harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
M = \pm \; \left( \begin{array}{l l l}{{\lambda}}&{{\quad}}&{{0}}\\{{\; 0}}&{{\quad}}&{{\lambda^{- 1}}}\\ \end{array}\right) \ \ \mathrm{o r}\M^{\prime}= \pm \; \left( \begin{array}{l l l}{{\; 0}}&{{\quad}}&{{\lambda^{- 1}}}\\{{\lambda}}&{{\quad}}&{{0}}\\ \end{array}\right) \ ,
\begin{array}{r c l}{{H^{( - 1 )}( \lambda ) [ \psi ]}}&{{=}}&{{\int_{x_{-}}^{x_{+}}d x \, \psi ( \lambda )^{2}\, ,}}\\{{H^{( - 2 )}( \lambda ) [ \psi ]}}&{{=}}&{{\int_{x_{-}}^{x_{+}}d x \, \psi ( \lambda ) \, \bar{\psi}( \lambda ) \, ,}}\\{{H^{( - 3 )}( \lambda ) [ \psi ]}}&{{=}}&{{\int_{x_{-}}^{x_{+}}d x \, \bar{\psi}( \lambda )^{2}\, . \}}\\ \end{array}
\sum_{n\geq1}\nu_{n}^{v}\left( \sum_{i=1}^{n}X\left( s_{i}\right)=z-y|s_{b},s_{f}\right) =\frac{\Theta_{q,p}\left( \hat{x}\right)}{\sqrt{\left( 2\pi\left\Vert x\right\Vert \right)^{d-1}}}F\left(s_{b}\right) F\left( s_{f}\right) \left( 1+o\left( 1\right) \right) \ ,
g_m(t)=\begin{dcases}e^{-t}\sum_{j=0}^m(m)_{j+1}t^{m-j}, m\geq0, e^{-t}\sum_{j=1}^{-m-1}\frac{t^{m+j}}{(-m-1)_{j}}-\frac{\gamma+\log(t)+\sum_{j=0}^{+\infty}(-1)^j\frac{t^j}{j\cdot j!}}{(-m-1)!}, m<0.\end{dcases}
Q=\sum_\sigma \int \frac{d^3{\bf p}}{(2\pi)^3}\, \left [a_\sigma^\dagger ({\bf p}) a_\sigma ({\bf p}) -b_\sigma ({\bf p}) b_\sigma^\dagger ({\bf p}) +c_\sigma^\dagger ({\bf p}) c_\sigma ({\bf p}) -d_\sigma ({\bf p}) d_\sigma^\dagger ({\bf p})\right ]\quad,
\gamma_F(-tJ_2 \psi) \cdot \gamma_F(tJ \psi) = \gamma_F(-2tJ_2, \frac{1}{2}\psi) \cdot \gamma_F(2tJ, \frac{1}{2}\psi) \cdot \gamma_F(\frac{1}{2}\psi)^2 = \gamma_F(-J_1, \frac{1}{2}\psi) \cdot (-2tJ_2, 2tJ)_F \cdot \gamma_F(-1, \frac{1}{2}\psi)^{-1}= \gamma_F(J_1, \frac{1}{2}\psi) \cdot (2tJ_2, J_1)_F.
\mbox{For}x_0 \notin N,\tilde{N}= N \cup \{x_0\}, \quad \tilde{h_0}= h_0 \cup\{x_0\}, \mbox{For}\{x_1, ..., x_n\}\subseteq N \mbox{such that}\quad \forall j=1,...,n \mbox{and}\forall y\in N, x_j\leq y. \quad\mbox{Then}\tilde{E}= E \cup \{(x_1, x_0), ..., (x_n, x_0) \}.
\begin{array}{ll}\delta^{S}_{(\xi)}M^{\mu\nu}={\cal L}_{\xi}( \sqrt{g}g^{\mu\nu}), \delta^{S}_{(\xi)}L^{\mu\nu}={\cal L}_{\xi}\lbrack M^{\mu\nu}- s ( \sqrt{g}g^{\mu\nu}) \rbrack.\end{array}
\begin{array}{r c l}{{\displaystyle \left[ a_{q}, a_{q}^{\dagger}\right]}}&{{=}}&{{\left[ N + 1 + b \gamma \right]_{q}- \left[ N + b \gamma \right]_{q}~ ,}}\\{{\displaystyle \left[ N , a_{q}\right]}}&{{=}}&{{- a_{q}~ , \displaystyle \left[ N , a_{q}^{\dagger}\right] = a_{q}^{\dagger}~ .}}\\ \end{array}
|\Sigma_k(a_1,\cdots,a_k)| =\sum_{i_1,\cdots,i_{k}\in\Lambda(I)\}\prod_{j=1}^k(\lambda_{i_j+1}-\lambda_{i_j}-a_j)_+ =\sum_{i_1,\cdots,i_{k}\in\Lambda(I)\}\prod_{j=1}^k({m}^*_{i_j}-a_j)_+.
S ={\frac{k A_{H}}{4 \ell_{P}^{2}}}+ 4 \pi{\frac{k}{\hbar}}\int_{H}\;{\frac{\partial{\cal L}}{\partial R_{\mu \nu \lambda \rho}}}\; g_{\mu \lambda}^{\perp}g_{\nu \rho}^{\perp}\; \sqrt{{}_{2}g}d^{2}x .
w=\frac{1}{f\rho_{o}}[-(\frac{\partial\tau^{x}}{\partial x}+\frac{\partial\tau^{y}}{\partial y})e^{z/d}sin(z/d)+(\frac{\partial\tau^{y}}{\partial x}-\frac{\partial\tau^{x}}{\partial y})(1-e^{z/d}cos(z/d))].
