harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\Vert (I-R_{A-q,B-q,\beta})(x_{n})\Vert =2\beta\Vert (P_{A-q}-P_{B-q}(2\beta P_{A-q}-I))(x_{n})\Vert &=2\beta\Vert P_{A-q}(x_{n})-P_{B-q}(2\beta P_{A-q}(x_{n})-x_{n})\Vert &=2\beta\Vert P_{A}(x_{n}+q)-q-P_{B}(2\beta P_{A}(x_{n}+q)-q-x_{n}+q)+q\Vert &=2\beta\Vert y_{n}-P_{B}(2\beta y_{n}-x_{n})\Vert.
\mathbb{P}_{couple}\left(\sum_{i=1}^{M_{1}\left(n\right)}n^{-\frac{1}{\alpha}}W_{i}\leq T\right) \leq Cn^{-c'}\mathbb{P}_{couple}\left(\sum_{i=1}^{M_{2}\left(n\right)}n^{-\frac{1}{\alpha}}W_{i}\leq n^{-c'}\right) \leq Cn^{-c'}.
\left ( i\frac{\partial}{\partial t}+ \frac{\nabla^2}{2m}\right )G_{\alpha\beta}(x,x') + i\lambda \langle T \left ( \hat\psi^+_\gamma (x)\hat\psi_\gamma (x) \hat\psi_\alpha (x) \hat\psi^+_\beta (x') \right )\rangle = \delta (x - x') .
\sum_\pi q^{\vert\pi\vert}=\prod_{i=1}^{a}\frac{1-q^{c+2i-1}}{1-q^{2i-1}}\prod_{1\le i<j\le a}^{}\frac{1-q^{2(c+i+j-1)}}{1-q^{2(i+j-1)}},
\sum_{l>n}C_{l}\,(b_{l+1}-b_{l})^{1-\frac1{p}}\Bigl(\ \sup_{b_{l+1}-2(b_{n+1}-b_n)\le k<b_{l+1}}\prod_{s=k+1}^{b_{l+1}-1}|w_s|\Bigr) =\sum_{k>k_0}2^{k-1}2^{\delta^{(k-1)}-\tau^{(k)}}(\Delta^{(k)})^{1-\frac1{p}}&=\sum_{k>k_0}2^{k-1}\gamma_k,
\langle M \rangle_s = \sum_{i=1}^2 \int_{(t^i)^{-1}_{\tau_0}}^{(t^i)^{-1}_s}\frac{|\dot{t}_u^i|^2}{|\dot{t}_u^i|^2-1}du \geq (t^1)^{-1}_s- (t^1)^{-1}_{\tau_0}.
G_m (\pi;x)=1+ \sum_{k=1}^{\infty}\frac{(-1)^k}{k!}\int_{x}^{\infty}\cdots \int_{x}^{\infty}\det[{K_{\mathrm{GUE}}(\pi; u_i, u_j)}]_{i,j = 1}^{k}du_1 \cdots du_k.
u_{n,k}(t)=\begin{cases}v_n(t)\qquad\enskip0\leq t\leq \zeta_{n,k}\,, e^{-(t-\zeta_{n,k})\Delta^2}v_n(\zeta_{n,k})\qquad\enskip\zeta_{n,k}<t\leq T\,,\end{cases}
\frac{1}{n}(2+\varepsilon)\left[\sqrt{\frac{2\Delta}{(1-\varepsilon)}}(2+\varepsilon)\right]^{n-1}|\mathcal{E}_x|_{n-1}=\frac{(2\Delta)^{\frac{n+1}{2}}|B_2^{n-1}|}{\kappa(x)^{1/2}}\frac{(1+\frac{\varepsilon}{2})^{n}}{n(1-\varepsilon)^{\frac{n-1}{2}}}\quad.
\alpha_a = \left( \int_{D_{R|a|}}|(i\nabla+A_0)v_{R,a}^{int}|^2 \,dx- \int_{D_{R|a|}}|(i\nabla+A_a)\varphi_a|^2\,dx \right) +(\lambda_a-\lambda_0) = |a|^k\left( \int_{D_{R}}|(i\nabla+A_0)Z_a^R|^2 \,dx- \int_{D_{R}}|(i\nabla+A_p)\tilde\varphi_a|^2\,dx \right) +(\lambda_a-\lambda_0)= O(|a|^k),
M = \left( \begin{matrix}{G^{- 1}}&{- G^{- 1}C}&{- G^{- 1}a^{T}}\\{- C^{T}G^{- 1}}&{G + C^{T}G^{- 1}C + a^{T}a}&{C^{T}G^{- 1}a^{T}+ a^{T}}\\{- a G^{- 1}}&{a G^{- 1}C + a}&{I + a G^{- 1}a^{T}}\\ \end{matrix}\right) ,
ds^2/\alpha^\prime = \frac{U}{R_5 f^{1/2}_1}( -d t^2 + d x^2_1) + \frac{U f^{1/2}_1}{R_5}( d x^2_2 + \cdots + d x^2_5 ) + U R_5 f^{1/2}_1 \left(\frac{d U^2}{U^2}+ d \Omega^2_3 \right).
\frac{\partial}{\partial t}(u+\frac{1}{2}\rho V_{i}V_{i})+\frac{\partial}{\partial x_{j}}(uV_{j}+\frac{1}{2}\rho V_{i}V_{i}V_{j}+J_{qj}+P_{ij}V_{i})-nF_{i}V_{i}=0,
\sum^k_{i=1}\sum^i_{l=1}|\mathcal{B}_{i,i-l}|+\sum^k_{i=1}\sum^i_{l=1}|\mathcal{B}_{i-l,i}|=|\{(i,i,t,p)\mid(i,i,t,p)\in\mathcal{I}'_{k-1}\}|+|\mathcal{I}'_{k-1}|.
