harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
H_{g o n i m e t r i c}^{2 d}= - k \sum_{\vec{r}, \vec{\alpha}}\sigma_{\vec{r}}\sigma_{\vec{r}+ \vec{\alpha}}+ \frac{k}{2}\sum_{\vec{r}, \vec{\alpha}, \vec{\beta}}\sigma_{\vec{r}}\sigma_{\vec{r}+ \vec{\alpha}+ \vec{\beta}}- \frac{1 - k}{2}\sum_{\vec{r}, \vec{\alpha}, \vec{\beta}}\sigma_{\vec{r}}\sigma_{\vec{r}+ \vec{\alpha}}\sigma_{\vec{r}+ \vec{\alpha}+ \vec{\beta}}\sigma_{\vec{r}+ \vec{\beta}}
\begin{array}{rl}= \{\strut \displaystyle -c_{1}\over\displaystyle c_{1}+c_{0}(s-1)-{\strut \displaystyle 2 c_{0}c_{2}(s-1) \over\displaystyle 2 c_{2}+c_{1}(s+1)-{\strut \displaystyle 4 c_{1}c_{3}(s+1) \over\displaystyle 4 c_{3}+c_{2}(s+3)}}}& \ddots\end{array}
{\cal A}= h^{( \alpha )}d y^{0}\wedge d y^{1}\wedge . . . \wedge d y^{q - 1}\wedge [ d z_{1}^{1}\wedge . . . \wedge d z_{1}^{r}H_{1}^{- 1}+ . . . + d z_{n}^{1}\wedge . . . \wedge d z_{n}^{r}H_{n}^{- 1}]
{\mathcal{L}}_{OB,m,i}^{(\alpha_k^*)}(x) = \frac{{\xi_{m,k,i}^{(\alpha_k^*)}}}{{x - z_{m,k,i}^{(\alpha_k^*)}}}/\sum\limits_{j = 0}^m{\frac{{\xi_{m,k,j}^{(\alpha_k^*)}}}{{x - z_{m,k,j}^{(\alpha_k^*)}}}},i = 0, \ldots ,m;\,k = 0, \ldots ,n,
\begin{array}{r c l}{\widetilde \phi_{\mathrm{f i x e d}}( u )}{=}{\widehat \phi_{\mathrm{f r e e}}( u / \sqrt{2}) \ ,}{\widetilde \phi_{\mathrm{f r e e}}( x )}{=}{\widehat \phi_{\mathrm{f i x e d}}( \sqrt{2}x ) \ .}\end{array}
\begin{aligned}\tilde{W}_1 = X_1 M_1^2 X_1^t - Y_1 M_1^2 Y_1^t+ i 2 X_1 M_1^2 Y_1^t &= (X_1 + i Y_1) M_1^2 (X_1^t + i Y_1^t),\end{aligned}
{\cal V}( x_{0}, \theta_{0}, \bar{\beta}, \bar{\zeta}, \bar{\omega}, \rho ) = e^{i P x_{0}}e^{- i Q \theta_{0}}e^{- i \bar{S}\bar{\beta}}e^{- i \bar{Q}\bar{\zeta}}e^{i \bar{M}\bar{\omega}}e^{i D \ln \rho}.
\begin{aligned}P1: \sum_{i=0}^{M-1}\mathcal{A}(\mathbf{c}_i^e)(\tau)=0, |\tau| \in \left(\mathcal{T}_{1}\cup \mathcal{T}_{2}\right)\cap \mathcal{T}; & P2: \sum_{i=0}^{M-1}\mathcal{C}(\mathbf{c}_i^e, \mathbf{c}_i^{e'})(\tau)=0, |\tau| \in\{0\}\cup \mathcal{T}_{1}\cup \mathcal{T}_{2}; & \end{aligned}
\int\limits_{a+\pi^e O_v}\chi(ac(x))^N|x|^{sN+n-1}\dx =\begin{cases}\frac{(1-q^{-1})(q^{-en-eNs})}{(1-q^{-n-Ns})}a \in \pi^eO_v, \chi^N=\chi_{triv}q^{-e}\chi(ac(a))^N|a|^{sN+n-1}a \notin \pi^eO_v, \chi^N|_{1+\pi^ea^{-1}O_v}=\chi_{triv}0 \textit{all other cases}.\end{cases}
H_{1}= p_{x}^{2}/ 2 \gamma + \gamma ( \omega_{1}^{2}+ \omega_{2}^{2}) x^{2}/ 2 - \gamma \omega_{1}^{2}\omega_{2}^{2}q^{2}/ 2 + \lambda x + u_{1}( p_{q}- \lambda ) + u_{2}p_{\lambda}.
\big\|\mathcal{N}(U^{1}) - \mathcal{N}(U^{2})\big\|_{\mathcal{H}_{\kappa}}=\Big(\rho \int_{\Omega}\big|E(|v^{1}|) v^{1}- E(|v^{2}|) v^{2}\big|^{2}\, \mathrm{d}x\Big)^{1/2}&\leq L \sqrt{\rho}\|v^{1}- v^{2}\|_{L^{2}}\leq L \big\|U^{1}- U^{2}\|_{\mathcal{H}_{\kappa}}
\frac{1}{\sqrt{- \overline{{g}}}}\partial_{\mu}[ \pm \sqrt{- \overline{{g}}}\enspace \overline{{g}}^{\mu \gamma}\overline{{g}}^{\nu \delta}\tilde{F}_{\gamma \delta}] - \frac{\tilde{e}^{2}}{8 \omega}\varphi^{2}\overline{{g}}^{\alpha \nu}\tilde{A}_{\alpha}= 0
[\![\phi_h]\!]_{i+\frac{1}{2},v}:= \left(\phi_h\right)^+_{i+\frac{1}{2},v}- \left(\phi_h\right)^-_{i+\frac{1}{2},v}\quad\forall \, \phi_h \in \mathcal{Z}_h, \{\phi_h\}_{i+\frac{1}{2},v}:= \frac{1}{2}\left(\left(\phi_h\right)^+_{i+\frac{1}{2},v}+ \left(\phi_h\right)^-_{i+\frac{1}{2},v}\right) \quad\forall \, \phi_h \in \mathcal{Z}_h.