u(t,x;a,\nu)=\frac{\nu}{a}\,\ln <1>\equiv \frac{\nu}{a}\,\ln\left\{\frac{1}{\sqrt{4\pi\nu t}}\int\limits_{-\infty}^{\infty}\,dy\, \exp \left(-\frac{(x-y)^2}{4 \nu t}+ \frac{a f(y)}{\nu}\right) \right\}
\left( \Gamma^{\mu}(i\partial_{\mu}\,+\, eA_{\mu}) \,-\, d\left(\begin{array}{cc}0 \gamma^{\mu}\gamma^{\nu}\gamma_{5}\gamma^{\mu}\gamma^{\nu}\gamma_{5}0\end{array}\right)F_{\mu \nu}\,-\, m\right) \psi = 0
[\begin{matrix}\tilde{K}&B^{T}+S_{12}^{T}\\ \tilde{B}&S_{22}\end{matrix}][\begin{matrix}U^{n+1}\\P^{n+1}\end{matrix}]=[\begin{matrix}F^{n}+\frac{1}{2\delta t}M(4U^{n}-U^{n-1})+F_{v}\\F_{q}\end{matrix}],
\xi^{-\mu}I_{\mu+n}(\xi) \leqq \frac{\xi^n}{2^{\mu+n}}\sum_{n=0}^\infty \frac{(\xi/2)^{2n}}{(\Gamma(m+1))^2\Gamma(\mu+n+1)}\leqq \frac{\xi^n}{2^{\mu+n}\Gamma(\mu+n+1)}\Bigl( \sum_{m=0}^\infty \frac{(\xi/2)^m}{m!}\Bigr)^2,
L \Longrightarrow\Sigma_{0}(l_{1,}l_{2},l_{3})=0\mbox{(cf. (\ref{vanishSigma0}))}L \Longrightarrow\Sigma_{1}(l_{1},l_{2},l_{3})=\frac{L}{2}\Sigma_{0}(l_{1,}l_{2},l_{3})
I= \frac{1}{2\pi i}\int_{\alpha-i\infty}^{\alpha+i \infty}\frac{L(g \otimes g,s + \beta)x^s}{s \prod_{r=2}^{10}(s+r)}ds + \frac{L(^2g,1)x^{1-\beta}}{(1-\beta)\prod_{r=2}^{10}(1-\beta+r)}&+\frac{L(g \otimes g,\beta)}{10!}-\frac{L(g \otimes g,-2+\beta)x^{-2}}{2 \cdot 8!}.
\begin{array}{@{}l@{}}{\displaystyle H = \mathbb{C}^{N}, \langle \vec{e}_{1}, \vec{e}_{2}\rangle = (\vec{e}_{2})^{\ast}W\vec{e}_{1}, \forall \vec{e}_{1}, \; \vec{e}_{2}\in H,}begin{align*}2mm]{\displaystyle g(\vec{e}) = \vec{e}-\vec{c}+{\rm i}\frac{\tau}{4}W^{-1}S\vec{e}+{\rm i}\frac{\tau}{2}W^{-1}Q(\vec{e})\vec{e}, \forall \vec{e}\in H,}\end{array}
\begin{gathered}\sigma(t)=qt, \mu(t)=(q-1)t, \eta^\Delta(t)=\frac{\eta(qt)-\eta(t)}{(q-1)t}, \int_{a}^{b}\eta(t)\Delta t=(q-1)\sum_{t=\log_{q}{a}}^{\log_{q}{b}-1}q^t \eta(q^t).\end{gathered}
\sum_{i=1}^{n}\frac{\partial f}{\partial\kappa_{i}}\kappa_{i}^{2}=\frac{1}{k}H_{k}^{\frac{1-k}{k}}\sum_{i=1}^{n}\frac{\partial H_{k}}{\partial\kappa_{i}}\kappa_{i}^{2}&=\frac{n}{k}H_{1}H_{k}^{\frac{1}{k}}-\frac{n-k}{k}H_{k}^{\frac{1-k}{k}}H_{k+1}.
\frac{\check{w}_n(\xi)}{w_n(\xi)}=\Pi_{l=1}^{s_\ell}\left(-\bar z_{l,\ell}\right)\,\xi^{-N}\,\frac{\check{u}_{j,\ell}(\xi)}{u_{j,\ell}(\xi)}\,\frac{\widetilde{Q}_{N-s_\ell,\ell}(\xi)}{Q_{N-s_\ell,\ell}(\xi)},
= [ X_{\beta_{m-k+1}}, X_{- \beta_{m-k+1}}] v_{k-1}+ \sum_{l=1}^{k-1}(-)^{k-l}X_{\beta_{m-k+1}}\cdots X_{\beta_{m-l}}[ X_{\beta_{m-l+1}}, X_{- \beta_{m-k+1}}] v_{l-1}
{j^W}^{(L)}(x) = \left( \begin{array}{l}\theta(x_1 + L) \theta(L-x_1) \; [\delta(x_2 - L) - \delta(x_2 + L) ] \theta(x_2 + L) \theta(L-x_2) \; [\delta(x_1 + L) - \delta(x_1 - L) ]\end{array}\right) \; .
M_{0, \varphi}(x) = 6 i \left(\begin{array}{cc}\frac{\sin^2 \frac{\varphi x}{2}}{x^2 + 16}- \frac{2 \sin\,{\varphi x}}{(x + 2 i)(x - 4 i)}\frac{2 \sin\,{\varphi x}}{(x - 2 i)(x + 4 i)}- \frac{\sin^2 \frac{\varphi x}{2}}{x^2 + 4}\end{array}\right).