-(p_i-2)F^{p_i-4}(\nabla \phi(x_i)) \mathcal{Q}_\infty \phi (x_i) - F^{p_i-2}(\nabla \phi(x_i)) \Delta_F ( \phi(x_i)) \ge\qquad \Lambda_{p_i}^{p_i}(\Omega) |u_{p_i}(x_i)|^{p_i-2}u_{p_i}(x_i).
{\cal H}_{\mathrm{6 - v e r t e x}}= \sum_{j = 0}^{l_{1}}\left[ \left( \begin{array}{c}{{2 l_{1}}}\\{{l_{1}- j}}\\ \end{array}\right) - \left( \begin{array}{c}{{2 l_{1}}}\\{{l_{1}- j - 1}}\\ \end{array}\right) \right] \rho_{j},
U_{k}=\{m \in \bar{I}_{k-1}\mid j_{2p+2}<m<j_{2p+1}\mbox{for some}p\geq 0\}, begin{align*}3pt]V_{k}=\{m \in \bar{I}_{k-1}\mid j_{2p+1}<m<j_{2p}\mbox{for some}p\geq 0\}.
\begin{array}{ll}2I=\int_{\mathbb R}[(1-\gamma)-|\omega|]\xi^2(x)+\mu b(\xi'(x))^2 (\frac{|c|}{b}(1-\gamma)-|\omega|)dx \geqq \Big[\frac{|c|}{b}(1-\gamma)-|\omega| \Big]\int_{\mathbb R}\xi^2+\mu b(\xi')^2dx=\Big[\frac{|c|}{b}(1-\gamma)-|\omega|\Big]\|J_b^{1/2}\xi\|^2.\end{array}
\int \frac{\delta \hat{\Gamma}_{cl}}{\delta \hat{\varphi}_0}\; ,\; \int \left\{A \frac{\delta}{\delta A}+ c \frac{\delta}{\delta c}\right\}\hat{\Gamma}_{cl}\; ,\; m_H \partial_{m_H}\hat{\Gamma}_{cl}\; , \; e \partial_e \hat{\Gamma}_{cl}\; ,
\log \Gamma ( z ) = ( z -{\frac{1}{2}}) \mathrm{l o g}z - z +{\frac{1}{2}}\mathrm{~ l o g \}2 \pi + 2 \int_{0}^{\infty}{\frac{\mathrm{t a n}({\frac{t}{z}})}{e^{2 \pi t}- 1}}d t ,
\begin{aligned}\sup_{B_{R}(O)}G_{1}\leq G_{1}\phi \leq \frac{((n-1)(1+\sqrt{K}R)+2)C_{1}^{2}+C_2}{(\frac{2}{n}-C_{5})R^{2}}+\frac{2K\phi}{(\frac{2}{n}-C_{3})}+\frac{C_{1}^{2}\frac{1}{C_{5}}}{(\frac{2}{n}-C_{5})R^{2}}\leq \frac{((n-1)(1+\sqrt{K}R)+2+\frac{1}{C_{5}})C_{1}^{2}+C_2+2KR^{2}}{(\frac{2}{n}-C_{5})R^{2}},\end{aligned}
\|(D_j x_j)\|_{l_{q}(Y)}=\left ( \sum \|D_j x_j\|_Y^q\right )^{1/q}\leq\left ( \sum\|D_j\|^q \|x_j\|_X^q\right )^{1/q}\le (\sup_j \|x_j\|_X) \|(D_j)\|_{l_q}.
\int_{t^*}^{2a_N/k_N}G_{k^*}(v) \: dv = s k^* (2a_N/k_N - t^*) - \mu(2a_N/k_N - t^*) - \int_{t^*}^{2a_N/k_N}s M(v) \: dv. \geq sk^*(2a_N/k_N - t^*) - 4.
\frac{d}{dr}\left( \begin{array}{c}U^{+}_{k,l}U^{-}_{k,l}\end{array}\right)=\left( \begin{array}{cc}\frac{(k-\frac{1}{2})}{r}-2l -2l -\frac{(k+\frac{1}{2})}{r}\end{array}\right)\left( \begin{array}{c}U^{+}_{k,l}U^-_{k,l}\end{array}\right).
{\cal L}= \sum_{j = 1}^{N}i \bar{\Psi}_{j}\gamma^{\mu}D_{\mu}\Psi_{j}+ \frac{G}{2}\sum_{j = 1}^{N}\left[ ( \bar{\Psi}_{j}\Psi_{j})^{2}+ ( \bar{\Psi}_{j}i \gamma_{5}\Psi_{j})^{2}\right] ,
s_\mathcal{A}(r+2,1)-s_\mathcal{A}(r+1,1)-s_\mathcal{A}(r,1) =c_13^r(3^2-3-1)+c_2\phi^r(\phi^2-\phi-1) &\quad+c_3\bar{\phi}^r(\bar{\phi}^2-\bar{\phi}-1)+c_4(1^2-1-1) &\quad+c_5(r+2-(r+1)-r) &=c_13^r\cdot 5-c_4-c_5(r-1).
\left[\frac{\AA (n+1)}{\AA (n)}\right]^n= \left[1 + \frac{A^{-1}(n+1)}{\sum\limits_{j=1}^n A^{-1}(j)}\right]^n \leq \left[1 + \frac{\varepsilon}{n}\right]^n\leq e^\varepsilon.
\{k_1,\ldots,k_N\}\backslash\{k_{d_1},\ldots,k_{d_M}\},c^{(M)}_m=c_m\times \prod_{j=1}^M\left(\frac{k_{d_j}-k_m}{k_{d_j}+k_m}\right)^\Xi,m\in\{1,\ldots,N\}\backslash\mathcal{D},
(\phi_1,\dots,\phi_n) \mapsto \sum_{i=1}^n (-1)^{n-i}\Big[\alpha(\phi_1(0),\dots,\phi_{i-1}(0), \widehat{\phi_i(0)},\phi_{i+1}(0),\dots,\phi_n(0))\Big]\big(\dot{\phi_i}(0)\big)\, .
