harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
t_1^{\lambda_1}\cdots t_r^{\lambda_r}\frac{\prod_{(i,j)\in \Delta_r}(1- \bar t_i/\bar t_j)}{\prod_{(i,j) \in C(\lambda)}(1- t_i/\bar t_j)}= \left.t_1^{\lambda_1}\cdots t_{{r+1}}^{\lambda_{{r+1}}}\frac{\prod_{(i,j)\in \Delta_{{r+1}}}(1- \bar t_i/\bar t_j)}{\prod_{(i,j) \in C(\lambda)}(1- t_i/\bar t_j)}\right|_{t_{{r+1}}=-\beta^{-1}}.
J_{j}^{i}(Q_{h})=\frac{q_{j}^{i+1}-2q_{j}^{i}+q_{j}^{i-1}}{h^{2}}+\sum\limits_{m=0}^{N-1}\sum\limits_{n=0}^{N-1}F_{jmn}\frac{(q_{m}^{i+1}-q_{m}^{i})(q_{n}^{i+1}-q_{n}^{i})}{h^{2}},
\hat{Q}= \int d x \, \colon \left[ \gamma^{\mu}\gamma^{0}\hat{\vec{\psi}}( x^{\mu}) \partial_{\mu}\hat{\vec{\psi}}( x^{\mu}) + i \gamma^{0}\hat{\vec{\psi}}( x^{\mu}) \vec{\nabla}\hat{W}( x^{\mu}) \right] \colon \qquad .
\| \mathcal{R}[f] \|_{L^{\infty}_{1+\gamma}(B^{{\rm c}}_{2R})}\le C_1 \big( |\alpha|^{-\frac{1+\gamma}{2}}\| f \|_{L^1 (B_{R})}+ \| |y|^{1+\gamma}f\|_{L^{1}(B_{R})}\big)\,.
\rho\overline{u}_{j}\frac{\partial\overline{u}_{i}}{\partial x_{j}}=\rho\overline{f}_{i}+\frac{\partial}{\partial x_{j}}[-\overline{p}\delta_{ij}+\mu(\frac{\partial\overline{u}_{i}}{\partial x_{j}}+\frac{\partial\overline{u}_{j}}{\partial x_{i}})-\rho\overline{u_{i}^{\prime}u_{j}^{\prime}}].
\left[ \begin{array}{c}{{X_{\mu_{1}}^{A_{1}}}}\\{{X_{\mu_{2}}^{A_{2}}}}\\ \end{array}\right] \rightarrow \left[ \begin{array}{c}{{X_{\mu_{1}}^{A_{1}}}}\\{{X_{\mu_{2}}^{A_{2}}}}\\ \end{array}\right]^{\prime}= \sqrt{( X_{\mu_{1}}^{A_{1}})^{2}+ ( X_{\mu_{2}}^{A_{2}})^{2}}\left[ \begin{array}{c}{{\sin ( \phi )}}\\{{\cos ( \phi )}}\\ \end{array}\right] ,
T_{\alpha}\Phi_{\alpha}(n^3 f(z)) D_0^n(z) = \frac{2\pi}{\sqrt{3}}\left(n^3 f(z)\right)^{-\frac{2\beta}{3}}\left( \mathbb{I}+ \frac{A_{\alpha}}{n^3 f(z)}+ O\left(\frac{1}{n^6 f(z)^2}\right)\right)L_{\alpha}(n^3 f(z)) D_2^n(z)
| \psi(w) | \le 3 \|f''\| \mathrm{e}^{\frac12 w^2}\int_{-\infty}^w (-s) \mathrm{e}^{-\frac12 s^2}\, \mathrm{d}s + 2 \| f^{(3)}\| \mathrm{e}^{\frac12 w^2}\int_{-\infty}^w (- s)^3 \mathrm{e}^{-\frac12 s^2}\, \mathrm{d}s &= 3 \|f'' \| + 2 (w^2+2) \| f^{(3)}\| ,
y_{2\ell}\overline{w}_j=y_{\ell+\frac{1}{2}}\overline{w}_j=y_{\ell-\frac{1}{2}}\overline{w}_j=0, &\y_{2\ell-2}^{(\xi_{\ell-1}-\xi_{\ell}+\delta_{j,2}+1)}\overline{w}_j= \delta_{\xi_\ell,\xi_{\ell-1}}\delta_{p,1}\delta_{j,0}\y_{\ell-\frac{3}{2}}^{(\xi_{\ell})}\overline{w}_j=0.
\sum_{r,s\geq 0}(-1)^{r+s}q^{4\binom{r}{2}+4\binom{s}{2}+12rs}q^{2r}q^{2s}&+ \sum_{r,s\geq 0}(-1)^{r+s}q^{4\binom{r}{2}+4\binom{s}{2}+12rs}q^{6r}q^{6s}q +\sum_{r,s\geq 0}(-1)^{r+s}q^{4\binom{r}{2}+4\binom{s}{2}+12rs}q^{10r}q^{10s}q^{4}&+ \sum_{r,s\geq 0}(-1)^{r+s}q^{4\binom{r}{2}+4\binom{s}{2}+12rs}q^{14r}q^{14s}q^9.