\sum_{l = 1}^{n}\mathbf{S}^{\top}_{ki}\mathbf{S}^{*}_{jk}= \sum_{l = 1}^{n}\mathbf{S}^{\top}_{li}\mathbf{S}^{*}_{jl}= \left(\mathbf{S}_t\mathbf{S}^{\dagger}\right)_{i,j}= \left(I\right)_{i,j}= 0, \; \forall i \neq j ,
\phi_{A/R}\circ \chi^{\otimes n}\colon F_{ij}\mapsto \begin{cases}f(a_i,a_j) i<j Q(a_i)=f(a_i,a_i) i=j B(a_i,a_j)-f(a_j,a_i)=f(a_i,a_j) i>j. \end{cases}
\widetilde D_g (r a_n) = r\pi_h h^{n-1}\sum_{k=1}^n{n \choose k}(gh^{-1})^{(k-1)}y^{n-k}= r\pi_h (gh^{-1})^{(n-1)}h^{n-1}+ \sum_{k=1}^{n-1}{n \choose k}(gh^{-1})^{(k-1)}h^k r \pi_h h^{n-k-1}y^{n-k}&= r \pi_h (gh^{-1})^{(n-1)}h^{n-1}+ \sum_{k=1}^{n-1}{n \choose k}(gh^{-1})^{(k-1)}h^k r a_{n-k}.
\varphi:U \to A x \mapsto\begin{cases}\varphi_1(x) x\in U_1 \varphi_2(x) x\in U_2\setminus U_1 \vdots \vdots \varphi_n(x) x\in U_n\cap U_1^c\cap\cdots \cap U_{n-1}^c 0 x\in U_1^c\cap\cdots\cap U_n^c \end{cases}
V_g(M_\eta T_\xi f)(x, \omega) = \langle M_\eta T_\xi f, M_\omega T_x g \rangle & = \langle f, T_{-\xi}M_{-\eta}M_\omega T_x g \rangle & = \langle f, e^{2 \pi i \xi \cdot (-\eta + \omega)}M_{\omega-\eta}T_{x-\xi}g \rangle & = e^{-2 \pi i \xi \cdot (\omega - \eta)}V_g f(x-\xi,\omega-\eta)
\dot{A}_{1}=(\beta+\frac{1}{2}b_{x}(1))A_{1}-\frac{1}{2}b_{xx}(1)B_{1}=(\beta+\frac{1}{2}b_{x}(1))A_{1}+H(b_x(1)+Hb(1)) B_{1}, \dot{B}_{1}=(\beta-\frac{1}{2}b_{x}(1))B_{1}+b(1)A_{1}=(\beta-\frac{1}{2}b_{x}(1))B_{1}-\frac{1}{H^2}(Hb_{x}(1)+\frac{1}{2}b_{xx}(1))A_{1}.
\mathtt{g}\cdot\mathtt{h}=g_1^{s_1}\cdots g_k^{s_k}h_1^{t_1}\cdots h_l^{t_l}, \bar{\mathtt{g}}=g_k^{-s_k}\cdots g_1^{-s_1}, \mathtt{g}^{(\lambda)}=g_1^{\lambda s_1}\cdots g_k^{\lambda s_k}.
T_{\mu \nu}= e^{- 2 \phi}[ F_{\mu \rho}F_{\nu}{}^{\rho}-{}^{\star}F_{\mu \rho}{}^{\star}F_{\nu}{}^{\rho}] - 2 [ \nabla_{\mu}\phi \nabla_{\nu}\phi -{\textstyle \frac{1}{2}}g_{\mu \nu}( \nabla \phi )^{2}] ,
\tilde{y}_{s}=\zeta +\int_{s}^{T}\tilde{f}\left( s,\tilde{y}_{s},\tilde{z}_{s},\mathbb{P}_{\left( \tilde{y}_{s},\tilde{z}_{s}\right)}\right) \mathrm{d}s-\int_{s}^{T}\tilde{z}_{s}\mathrm{d}W_{s}^{t_{0}}.
\begin{array}{r c l}{\frac{\partial^{n}}{\partial \lambda^{n}}\psi}{=}{( - 1 )^{n}\, n ! \, L ( \lambda )^{- n}\, \psi \, ,}{\frac{\partial^{n}}{\partial \lambda^{n}}\bar{\psi}}{=}{( - 1 )^{n}\, n ! \, L ( \lambda )^{- n}\, \bar{\psi}\, , \quad n \geq 1 \, ,}\end{array}
\mathit{\Omega}_{\mathfrak{m}}\left( c_{+}\right) \doteq \left\{\omega \in \mathit{\Omega}_{\mathfrak{m}\left( c_{+}\right)}^{\sharp}:e_{(\cdot)}\left( \omega \right) =c_{+}\mathfrak{a}_{+}\right\}\subseteq E_{1}\ ,
\begin{array}{r c l}{{\mathrm{i n t e g r a l}}}&{{=}}&{{\displaystyle - 2 m^{2}( 4 \pi )^{- 2}\int_{0}^{1}d x \int \frac{d^{4}k_{1}}{( 2 \pi )^{4}}D_{m}^{2}( k_{1}) \overline{{{D ( k_{1}+ p )}}}K_{0}\left( \sqrt{x m^{2}\tilde{k_{1}}^{2}}\right) .}}\\ \end{array}
\begin{aligned}K_{F,4}(\tau) \coloneqq 2\left(\vphantom{\frac{C_{F,3}(T_0)}{e^3\tau}}A_Fe\tau\log{(e^2\tau)}+|B_F|e\tau \right. & \left.+\frac{C_{F,1}(T_0)\log{(e^2\tau)}+C_{F,2}(T_0)}{3}+\frac{C_{F,3}(T_0)}{9e\tau}\right).\end{aligned}
\frac{1}{\pi}\int e^{-2 |w|^2 + aw + b\overline{w}+ cw^2 + d\overline{w^2}}P(w,\overline{w})\,dL(w) = P(\partial_a,\partial_b) \frac{1}{2 \sqrt{1-cd}}e^{\frac{da^2+cb^2+2ab}{4(1-cd)}}.
e^{it\sqrt{\mathcal{L}_V}}u_0 =\sum_{\nu\in\chi_\infty}\sum_{\ell=1}^{d(\nu)}\varphi_{\nu,\ell}(y)\int_0^\infty(r\rho)^{-\frac{n-2}2}J_{\nu}(r\rho)e^{it\rho}b_{\nu,\ell}(\rho)\rho^{n-1}d\rho &=\sum_{\nu\in\chi_\infty}\sum_{\ell=1}^{d(\nu)}\varphi_{\nu,\ell}(y)\mathcal{H}_{\nu}\big[e^{it\rho}b_{\nu,\ell}(\rho)\big](r).