S[\vec{n}]=-\frac{1}{4}\int d^4x (\vec{\Gamma}_\mu \wedge \vec{\Gamma}_\nu) (\vec{\Gamma}^\mu \wedge \vec{\Gamma}^\nu)=-\frac{1}{4}\int d^4x (\partial_\mu\vec{n}\wedge \partial_\nu\vec{n})(\partial^\mu\vec{n}\wedge \partial^\nu\vec{n}).
a_{\lambda,n}:= \frac{1}{\beta}\int_0^{\beta}e^{-t\lambda}e^{-\pi i n t/\beta}\, dt= \int_0^1 e^{-(\beta\lambda + \pi i n) t}\, dt= \frac{1 - e^{-(\beta\lambda + \pi i n)}}{\beta \lambda + \pi i 2n}= \frac{1 - (-1)^ne^{-\beta\lambda}}{\beta \lambda + \pi i 2n}.
\|\nabla \varphi(t,\cdot)\|_{q}\leq \|\nabla\varphi_o\|_{q}&+C_{n}\Big(1+\frac{1}{t^{(n-q')/2q'}}\Big)\Big(1+\sup_{0\leq s \leq t}\|f(s,\cdot)\|_{1}\Big)\sup_{\tfrac{t}{2}\leq s \leq t}\|f(s,\cdot)\|^{\frac{\theta}{n}\frac{q(n-1)-n}{\theta q -2}}_{q\theta/2}\,.
\left( \mathbf{f},\mathbf{g}\right)_{\operatorname{curl},\Omega,k}:=\left(\operatorname{curl}\mathbf{f},\operatorname{curl}\mathbf{g}\right)+k^{2}\left( \mathbf{f},\mathbf{g}\right) \quad\left\Vert\mathbf{f}\right\Vert_{\operatorname{curl},\Omega,k}:=\left( \mathbf{f},\mathbf{f}\right)_{\operatorname{curl},\Omega,k}^{1/2},
F(X_{t})= \F(X_{0})+\int_0^t \Delta_{s}F(X_{s}) ds+\int_0^t \Delta_{x}F(X_{s}) \psi(s) ds + \int_0^t \Delta_{x}F(X_{s})\varphi(s) dW(s) +\frac{1}{2}\int_0^t \Delta_{xx}F(X_{s})\varphi(s)^{2}ds.
0 \leq \varphi_1(p) - \lfloor \varphi_1(p) \rfloor= \varphi_1(p) -1 - \lfloor \varphi_1(p) - \psi(p) \rfloor & < \varphi_1(p) - 1 + 1 + \psi(p) - \varphi_1(p) = \psi(p).
{}_2 F_1\bigg(\frac{d-2}{2},\frac{d-1}{2};\frac{d}{2};\cos^2\bigg(\frac{\zeta}{2}\bigg)\bigg) + \frac{d-1}{d}\, \cos^2\bigg(\frac{\zeta}{2}\bigg) \,{}_2 F_1\bigg(\frac{d}{2},\frac{d+1}{2};\frac{d}{2}+1;\cos^2\bigg(\frac{\zeta}{2}\bigg)\bigg) = \sin^{-(d-1)}\bigg(\frac{\zeta}{2}\bigg).
(fh-f^{p+1}v_0)-\alpha fg =\sum_{j=1}^pf^{p-j+1}v_j\big(Q_j(\alpha)-\alpha j Q_{j-1}(\alpha)\big) &=(1-\alpha)\cdot\sum_{j=2}^p (j-1)Q_{j-1}(\alpha)f^{p-j+1}v_j.
\left|(Vu,\varphi_{\alpha}v)_{L^2(P;d\mu)}\right| \leq C\int_{S_{n_{\alpha},m_{\alpha}}}\left(1+\sum_{j\notin J_{\alpha}}|\ln x_j|\right) \left(\sum_{i=1}^{n_{\alpha}}x_i|u_{x_i}|+\sum_{l=1}^{m_{\alpha}}|u_{y_l}|\right) |\varphi_{\alpha}v|\, d\mu,
|\alpha-\frac{p_{i}}{q_{i}}|<\frac{1}{\sqrt{5}q_{i}^{2}},|\alpha-\frac{p_{i+1}}{q_{i+1}}|<\frac{1}{\sqrt{5}q_{i+1}^{2}},|\alpha-\frac{p_{i+2}}{q_{i+2}}|<\frac{1}{\sqrt{5}q_{i+2}^{2}}.
W ( k_{1}, n_{2}) = \frac{1}{4 \pi^{2}R_{n c}^{2}}\int_{0}^{2 \pi R_{n c}}d^{2}x \, \Omega ( x_{1}; x_{2}\to x_{2}+ 2 \pi R_{n c}n_{2}- 2 \pi R_{n c}\frac{c}{N}k_{1}) \star \exp ( i \frac{k_{1}x}{R_{n c}}) .
\begin{array}{l}{\cal O}[V]=V^{i_1\ldots i_s}\sum\limits_{k=0}^s{(-1)^k\over k!\left(k+{D-4\over 2}\right)! (s-k)!\left(s-k+{D-4\over 2}\right)!}\partial_{i_1}\cdots\partial_{i_k}\phi^*\;\partial_{i_{k+1}}\cdots\partial_{i_s}\phi, begin{align*}5pt]g_{i_1 i_2}V^{i_1\ldots i_s}=0\end{array}
-u Q(X)^{-1}v = - \begin{pmatrix}0 u_1\end{pmatrix}\begin{pmatrix}0 Q_2(X)^{-1}Q_1(X)^{-1}- Q_1(X)^{-1}v_1u_2 Q_2(X)^{-1}\end{pmatrix}\begin{pmatrix}0 v_2 \end{pmatrix}&= u_1 Q_1(X)^{-1}v_1 u_2 Q_2(X)^{-1}v_2 &= r_1(X) r_2(X).