\tilde{\mathbf{x}}_{\mathrm{p}}^{(l-1)}= \begin{Bmatrix}\cdots \, \mathbf{y}_{i}^{(l)}\, \cdots \end{Bmatrix}^{\mathrm{T}}\forall \, \Omega_{i}^{(l)}\in \mathcal{C}_{\mathrm{p}}^{(l-1)}
\lVert AB-A_0B_0 \rVert^2 = \sum_{j=1}^N \sum_{i=1}^{M-1}\Biggl[(a_{iH}- a^0_{iH})+\sum_{k=1}^{H-1}\{(a_{ik}-a_{iH})b_{kj}- (a^0_{ik}-a^0_{iH}) b^0_{kj}\}\Biggr]^2 &+ \sum_{j=1}^N \Biggl[\sum_{i=1}^{M-1}\sum_{k=1}^{H-1}\{(a_{ik}{-}a_{iH})b_{kj}{-}(a^0_{ik}{-}a^0_{iH})b^0_{kj}\}{+}\sum_{i=1}^{M-1}(a_{iH}{-}a^0_{iH}) \Biggr]^2
A_3= 18t\theta_2-9ad=\dfrac{9bc(3ac-b^2)(3bd-c^2)}{3 a c^3 +3 b^3d-2 b^2 c^2}-9ad = \dfrac{9 (b^3 c^3 - 3 a b c^4 - 3 b^4 c d + 11 a b^2 c^2 d - 3 a^2c^3 d - 3 a b^3 d^2)}{3 a c^3 +3 b^3 d-2 b^2 c^2}.
0 = \mathbf E[kp (1-X_\infty)X^k_\infty + X^k_\infty(p^k -1)] = -pk \mathbf E[X_\infty^{k+1}] + (pk + p^k -1)\mathbf E[X^k_\infty] \intertext{or}\mathbf E[X_\infty^{k+1}] = \frac{pk + p^k-1}{pk}\mathbf E[X_\infty^k] = \Big(1 - \frac{1-p^k}{pk}\Big) \mathbf E[X_\infty^k].
K^\mu \; := \; \mbox{diag}\left(\frac{\pi}{\pi + gN}\frac{1}{-\triangle+\frac{e^2N}{\pi + gN}}\; , \;\frac{1}{-\triangle +\mu}\; , \; . \; . \; . \; . \; , \;\frac{1}{-\triangle +\mu}\right) \; ,
\partial_{t}\mathcal{F}(t)\leq -2C_{\min}\cdot \tfrac{\min\big\{1, C_{\mathcal{K}}\big\}}{\max\{1, \rho_{1}, \rho_{2}, \rho_{3}, \tau{0}\}}\mathcal{E}(t) =: - \tilde{C}\mathcal{E}(t).
\sup_{m> R+1}\sup_{\alpha \in I_{j,m}}| (\alpha ! \beta !)^{\frac 1{4\sigma}}R^{|\alpha + \beta|}a_{j, \alpha ,\beta}| <\infty ,j=1,2,\intertext{and}\sup_{(m_1,m_2)\in M_k}\sup_{\alpha \in I_{2,m_2}}\sup_{\beta \in I_{1,m_1}}| R^{|\alpha | + |\beta|}a_{0, \alpha ,\beta}| <\infty ,k=1,2,3,
\limsup_{t\to+\infty}\|\theta(t)\|_{L^1(B_{A\sqrt t}^c)}\le \kappa A^{-1}&\limsup_{t\to+\infty}\|u(t)\|_{L^3(B_{A\sqrt t}^c)}\le \kappa A^{-2}\qquad \limsup_{t\to+\infty}\sqrt t \|u(t)\|_{L^\infty(B_{A\sqrt t}^c)}\le \kappa A^{-3}.
V_n(\varphi) = \begin{bsmallmatrix}x \Phi_n\end{bsmallmatrix}^{\top}\mathbf{P}_n\begin{bsmallmatrix}x \Phi_n\end{bsmallmatrix}\leq 0, \; \forall n\geq \mathcal{N}(\mathcal{E}(\eta_0)),
I_4= \frac{n-2}{2}\Delta(\sigma_1(A)\Delta(r^{6-n})) = 2(n-2)(6-n)\Delta(\sigma_1(A)r^{4-n})+O(r^{N+4-n}) = 2(n-2)(6-n)r^{2-n}[\Delta\sigma_1(A)r^2+2(4-n)\sigma_1(A)_{,i}x^i+2(4-n)\sigma_1(A)]+O(r^{N+2-n}) = 2(n-2)(n-6)r^{2-n}\Big[\frac{1}{12(n-1)}|W(p)|^2r^2+3(n-4)\sigma_1(A)_{,ij}(p)x^i x^j\Big]+O(r^{5-n}).
\Big\langle \Lambda^TE(\cdot,s_1) , \Lambda^TE(\cdot,s_2) \Big\rangle &= \frac{T^{s_1+\bar{s}_2-1}}{s_1+\bar{s}_2-1}+ \overline{m(s_2)}\frac{T^{s_1-\bar{s}_2}}{s_1-\bar s_2}+ m(s_1) \frac{T^{\bar{s}_2-s_1}}{\bar s_2- s_1}+ m(s_1)\overline{m(s_2)}\frac{T^{1-s_1-\bar{s}_2}}{1-s_1-\bar s_2}.
\Phi_N(\phi_1,\ldots\phi_{\frac{N(N-1)}{2}}):=\left(\prod_{1\leq k\leq N-1}e^{\phi_k \lambda_{(k+1)^2-2}}\right)\cdot\begin{pmatrix}\Phi_{N-1}(\phi_{N},\ldots,\phi_{\frac{N(N-1)}{2}}) 0 0 1\end{pmatrix},
{\rm Re}\,\Pi^{\rm (non\hbox{-}planar)}=\frac{2\pi^{\frac{n+1}{2}}}{(2\pi)^{n}}\left(|\bar{\theta}|\right)^{\frac{3-n}{2}}\int_{0}^{\infty}dz\,\frac{1}{z^{2}-p^{2}}\,z^{\frac{n-3}{2}}J_{\frac{n-3}{2}}(\frac{|\bar{\theta}|z}{2})
|I_2| \leq C\int_{(t-s)^{\frac{1}{\alpha}}\leq|z| \leq \frac{|x-y|}{2}}(t-s)^{-\frac{1+j}{\alpha}}|z| \left[\rho^j(t-s,y-x-z) + \rho^j(t-s,y-x) \right] \, \frac{dz}{|z|^{d + \alpha}}+ C\int_{|z| > \frac{|x-y|}{2}}(t-s)^{-\frac{1+j}{\alpha}}|z| \left[\rho^j(t-s,y-x-z) + \rho^j(t-s,y-x) \right] \, \frac{dz}{|z|^{d + \alpha}}=: I_{2,1}+ I_{2,2}.