\bar{\cal R}^{\frac{1}{2}z \omega , \frac{1}{2}z \omega}\sim \left( \begin{array}{c c c c}{t}&{0}&{0}&{0}\\{0}&{s^{- 1}}&{t - t^{- 1}}&{0}\\{0}&{0}&{s}&{0}\\{0}&{0}&{0}&{- t^{- 1}}\\ \end{array}\right) .
K = 6 d C - \frac{\beta}{\sqrt{2}}\frac{\bar{\kappa}^{2}}{\lambda^{2}}\delta ( x^{1 1}) d x^{1 1}\wedge I_{3}+ \frac{1}{2 \sqrt{2}}( 1 - \beta ) \frac{\bar{\kappa}^{2}}{\lambda^{2}}\epsilon ( x^{1 1}) I_{4}
\mathrm{H}=\Big\{\mathfrak{g}_0\sigma_n\mathfrak{g}_0^{-1}\,\mathfrak{g}_1\sigma_n\mathfrak{g}_1^{-1}\,\ldots\,\mathfrak{g}_\ell\sigma_n\mathfrak{g}_\ell^{-1}\,\vert\,\mathfrak{g}_i\in\mathrm{PGL}(n+1;\mathbb{C}),\,\ell\in\mathbb{N}\Big\}.
\begin{cases}\forall s \mbox{a.e.}\in [0,t], \: \forall q \geq 0, \: (\dot z_\omega^{x,t}(s)-\omega(s))(q - z_\omega^{x,t}(s)) \leq 0 z_{\omega}^{x,t}(t) = x, \end{cases}
\sigma \partial_t \eta - \emph{div}\, \boldsymbol{\tau}+ \lambda^{-1}\zeta = 0, \boldsymbol{\tau}= \nu \nabla \eta, \sigma \partial_t \zeta - \emph{div}\, \boldsymbol{\rho}+ \eta = y_d, \boldsymbol{\rho}= - \nu \nabla \zeta.
\hat f(r) = \frac\pi{\sinh(\pi r)}\int_0^\infty \! \frac{J_{2ir}(\xi) - J_{-2ir}(\xi)}{2i}f(\xi) \, \frac{d\xi}\xi, \check f(r) = \frac4\pi \cosh(\pi r) \int_0^\infty \! K_{2ir}(\xi) f(\xi) \, \frac{d\xi}\xi, \tilde f(\ell) = \int_0^\infty \! J_\ell(\xi) f(\xi) \, \frac{d\xi}\xi.
\sum_{\ell =0}^L s^{-\frac{\beta-\gamma \mu_2}{\mu_2+1}\ell}= \frac{1-s^{-\frac{\beta-\gamma \mu_2}{\mu_2+1}L}}{1-s^{-\frac{\beta-\gamma \mu_2}{\mu_2+1}}}< \frac{1}{1-s^{-\frac{\beta-\gamma \mu_2}{\mu_2+1}}}.
A_\Pi(Y_{I_w}^* \cdot \zeta_{I_v}) =\delta_{w,v}\mathbf{1}, \quad \mathbf{1}=\sum_u f_u,A_\Pi (\sum_v p_vf_v)=(\sum_{v}\tfrac{p_v}{v(x_\Pi)})\cdot\mathbf{1}.
+{\cal C}^{1\mu}S_{\mu\nu}\bar{\cal P}^\nu_2 +\frac{1}{M^2}{\cal C}^{2\mu}P_\mu P^\nu\bar{\cal P}_{1\nu}+\frac{1}{M^2}{\cal C}^{2\mu}P_\mu S^{\nu\sigma}P_\sigma\bar{\cal P}_{2\nu}-{\cal C}'P'\bar{\cal P}'.
\mathbb{H}_p^{-m}((S,T)\times \mathbb{R}^d)=(1-\Delta)^{\frac{m}{2}}L_p((S,T)\times \mathbb{R}^d), &\|f\|_{\mathbb{H}_p^{-m}((S,T)\times \mathbb{R}^d)}=\|(1-\Delta)^{-\frac{m}{2}}f\|_{L_p((S,T)\times \mathbb{R}^d)},
T_1 :=\{(s,t)\in\mathbb{R}^2\mid 0<s<t<z\}T_2 :=\{(s,t)\in\mathbb{R}^2\mid 0<t<s<z\}T_3 :=\{(s,t)\in\mathbb{R}^2\mid 0<s=t<z\}.
H \; = \; - \frac{2}{3 \sqrt{3}}\; \sum_{j = 1}^{M}\; \Gamma_{\, j}+ \Gamma_{\, j}^{\, 2}+ \lambda \, ( \sigma_{\, j}\sigma_{\, j + 1}^{\, 2}+ \sigma_{\, j}^{\, 2}\sigma_{\, j + 1})
\left( 2 \dot{X}_{0}^{\mu}\dot{X}_{1}^{\nu}+ X_{0}^{\mu}X_{0}^{\nu}\right) G_{\mu \nu}+ G_{\mu \nu , \alpha}\dot{X}_{0}^{\mu}\dot{X}_{0}^{\nu}X_{1}^{\alpha}= 0 ,
\epsilon_{1,n}:= C (1+ \|(y^n,p^{1,n},p^{2,n},f^n)\|_Y^2 + \|(y,p^1,p^2,f)\|_Y^2) \|(y^n,p^{1,n},p^{2,n},f^n)-(y,p^1,p^2,f)\|_Y^2
C^m_0(t) = -(1-e^{-mt}) \langle \eta,\psi\rangle -\int_0^t me^{-m(t-s)}\langle \psi(\cdot-cs),W_s\rangle ds & - c \int_0^t (1-e^{-m(t-s)}) \langle \psi_x(\cdot-cs),W_s\rangle ds.