\mu\phi(x) = (-\Delta_x + V_\mathrm{trap}(x))\phi(x) + (\upsilon_N\ast |\phi|^2)(x)\phi(x) + \frac{1}{N}(\upsilon_N\ast \rho^\mathrm{pair})(x)\phi(x) &+ \frac{1}{N}\int{dy\{\upsilon_N(x-y)n^\mathrm{pair}(x,y)\phi(y)\}}+ \frac{1}{N}\int{dy\{\upsilon_N(x-y)m^\mathrm{pair}(x,y)\overline{\phi(y)}\}}.
J_\alpha^0(s) = I(s,n) + 8 \Bigg( \sum_{m=0}^{N-1}n^{-2m}\sum_{j=0}^m W_{m,j}^1(s,n) + n^{-2N}R_N^1(s,n) \Bigg) + 4 \Bigg( \sum_{m=0}^{N'-1}n^{-2m}\sum_{j=0}^m W_{m,j}^1(s,n) + n^{-2N'}R_{N'}^2(s,n) \Bigg).
E_{C}( R , T = 0 , m ) = = - \frac{R}{\pi}\int_{0}^{\infty}\frac{\omega \, d \omega}{g ( \omega ) - 1}\left( 1 - \frac{2 m M_{0}^{2}}{R ( \omega^{2}m^{2}- M_{0}^{4})}\right) \,{,}
H^{\frac{n}{2}}(\mathbb{S}^n)=\{u\in L^{2}(\mathbb{S}^n): \; \|u\|_{H^{\frac{n}{2}}(\mathbb{S}^n)}^2:=\|u\|_{L^{2}(\mathbb{S}^n)}^2+\|(P_{g_0}^n)^{\frac{1}{2}}u \|_{L^{2}(\mathbb{S}^n)}^2<\infty\},
\sum_{l_1,l_2,\dots, l_r \atop{0 \leq l_i \leq L}}U(l_1)U(l_2)\dots U(l_r) \sum_{l_1',l_2',\dots, l_r' \atop{0 \leq l_i' \leq L}}U(l_1')U(l_2')\dots U(l_r') &\sum_{(s_1,s_2,\dots ,s_{t})}\sum_{(b_1,b_2,\dots,b_{t}) \atop{b_u \in \mathcal J_{\mathcal I(s_u)}}}\frac{D(b_1)D(b_2)\dots D(b_{t})}{s_1^{b_1}s_2^{b_2}\dots s_{t}^{b_{t}}}.
( S f ) ( y , x_{0}, \dots , x_{d - 1}) = \int \frac{\prod_{i = 0}^{d - 1}d p_{i}}{( 2 \pi y )^{d / 2}}f ( y , p_{0}, \dots , p_{d}) e^{\frac{i}{y}\sum_{i = 0}^{d - 1}p_{i}x_{i}}
Y_{i}(\lambda):=\frac{(e^{\lambda(v_{i})}-e^{\lambda(v_{i+1})})T_{i}-e^{\lambda(v_{i})}(\alpha\,e^{\lambda(v_{i+1})}+1-q)}{\alpha\,e^{\lambda(v_{i}+v_{i+1})}+e^{\lambda(v_{i})}-qe^{\lambda(v_{i+1})}}.
0 =\left[-\widetilde{\Delta}_ju+\frac{\partial^2u}{\partial\xi_{j+1}\partial\xi_j},\phi\otimes\mu_k\right] &=[u,-\widetilde{\Delta}_j\phi\otimes\mu_k-k_jk_{j+1}\phi\otimes\mu_k] &=[u,(-\widetilde{\Delta}_j\phi-k_jk_{j+1}\phi)\otimes\mu_k] &=[c_k,-\widetilde{\Delta}_j\phi-k_jk_{j+1}\phi] &=[-\widetilde{\Delta}_jc_k-k_jk_{j+1}c_k,\phi],
\langle N\rangle_T^{(d)}\sim -\frac{mT}{\pi}\frac{1}{d! (2T)^d}\,e^{-\frac{m}{T}}\,\int d^2 x\,\frac{q(\vec{x})}{|\vec{v}(\vec{x})|}\left[\left(\partial_{i}\hat{n}^a \,\partial_{i}\hat{n}^a+|\vec{v}(\vec{x})|\right)^{\frac{d}{2}}-\left(\partial_{i}\hat{n}^a \, \partial_{i}\hat{n}^a-|\vec{v}(\vec{x})|\right)^{\frac{d}{2}}\right]
(d f)(x_1\otimes x_2\otimes \dots\otimes x_{n+1}) = x_1f(x_2\otimes x_3\otimes \dots \otimes x_{n+1}) &+\sum_{i=1}^{n}(-1)^i f(x_1\otimes \dots\otimes x_ix_{i+1}\otimes \dots\otimes x_{n+1}) &+(-1)^{n+1}f(x_1\otimes x_2\otimes \dots \otimes x_{n})x_{n+1},
0 = \delta \psi_{\mu}= \partial_{\mu}\epsilon + \frac 1 4 \omega_{\mu}^{a b}\Gamma_{a b}\epsilon + \frac 1{2 8 8}e_{\mu}{}^{a}\left( \Gamma_{a}{}^{b c d e}- 8 \delta_{a}^{[ b}\Gamma^{c d e ]}\right) F_{b c d e}\epsilon \, .