\begin{pmatrix}\alpha Y \beta \end{pmatrix}\mapsto \begin{pmatrix}\widetilde{\alpha}\widetilde{Y}\widetilde{\beta}\end{pmatrix}= \begin{pmatrix}e^{- \sigma}(\alpha - Y(\sigma) - \frac{1}{2}\beta ||^g \sigma ||^2_g e^{- \sigma}(Y + \beta^g \sigma) e^{\sigma}\beta \end{pmatrix}.
\lambda \sum_{j=1}^m\langle X_j(x),\xi\rangle^2\leq \langle B(x)\xi,\xi\rangle\leq \Lambda\sum_{j=1}^m \langle X_j(x),\xi\rangle^2\quad x\in \Omega\xi\in\mathbb{R}^N \langle b(x),\xi\rangle^2\leq \gamma^2(x)\sum_{j=1}^m\langle X_j(x),\xi\rangle^2 \quad x\in \Omega\xi\in\mathbb{R}^N
\begin{cases}\dot{z}_1(t)=\bar{F}(z_1(t),t)+\frac{\lambda_2 i}{2\pi}\frac{1}{\overline{z_1(t)-z_2(t)}}=\bar{F}(z_1(t))-\frac{|\lambda|i}{4\pi x(t)}\dot{z}_2(t)=\bar{F}(z_2(t),t)+\frac{\lambda_1 i}{2\pi}\frac{1}{\overline{z_2(t)-z_1(t)}}=\bar{F}(z_2(t))-\frac{|\lambda|i}{4\pi x(t)}.\end{cases}
\sum_{k\ge 0}g_k(x^{-1})\, f_k(y)=\sum_{k\ge 0}\frac{y^{k}}{x^{k}}=i_{x,y}\left(\frac{x}{x-y}\right), \sum_{k<0}g_k(x^{-1})\, f_k(y)=\sum_{k\ge 0}\frac{x^k}{y^{k}}=i_{y,x}\left(\frac{x}{y-x}\right),
\begin{aligned}\delta(x) = \left[\left(0,\, \dfrac 1{2\pi\sqrt{-1}}\frac{\varphi_{\Omega_+}}z\right)\right]\otimes [\nu_{\Omega_+}] -\left[\left(0,\, \dfrac 1{2\pi\sqrt{-1}}\frac{\varphi_{\Omega_-}}z\right)\right]\otimes [\nu_{\Omega_-}] & &=\dfrac{-1}{2\pi\sqrt{-1}}\left(\dfrac{1}{x + \sqrt{-1}\,0}- \dfrac{1}{x - \sqrt{-1}\,0}\right).\end{aligned}
\big( 1 - (C_{v_\Sigma}- C_{\tilde{v}_\Sigma}) (1 - C_{\tilde{v}_\Sigma})^{-1}\big)^{-1}=\sum_{k = 0}^\infty \big[ (C_{v_\Sigma}- C_{\tilde{v}_\Sigma}) (1 - C_{\tilde{v}_\Sigma})^{-1}\big]^k
u(x,t) ={C_{\nu}}\int_0^\infty(r\rho)^{-\frac{n-1}2}\big(e^{i(r\rho-\frac{\nu\pi}2-\frac{\pi}4)}-e^{-i(r\rho-\frac{\nu\pi}2-\frac{\pi}4)}\big)e^{it\rho^2}\chi_{N}(\rho)\rho^{n-1}\mathrm{d}\rho &+C_{\nu}\int_0^\infty(r\rho)^{-\frac{n-2}2}O_{\nu}\big((r\rho)^{-\frac32}\big)e^{it\rho^2}\chi_{N}(\rho)\rho^{n-1}\mathrm{d}\rho.
\lim\limits_{q\to\infty}\frac{2^{m-1}(q-m+1)\pm 2^{2m}(1+2^m\cdot 3m\sqrt{q})(3^{m-1}-1)}{m\times 4^{m-1}q}&=\frac{1}{m\times 4^{m-1}}2^{m-1}&=\frac{1}{m\times 2^{m-1}},
I(p;m_0,r)=\left(\prod_{\mu=1}^{d}c_\mu \right)\left[m_0+r\sum_\rho (c_\rho -1) +r\sum_\rho \frac{s_\rho^2}{c_\rho}\right](H_0^2)^{-n-\frac{1}{2}}.
G^{+}(z_1)\gamma(z_2)=-\frac{(\gamma\gamma )}{z_{12}^2}+ \left[ \frac{2}{3(1+c_1)}(G^{-}\gamma\gamma\gamma )- \frac{2}{1+c_1}(J_w\gamma\gamma )+ (\gamma'\gamma )\right] \frac{1}{z_{12}}\; .
\Lambda(R,R,R) = \Lambda(R^\flat, R, R) + \Lambda(R^\sharp, R^\flat, R) + \Lambda(R^\sharp, R^\sharp, R^\flat) + \Lambda(R^\sharp, R^\sharp, R^\sharp) \le \Lambda(R^\sharp, R^\sharp, R^\sharp) + 3 \|R\|_2^2 \|R^\flat\|_1.
f\big[U\to Y\big] = \sum_{i=1}^{2k-1}f\Big[\big((z_i,2),(z_i,1)\big)\to \big((z_{i+1},2),(z_{i+1},1)\big) \Big] \mathbf{1}\big\{(z_i,z_{i+1}]\textrm{is type 2}\big\}&+\sum_{i=1}^{2k-1}f\big[(z_i,2)\to (z_{i+1},1)\big] \mathbf{1}\big\{(z_i,z_{i+1}]\textrm{is type 1}\big\}.