B^{n,m}_t = G(M^{n-1}(t,x)) + 2 \sigma \frac{A_m(v)}{\sqrt{1+v^{2m}}}\,,\;\; R^{n,m}_t = \left[A_m(v)(B^{n,m}_t - v) - \sigma A_m'(v)\right] f^{n}_t \, ,
\Omega^{(1)}(E)=\sum_{n=1}^{\infty}\sum_{p=0}^{1}\sum_{m=0}^{\infty}A(m,n,p)\frac{e^{\beta_nE}}{E^{27/2+m+p}}{\cal H}\left(E-\frac{8\pi n}{\beta_H \varepsilon}\right)
\lim_{n\rightarrow +\infty}\gamma_{n}\,=\,A\lim_{n\rightarrow +\infty}\int_{0}^{\pi}{\phi^{1-\frac{2}{n}}(\theta)}\,\sin{\theta}d\theta\,=\,\int_{0}^{\pi}{\phi(\theta)}\,\sin{\theta}d\theta\,\lesssim 1.
\hbar \sum_{r \geq 0}X_{i,r}^{\pm}u^{-r-1}= x_i^{\pm}(u - \dfrac{1}{2}i(\varepsilon_1 - \varepsilon_2)), 1 + \hbar \sum_{r \geq 0}H_{i,r}u^{-r-1}= h_i(u - \dfrac{1}{2}i(\varepsilon_1 - \varepsilon_2)).
\frac{e^{-2\pi m \alpha w^{-1}}}{1-e^{-2\pi m w^{-1}}}-\frac{1}{2 \pi w^{-1}m}= -\frac{1}{2 \pi i}\sum_{n=-\infty}^{\infty}\frac{e^{2\pi i \alpha n}}{w^{-1}mi -n}.
\hat{\gamma}= \frac{H}{\sqrt{H^{2}}}= \sum_{m = 0}^{\infty}( - 1 )^{m}\int_{t}\left( \sum_{i = 0}^{\infty}H_{i}\right) P_{0}\left( \sum_{j = 1}^{\infty}( H^{2})_{j}P_{0}\right)^{m}.
\mathcal{A}_{p}= \big[A^{\frac{1}{2}}_{p'}\big]^* \Phi^{-1}\circ A_{p}\circ A^{-\frac{1}{2}}_{p}\Phi = \big[A^{\frac{1}{2}}_{p'}\big]^* \Phi^{-1}\circ A_{p}\circ \Phi \big[A^{-\frac{1}{2}}_{p'}\big]^*.
p_2(s) = (1+e^{-hs})d + (1-e^{-hs})k_{p}\;, p_1(s) = (1+e^{-hs})(bk_p+a)d + (1-e^{-hs})(bd^2+ak_p+k_i) \;, p_0(s) = (1+e^{-hs})bk_i d + (1-e^{-hs}) a k_i \;.
\theta(E,x) =\lim_{\rho \rightarrow 0^{+}}\frac{\mathcal{H}^{n}( E \cap B_{\rho}(x))}{\omega_{n}\rho^{n}}, \mbox{for $\mathcal{H}^{n}$-a.e.}x \in \mathbb{R}^{n}.
V^{a{\mu}_{1}{\cdots}{\mu}_{q+1}}=\frac{(-1)^p}{(2p-q+1)!}{\Gamma}^{ab}{\epsilon}^{{\mu}_{1}{\cdots}{\mu}_{q+1}{\nu}_{1}{\cdots}{\nu}_{2p-q+1}}{\partial}_{[{\nu}_{1}}Z^{b}_{{\nu}_{2}{\cdots}{\nu}_{2p-q+1}]},
K_{n,m}^{*}(f;x,p,q):=K_{n,m}^{\alpha,\beta}(f;x,p,q)-f\left(\dfrac{[2]_{p,q}\alpha+(p\frac{x}{b_n}+1-\frac{x}{b_n})_{p,q}^{n+m}}{[2]_{p,q}([n+1]_{p,q}+\beta)}b_n+\dfrac{(p+2q-1)[n+m]_{p,q}}{[2]_{p,q}([n+1]_{p,q}+\beta)}x\right)+f(x).
\widehat{\phi}(\mathfrak{C}((\omega^1,y^1),1_{H_1},1_{H_2}))(z_1,z_2) = \widehat{\Phi}(\mathfrak{C}((\omega^1,y^1),1_{H_1},1_{H_2})(z_1,z_2)) & = \widehat{\Phi}(y^1([\omega]((z_1,z_2),(1_{H_1},1_{H_2})))) & = \widehat{\Phi}(y(s^{-1},[\omega]((z_1,z_2),(1_{H_1},1_{H_2})))).
\omega_{\mathrm{v}}^\mathrm{DL}= 2\pi\frac{d_\mathrm{v}}{\lambda^\mathrm{DL}}\cos{\theta}= \frac{\lambda^\mathrm{UL}}{\lambda^\mathrm{DL}}\omega_{\mathrm{v}}^\mathrm{UL}= \frac{F_\mathrm{c}^\mathrm{DL}}{F_\mathrm{c}^\mathrm{UL}}\omega_{\mathrm{v}}^\mathrm{UL},
\mathcal{I}_\alpha(\vec{f})(x) \leq \frac{1}{(m|x-x_0|)^{mn-\alpha}}\prod_{i=1}^m \int_Q f_i(y_i)dy_i\leq \left( \frac{L}{m^n}\right)^{m-\alpha/n}\mathcal{M}_\alpha(\vec{f})(x).