d e t^{\prime}\Bigl ( \ast d_{\scriptscriptstyle 1}+ ( \ast d_{\scriptscriptstyle 1})^{\scriptscriptstyle 2}\Bigr ) = ( - 1 )^{\psi}d e t^{\prime}( \ast d_{\scriptscriptstyle 1}) d e t^{\prime}( \mathrm{1 \m k e r n - 5 m u I}+ \ast d_{\scriptscriptstyle 1})
\mathcal{T}_{V_{N,\beta,a}}\big(n_N(t),m_N(t)\big)\leq \int_{\mathcal{C}_N\times\mathcal{C}_N}V_{N,\beta,a}([s]_N,[s']_N)\omega_N(t,d[s]_N,d[s']_N)=:T_{V_{N,\beta,a}}(\omega_N),
\begin{array}{c}{\theta = B^{- 1}, \quad G = - ( 2 \pi l_{s}^{2})^{2}B g^{- 1}B , \quad G_{s}= g_{s}\det ( 2 \pi l_{s}^{2}B g^{- 1})^{\frac{1}{2}}.}\\ \end{array}
c) \U_{j}(N_{Z_{D}/S^{i}X}(\nu^{j}))=\left(1-\frac{1}{\nu^{j}}\right)^{A}\left(1-\frac{e^{-x}}{\nu^{j}}\right)^{-A}exp\left(\frac{\theta e^{-x}}{\nu^{j}-e^{-x}}\right)\cdot \left(\frac{1-e^{-x}/\nu^{j}}{1-\nu^{-j}}\right)^{-n_{j}},
\xi^{(m)}(t)=\Big\{x\in B_1\, :\, \Theta(\|V_t^{(m)}\|,x)=\frac32\Big\},\ \ \tilde\xi^{(m)}(t)=\Big\{\frac{x}{\mu^{(m)}}\ : \x\in \xi^{(m)}(t)\Big\}.
\sup_{g\in V,\Vert g\Vert\leq 1}|\Vert g\Vert_n^2-\Vert g\Vert^2|=\sup_{x\in \mathbb{R}^d,\Vert x\Vert_2 \leq 1}|x^T(B_n-I)x|=\Vert B_n-I\Vert_{\operatorname{op}},
\psi(\xi)\psi(\eta)-\psi(\eta)\psi(\xi) =\sum_{i,j}\xi(a_i)\eta(a_j)\varphi(b_i)\varphi(b_j)-\sum_{i,j}\eta(a_j)\xi(a_i)\varphi(b_j)\varphi(b_i) &=\sum_{i,j}\xi(a_i)\eta(a_j)\varphi([b_i,b_j]).
\begin{array}{c}\displaystyle(C \Gamma^{\mu_1\dots \mu_p}) = \epsilon (-1)^{(p-1)(p-2)/2}(C\Gamma^{\mu_1\dots \mu_p})^T \quad,\quad\mu=0,1,\dots,D-1 begin{align*}0.3cm]\displaystyle\epsilon = -\sqrt{2}\cos{\pi\over 4}(D+1)\quad.\end{array}
\lambda_1^2 = \sum_{y\sim x}\sum_{z\sim y}\mathbf{v}_z \leq \left(\sum_{uv\in E(G)}\mathbf{v}_u+\mathbf{v}_v\right) - \sum_{y\sim x}\mathbf{v}_y = \left(\sum_{uv\in E(G)}\mathbf{v}_u+\mathbf{v}_v\right) - \lambda_1.
\sum_{s_1\geq\dots\geq s_{r-1}\geq0}\frac{q^{s_1^2+\dots+s_{r-1}^2-s_1-\dots-s_i}}{(q)_{s_1-s_2}\dots(q)_{s_{r-2}-s_{r-1}}(q^2;q^2)_{s_{r-1}}}=\sum_{k=0}^{i}\frac{(q^{2r},q^{r-i+2k},q^{r+i-2k};q^{2r})_\infty}{(q)_\infty},
\ast_{\sigma\circ\tau}= \sigma\circ\tau\circ\ast\circ((\sigma\circ\tau)^{-1})^{\times k}&= \sigma\circ\tau\circ\ast\circ(\tau^{-1})^{\times k}\circ(\sigma^{-1})^{\times k}&= \sigma\circ\ast_{\tau}\circ(\sigma^{-1})^{\times k}&={(\ast_{\tau})}_{\sigma}\sigma,\tau\in S_n,
c_o(m,n;s) ={(-b/a;q)_m(qs/cd;q)_m\over (q;q)_m(-qs/ac;q)_m}{(s;q)_{m+n}(-qs/ac;q)_{m+n}(qs^2/a^2c^2;q)_{m+n}\over (q^2s^2/abcd;q)_{m+n}(q^{1/2}s/ac ;q)_{m+n}(-q^{1/2}s/ac ;q)_{m+n}}(q/b)^m &\times{(-d/c;q)_n (qs/ab;q)_n\over (q;q)_n (-qs/ac;q)_n}(q/d)^n.
\begin{array}{l}d_X(y\circ \alpha_n,\alpha_n)= E\min\{\|y\circ \alpha_n-\alpha_n\|_X; 1\}=E\min\{\|y\circ \alpha_n-x^*\circ\alpha_n\|_X; 1\}=E^n \min\{\|y-x^*\|_X; 1\}<\delta \ \ \ \mbox{for all}\ \ \n\ge n_2\end{array}
\chi (Z, D) = \chi (Z', D)+ \chi (E_i, D-Z') &= \frac{(2D-K_Y-Z') \cdot Z'}{2}+ \frac{(2(D-Z')-K_Y-E_i) \cdot E_i}{2}&=\frac{(2D-K_Y-Z) \cdot Z}{2}.