\kappa_j(x_1,\cdots,x_{j-1},B)= \frac{\overline{V}(X_1)-\bar{f}(\frac{j-1}{n},\hat{W^{(n)}})^2}{\overline{V}(X_1)-\underline{V}(X_1)}\underline{P}(B) &+\frac{\bar{f}(\frac{j-1}{n},\hat{W^{(n)}})^2-\underline{V}(X_1)}{\overline{V}(X_1)-\underline{V}(X_1)}\overline{P}(B),
f(\lambda, i) \widetilde{\pi}_{i,j}(\lambda) - \lambda^{-1}f(\lambda, i) \pi_{i,j}(0) \, = \, -\theta_i \widetilde{\pi}_{i,j}(\lambda) + \sum_{k \in \mathcal{S}}\theta_i h_{ik}\widetilde{\pi}_{k,j}(\lambda).
I(\phi) := \int_M\phi\frac{\omega^n}{n!}- \sum_{i=0}^{n-1}\frac{i+1}{n+1}\int_{M}\partial\phi\wedge\bar\partial\phi\wedge\frac{\omega^{i}}{i!}\wedge\frac{\omega^{n-1-i}_{\phi}}{(n-1-i)!}= 0.
\begin{aligned}L_{i+1}=X_{i+1}+Y_{i+1}^{1+\kappa}\leq C16^{i}\left(X_i^{1+\frac{2}{n+2}}+X_i^{\frac{2}{n+2}}Y_i^{1+\kappa}\right)+C^{1+\kappa}16^{i(1+\kappa)}L_i^{1+\kappa}&\leq C^{1+\kappa}16^{i(1+\kappa)}\left[X_i^{1+\frac{2}{n+2}}+X_i^{\frac{2}{n+2}}Y_i^{1+\kappa}+L_i^{1+\kappa}\right].\end{aligned}
\frac{1}{2 Q \cdot k_{i}+ k_{i}^{2}}= \frac{1}{2 Q \cdot k_{i}}- \frac{k_{i}^{2}}{\left( 2 Q \cdot k_{i}\right)^{2}}+ \frac{\left( k_{i}^{2}\right)^{2}}{\left( 2 Q \cdot k_{i}\right)^{3}}+ \cdots .
{\cal H}_{e f f}= \frac 1 2 ( \partial_{i}C_{j})^{2}- \Pi [ \Delta_{\perp}^{d}]^{- 1}* \left[ \partial_{i}j^{i}+ \partial_{-}j^{-}\right] - C_{i}\left[ j^{i}- 2 \partial_{i}[ \Delta_{\perp}^{d}]^{- 1}* ( \partial_{-}j^{-}+ \partial_{j}j^{j}) \right] - C_{-}j^{-},
\langle f | g \rangle_{{\cal L}^{1 | 2}}= \langle f_{0}| g_{0}\rangle_{\cal L}^{s}+ \langle f_{1}| g_{1}\rangle_{\cal L}^{s + 1 / 2}+ \langle f_{2}| g_{2}\rangle_{\cal L}^{s + 1 / 2}+ \langle f_{3}| g_{3}\rangle_{\cal L}^{s + 1},
\#J_{ns}(13)(\mathbb{F}_{5}) = 3 \cdot 7 \cdot 11 \cdot 13^2 \cdot 29, \#J_{ns}(13)(\mathbb{F}_{19}) = 2^2 \cdot 7 \cdot 83 \cdot 97 \cdot 113 \cdot 883,
(\tfrac{\beta}{2}-m+j+1)_{2m-2j}=(-1)^j\tfrac{(\tfrac{\beta}{2}-m+1)_{2m}}{(\tfrac{\beta}{2}-m+1)_j(-\tfrac{\beta}{2}-m)_j}, \prod_{l=1}^{j-1}\big[(m-l+i)(m-l-i+1)\big] =(-m-i+1)_{j-1}(-m+i)_{j-1}
\int^t_0 (t-s)^{\nu-1}d(s)ds =\int^t_0 (t-s)^{\nu-1}\left(\int^s_0(s-\xi)^{\alpha-1}|f(\xi)|\|R(\xi)\|_{D(A)}d\xi\right)ds &=\int^t_0 \left( \int^t_{\xi}(t-s)^{\nu-1}(s-\xi)^{\alpha-1}ds\right)|f(\xi)|\|R(\xi)\|_{D(A)}d\xi &=B(\nu,\alpha)\int^t_0 (t-\xi)^{\nu+\alpha-1}|f(\xi)|\|R(\xi)\|_{D(A)}d\xi
E^{\underline{a}}= (E^a, E^i)= dX^{\underline{m}}u_{\underline{m}}^{\underline{a}}= ( dX^{\underline{m}}u_{\underline{m}}^{a},dX^{\underline{m}}u_{\underline{m}}^{i})\qquad{\underline{a}= 0, \ldots , (D-1); ~~ a = 0, \ldots , p}
\begin{aligned}w = -\Big(\frac{\nabla u}{u}+ \nabla(U + \psi)\Big) \ \mbox{and}\ \omega = -\Big(\frac{\nabla v}{v}+ \nabla(V - \psi)\Big) \ \mbox{so that}& u_t + \mbox{div}(wu) = 0 \ \mbox{and}\v_t + \mbox{div}(\omega v) = 0.\end{aligned}
D (U\otimes v_1) = (-\sqrt{n}+1)U\otimes v_1, D (U\otimes v_2) = (-\sqrt{n}+1)U\otimes v_2, D (U \otimes u_i) = (-\sqrt{n}+1)(U \otimes u_i), \mbox{for any}2\leq i\leq n D (V\otimes v_1) = -V\otimes v_1, D (V\otimes v_2) = -V\otimes v_2, D (V \otimes u_i) = -(V \otimes u_i), \mbox{for any}2\leq i\leq n.
a_{11}u_{1}+a_{12}u_{2}+\cdots+a_{1M}u_{M}<v_{1}\shortintertext{and}\tilde{a}_{i1}u_{1}+\tilde{a}_{i2}u_{2}+\cdots+\tilde{a}_{iM}u_{M}=v_{i},i=2,3,\ldots,N.