\mathcal{L}\left(\boldsymbol{x},\boldsymbol{\lambda}\right) =\sum_{t\in \mathcal{T}}C^{t}\left(x^{t}_{0}\right)+\sum_{t\in \mathcal{T}}\lambda^{t}\left(\sum_{i\in \mathcal{I}\setminus 0}x^{t}_{i}- x^{t}_{0}\right),
\mathrm{DTE}_{(p,q)}(X)=\mu+\sigma\frac{\phi(\xi_{p})(1+\sqrt{2\pi}\phi(\xi_{q}))-\phi(\xi_{q})(1+\sqrt{2\pi}\phi(\xi_{p}))}{\Psi_{2}^{\ast}(-1,\frac{1}{2},1)F_{Y}(\xi_{p},\xi_{q})(1+\sqrt{2\pi}\phi(\xi_{p}))(1+\sqrt{2\pi}\phi(\xi_{q}))},
\frac{{{d^2}{{{\cal L}}\left( \omega,D \right)}}}{{d{\omega^2}}}= \frac{{2\left({{B_1}+{C_1}}\right)}}{{{\omega^3}{{\bar \gamma}_2}}}+ \frac{{2\left({{B_2}+{C_2}}\right)}}{{{{\left({1 - \omega}\right)}^3}{{\bar \gamma}_1}}}> 0
\int_{0}^{b}\! g(s)\mkern+1.5mu df(s) , \lim_{v \downarrow 0}f(v) g(v) = 0, \lim_{v \downarrow 0}\frac{1}{f(v)g(v)}\! \int_{0}^{v}\! g(s)\mkern+1.5mu df(s) = \frac{\alpha}{1+\alpha}.
\begin{array}{c}\forall p \in \mathbb{C}^n, ~ \exists ! z \in \mathbb{C}^n: ~ Mz + N \bar{z}= p begin{align*}.2cm] \Longleftrightarrow begin{align*}.2cm] \begin{pmatrix}M N begin{align*}.1cm]\overline{N}\overline{M}\end{pmatrix}~\end{array}
A^{2}= 1 \quad \mathrm{f o r}\quad D_{2 n}^{( 1 )}\to A_{2 n - 1}^{( 2 )}, \quad D_{n + 2}^{( 1 )}\to D_{n + 1}^{( 2 )}, \quad E_{7}^{( 1 )}\to E_{6}^{( 2 )},
Z(\beta)=\frac{1}{2^N}\sum_{\vec{n}}\mathrm{e}^{-\beta \,E_{\vec{n}}}= \prod_j \cosh \left ( \tfrac{1}{2}\beta e_j \right )=\exp \left [ \sum_j \ln \cosh \left ( \tfrac{1}{2}\beta e_j \right )\right ]\ .
\begin{cases}\frac{\partial\phi_{i}}{\partial t}\left(z,t\right)+H\left(t,\nabla_{z}\phi_{i}\left(z,t\right)\right)=0 \,\mathbb{R}^{n}\times\left(0,+\infty\right), \phi_{i}\left(z,0\right)=J_{i}\left(z\right) \forall z\in\mathbb{R}^{n},\end{cases}
J_1(k)=J(k)T_6=kM_1^{-1}\begin{bmatrix}I_\mu+o(1) o(1) o(1) \frac{1}{k}(I_{n-\mu}+o(1))\end{bmatrix}T_1^{-1}T_6, k\to0,\;\;k\in \overline{\mathbb{C}}^+.
\int{\cal D}[ \phi_{f a s t}] e^{- ( S_{0}( f a s t ) + \delta S [ \Phi_{s l o w}, \Phi_{f a s t}] )}\equiv \langle e^{- \delta S [ \Phi_{s l o w}, \Phi_{f a s t}]}\rangle_{f}= e^{- S_{s l o w}^{\prime}}
E = \mathrm{T r}\; \sqrt{\det ( \delta_{\mu \nu}+ 2 \pi \alpha^{\prime}F_{\mu \nu})}= 2 \sqrt{( 1 + 4 \pi^{2}\alpha^{2}F_{1 2}^{2}) ( 1 + 4 \pi^{2}\alpha^{2}F_{3 4}^{2})}= 2 ( 1 + 1 6 \pi^{4}\alpha^{2})
\frac{\partial^{2}S_{n}}{\partial t_{1}^{+2}}=-\frac{S_{n}}{1-S_{n}^{2}}(\frac{\partial S_{n}}{\partial t_{1}^{+}})^{2}-\frac{1}{t_{1}^{+}}\frac{\partial S_{n}}{\partial t_{1}^{+}}+\frac{(n+1)^{2}}{t_{1}^{+2}}\frac{S_{n}}{1-S_{n}^{2}}-4S_{n}(1-S_{n}^{2}).
F^1_{n,\underline{k}}(g)(x):=C^{n+1}\int_{\mathbb{R}^{n+2}}g(x-\rho)\prod_{l=1}^{n+1}P'_{k_l}h(x-\rho-y_l)\cdot \widetilde{K}_{k_{n+1}}(y_{n+1},\rho)\prod_{l=1}^{n}K_{k_l}(y_l,\rho)\,d\rho d\underline{y},
\vec{A_{1}}=\left( \begin{array}{c}x 0 0 \end{array}\right) \;\;\;\;\vec{A_{2}}=\left( \begin{array}{c}0 x 0 \end{array}\right) \;\;\;\;\vec{\phi}=\left( \begin{array}{c}-x x 0 \end{array}\right) \; \;\;\;\dot{\vec{\phi}}=\frac{1}{2}\left(\begin{array}{c}x x 0 \end{array}\right) .