= l o g ( \rho_{0}) + l o g ( 1 + \frac{1}{N}\sum_{{\bf{q}}\neq 0}( e^{i{\bf{q}}. ({\bf{x}}-{\bf{x}}^{'})}- 1 ) \theta ( k_{f}- |{\bf{q}}| ) ) \approx l o g ( \rho_{0}) + \frac{1}{N}\sum_{{\bf{q}}\neq 0}( e^{i{\bf{q}}. ({\bf{x}}-{\bf{x}}^{'})}- 1 ) \theta ( k_{f}- |{\bf{q}}| )
\frac{\left[\left(\begin{array}{c}\psi 0\end{array}\right),\widetilde{A}\left(\begin{array}{c}\psi 0\end{array}\right)\right]_{{\cal K}_2}}{\left[\left(\begin{array}{c}\psi 0\end{array}\right),\left(\begin{array}{c}\psi 0\end{array}\right)\right]_{{\cal K}_2}}=\frac{\langle \psi\mid A\psi\rangle}{\langle\psi\mid\psi\rangle},
\widetilde{u}_{M}^{N}(x):=\frac{1}{N}\sum_{i=1}^N\left[ \phi_g(X^{x,i}_{M})+\frac{1}{d}\sum\limits_{k=1}^M \Pi\left(\Pi\left(\widetilde{r}(X^{x,i}_{k-1}),\widetilde{r}(X^{x,i}_{k-1})\right),\phi_f\left(X^{x,i}_{k-1}+\widetilde{r}(X^{x,i}_{k-1})Y^i\right)\right)\right], x\in D.
a^{( e )}( v ) = a_{g r a v}^{( s )}( v ) + \frac{8}{7}\,_{2}F_{1}( 4 , 5 ; 9 ; 1 - v ) - \frac{1 6}{2 1}\,_{2}F_{1}( 4 , 4 ; 8 ; 1 - v ) .
\begin{cases}X^{\delta}(0)=(x_0,Z_{0}) dx^{\delta}(t) = b_{i^{\delta}(t)}(t,x^{\delta}(t))dt + \sigma_{i^{\delta}(t)}(t,x^{\delta}(t))dW(t) + \delta d(\displaystyle\sum_{0\leq s\leq t}\mathbf{1}_{\{x^{\delta}(s-)=0\}}) i^{\delta}(t)=Z_{\displaystyle\sum_{0\leq s\leq t}\mathbf{1}_{\{x^{\delta}(s-)=0\}}}\end{cases},~~\mathbb{P}
\frac{[-q^{l+j},q^{2l};q^{6l}]_\infty(q^{6l};q^{6l})^2_\infty}{[-q^{j},q^{l},-q^{2l+j};q^{6l}]_\infty}=\sum_{n=-\infty}^\infty \frac{(-1)^nq^{3ln(n+1)+ln+l}}{1+q^{6ln+j+2l}}+\sum_{n=-\infty}^\infty \frac{(-1)^nq^{3ln(n+1)-ln}}{1+q^{6ln+j}}
\delta H(t,\Delta)+\frac{1}{2}P(t)\delta\sigma(t,\Delta)^{2}& =\mathcal{H}(t,\bar{X}(t),\bar{Y}(t),\bar{Z}(t),u(t),p(t),q(t),P(t))-\mathcal{H}(t,\bar{X}(t),\bar{Y}(t),\bar{Z}(t),\bar{u}(t),p(t),q(t),P(t)).
\begin{aligned}\mathsf{T}_{2}(t_{k}) =\frac{\phi\lambda-\sum_{i=1}^{n}\beta_{i}u_{i}(t_{k})}{\sum_{i=1}^{n}\alpha_{i}}, u_i(t_{k+1}) =\beta_{i}u_i(t_{k})+\alpha_{i}\mathsf{T}_{2}(t_{k}).\end{aligned}
S_{m,n}(i,k)=\sum_{\rho,\sigma,\mu,\nu}\frac{(-m)_{\rho+\mu}(-n)_{\nu+\sigma}(-i)_{\rho+\sigma}(-k)_{\mu+\nu}}{\rho!\sigma!\mu!\nu!}\,u_{11}^{\rho}u_{12}^{\sigma}u_{21}^{\mu}u_{22}^{\nu},
\left( \begin{array}{c}a_{1},\ldots, a_{m}b_{1}, \ldots, b_{n}\end{array}; z \right)_{\infty}= \prod_{n=1}^{\infty}\frac{(a_{1}; z)_{\infty}\cdots (a_{m}; z)_{\infty}}{(b_{1}; z)_{\infty}\cdots (b_{n}; z)_{\infty}}, (a; z)_{\infty}= \prod_{n=0}^{\infty}(1 - az^{n}).
\int_{0}^{\infty}e^{-s x}F_1^n(x)dx=\frac{1}{s}\left(\frac{\mathbb E\left[e^{-s \overline{X}^n_{e(q)}}\right]}{\mathbb E\left[e^{-s \overline{X}^n_{e(p+q)}}\right]}-1\right),\ \s > 0.
\sum_{k=1}^2 \lambda_k \langle B_k(u_k),(u_k-\overline{u}_k)_+\rangle_{\mathcal{H}_k}\leq \sum_{k=1}^2 \langle A_k(\overline{u}_k)-A_k(u_k),(u_k-\overline{u}_k)_+\rangle_{\mathcal{H}_k}\leq 0.
\big\langle \mu_{\mathbf{x}^N}^{\otimes k}, \varphi_k \big\rangle = \frac{1}{N^k}\sum_{i_1, \ldots , i_k}\varphi_k{\left( x^{i_1}, \ldots , x^{i_k}\right)}&= \frac{1}{N^k}\sum_{\substack{i_1, \ldots , i_k}}\varphi_k{\left( x^{i_1}, \ldots , x^{i_k}\right)}+ R_{k,N},
\tau_{k,0}=\max\Big\{\min\Big\{\frac{\|\Delta z^k\|^2}{\langle\Delta z^k,\Delta\zeta^k\rangle}, \frac{\langle\Delta z^k,\Delta\zeta^k\rangle}{\|\Delta\zeta^k\|^2},{\tau}_{\rm max}\Big\},{\tau}_{\rm min}\Big\},
J_{2,4}=6T_{xy^2}^2+T_{xy}^2T_{y^2}-3T_{xy}^2T_{y}^2-6T_{y^3}T_{x^2y}+2T_{y^2}T_{x^2y}T_{y}&+4T_{y^3}T_{xy}T_{x}-2T_{y^2}T_{xy}T_{x}T_{y}+2T_{xy}T_{y}^3T_{x}-4T_{xy^2}T_{y^2}T_{x}-T_{y^2}^2T_{x^2}&+T_{y^2}^2T_{x}^2+4T_{y^2}T_{y}^2T_{x^2}-T_{y^2}T_{y}^2T_{x}^2-T_{y}^4T_{x^2}-2T_{y^3}T_{y}T_{x^2}.