\psi_{\mathrm{A B}}( r , \theta ) ={\frac{i}{4 \pi}}e^{( N +{\frac{1}{2}}) \theta}\int_{\cal C}d t e^{- i k r \cos t}\left[{\frac{e^{i \gamma t}}{\sin{\frac{1}{2}}( t - \theta )}}+{\frac{e^{- i \gamma t}}{\sin{\frac{1}{2}}( t + \theta )}}\right] .
S^{G}_{,ij}=-{\cal G}^{-1}_{ij}+\left\{\begin{array}{c}k i\j \end{array}\right\}{\cal E}_{k}+\frac{\delta\chi^{\alpha}}{\delta\Phi^{i}}c_{\alpha\beta}\frac{\delta\chi^{\beta}}{\delta\Phi^{j}}.
\Gamma_{N \rightarrow \infty}[ \sigma , \rho , \pi ] = \int d x \, \sqrt{g}\, \frac{N}{2 \lambda_{B}}( \sigma^{2}+ \rho^{2}+ \pi^{2}) - N \ln \det \left( \not{\! \! D}+ \sigma + i \tau^{1}\rho + i \gamma^{5}\pi \right) .
\Phi_{n}^{(m)}(X_n)=\sum_{a=m}^{n}e_a(X_n) \beta^{a-m}, A_{i,j}(X_{n+k-1})={1 \over (i-j)!}\left({\partial \over \partial \beta}\right)^{j-1}\Phi_{k+n-i}^{(n+1-i)}(X_{k+n-i}) 1 \le i \le j \le n,
\partial_{\mathrm{ij}}\Big(\mathbb{G}^{(\mathrm{k})}-\mathbb{G}^{(0)}\Big)(\mathrm{x},\mathrm{y}) = 2a \delta_{\mathrm{ij}}\log|x-y| + 2\mathrm{a}\dfrac{(\mathrm{x}-\mathrm{y})_{\mathrm{i}}(\mathrm{x}-\mathrm{y})_{\mathrm{j}}}{|\mathrm{x}-\mathrm{y}|^{2}}+ 2\mathrm{b}\delta_{\mathrm{ij}}+ \mathcal{O}(|\mathrm{x}-\mathrm{y}|),\mathrm{x}\sim \mathrm{y},
h^0(N_Y(-1) (y)[x \to y][2y \to x][w_1 \to x][w_2 \to x][z_3 \to w_3][w_3 \to z_3]) & =h^0(N_X(-1)[z_1 \to w_1][w_1 \to z_1][z_2 \to w_2][w_2 \to z_2][z_3 \to w_3][w_3 \to z_3]) & = h^0(N_{C}(-1)(-q_1-\cdots-q_5)).
n>\frac{4\left( R+1\right) p}{\gamma\left( p-1\right)}A^{-p/\left(\gamma p-\gamma\right)}=\frac{r^{4}\left( r+4\right) p}{p-1}\left(\frac{r}{r-1}\right)^{r^{4}p/\left( p-1\right)}.
4 \Lambda^{2n_{c}-n_{f}}\prod_{i=1}^{n_{f}}(\Lambda \omega^{2\ell-1}+ \delta_{\ell}+ m_{i}) = 4 \Lambda^{2n_{c}}(\omega^{2\ell-1})^{n_{f}}\left[ 1 + \sum_{i=1}^{n_{f}}\frac{\delta_{\ell}+ m_{i}}{\Lambda \omega^{2\ell-1}}\right].
\frac{T}{2}\int_0^L\left( \varphi_0^2(x) +\frac{b}{c}\psi_0^2(x)\right)dx= \frac{1}{2}\int_0^T\int_0^L \left(\varphi^2(x,t)+\frac{b}{c}\psi^2(x,t)\right)dxdt + \frac12\int_0^T(T-t)\left[\varphi_x^2(0,t)+\frac{2ab}{c}\psi_x(0,t)\varphi_x(0,t)+\frac{b}{c^2}\psi_x^2(0,t)\right]dt.
(T_{\theta}u)_n=\left\lbrace \begin{matrix}\sqrt{\frac{\prod_{j=0}^{n-1}\tilde{c}(\theta+j\alpha)}{\prod_{j=0}^{n-1}c(\theta+j\alpha)}}u_n\ \ n\geq 1, u_n\ \ n=0, \sqrt{\frac{\prod_{j=n}^{-1}c(\theta+j\alpha)}{\prod_{j=n}^{-1}\tilde{c}(\theta+j\alpha)}}u_n\ \ n\leq -1,\end{matrix}\right.
\Psi(x)z(e) = \sum_{i=1}^\infty (\rho_ix)^{1/2}(e) \sum_{s(f) = s(e)}(\rho_ix)^{1/2}(f)z(f) = \sum_{i=1}^{\infty}\rho_i(e) x(e) z(e) = x(e)z(e).
\left(-\frac{1}{n}+\frac{1}{n^2}\right) e_{i,g+1}^{n-2}+\frac{1}{n}e_{i,g+2}^{n-2}+\frac{1}{n}\sum_{\tau=2}^{n-s-2}\left[e^{n-\tau-1}_{i,g+\tau+1}-e_{i,g+\tau}^{n-\tau-1}\right] &+\left(1-\frac{1}{n}\right)\left[\sum_{\tau=0}^{n-s-3}\frac{1}{n^{n-\tau-s-1}}e^{s}_{i,g+\tau+1}\right] -\frac{1}{n^2}e_{i,g+n-s-1}^s .
