2}-c_{n}c_{n-1}}& \ = \ \frac{c_{n-1}}{c_{n-1}(c_{n+1}-c_{n})+t^2(-r)^{n-1}}\ >\ \frac{1}{c_{n+1}-c_{n}}.
\begin{aligned}d\in\lbrace 214\,712 , 943\,077 , 1\,618\,493 , 2\,374\,077 , 3\,472\,653 , 4\,026\,680, 4\,628\,117 , 5\,858\,753 , 6\,405\,317 , 7\,176\,477 , 7\,582\,988 \rbrace\end{aligned}
|(\nabla_\xi^2\omega_\rho)(v+\theta\xi)|=\max_{1\leqslant j\leqslant n}\sum_{k=1}^n|(\partial_{\xi_j}\partial_{\xi_k}\omega_\rho)(v+\theta\xi)|\leqslant2n(1-\rho)(|v|-\eta)^{2\rho-2}.
R(\rho,\alpha_a,\alpha_b,\gamma,d_b) =\gamma R_a +(1-\gamma) R_b =\gamma \log(1+hP_a)+(1-\gamma)\log(1+hP_b) &\stackrel{}{\leq}\log(1+h\tau/N_s\left(\rho(\alpha_a c-p) +(1-\rho)(\alpha_b c-p+d_b)\right)) &\stackrel{}{\leq}\log(1+h\tau/N_s\left(c-p+ \hat{d}_b\right))= \left.R_b\right\vert_{\rho=0}
ne^{-\alpha R}\kappa\sqrt{\ss}\int_{\widehat{r}}^R e^{-\beta r_0}\sinh(\alpha r_0)dr_0 =\begin{cases}O\Big(n\Big(\frac{\kappa\sqrt{\ss}}{e^{\beta R}}\Big)^{1\wedge \frac{\alpha}{\beta}}\Big), begin{align*}8pt]\Theta\Big(n\frac{\kappa\sqrt{\ss}}{e^{\beta R}}\log\big(\frac{e^{\beta R}}{\kappa\sqrt{\ss}}\big)\Big), \end{cases}
E_{i}^{\Lambda_{i}+ 1}( z ) \Phi_{L . W .}^{( \lambda , \mu )}( w ) = \frac{- [ \Lambda_{i}+ \nu ( \vec{\alpha}_{i}, \vec{\beta}) ]}{( z - w )}a_{i i}^{*^{\Lambda_{i}}}\Phi_{L . W .}^{( \lambda , \mu )}( w ) + R . T . .
d s^{2}= d U d V + d X^{+}d X^{-}+ d X^{i}d X^{i}+ R^{d}\Big ( \frac{{d X^{+}}^{2}}{U^{d}}+ \frac{{X^{+}}^{2}}{U^{d + 2}}d u^{2}- 2 \frac{X^{+}}{U^{d + 1}}d U d X^{+}\Big )
\int_{t}^{\infty}\left| \frac{\widetilde{B(t-\tau)}}{(t-\tau)}\right| d\tau\leq \int_{t}^{\infty}\sum_{m=0}^{n_{\tilde{r}}-1}\frac{1}{p^{m+1}}M(\lambda_{\tilde{r}},m+1) \frac{(\tau - t)^{m+1}}{(\tau-t)}e^{-(\tau -t) \alpha}~d\tau,
\left. \phantom{{\frac{A}{B}}}{\epsilon^{i_{1}\ldots i_{n}}\phi_{i_{1}}d \phi_{i_{2}}\cdots d \phi_{i_{n}}}\right|_{\Sigma}= \left.{\frac{\overline{{\Delta}}}{\Delta}}{\epsilon^{j_{1}\ldots j_{n}}\overline{{\phi}}_{j_{1}}d \overline{{\phi}}_{j_{2}}\cdots d \overline{{\phi}}_{j_{n}}}\right|_{\Sigma}
\left( \begin{array}{c c}{{l_{1 1}}}&{{l_{1 2}}}\\{{l_{2 1}}}&{{l_{2 2}}}\\ \end{array}\right) = \left( \begin{array}{c c}{{1}}&{{0}}\\{{1}}&{{0}}\\ \end{array}\right) , \hspace{3 m m}\left( \begin{array}{c c}{{1}}&{{1}}\\{{0}}&{{0}}\\ \end{array}\right) \hspace{3 m m}\mathrm{e t c .}
{\check{\delta}}^{( m )}\ast{\check{\delta}}^{( n )}= \frac{i}{4 \pi}\int_{{\Gamma}_{1}}d k_{1}i^{m}k_{1}^{\lambda + m}\frac{\epsilon ( k_{1})}{2}\int_{{\Gamma}_{2}}d k_{2}\frac{i^{n}k_{2}^{\lambda + n}\epsilon ( k_{2})}{z - k_{1}- k_{2}}=
\sum_{z\in S^1_N}\exp\Big( -\kappa (z-w -y)^2 \Big) \leq \sum_{z= w-1}^{w+1}\exp\Big( -\kappa (z -w - y )^2 \Big) &+ \int_{-\infty}^{\infty}\exp\Big( -\kappa (z-w - y)^2 \Big)dz.
\begin{cases}\hat{y}:=\rho_0^{-2}(L^*\hat{u}- \alpha_1 \hat{z}^1{1}_{\mathcal{O}_{1,d}}- \alpha_2 \hat{z}^2{1}_{\mathcal{O}_{2,d}})\ \ \ \Q, \hat{p}^i := \rho_0^{-2}(L\hat{z}^i + \dfrac{1}{\mu_i}\hat{u}{1}_{\mathcal{O}_i})\ \ \ \Q, \hat{f}:= -\rho_1^{-2}\ \hat{u}\ \ \ \ \mathcal{O}\times (0,T).\end{cases}
\bigl(x^{-\rho_2+1}y_1'(x)\bigr)'+ \Bigl[z\kappa_3\Bigl(x^{\frac{\rho_1-\rho_2}{2}}\Bigr)'+ z^2\bigl(\kappa_1\kappa_2-\kappa_3^2\bigr)x^{\rho_1-1}\Bigr]y_1(x) = 0.