{\bar \alpha}^{A}{\bar \pi}_{A}=\pm{\sqrt E}e^{{i\varphi \over 2}}e^{-i\psi}\sin{\theta \over 2},\qquad \qquad\alpha^{B^\prime}\pi_{B^\prime}=\pm{\sqrt E}e^{-{i\varphi \over 2}}e^{i\psi}\sin{\theta \over 2},
\phi'_{e_n}(\lambda P') = \phi(\lambda P' + \overline{e_n}) = \phi(\lambda^{\frac{k-1}k}P' + \lambda^{\frac{k-1}k}\overline{e_n}) = \lambda^{k-1}\phi(P' + \overline{e_n}) = \lambda^{k-1}\phi'_{e_n}(P') ,
\begin{array}{l c l}{{\tilde{\mathcal L}_{- 2}^{( - 3 , - 1 )}}}&{{=}}&{{{v_{1}\star p}^{- 3}+ v_{0}\star p^{- 2}+ v_{- 1}\star p^{- 1}}}\\{{}}&{{=}}&{{v_{- 1}{p}^{- 1}+ ( v_{0}+ \theta v_{- 1}^{\prime}) p^{- 2}+ ( v_{1}+ 2 \theta v_{0}^{\prime}+ \theta^{2}u_{- 1}^{\prime \prime}) p^{- 3}}}\\ \end{array}
\frac{\overline w_{2p}'(t)}{\overline w_{2p}(t)}\,=\, B_{2p}\alpha^2 - \frac{p-1}{2p}\frac{V'(t)}{V(t)}~, \hbox{and}\Bigl(\frac{\overline w_{2p}(t)}{\overline w_p(t)}\Bigr)^{\beta p}\,=\, \Bigl(\frac{A_{2p}}{A_p}\Bigr)^{\beta p}\,\frac{e^{\beta p(B_{2p}-B_p)\alpha^2 t}}{V(t)^{\beta/2}}~.
\lambda = (4,4,4,4,3,3,2,2,2), \mu = (2,0,0,0,0,0,0,0,0) w = (2, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 16, 18, 20, 22, 23, 25, 27, 28).
\begin{array}{lll}X_{i}=\textstyle{\frac{\mu}{3}}J_{i}\,,~~~ ~~~[J_{i},J_{j}]=i\epsilon_{ijk}J_{k}\,,~~~ ~~~A_{0}=0\,,{}{}{}Z_{1}(t)=e^{-i\frac{\mu}{6}t}z_{1}{1}\,,~~~ ~~~Z_{2}(t)=e^{-i\frac{\mu}{6}t}z_{2}{1}\,,~~~ ~~~Z_{3}(t)=e^{-i\frac{\mu}{6}t}z_{3}{1}\,,\end{array}
\begin{array}{l c r}{( | X_{1}|^{2}- | X_{2}|^{2}) - ( | Y_{1}|^{2}- | Y_{2}|^{2}) = R_{2}}\\{X_{1}\overline{{Y}}_{1}- X_{2}\overline{{Y}}_{2}= \overline{{X}}_{1}Y_{1}- \overline{{X}}_{2}Y_{2}= 0 .}\\ \end{array}
\exp\left(\int\limits_0^\infty\frac{dx}{x}\frac{e^{-bx}(1-e^{-cx})(1-e^{-dx})}{(1-e^{-x})(1-e^{-x\tau})}(e^{xq}-1)\right) = \frac{G(b\,|\,\tau)}{G(b-q\,|\,\tau)}\frac{G(b-q+c\,|\,\tau)}{G(b+c\,|\,\tau)}\times &\times \frac{G(b-q+d\,|\,\tau)}{G(b+d\,|\,\tau)}\frac{G(b+c+d\,|\,\tau)}{G(b-q+c+d\,|\,\tau)}.
\begin{array}{cccc}\sigma : \Gamma \to \operatorname{SL}_2(\mathbb{F}_\ell) \otimes \operatorname{SO}_3(\mathbb{F}_\ell) & \gamma=c+di \mapsto \sigma_{\gamma}:=\left(\begin{matrix}a 0 0 a^{-1}\end{matrix}\right) \otimes \left(\begin{matrix}c d 0 -d c 0 0 0 1 \end{matrix}\right),\end{array}
+ \nabla_{m}( \pi^{i j}N^{m}) - \pi^{j m}\nabla_{m}N^{i}- \pi^{i m}\nabla_{m}N^{j}+ N \sum_{n}\delta^{2}( x - q_{n}) \frac{P_{n a}P_{n b}g^{a i}g^{b j}}{2 \sqrt{m_{n}^{2}+ P_{n a}P_{n b}g^{a b}}}
\rho\partial_{t}u-\nu\Delta u+\rho u\cdot\nabla u+\nabla p=0,{\rm in}\ \Omega, &\partial_{t}\rho+u\cdot\nabla \rho=0,{\rm in}\ \Omega, &\nabla\cdot u=0,{\rm in}\ \Omega, &u(0,x)=u_0(x),\rho(0,x)=\rho_0(x),{\rm in}\ \Omega,
\bar{\Omega}:= \mathbb{B}\times \mathbb{W}_0,\,\bar{\mathcal{F}}^{\omega}:= \overline{\mathcal{B}(\mathbb{B})\otimes \mathcal{B}(\mathbb{W}_0)}^{\mathbb{P}_{\omega}},\,\bar{\mathcal{F}}^{\omega}_t := \bigcap_{\epsilon > 0}\sigma\big(\mathcal{B}_{t+\epsilon}(\mathbb{B})\otimes \mathcal{B}_{t+\epsilon}(\mathbb{W}_0),\mathcal{N}_{\omega}\big),
f^{-1}(a_if(x)) =g^{-1}(p_i g(x)+q_i), g\circ f^{-1}(a_iy) =p_ig\circ f^{-1}(y)+q_i, m(a_iy) =p_im(y)+q_i \textrm{, where}m(y)=g\circ f^{-1}(y).