(-i\nabla-\epsilon A^0(x)-\kappa A(\epsilon x))\widetilde{\Lambda}^{\epsilon,\kappa}(x,\tilde{\beta})=\widetilde{\Lambda}^{\epsilon,\kappa}(x,\tilde{\beta})\left\{(-i\nabla-\epsilon A^0(x))\,+\,\kappa a_\epsilon(x,\tilde{\beta})\right\}\,,
\tilde{\tau}_{\mathcal{C}}(x,y,t) \,=\, \frac{1}{E_2E_4\cdots E_{2g}}\biggl(\displaystyle\sum_{I}\displaystyle\prod_{i< j \in I}(\kappa_i - \kappa_j) \displaystyle\prod_{\{i\,:\, c_i = 1\}}\lambda_i \prod_{i\in I}E_i \biggr),
( \Gamma_{1}^{\mathrm{s c a l a r s}})^{\wedge \wedge \vee}= - \frac{N_{s}g^{3}}{4 \pi^{2}}\frac{a}{m}\ln ( 1 / a ) \frac{p^{+}K^{\wedge}}{p_{1}^{+}p_{2}^{+}}\sum_{l = 1}^{M_{1}- 1}\frac{f_{s}( l / M )}{M}- ( 1 \leftrightarrow 2 ) .
[\textbf{K}^X_{t,t}]_{i,j}= k_{S}(\bar x^i|_{I_t}, \bar x^j|_{I_t}), [\textbf{K}^{X,Y}_{T,T}]_{i,j}= k_S(\bar x^i,\bar y^j), &[\textbf{K}^Y_{s,s}]_{i,j}= k_S(\bar y^i|_{I_s}, \bar y^j|_{I_s}).
[\omega^{[1]}_{k,k},\dots,\omega^{[1]}_{k,k+N-1}]\begin{bmatrix}P^{[1]}_k(x) \vdots P^{[1]}_{k+N-1}(x)\end{bmatrix}+P^{[1]}_{k+N}(x)=\hat P^{[1]}_k(x)W(x).
\left(\frac{d\psi}{d\zeta}+\frac{i\zeta}{2}\sigma_3\psi\right)\psi^{-1}= \left[\frac{dm^{PC}_{\eta}}{d\zeta}+m^{PC}_{\eta}\frac{i\nu}{\zeta}\sigma_3\right]\left(m^{PC}_{\eta}\right)^{-1}+\frac{i\zeta}{2}\left[\sigma_3,m^{PC}_{\eta}\right]\left(m^{PC}_{\eta}\right)^{-1},
\lim_{\varepsilon \to 0}\frac{E_{\varphi}(F_{\varepsilon,h}) - E_{\varphi}(F)}{\varepsilon}= \lim_{\varepsilon \to 0}\frac{E_{\varphi}[\mu^F_{\varepsilon,h}] - E_{\varphi}[\mu^F]}{\varepsilon}
\widetilde{X}_{ij}=\begin{pmatrix}\lambda_p^{(i)}I_{s_{ip}}B_{pr}^{(i)}(B_{pr}^{(i)})^{t}- \lambda_p^{(i)}I_{v_{ir}}\end{pmatrix}\quad \widetilde{Y}_{ij}=\begin{pmatrix}\mu_i I_{s_{ip}}0 0 -\mu_i I_{v_{ir}}\end{pmatrix}
v_{f}^{i}( \sigma , t ) = \sum_{n = 1}^{\infty}W ( n + \frac{\epsilon}{\tau}) \Biggl [ \frac{c_{n}^{i}}{\sqrt{n}}\, \dot{x}_{n}^{+}+ \frac{\tilde{c}_{n}^{i}}{\sqrt{n}}\, \dot{x}_{n}^{-}+ h . c . \Biggr ] \, .
\left\langle r_{p}\right\rangle_{n,\omega\gamma,\overline{Q}}^{(m+\gamma,\frac{m+\gamma}{m})}=\left\langle w_{p}-Q_{p}(y,\tau)\right\rangle_{n,\omega\gamma,\overline{Q}}^{(m+\gamma,\frac{m+\gamma}{m})}\leq C.
\begin{array}{l c r}{{U^{+{\frac{n ( n + 1 )}{2}}}= u M^{+{\frac{n ( n + 1 )}{2}}}}}\\{{V^{+{\frac{n ( n + 1 )}{2}}}= v N^{+{\frac{n ( n + 1 )}{2}}}}}\\{{Z^{+{( n + 1 )}}= z S^{+{( n + 1 )}};}}\\ \end{array}
\langle P_{\lambda}, P_{\lambda}\rangle_F = \displaystyle \prod_{\substack{s \in \lambda h_{\lambda}(s) \equiv 0 \bmod N}}\dfrac{\varepsilon_1 l_{\lambda}(s) - \varepsilon_2 (a_{\lambda}(s) + 1)}{\varepsilon_1 (l_{\lambda}(s) + 1) - \varepsilon_2 a_{\lambda}(s)}
W(\vec{k}_{1},\vec{k}_{2},t)=\sum_{M_{i},M_{f},\sigma_{1},\sigma_{2}}|\sum_{M_{a}}\langle M_{f}|H_{2}\Lambda(t)|M_{a}\rangle\langle M_{a}|H_{1}|M_{i}\rangle|^{2}=\langle\rho(\vec{k}_{2})\rangle_{t}
\P\big( \|X\|_{\infty}\vee \|X^k\|_{\infty}\geq n \big) \leq \P\Big( \lbrace \|X\|_{\infty}\geq n\rbrace \cup\lbrace \|X^k-X\|_{\infty}+\|X\|_{\infty}\geq n\rbrace \Big) \leq \P\big( \|X\|_{\infty}\geq n/2\big) +\P\big( \|X^k-X\|_{\infty}\geq n/2 \big) \leq\delta/4+\delta/4=\delta/2.