R^{(+)k}_{lij}= \partial_i \Gamma^{(+)k}_{jl}-\partial_j \Gamma^{(+)k}_{il}+\Gamma^{(+)k}_{im}\Gamma^{(+)m}_{jl}-\Gamma^{(+)k}_{jm}\Gamma^{(+)m}_{il},\ \R^{(+)}_{ij}= R^{(+)k}_{ikj}.
\Omega:\Gamma (\mathcal{H}\times \mathbb{R})\to \mathbb{C},\,\, |\Psi(0)\rangle\mapsto |\langle \Psi(0)|\Phi_\mathcal{H}(t,|\Psi (0)\rangle )\rangle |^2=\,|\langle\Psi(0)|\mathcal{U}(t)|\Psi (0)\rangle |^2.
\check{H}_{\bullet}\left(X;G\right) :=\varprojlim_{\lambda\in\mathrm{Cov}_{X}}H_{\bullet}\left(V\left(\lambda\right);G\right). \check{H}^{\bullet}\left(X;G\right) :=\varinjlim_{\lambda\in\mathrm{Cov}_{X}}H^{\bullet}\left(V\left(\lambda\right);G\right).
\lambda(\beta)=\lambda_{0}-\frac{3}{2}\frac{\lambda^{2}_{0}}{(2\pi)^{D/2}}\sum^{\infty}_{n=1}\biggl(\frac{m_{0}}{\beta n}\biggr)^{\frac{D}{2}-2}K_{\frac{D}{2}-2}(m_{0}n\beta),
U(t)=e^{\frac{1}{2n}t\Lambda_{1}-t\Lambda}u_{0}-\int_{0}^{t}\left[e^{\frac{1}{2n}t\Lambda_{1}-(t-\tau)\Lambda}\nabla\cdot \left (e^{-\frac{1}{2n}\tau\Lambda_{1}}U\cdot e^{-\frac{1}{2n}\tau\Lambda_{1}}\nabla(-\Delta)^{-1}U\right)\right](\tau)d\tau.
({\rm d}z_x)^2=l_x^2\widehat\theta^2|F^\varepsilon_1|^2{\rm d}t+2l_x\widehat\theta^2F^\varepsilon_1F^\varepsilon_{1,x}{\rm d}t+\widehat\theta^2|F^\varepsilon_{1,x}|^2{\rm d}t,
z^{\{w\}}_v=\rho(A)^{-1}\big(1-\rho(A)^{-1}A_{\{v\}}\big)^{-1}A(v,w)x^{\{w\}}_w=3^{-1}(1-3^{-1}.2)^{-1}1.1=3^{-1}.3=1.
\sum_{1\leq j,k \leq n}f_j(\tau_k)f_k(\tau_j)=\sum_{j=1}^n\sum_{k=1}^nf_j(\tau_k)f_k(\tau_j)\geq \frac{1}{(G_{f,\tau})}\left(\sum_{j=1}^nf_j(\tau_j)\right)^2.
h_{H}(f^{n}(P))=\delta_{f}^{n}h_{H}(P)+ \sum_{k=0}^{n-1}\delta_{f}^{n-1-k}h_{E'_{1}}(f^{k}(P))-\sum_{k=0}^{n-1}\delta_{f}^{n-1-k}h_{Z_{1}}(p^{-1}f^{k}(P)).
k^L_{\gamma,\Sigma}:=\sqrt{\frac{||\nabla^{\Sigma,L}_{\dot{\gamma}}{\dot{\gamma}}||_{\Sigma,L}^2}{||\dot{\gamma}||^4_{\Sigma,L}}-\frac{\langle \nabla^{\Sigma,L}_{\dot{\gamma}}{\dot{\gamma}},\dot{\gamma}\rangle^2_{\Sigma,L}}{||\dot{\gamma}||^6_{\Sigma,L}}}.
a^{\frac 1 2}_A := \lim_{\varepsilon\to 0}\ \left(a^{\frac 1 2}_3(A_\varepsilon) +\frac{1}{\pi i}\log(\varepsilon)\t_{12}\right) a^{\frac 1 2}_B :=\lim_{\varepsilon\to 0}\ \left(a^{\frac 1 2}_3(B_\varepsilon) +\frac{1}{\pi i}\log(\varepsilon)\t_{23}\right)
\mathbf E\left[\phi\big(0,X^{x,\alpha}(\tau_{t,x})\big) +\int_0^{\tau_{t,x}}f_\phi\left(-X^{t,\beta}(s),X^{x,\alpha}(s)\right)ds\right]=\mathbf E\left[ \phi\left(-X^{t,\beta}(\tau_{t,x}),X^{x,\alpha}(\tau_{t,x})\right)\right].
E = O + \bar{z}_1z_2 \frac{\beta(l_j)}{2}(\mu_{\beta}-\frac{\tanh(\beta(a)+\coth(\beta(a))}{2}\tau_{\beta}) & +z_1\bar{z}_2\frac{3\beta(l_j)}{4}(\coth(\beta(a))-\tanh(\beta(a)))\theta(\tau_{\beta})).
\left( \frac{\delta A}{\delta y(s)}\right)^{2}+\left( \frac{\delta A}{\delta \vec{x}(s)}\right)^{2}=\frac{1}{y^{4}(s)}\left\{\left( \frac{dy}{ds}\right)^{2}+\left( \frac{d\vec{x}}{ds}\right)^{2}\right\}\; .