\Theta (\mathbf{x}\vert v)=\vartheta_{3}\left( \dfrac{\pi x_{1}}{2a}\right. \left| i\dfrac{\pi v}{a^{2}}\right)\ldots \vartheta_{3}\left( \dfrac{\pi x_{n}}{2a}\right. \left| i\dfrac{\pi v}{a^{2}}\right) \ \ , \ \mathbf{x}=\left( x_{1},\ldots x_{n}\right) \in \mathbb{R}^{n}\ \ , \v>0
q\cdot N_d =\sum_{x,y,z\in\mathbb{F}_q}\theta(zE_d(x,y)) &=q^2+\sum_{z\in\mathbb{F}_{q}^{\times}}\theta(zb)+\sum_{y,z\in\mathbb{F}_{q}^{\times}}\theta(bz)\theta(-zy^2)+\sum_{x,z\in\mathbb{F}_{q}^{\times}}\theta(zx^d)\theta(zax)\theta(zb) &+\sum_{x,y,z\in\mathbb{F}_{q}^{\times}}\theta(x^dz)\theta(axz)\theta(bz)\theta(-zy^2) &=q^2+A+B+C+D.
\phi(x,y):=\begin{pmatrix}-1+(1+2\tau)(1+x) \frac{1-|M|^{1/2}+x}{x|M|^{1/2}}y + \frac{1+x}{|M|^{1/2}}\phi_2 \left(-1+|M|^{1/2},\frac{-1+|M|^{1/2}}{x}y\right)\end{pmatrix}.
C^*_2(g,u) = \Psi\left( \frac{u}{\sigma_g}\right), C^*_1(g,u) = - \frac{\sqrt{2\pi \lambda}}{4\, \sigma_g}\Psi'\left ( \frac{u}{\sigma_g}\right ), C^*_0(g,u) = \frac{\lambda}{2\pi\, \sigma_g^2}\Psi^{''}\left( \frac{u}{\sigma_g}\right).
dX^i(t) =f^i(t,X_t)dt+ \sum_{j=1}^m g^{i,j}(t,X_t)dW^j (t)+ \sum_{j=1}^n\int_{\mathbb{R}_0}h^{i,j}(t,X_t,z)\tilde N^j(dt,dz), X^i_0 =\eta^i \, , i = 1, \dots, d \, .
\hat{\Pi}^{\underline{{{m}}}}= d \hat{X}^{\underline{{{m}}}}- i d \hat{\Theta}^{1}{\sigma}^{\underline{{{m}}}}\hat{\Theta}^{1}- i d \hat{\Theta}^{2}\tilde{\sigma}^{\underline{{{m}}}}\hat{\Theta}^{2}= d \xi^{m}\hat{\Pi}_{m}^{~ \underline{{{m}}}}( \xi )
{}\sum_{n=1}^{\infty}\tfrac{1}{n^{1+\delta}}\int_{T}^{2T}\left({\tfrac{t}{2\pi}}\right)^{{1}/{4}+{\delta}/{2}+{it}/{2}}e^{-{it}/{2}-it \log n}dt{}= \sum_{n=1}^{\infty}\tfrac{1}{n^{1+\delta}}\int_{T}^{2T}\left({\tfrac{t}{2\pi}}\right)^{{1}/{4}+{\delta}/{2}}e^{\frac{it}{2}\log ({t}/{2\pi en^{2})}}dt.
c(I):=(\beta/q \alpha)^{d}q^{\sum_{p=1}^{d}i_{p}}, \widehat{Q}_{I}:=\widehat{X}_{l-d+1}\cdots \widehat{X}_{l}(\widehat{T}_{l-d}\cdots \widehat{T}_{\nu_{i_{1}}})\cdots (\widehat{T}_{l-1}\cdots \widehat{T}_{\nu_{i_{d}}}).
\begin{Vmatrix}1_{S^n}-{\frac 1{(2k)^l}}\sum_{s\in{\hat S}^l}H_t(x_0s,x)\end{Vmatrix}_{L^2}\leq{\frac{r^{\frac n2}}3}\sqrt{\frac{\Gamma\begin{pmatrix}{\frac{n+1}2}\end{pmatrix}}{\Gamma\begin{pmatrix}{\frac n2}+1\end{pmatrix}}}
\sum_{n=1}^{\infty}\frac{(-1)^{n}}{(x^{2}+n^{2})^{s}}=-\frac{x^{-2s}}{2}+\frac{2^{\frac{3}{2}-s}\pi^{s}x^{\frac{1}{2}-s}}{\Gamma(s)}\sum_{n=1}^{\infty}\left(2n-1\right)^{s-\frac{1}{2}}\,K_{s-\frac{1}{2}}\left(\pi(2n-1)x\right).
\hat\tau=(1+O(\varepsilon))(x-x_*)^4 + (1+O(\varepsilon))(y-y_*)^2+\cr (4x_*+O(\varepsilon))(x-x_*)^3+(6x_*^2-b_2+O(\varepsilon))(x-x_*)^2+const+O_{x,y}(1)\varepsilon.
|X| \leq |\tilde{J}(9,4) \cup X^{(i)}\cup X^{(iii)}|+t+\mathfrak{M}_{6-t}+t \binom{9-t}{2}&= \frac{t^3}{3}-\frac{9 t^2}{2}+\frac{31 t}{6}+257\leq 258,
\mathbf{H}=\begin{pmatrix}H_{1, 1}H_{1, 2}\dots H_{1, n}H_{2, 1}H_{2, 2}\dots H_{2, n}\vdots \vdots \vdots \vdots H_{r, 1}H_{r, 2}\dots H_{r, n}\end{pmatrix}\in \mathbb{B}^{r\ell \times n\ell}.