-\int\limits_{X_{3T_1}}\frac{\partial}{\partial s}\Big(\frac{\partial}{\partial\tau}+iA_-'\Big)u^f\overline{v^g}dsd\tau &=\int\limits_{X_{3T_1}}\Big(\frac{\partial}{\partial\tau}+iA_-'\Big)u^f\frac{\overline{\partial v^g}}{\partial s}dsd\tau-\int\limits_{y_n=0}\Big(\frac{\partial}{\partial\tau}+iA_-'\Big)u^f\overline{v^g}d\tau.
\Vert C_{\varphi}(f) - A (f) \Vert = \Vert f \circ \varphi - I_1\circ \varphi \Vert \leq \Vert f \circ \varphi - g\circ \varphi \Vert + \Vert g \circ \varphi - I_1 \circ \varphi \Vert \,,
D + \rho = \left( \begin{matrix}{\partial \kern - 0 . 5 e m / \otimes{1 \kern - . 4 e m 1}\otimes{1 \kern - . 4 e m 1}}&{\gamma^{5}\otimes H \otimes K}\\{\gamma^{5}\otimes H^{\dag}\otimes K^{\dag}}&{\partial \kern - 0 . 5 e m / \otimes{1 \kern - . 4 e m 1}\otimes{1 \kern - . 4 e m 1}}\\ \end{matrix}\right)
G_{( 1 + 2 )}^{\mathrm{D}}( \tau \rightarrow 0 ) = G_{( 3 )}^{\mathrm{D}}( \tau \rightarrow 0 ) = 1 2 0 \tau^{4}\int_{0}^{\infty}d \Omega^{\prime}\frac{\Omega^{3}}{e^{\Omega^{\prime}}- 1}.
d{\cal O}/ d t = -{\frac{i}{\hbar_{1}}}[{\cal O},{\cal H}_{1}] -{\frac{i}{\hbar_{2}}}[{\cal O},{\cal H}_{2}] -{\frac{i}{\hbar_{\mathrm{a v}}}}[{\cal O},{\cal H}_{I}] -{\frac{i \Delta \hbar^{- 1}}{2}}( J_{2}{\cal O}J_{1}- J_{1}{\cal O}J_{2})
a_{d,k}(\mathcal{D})=\left[ \int_{\mathcal{D}}\int_{\mathcal{D}}\frac{1}{\Vert \mathbf{x}-\mathbf{y}\Vert^{k\delta}}d\mathbf{x}d\mathbf{y}\right]^{1/2},k\geq 1.
X_z=a_1z_2^m\frac{\partial}{\partial z_1}+a_2z_2\frac{\partial}{\partial z_2}, Y_z=b_1z_2^m\frac{\partial}{\partial z_1}+b_2z_2\frac{\partial}{\partial z_2}{\rm or}X_z=(a_1z_1+a_2z_2^m)\frac{\partial}{\partial z_1}, Y_z=(b_1z_1+b_2z_2^m)\frac{\partial}{\partial z_1}.
v_{\lambda_i}(x)=\left(\frac{C_{\ast}}{|x|^2}\right)^{\frac{1}{1-m}}\left(1-B_{\lambda_i}|x|^{-\gamma_1}+o(|x|^{-\gamma_1})\right)\quad\forall i=1,2\quad\mbox{as}|x|\to\infty.
|\Lambda| =|A|+|B|,{\rm size}(\Lambda) = (|A|,|B|),{\rm rk}(\Lambda) =\sum_{i=1}^{m_1}a_i+\sum_{j=1}^{m_2}b_j-\left\lfloor\biggl(\frac{|A|+|B|-1}{2}\biggr)^2\right\rfloor,{\rm def}(\Lambda) =|A|-|B|.
\left\{v : \exists \, t \in I_j^{(k)}\textrm{with}\delta(\phi_t(v)) = T_k^{-\xi}\right\}\subset \left\{v : \delta(\phi_{a_j^{(k)}}(v))\in \left[\frac{1}{2}T_k^{-\xi}, 2T_k^{-\xi}\right]\right\}.
\frac{\left(\lambda_1\lambda_2\right)^{-\frac{1}{2}n\left(n-1\right)}\left(\lambda_3\lambda_4\right)^{-\frac{1}{2}n}}{\sqrt{{\rm det}\ \bold{C}_n}}= (\alpha^2 - \mu^2)^{-\frac{n}{2}}.
\begin{aligned}G(t)=\int_{0}^{t}\int_{B}\mathcal{\mathcal{P}}\left[\theta(\tau),\Theta{}_{,i}(\tau)\right]dvd\tau \quad+\int_{B}\left[\rho u_{i}(t)u_{i}(t)+\beta k_{ji}\Theta_{,j}(t)\Theta_{,i}(t)\right]dv,\end{aligned}
C=\frac{1}{(m|g|)^2}\sum_{\substack{\ell=1 (\ell\neq j)}}^n\sum_{i,h,h'=0}^{m-1}[\tilde{v}_i\tilde{w}_i]_{\ell,k}[\tilde{v}_{i+h+1}\tilde{w}_{i+h+1}]_{j,\ell}[\tilde{v}_{i+h'+1}\tilde{w}_{i+h'+1}]_{k,j}\frac{\lambda_\ell^{m-h-1}\lambda_j^{m-h'-1+h}\lambda_k^{h'}}{(\lambda_\ell^m-\lambda_j^m)(\lambda_j^m-\lambda_k^m)},
\kappa\|\Lambda\Theta^{i+1}\|_2^2 =-(U^i\cdot\nabla\Theta^{i+1},\Lambda\Theta^{i+1})+(f,\Lambda\Theta^{i+1}) &\leq \|U^i\|_\infty\|\Lambda\Theta^{i+1}\|_2^2+C\kappa^{-1}\|f\|_2^2+\frac{\kappa}{4}\|\Lambda\Theta^{i+1}\|_2^2.