\overline{{{x}}}= \left[ \frac \alpha{1 - \alpha^{2}+ 6 \gamma ( \gamma - 1 )}\frac{1 2 \gamma}{\alpha + 2 \gamma}\left( \frac{g_{i}^{N}- \gamma / \alpha f_{i}^{N}}{g_{i}^{N}+ \gamma / \alpha f_{i}^{N}}\right) \right] \ .
f_{n}(re^{i\theta})=\frac{1}{2\pi}\int_{0}^{2\pi}\frac{1-r^{2}}{1-2rcos(\theta-\varphi)+r^{2}}f_{n}(\varphi)d\phi=\int_{0}^{2\pi}\frac{1-r^{2}}{1-2rcos(\theta-\varphi)+r^{2}}d\mu_{n}(\varphi)
v^{T}\cdot \tilde{P}\cdot v = \sum_{j = 1}^{l}\frac{n_{j}( n_{j}- 1 )}{2}- Q_{R}\sum_{j = 1}^{k}\frac{c_{j}( c_{j}- 1 )}{2 Q_{R}}= \frac{\tilde{C}( R )}{2},
\mu_i(\tilde{H}') = \mu_i(H')\le \frac{1}{d}\cdot\sum_{j=1}^{d+1}\mu_j(H') = \frac{1}{d}\cdot\left(\sum_{j=1}^{d+1}\mu_j(\tilde{H}')\right) + (\lceil\alpha\rceil-\alpha) &=(n-\lceil\alpha\rceil) + (\lceil\alpha\rceil-\alpha) = n-\alpha.
\implies\begin{cases}a\left(r^k-\dfrac{1}{r^k}\right)=a\left(r^k+\dfrac{(-1)^{k+1}}{r^k}\right)=F_k \left(r^k+\dfrac{1}{r^k}\right)=\left(r^k+\dfrac{(-1)(-1)^{k+1}}{r^k}\right)=L_k\end{cases}
\exp \left( - \mathrm{i}\omega \left( N + \sigma^3 / 2 \right) t \right) \widetilde{\varrho}_{\infty}\exp \left( \mathrm{i}\omega \left( N + \sigma^3 / 2 \right) t \right) = \varrho_c \left( \omega t - \theta_{\infty}\right) ,
\frac{\partial V_{eff}}{\partial \sigma}=0 \Longrightarrow-m_{B}^{2}+\lambda_{B}v^{2}=2g_{B}^{2}\int^{\Lambda}\frac{d^{d}q}{(2\pi)^{d}}\frac{1}{q^{2}+g_{B}^{2}v^{2}}.
\frac{d e}{d \omega}= \frac{\hbar}{\pi^{2}c^{3}}\Big [ \frac{\omega^{3}}{2}+ \gamma \omega_{c}^{3}x^{2}\sum_{n = 0}^{\infty}\frac{\beta^{2 n}}{2 n + 1}+ \sum_{k = 0}^{n}( - 1 )^{k}\frac{n - k - x}{k ! ( 2 n - k )}\Theta [ n - k - x ] \Big ]
e^{-\beta t}=\frac{1}{\sqrt\pi}\int_0^{\infty}\frac{e^{-u}}{\sqrt u}e^{-\frac{t^2 \beta^2}{4u}}\,du= \frac{t}{2\sqrt\pi}\int_0^{\infty}\frac{e^{- t^2/(4v)}}{\sqrt{v}}e^{-v\beta^2}\,\frac{dv}{v},
\int\limits_{V}\left(\overline{\rho_R \log \rho_R}-\rho_R \log \rho_R\right)(\tau,\cdot)\dx = \frac{1}{2\mu+\lambda}\int\limits_0^{\tau}\int\limits_{V}\left(\overline{p(\rho_R)}\rho_R-\overline{p(\rho_R) \rho_R}\right)\dx\dt .
\mathcal{L}= -\int_1^{\xi}\left(\int_{1}^{\hat{X}}\frac{(X-\lambda/3)dX}{2\sqrt{X(X-1)(X-\lambda)}}\right)\frac{d\hat{X}}{2\sqrt{\hat{X}(\hat{X}-1)(\hat{X}-\lambda)}}- \lambda\int_1^{\xi}\frac{R_{\varphi}d\hat{X}}{2\sqrt{\hat{X}(\hat{X}-1)(\hat{X}-\lambda)}}+z\pi i/\omega_1 -\pi i.
\textstyle \textstyle \sum_{j=1}^t j(p-1){r \choose j(p-1)}\textstyle \equiv \textstyle r \sum_{j=1}^t{r-1 \choose j(p-1)-1}\equiv \textstyle r \sum_{j=1}^t{r \choose j(p-1)}- r \sum_{j=1}^t{r-1 \choose j(p-1)}\equiv 0 \pmod{p^2},
\alpha_1 := \frac{1}{2}(e_1+e_8 - e_2 - e_3 - e_4 - e_5 - e_6 - e_7), alpha_2 := e_1+e_2, \ \alpha_3 := e_2-e_1, \alpha_4 := e_3-e_2, \ \alpha_5 := e_4-e_3, \ \alpha_6 := e_5-e_4,\ \alpha_7 := e_6-e_5, \ \alpha_8 := e_7-e_6,
\zeta_{j}\left( \beta\right) =-\mathrm{i}b_{j}\beta+\rho_{j}+O\left(\beta^{-1}\right) ,b_{j}>0,\rho_{j}\in\mathbb{R},w_{j}\left( \beta\right) =\mathring{w}_{j}+O\left( \beta^{-1}\right) ,1\leq j\leq N_{R}.
\gamma_F(\frac{1}{2}\psi \circ q_8) = \gamma_F(\frac{1}{2}\psi)^2 \cdot \gamma_F(-J_1, \frac{1}{2}\psi) \cdot (J_1, - 2 \nu r u)_F = \gamma_F(J_1, \frac{1}{2}\psi) \cdot (J_1, 2 \nu r u)_F.