\left(\begin{array}{cc}L_{_{1}}^{(L) \;2}+L_{_{2}}^{(L) \;2}+\alpha^2L_{_{3}}^{(L) \;2}+\frac{3}{4}\alpha^2 +\alpha^2 L_{_{3}}^{(L)}\alpha(L_{_{1}}^{(L)}-iL_{_{2}}^{(L)}) \alpha(L_{_{1}}^{(L)}+iL_{_{2}}^{(L)}) L_{_{1}}^{(L) \;2}+L_{_{2}}^{(L) \;2}+\alpha^2L_{_{3}}^{(L) \;2}+\frac{3}{4}\alpha^2 -\alpha^2 L_{_{3}}^{(L)}\end{array}\right),
- \delta J_{n}( z_{1}) = \frac{2}{\gamma^{2}}\, \int_{\tau}^{\tau + 2 \pi}\, \, \langle ( g^{- 1}( z ) \, \delta g ( z ) )^{\prime}\, g^{- 1}( z ) \, \{\, J_{n}( z_{1}) , g ( z ) \, \}\rangle \, d z ,
\partial_s \Sigma(t,s) = - \frac{1+t L_1(s)}{L_1(s)}\left[ -L_0'(s) + kt\ \frac{L_1'(s)}{1+t L_1(s)}\right] + \frac{L_1'(s)}{L_1(s)^2}\ \left( 1 - L_0(s) + k \ln{(1+t L_1(s))}\right) = \frac{L_0'(s)}{L_1(s)}+ t L_0'(s) - kt \frac{L_1'(s)}{L_1(s)}- \frac{s L_1'(s)}{L_1(s)}+ k\ \frac{L_1'(s)}{L_1(s)^2}\ln{(1+t L_1(s))}\ ,
\Delta Y_t =\int_{t}^{T}(f(s,\tilde{Z}_s)-f(s,Z_s))ds-\int_{t}^{T}\Delta Z_sdW_s &=\int_{t}^{T}(b_s\Delta Z_s)ds-\int_{t}^{T}\Delta Z_sdW_s=-\int_{t}^{T}\Delta Z_sd\tilde{W}_s
H = \frac{\sqrt{\alpha^{\prime}}}{2}g^{\mu \nu}( P_{\mu}- Q_{_{E}}{\cal A}_{\mu}- Q_{_{M}}{\tilde{\cal A}}_{\mu}) ( P_{\nu}- Q_{_{E}}{\cal A}_{\nu}- Q_{_{M}}{\tilde{\cal A}}_{\nu}) + \frac{\sqrt{\alpha^{\prime}}}{2 k^{2}}({Q_{_{E}}}^{2}+{Q_{_{M}}}^{2}) ,
\Big ( \frac{\theta_{2}( z | \tau )}{\theta_{1}( z | \tau )}\Big )^{2}= \Big ( \frac{\theta_{2}( 0 | \tau )}{\theta_{1}^{\prime}( 0 | \tau )}\Big )^{2}({\cal P}( z | \tau ) - e_{1}( \tau ) ) ,
A_{(ij)k}=\frac{1}{2}\left(A_{ijk}+A_{jik}\right)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,A_{[ij]k}=\frac{1}{2}\left(A_{ijk}-A_{jik}\right)
u=\frac{dx}{d\tau}=\frac{dx^{0}}{d\tau}+\frac{d}{d\tau}(x^{1}e_{1}+x^{2}e_{2}+x^{3}e_{3})=\frac{dx^{0}}{d\tau}[1+\frac{d}{dx^{0}}(x^{1}e_{1}+x^{2}e_{2}+x^{3}e_{3})].
S_{e f f}= \int \left( - \frac{1}{4}\sum_{n}( \partial_{\mu}b_{\rho \sigma}^{n}\partial^{\mu}b^{n , \rho \sigma}+{m_{n}^{2}}b_{\rho \sigma}^{n}b^{n , \rho \sigma}) - \frac{1}{2}\partial_{\mu}\phi \partial^{\mu}\phi \right) d x .
X=x^1\left(\frac{\partial v^2}{\partial x^2}({\bf x})+\frac{\partial v^3}{\partial x^3}({\bf x})\right)\frac{\partial}{\partial x^1}+v^2({\bf x})\frac{\partial}{\partial x^2}+v^3({\bf x})\frac{\partial}{\partial x^3},
\Im\frac{1}{1-\eta_{\mu_{1}}(re^{if(r)})}=\Im(1+\psi_{\mu_{1}}(re^{if(r)}))=\int_{\mathbb{R}_{+}}\frac{tr\sin(f(r))}{|1-tre^{if(r)}|^{2}}\,d\mu_{1}(t).