(b\gamma+c)a+\gamma(b\gamma+c+1) =a(a^{-1}\gamma+c(\gamma+1))+a^{-1}\gamma^2+c\gamma(\gamma+1)+\gamma &=a^{-1}\left(c(\gamma+1)a^2+c\gamma(\gamma+1)a+\gamma^2\right) (b\gamma+1)a+\gamma(b\gamma+c+1) =a(a^{-1}\gamma+c\gamma+1)+a^{-1}\gamma^2+c\gamma(\gamma+1)+\gamma &=a^{-1}\left((c\gamma+1)a^2+c\gamma(\gamma+1)a+\gamma^2\right)
\frac{1}{\Delta t}(u_i^{n+1}-u_i^n)+\frac{u_i^n}{2 \Delta x}(u_{i+1}^n-u_{i-1}^n)-\frac{\nu}{(\Delta x)^2}(u_{i+1}^n-2u_i^n+u_{i-1}^n)=g(i\Delta x,n \Delta t)
\sum_{n=1}^{\infty}\frac{1}{n\left(e^{2\alpha n}+1\right)}+2\,\sum_{n=1}^{\infty}\frac{1}{(2n-1)}\,\frac{1}{\left(e^{\beta(2n-1)}-1\right)}=\frac{\alpha}{4}-\log(2)-\frac{1}{4}\log\left(\frac{\alpha}{\beta}\right).
\mathcal{P}_{n}^{\left( +\right)}\left( \mathcal{E}\right) =\frac{\left(-1\right)^{n}\left( \gamma +\frac{3}{2}-D)\right)_{n}}{\sqrt{n!\left(2\gamma +1\right)_{n}}}_{3}F_{2}\left(\begin{array}{ccc}-n 1-D+i\lambda 1-D-i\lambda . \gamma +\frac{3}{2}-D \gamma +\frac{3}{2}-D\end{array}\mid 1\right)
w_2(x, x \tan \phi , z)= x^k(\cos \phi)^{-k}\cos k\phi+ O(|x|^{k+1})>0, \ |\phi|\le\phi_1, w_2(x, \pm x \tan \phi_2 , z)= x^k(\cos \phi_2)^{-k}\cos k\phi_2+ O(|x|^{k+1})<0, \{}\partial_x w_2(x,y,z)\ge c(|x|^2+|y|^2)^{(k-1)/2}>0\ \Omega.
\|B_{3,M_n,n}\|^2_{L^2(\Omega)}= \sum_{p=0}^{M_n}e^{2(T-t)p^{2\beta}}\Big| \widetilde{G}_{np}\Big|^2 \le C_1^2(\beta, \widetilde E_1) \frac{\sum_{p=0}^{M_n}e^{2(T-t)p^{2\beta}}}{n^{4\beta}}.
\tilde{T}_{\mu\nu}=\rho\left(\begin{array}{cccc}1 0 e^{\alpha (t)x}0 & 0 0 \displaystyle{\frac{1/2e^{\alpha (t)x}\left(1+2\alpha (t)x\right)\alpha{'}(t)}{\alpha (t)^2}}0 & e^{\alpha (t)x}\displaystyle{\frac{1/2e^{\alpha (t)x}\left(1+2\alpha (t)x\right)\alpha{'}(t)}{\alpha (t)^2}}e^{2\alpha (t)x}0 & 0 0 0 0 \end{array}\right)
H_{2}^{'}( p ) = \int \frac{d^{3}q}{( 2 \pi )^{3}}\, \frac{1}{( p - q )^{2}}\frac{\left[ H_{1}^{'}( q ) \sin^{2}\theta + H_{2}^{'}( q ) \cos^{2}\theta \right]}{\left| - \frac{1}{2}\delta + i q_{3}+ \frac{\left|{\bf q}\right|^{2}}{2 m}+ \Sigma_{\pm}( q_{3}, \left|{\bf q}\right| ) \right|^{2}}.
\delta(\langle\prod_{i}\phi_{n_{i}}(r_{i},s_{i})|\hat{H}|\prod_{i}\phi_{n_{i}}(r_{i},s_{i})\rangle-\sum_{i}\epsilon_{i}\langle\phi_{n_{i}}(r_{i},s_{i})|\phi_{n_{i}}(r_{i},s_{i})\rangle)=0
\widetilde{Z}( R ( S ) , \langle \cdot \, , \cdot \rangle_{H}, \langle \cdot \, , \cdot \rangle ) = \widetilde{Z}( R ( S ) , \langle \cdot \, , \cdot \rangle ){\cdot}\Psi ( R ( S ) , \langle \cdot \, , \cdot \rangle_{H}, \langle \cdot \, , \cdot \rangle )
\Xi(x, y) = \left| w_-(x) - w_-(y) \right|^2 - \left| (1 - \eta(x)) \left( w_-(x) - w_-(y) \right) - \left( \eta(x) - \eta(y) \right) w_-(y) \right|^2 & \ge \left( 1 - 2 (1 - \eta(x))^2 \right) \left| w_-(x) - w_-(y) \right|^2 - 2 \left| \eta(x) - \eta(y) \right|^2 w_-(y)^2.
\delta_{k,\ell}:= \sum_{i=0}^{\ell-1}\varrho_0^{\ell-i-1}\eta_{k,i}+\frac{\beta_0}{2}\left( \sum_{i=1}^{\ell-1}\varrho_0^{i}\right)^2 +\frac{\beta_0}{2}\sum_{i=1}^{2\ell}i \varrho_0^{i-1},