\Vert y - (R(a)\, \cdot + c(a))\Vert^2_{L^2(Q^s(b))}\le Cm^2\Big(\sum\nolimits^m_{j=2}E_{j-1,j}\Big)^2 \le Cm^3\sum\nolimits^m_{j=2}\big(E_{j-1,j}\big)^2.
\begin{pmatrix}q_1 b_1 0 0 \cdots begin{align*}1mm] b_1 q_2 b_2 0 \cdots begin{align*}1mm] 0 b_2 q_3 b_3 0 0 b_3 q_4 \ddots \vdots \vdots \ddots \ddots \end{pmatrix}\,.
\sum_{\sigma\in \mathfrak{S}_{k}}{\sigma}\det\begin{bmatrix}1 z_{1}z_{1}^{2}\dots z_{1}^{k-1}z_{2}z_{2}^{2}z_{2}^{3}\dots z_{2}^{k}z_{3}^{2}z_{3}^{3}z_{3}^{4}\dots z_{3}^{k+1}\vdots \vdots \vdots \ddots \vdots z_{k}^{k-1}z_{k}^{k}z_{k}^{k+1}\dots z_{k}^{2k-2}\end{bmatrix}.
\underset{i}{\lim}\ (\Gamma^{A,B}(u_i)(X)\Gamma^{B,C}(v_j)(Y)Z) =(\Gamma^{A,B}(u)(X)\Gamma^{B,C}(v_j)(Y)Z) & = (\Gamma^{B,C}(v_j)(Y)Z\Gamma^{A,B}(u)(X))
\lambda\,\mathbb E \bigl[(\xi_i-\eta_i)^3 \bigr] =6\tilde\lambda\tilde\mu_1^3 \biggl(1-\frac{\mu_2}{\mu_1}+\frac{\mu_2^2}{\mu_1^2}-\frac{\mu_2^3}{\mu_1^3}\biggr).
\sum\limits_{b \in B(G, \mu)}\mathrm{Mant}_{b, \mu}( \mathrm{Red}_b( \pi)) = [ \pi ] [ r_{-\mu}\circ \mathrm{LL}( \pi)|_{W_{E_{\{\mu\}_G}}}\otimes | \cdot |^{\mathrm{- \sum_{\tau}p_{\tau}q_{\tau}/2}}].
\psi_j^l(x) = e^{\zeta_1^{k_l,t_{k_l}}\cdot x}+ \int_{\partial \Omega}G(x-y,\zeta_1^{k_l,t_{k_l}})(\Lambda_{q_j}- \Lambda_0)\psi_j^l(y) d\sigma (y),x \in \partial \Omega,
\mathrm{R}_{\digamma,N}(\mathrm{Z},\mathrm{A};\alpha) = \int_{\digamma}e^{-(A-zId)\rho(g)}\,\rho(g^{\alpha})\;d\nu(g_N)=(A-zId)^{-\alpha}\;,\;\;\;\;\;\;\;\;\alpha\in\langle0,\infty\rangle
- \sum_{1 \leq \nu \leq \mu}\binom{\mu}{\nu}\langle \partial_t^{\mu+1}v, \partial_t^\nu B_\lambda \partial_t^{\mu-\nu +1}v \rangle \leq C Z_0 \sum_{1 \leq \nu \leq \mu}\|\partial_t^\nu B_\lambda\|_\infty \leq C \lambda^2 Z_0,
|I^{1}_{2}| \leq C_{n}\Bigg(\begin{array}{cc}\ln(t/\varepsilon) \;n=2 begin{align*}.3cm]\varepsilon^{-\frac{n-2}{2}}\;n\geq 3\end{array}\Bigg)\times\sup_{0\leq s\leq t-\varepsilon}\|f_{+}(s)\|_{1}=: C_{n}\Phi(\varepsilon,t)\sup_{0\leq s\leq t-\varepsilon}\|f(s)\|_{1},
\Phi^{\dagger}\Psi \Psi^{\dagger}\Phi = \mathrm{T r}\left( \Psi^{\dagger}\Phi \Phi^{\dagger}\Psi \right) = \mathrm{T r}\left( \Psi \Psi^{\dagger}\Phi \Phi^{\dagger}\right) = \phi^{2}\left| \vec{\psi}_{1}\right|^{2}
A_\perp^{ic}(x)=\sum_\lambda\int_{k^+>0}{dk^+ d^2k_\perp \over 16\pi^3 k^+}\Bigl[a^c(k,\lambda) \epsilon_\perp^i(\lambda) e^{-ik\cdot x}+a^{c\dagger}(k,\lambda) \epsilon_\perp^{i*}(\lambda) e^{ik\cdot x}\Bigr] \;.
4i^2 (dn)^2 - 40 i^2 dn - 4 i (dn)^2 - 16 i^2 d^3 n - 8 i^2 \Delta d^2 n &= 4i^2(dn)^2 - 2i(dn)^3\left( 20 i/(d^2n^2) + 2/(dn) + 8i/n^2 + 4i\Delta/(dn^2)\right) &\geq 4i^2(dn)^2 - 2i(dn)^3\left( 13 \Delta^2/n^2 + 2/n + 6 d\Delta/n^2 \right).
\delta^N(t)\leq \, C \int_0^t dt_1 \int_0^{t_1}dt_2\left[ N^{3\alpha+2}\delta^N(t_2)+N^{3\alpha}e^{-\frac{\lambda}{2}N^2}\right]\leq & C \int_0^t dt_1 \int_0^{t_1}dt_2\left[ N^{3\alpha+2}\delta^N(t_2)+e^{-\frac{\lambda}{4}N^2}\right].