{\rm M.S}:~~~~\hbox{\bf Tr}_{{\cal H}_o}\left[\frac{\Omega}{2}\,(-1)^{F_s}e^{-2\pi t \, \hbox{H}_o}\right]-\hbox{\bf Tr}_{{\cal H}_o}\left[\frac{\Omega}{2}\,e^{-2\pi t \, \hbox{H}_o}\right]
\langle f^i_k,\bar u_k^i-\bar u_k^{i-1}-\bar w_k^i+\bar w_k^{i-1}\rangle=\langle \bar\varrho_k^i,E\bar u_k^i-E\bar u_k^{i-1}-E\bar w_k^i+E\bar w_k^{i-1}\rangle =\langle \bar\varrho_k^i,e_k^i-e_k^{i-1}\rangle +\langle \bar\varrho_k^i,p_k^i-p_k^{i-1}\rangle -\langle \bar\varrho_k^i,E\bar w_k^i-E\bar w_k^{i-1}\rangle
C_1=-2\pi i \sum_{j=1}^m \beta_{j}\mu(\tau_j),C_2=-\frac{3}{2}\sum_{j=1}^m\beta_j^2, &C_3=\sum_{j=1}^m\log G(1+\beta_j)G(1-\beta_j)-\frac{3}{2}\sum_{j=1}^m\beta_j^2\log(4|\tau_j|)-2\sum_{1\leq k<j\leq m}\beta_{j}\beta_{k}\log \widetilde{T}_{j,k}.
a^{\prime}( r_{2}) - a^{\prime}( r_{1}) = - \frac{1 5}{2}\pi^{2}\int_{S ( r_{1}, y )}^{S ( r_{2}, y )}\mathrm{d}^{4}x \ | x - y |^{4}\langle \Theta ( x ) \, T h e t a ( y ) \rangle .
{\cal L}_b=-\frac{1}{4}F_{\mu\nu}^{a}F^{a\mu\nu}+\frac{1}{2}(D^{a})^2-\frac{1}{2}(D_{\mu}C^a)^2-\frac{1}{2}(v^a_{\mu})^2+\frac{k}{2}C^a D^a-\frac{k}{2}A_{\mu}^a v^{a\mu},
\vec{E}=\frac{-q}{4\pi\epsilon_{\circ}}[\frac{\vec{e}_{r^{\prime}}}{r^{\prime2}}+\frac{r^{\prime}}{c}\frac{d}{dt}(\frac{\vec{e}_{r^{\prime}}}{r^{\prime2}})+\frac{1}{c^{2}}\frac{d^{2}}{dt^{2}}(\vec{e}_{r^{\prime}})]
h^p = \begin{pmatrix}1 p 0 1 \end{pmatrix}+ \begin{pmatrix}ap + \frac{c(p-1)p}{2}\frac{(a+d+c(p-1))(p-1)p}{2}- c \sum_{k=0}^{p-1}k^2 + bp cp dp + \frac{c(p-1)p}{2}\end{pmatrix}p &= \begin{pmatrix}1 p 0 1 \end{pmatrix}+ \begin{pmatrix}0 -c\sum_{k=0}^{p-1}k^2 0 0 \end{pmatrix}p.
\zeta(s,z) = \frac{z^{1-s}}{s-1}+ \frac{z^{-s}}{2}+ \frac{s}{12}z^{-s-1}- \frac{s(s+1)}{2!}\int_{0}^{\infty}\frac{\overline{B}_{2}(t)}{(z+t)^{s+2}}dt.
\Vert M_{\widetilde{m}\vert_{\Lambda_N \times \Lambda_N \times \Lambda_N}( \: \cdot \:, 0 \: \cdot \: )}: S_{1}^{N+1}\rightarrow S_{1}^{N+1}\Vert\leq \Vert M_{\widetilde{m}\vert_{\Lambda_N \times \Lambda_N \times \Lambda_N}}: S_{p_1}^{N+1}\times S_{p_2}^{N+1}\rightarrow S_{p}^{N+1}\Vert.
L^2(Q; x^0, 0_{\mathbb{R}^2})=L(Q^0; x^0)=\bigcap_{i=1}^3 T(M^i; x^0)=\bigcap_{i=1}^3 T^2(M^i; x^0, 0_{\mathbb{R}^2}), &L^2(Q; x^0, 0_{\mathbb{R}^2})=L(Q^0; x^0)\nsubseteq \bigcap_{i=1}^3 T(Q^i; x^0)= \bigcap_{i=1}^3 T^2(Q^i; x^0, 0_{\mathbb{R}^2}),
\left( \begin{matrix}{\Gamma_{\phi^{-}}^{1 1}}&{\Gamma_{\phi^{-}}^{1 2}}\\{\Gamma_{\phi^{-}}^{2 1}}&{\Gamma_{\phi^{-}}^{2 2}}\\ \end{matrix}\right) = M_{C}^{- 1}\left( \begin{matrix}{\Gamma_{R}^{\phi^{-}}( p )}&{0}\\{0}&{{\Gamma_{R}^{\phi^{-}}}^{*}( p )}\\ \end{matrix}\right) M_{C}^{- 1}
{Z}_1(x) ={{K_1(1-ax)}\over{(1-a)(1-x)}}\ln{{1-ax}\over{1-x}}-{{2a(1-ax)}\over{(1+3 \beta^2)(1-x)}}\int^x_0 dy \left[{{(1-y)^{{2\beta^2}\over{1+\beta^2}}}\over{1-ay}}\right] ,
\mathcal C := -\underset{x\in\mathbb R}{\min}\left(r_u(x)-\gamma_u(x)\left(\frac{q(x)}{K}+ C\right) -\mu (x),r_v(x)-\gamma_v(x)\left(\frac{p(x)}{K}+C\right) -\mu (x), 0\right)
\left |S_{\kappa,3}^A\right | =\sum_{i,j\in \mathbb N \colon\atop{3(i+j)\leq \kappa}}\left |\left \{[n_0,n_7] \colon n_0+n_7=\kappa-3(i+j) \right \}\right |=\sum_{i,j\in \mathbb N \colon\atop{3(i+j)\leq \kappa}}(\kappa-3(i+j)+1)=a_\kappa.