\frac{\tau_{w}}{\rho U^{2}}=\frac{1}{U^{2}}\frac{\partial}{\partial t}(U\delta_{1})+\frac{\partial\delta_{2}}{\partial x}+\frac{2\delta_{2}+\delta_{1}}{U}\frac{\partial U}{\partial x}+\frac{v_{w}}{U}
p(\mathbf{y}\vert\mathbf{U},\boldsymbol{\theta}) \approx\frac{1}{M}\sum_{m=1}^{M}p(\mathbf{y}\vert\mathbf{X}^{(m)},\boldsymbol{\theta}) =\frac{1}{M}\sum_{m=1}^{M}\mathcal{N}(\mathbf{y};\mathbf{\mathbb{\boldsymbol{\mu}}}(\mathbf{X}^{(m)}),\mathbf{K}^{(m)}),
\begin{cases}-\Delta v_1^+ \le - a_{1j}(v_1^+)^{p}(v_j^+)^{p+1}-\Delta v_j^+ \le -a_{1j}(v_j^+)^{p}(v_1^+)^{p+1}\end{cases}\begin{cases}-\Delta v_1^+ \le -a_{1j}(v_1^+)^{p}(v_j^-)^{p+1}-\Delta v_j^- \le -a_{1j}(v_j^-)^{p}(v_1^+)^{p+1}\end{cases}
\Lambda(t) = \sum_{j=1}^d \Lambda^j(t) := \sum_{j=1}^d \Bigl[ \delta\sigma^j(\bar{X}_t,\bar{u}_t) (y^{\varepsilon}_t)^T + y^{\varepsilon}_t \left(\delta \sigma^j(\bar{X}_t,\bar{u}_t) \right)^T \Bigr].
\frac{L(1,\pi_{\infty},\mathrm{Ad})L(1,\tau_{\infty}, \mathrm{Ad})}{L(1/2,\pi_{\infty}\times \mathrm{Ad}(g))}= \frac{2(2\pi)^{-k-1}\Gamma(k+1)\pi^{-1}\Gamma(1)\cdot 2(2\pi)^{-2k}\Gamma(2k)\pi^{-1}\Gamma(1)}{2^2(2\pi)^{-2k-1}\Gamma(2k)\Gamma(k-k+1)\cdot 2(2\pi)^{-k}\Gamma(k)}= \frac{k}{2\pi^2},
- g^{2}m : u ( x ) \Phi ( x ) [ \partial_{\mu}\phi ( x ) , \phi ( y ) ] A_{\nu}( y ) A^{\nu}( y ) : = i g^{2}m : u ( x ) \Phi ( x ) A_{\nu}( y ) A^{\nu}( y ) : \partial^{\mu}D_{m_{H}}( x - y ) .
(\phi_{2^{k_j}}- \phi_{2^{k_{j-1}}})(u) (\phi_{2^{k_j}}- \phi_{2^{k_{j-1}}})(v) = \big( \phi_{2^{k_{j-1}}}(u)\phi_{2^{k_{j-1}}}(v) - \phi_{2^{k_{j}}}(u)\phi_{2^{k_{j}}}(v)\big ) & + \phi_{2^{k_{j}}}(u) (\phi_{2^{k_{j}}}- \phi_{2^{k_{j-1}}})(v) & + (\phi_{2^{k_{j}}}-\phi_{2^{k_{j-1}}})(u)\phi_{2^{k_{j}}}(v) .
f^{even}_1(z_0,n)=\begin{cases}f_1(z_0,n) n>0 f_1(z_0,-n) n<0\end{cases}\intertext{and}f^{odd}_1(z_0,n)=\begin{cases}f_1(z_0,n) n>0 -f_1(z_0,-n) n<0\end{cases}
(\Phi^h)^m = i^m(-1)^{\frac{m(m-1)}{2}}m!\:h\:\mathcal{Z}^1\wedge\dots\wedge\mathcal{Z}^m\wedge\mathcal{Z}^{\bar{1}}\wedge\dots\wedge\mathcal{Z}^{\bar{m}}, (\Phi^v)^m = i^m(-1)^{\frac{m(m-1)}{2}}m!\:h\:\delta\mathcal{V}^1\wedge\dots\wedge\delta\mathcal{V}^m\wedge\delta\mathcal{V}^{\bar{1}}\wedge\dots\wedge\delta\mathcal{V}^{\bar{m}},
\delta_{\ell k}1(\ell>n_1)+B(\ell,k) &=-\frac 1{(2\pi i)^4}\int_{\Gamma_{d_2}}dz\int_{\Gamma_{d_4}}dw\int_{\gamma_{\tau_1}}d\zeta\int_{\gamma_{\tau_2}}d\omega\frac{G_{n_1+1,\mu_1,\xi_1}(z)G_{\Delta n-1,\Delta\mu,\Delta\xi}(w)}{G_{k,\mu_1,\xi_1}(\zeta)G_{n_2+1-\ell,\Delta\mu,\Delta\xi}(\omega)}\frac 1{(z-w)(z-\zeta)(w-\omega)}.
\varphi^{00}_{n,p}= \frac{\mu_n^{(12)}+\mu^{(1)}-\mu^{(2)}}{2\mu_n^{(12)}}\,\rho^{00}_{n,p}- \frac{1}{2\mu_n^{(12)}}\big( (\mu_n^{(12)}-\mu^{(1)}-\mu^{(2)}-\mu^{(0)})\mu_n^{(12)}+ \mu^{(0)}(\mu^{(1)}-\mu^{(2)}) \big)
\| \partial^{\alpha}f D^{s,\alpha}g \|_2 = \| \partial^{\alpha}D^{\gamma-1}\nabla^{\perp}\theta \cdot D^{s,\alpha}\nabla \theta\|_2 \lesssim \| \partial^{\alpha}D^{\gamma-1}\nabla^{\perp}\theta \|_{\infty}\cdot \| D^{s,\alpha}\nabla \theta\|_2 \lesssim \| \theta \|_{H^s}\cdot \| \theta \|_{H^s}.
[r_0,r_1, \ldots, r_{(k+1)n+1}] - [r_0,r_1, \ldots, r_{n+1}] < C \sum_{i= k n}^{(k+1)n}{1 \over{\rm Fib}_{i}\{\rm Fib}_{i+1}}< C \theta^{2 k n}, \left|[r_{(k+1) n-1}, r_{(k+1) n-2}, \ldots r_0]-[r_{k n-1}, r_{k n-2}, \ldots r_0] \right| <{1 \over{\rm Fib}_n \{\rm Fib}_{n+1}}< C \theta^{2 k n},