A^{0}( x ) = ( 2 \pi )^{- 3 / 2}\! \! \! \int \! \! \frac{d^{3}k}{k^{0}\sqrt{2}}i \left( a^{0}({\mathbf{k}}) e^{- i k x}+ a^{0}({\mathbf{k}})^{\dagger}e^{i k x}\right) . \nonumber \,
\psi_{1 \xi}^{\dagger}( \epsilon ) \psi_{2 \eta}( - \epsilon ) = i e^{\frac{i}{2}[ - \theta ( \eta ) - \theta ( \xi ) + \frac{1}{2}R_{2}( \epsilon , \xi , \eta ) ]}e^{- i \pi \int d^{2}y \frac{1}{e}\partial_{i}E_{i}( y )}e^{i F_{2}( A , E )}
\frac{\d L_h}{\d t}=-2\sum\limits_{j=1}^N\Big(\frac{4\sin^2\big(\frac{\alpha_j}{2}\big)}{q_j+q_{j+1}}-\frac{\pi}{L_h}\sin\alpha_j \Big)\le\frac{-2}{L_h}\Big(2 \big(\sum_{j=1}^N\sin\big( \frac{\alpha_j}{2}\big)\big)^{2}-\pi \sum_{j=1}^N\sin\alpha_j\Big)\le 0,
\mu(\sigma) \ll \begin{cases}10\bigr( \mu(3/5)-\mu(1/2) \bigl) \sigma + \bigl( 6\mu(1/2) - 5\mu(3/5) \bigr), \sigma\in[1/2, 3/5], 5\mu(3/5)(1-\sigma)/2 , \sigma\in[3/5,1-\eta], \end{cases}
T_c =\lim\limits_{\lambda_s^{}\to 1}p_{sd}\left(1-\pi_s(0) \right)+p_{sr}\left(1-\pi_s(0) \right) (1-\pi_r(B_r)) =p_{sd}\cdot \left(1-0 \right)+p_{sr}\cdot \left(1-0 \right) \cdot \left(1-\frac{n-2}{n-2+B_r}\right) & =p_{sd}+p_{sr}\frac{B_r}{n-2+B_r}.
\;(\|u_*\|_{L^2}^2+\alpha^2 \|A^{1/2}u_*\|_{L^2}^2)(t) &\;+2\nu\int_0^t(\|A^{s/2}u_*\|^2_{L^2}+\alpha^2\|A^{(1+s)/2}u_*\|^2_{L^2})(\tau)\,d\tau \leq \;\|u_0\|_{L^2}^2+\alpha^2 \|A^{1/2}u_0\|_{L^2}^2.
\begin{array}{c c c c}{{\mathrm{l o n g ~ r o o t s :}}}&{{\pm \, 2 \! e_{k}}}&{{}}&{{( 1 \leq k \leq 3 )}}\\{{\mathrm{s h o r t ~ r o o t s :}}}&{{\pm \, e_{k}\pm e_{l}}}&{{}}&{{( 1 \leq k < l \leq 3 )}}\\ \end{array}~ .
\sum_{b\ge 0}x^b\sum_{j=0}^a(-1)^j\binom{a}{j}\binom{2j}{b}=[1-(1+x)^2]^a=(-1)^a x^a(2+x)^a, \sum_{b\ge 0}x^b\sum_{j=0}^a(-1)^j\binom{a}{j}\binom{2j+1}{b}=(-1)^a\bigl(x^a+x^{a+1}\bigr)(2+x)^a.
\int_{\Gamma^t_{\theta}}\frac{(N\cdot \nu^t_{\theta})}{\frac{\partial\sigma}{\partial s}}(x) f(x) d\mathcal{H}^{n-1}(x) = \int_{Z}f(z,\theta\lambda^t(z)) \bar J(z,\theta\lambda^t(z)) dz.
D_1 :=\Lambda_1+\Phi_1+M_3, D_2 :=\Lambda_2, D_3 :=Q_1'+\Phi+N_3, D_{g,\i}:=N_1+N_2, D_{g,-\i}:=M_2, D_{g_1g, \i}:=0, D_{g_1g, -\i}:=M_1.
\frac{1}{2}\frac{\partial \phi}{\partial \nu}= u \frac{\partial u}{\partial \nu}+D^2u(\nu, Du) =\varphi(f+D^2u(\nu, \nu))+ D^2 u(\nu, \bar{\nabla}f) =\varphi(f+D^2u(\nu, \nu))+ \nabla_{\bar{\nabla}f}\nabla u \cdot \nu = \varphi(f+D^2u(\nu, \nu))+ \bar{\nabla}f\cdot \bar{\nabla}\varphi- h(\bar{\nabla}f, \bar{\nabla}f).
\partial_tv+v\cdot\nabla v+h(u_s\cdot\nabla v+v\cdot\nabla u_s)=\Delta v-\nabla p_v-hu_\infty\cdot\nabla v+g, &\mbox{div $v$}=0, &v|_{\partial\Omega}=0, &v\to 0 \quad\mbox{as $|x|\to\infty$}, &v(\cdot,0)=v_0:=u_0-h(0)u_s
g_{h,\mathrm{1c}}(x\lambda \partial_x + \omega\partial_y) = \lambda \frac{dx}{h^2x^5}+\frac{g_1(\omega\partial_y)}{hx^2}& = h^{-1/2}x^{-1}(\hat{\lambda}\frac{dx}{hx^3}+ \frac{g_1(\omega\partial_y)}{h^{1/2}x}).
\widehat{V}^{\dagger}( k_{1}, k_{2}) = e^{i \sqrt{2 | \theta^{1 2}|}\; \left( \kappa \hat{a}+ \overline{{{\kappa}}}\hat{a}^{\dagger}\right)}= e^{- | \theta^{1 2}| \kappa \overline{{{\kappa}}}}\, e^{i \sqrt{2 | \theta^{1 2}|}\; \overline{{{\kappa}}}\hat{a}^{\dagger}}\, e^{i \sqrt{2 | \theta^{1 2}|}\; \kappa \hat{a}}~ ,