\mathcal{H}_n \coloneqq \begin{pmatrix}t_p H^{(1)\prime}_n(t_p) -in H^{(1)}_n(t_s) in H^{(1)}_n(t_p) t_s H^{(1)\prime}_n(t_s)\end{pmatrix}Q \coloneqq \begin{pmatrix}\frac{1}{\sqrt{k_p}}0 0 \frac{1}{\sqrt{k_s}}\end{pmatrix}.
\Vert g_1+g_2\Vert_\infty \leq\varphi_1\sqrt{d_1}\Vert g_1\Vert+\frac{\varphi_2}{\sqrt{1-\epsilon_2}}\sqrt{\sum_{j=1}^{q-1}d_{2j}}\Vert g_2\| \leq\sqrt{\frac{\varphi_1\vee\varphi_2}{(1-\epsilon_2)(1-\rho_0)}}\sqrt{d_1+\sum_{j=1}^{q-1}d_{2j}}\Vert g_1+g_2\Vert.
\dot{x}(t) = \sum\nolimits_{k=1}^n \omega_k(x(t)) f_k \bigl(x(t), \hat{v}_k^{0}(x(t))\bigr) \triangleq f_{\rm av}^{\hat{v}}\bigl(x(t)\bigr), x(0) = x_0 \in \mathbb{R}^d,
X^{i}( \sigma , \tau ) = X_{u}^{i}+ p^{i}\tau + \frac{w^{i}\sigma}{2 \pi}+ \frac{i}{2 \pi}\sum_{n \neq 0}\left( \frac{\alpha_{n}^{i}}{n}e^{- i n ( \tau - \sigma )}+ \frac{{\tilde{\alpha}}_{n}^{i}}{n}e^{- i n ( \tau + \sigma )}\right) ,
\int_1^T|L_f(\sigma+it)|^2dt=\begin{cases}A_{f}T\log T+O(T), \sigma= 1/2, \displaystyle T\sum_{n=1}^\infty\frac{|\lambda_f(n)|^2}{n^{2\sigma}}+O(T^{2(1-\sigma)}), 1/2<\sigma<1, \displaystyle T\sum_{n=1}^\infty\frac{|\lambda_f(n)|^2}{n^{2\sigma}}+O(\log^2T), \sigma=1, \end{cases}
\Pi_{i}\left( r , K \right) = 8 ( T_{i}^{\mu \nu , \alpha \beta}T_{_j \mu \nu , \alpha \beta})^{- 1}\bar{P}_{j}\left( r , K \right) \equiv 8 \left( T_{i j}\right)^{- 1}\bar{P}_{j}\left( r , K \right) ,
\dfrac{\Phi(\theta)}{G(0)}= \dfrac{\theta \lambda \mu(\alpha_{\lambda}- \alpha_{\mu}) + \lambda \mu (\mu \alpha_{\lambda}- \lambda \alpha_{\mu})}{\theta^2 (\mu - \lambda) + \theta \left[ (\mu^2 - \lambda^2) + \lambda \mu (\alpha_{\mu}- \alpha_{\lambda}) \right] + \lambda \mu \left[ \mu (1 - \alpha_{\lambda}) - \lambda (1 - \alpha_{\mu})\right]}.
|K|\bar{\mathtt{f}}^n_K(\xi)\mathtt{f}^{n+1/2}_K(\xi)+\Delta t_n\bar{\mathtt{f}}^n_K(\xi)\sum_{L\in\mathcal{N}(K)}a_{K\to L}^n(\xi)=|K|\Delta t_n\,\bar{\mathtt{f}}^n_K(\xi)\partial_\xi m^n_K(\xi).
g_1^{(j)}(z) \quad=\left(-\frac{\|x\|^2}{2}\right)^j\frac{d-[p+k+1]}{d}\cdots \frac{d-[p+k+1+2(j-1)]}{d}&\quad\quad\times\left(1-\frac{\|x\|^2}{d}z\right)^{(d-[p+k+1+2j])/2}e^{\frac{k}{2}\|x\|^2}.
Q_m^m(z)=\frac{[(2m-1)!!]^2}{2P_m^m(z)}\int_{-1}^1\frac{(1-\mu^2)^m}{z-\mu}\,d\mu,Q_{m+1}^m(z)=(2m+1)zQ_m^m(z)-\frac{(2m)!}{P_m^m(z)}.
\mathrm{I = \left\{\int_{0}^{1}d x \int_{0}^{x}d G \int_{G}^{1 - x + G}d y \int_{0}^{x - G}d v \right\}\int_{0}^{\infty}d A A^{4}\int \frac{d^{4}q}{( 2 \p i )^{4}}{\it q}_{i}e^{- E},}
\left(\begin{matrix}x=0 \, \left( \frac13 \right) x=\infty \, \left( \frac13 \right) \overbrace{\begin{matrix}1 0 \omega 0 \omega^2 0 \end{matrix}}\overbrace{\begin{matrix}-t^{\frac13}0 -\omega t^{\frac13}0 -\omega^2t^{\frac13}0 \end{matrix}}\end{matrix}\right).
{1\over R}\log{\| \mathcal{A}(\phi\circ\omega,R)v \|\over \| v\|}-{1\over R}\log{\| \mathcal{A}(\phi\circ\gamma,R)v \|\over \| v\|}={1\over R}(f_{x_0,[v]}(x_R)-f_{x_0,[v]}(y_R) ) .
Q_{1}=\tilde{J}_{Z_1}\tilde{J}_{Z_2}\tilde{J}_{Z_3}\tilde{J}_{Z_4},~Q_{2}=\tilde{J}_{Z_1}\tilde{J}_{Z_2}\tilde{J}_{Z_5}\tilde{J}_{Z_6}, &Q_{3}=\tilde{J}_{Z_1}\tilde{J}_{Z_3}\tilde{J}_{Z_5}\tilde{J}_{Z_7},~Q_{4}=\tilde{J}_{Z_5}\tilde{J}_{Z_6}\tilde{J}_{Z_7}.
\frac{1}{48}\int_{\mathcal{N}}\tilde{R}^2d\mu_{\tilde{h}}+ \int_{\mathcal{N}}|\tilde{W}^{+}|_{\tilde{h}}^2d\mu_{\tilde{h}}= 8\pi^2\sum_{i=1}^{r}k_i + \frac{1}{4}\int_{\mathcal{N}}|\mathring{\widetilde{Ric}}|_{\tilde{h}}^2d\mu_{\tilde{h}},
\mathbf{H}_{k,}(m) =\frac{1}{N}\sqrt{\frac{\kappa}{\kappa+1}}\sum_{p=1}^{L_k}\sum_{n_1=0}^{N_1-1}\sum_{n_2=0}^{N_2-1}\alpha_{k,p}\chi_{k,p,n_1\cdot N_2+n_2}e^{-j2\pi (m-1) \phi^{}}
\overline{Y}=\frac{1}{n}\frac{K_{1}[X]+2K_{1}K_{2}[X]^{2}+...+n(K_{1}K_{2}...K_{n})[X]^{n}}{1+K_{1}[X]+K_{1}K_{2}[X]^{2}+...+(K_{1}K_{2}...K_{n})[X]^{n}}
\begin{aligned}d(\xi): =(12 c_4-24\,c_1^2)\,M_{\varphi, x}[\overline{v}_x^2]+\varepsilon^2(2 c_6-\frac{8}{3}c_2^2) M_{\varphi, x}[\overline{v}^2] &=(24 c_4-48 c_1^2) v_3\cdot \xi+(4 c_6-\frac{16}{3}c_2^2) v_1\cdot \xi\end{aligned}
[\pi^M,\partial^s(\partial\pi)] =\bigcap_{t\neq 0}[\pi^M, \partial^{s+t}\pi]=\bigcap_{t,u\neq 0}[\pi^M, \pi^{s+t+u}]\leq \bigcap_{t\neq 0}\pi^{s+t}=\partial^s\pi.
\sum_{2r_0+r_1=d}{d+1 \choose r_0}{d+1-r_0 \choose r_1}\frac{(d+2)!}{2^{r_2}}\left[{n_1 \choose d+1}{n_2 \choose d+2}+{n_1 \choose d+2}{n_2 \choose d+1}\right].
R(\theta)=\left(\begin{array}{cccc}1 0 0 0 0 \frac{\sinh{\gamma}}{\sinh{(\theta+\gamma)}}\frac{\sinh{\theta}}{\sinh{(\theta+\gamma)}}0 0 \frac{\sinh{\theta}}{\sinh{(\theta+\gamma)}}\frac{\sinh{\gamma}}{\sinh{(\theta+\gamma)}}0 0 0 0 1\end{array}\right)
{}_{A+1}F_{A}\left[\left.\begin{matrix}a_{1}, a_{2}, \dots, a_{A+1}a_{1}^{\prime}, \dots, a_{A}^{\prime}\end{matrix}\right| x\right]:= \sum_{n=0}^{\infty}\frac{(a_{1})_{n}(a_{2})_{n}\cdots (a_{A+1})_{n}}{(a_{1}^{\prime})_{n}\cdots (a_{A}^{\prime})_{n}}\frac{x^{n}}{(1)_{n}},
\begin{array}{ll}\displaystyle \int_{0}^{T}X(t) dt =\displaystyle \mathop{l.i.m}\limits_{\|\bigtriangleup_{n}\| \longrightarrow 0}\sum_{j=0}^{n-1}X(u_{j}) (t_{j+1}-t_{j}) =\displaystyle \mathop{l.i.m}\limits_{\|\bigtriangleup_{n}\| \longrightarrow 0}\sum_{\substack{j\in J\setminus J_{K}}}X(u_{j}) (t_{j+1}-t_{j}) \end{array}
\frac{[Ext^1_U(j_{*}\hat{T},\mathbb{G}_m)]}{[Ext^1_U(j_{*}\hat{T}',\mathbb{G}_m)]}=\frac{[Ext^1_U(\mathcal{Q},\mathbb{G}_m)][\mathrm{cok}(\hat{\lambda}(H^1_c))]}{[Ext^2_U(\mathcal{Q},\mathbb{G}_m)][Hom_U(\mathcal{Q},\mathbb{G}_m)] q(\lambda(O_{K,S}))}.
\Psi [ \xi , t ) = e x p \biggl [ i \int d^{2}x \biggl ( \xi ({\bf x}) \rho ({\bf x}, t ) + \int^{t}d \tau J_{i}({\bf x}, \tau ) \epsilon_{i j}{\frac{\partial_{j}}{\nabla^{2}}}\rho ({\bf x}, \tau ) \biggr ) \biggl ]
k[v] = -k_1[v]+k_1'[v] \lesssim{\sf R}_{\sf mid}^{-{\sf c}_2}{\sf R}_1^{-1}(T-t)^{{\sf d}_1}\eta^\frac{2q}{1-q}|\xi|^{\gamma-2-2{\sf c}_2}{\bf 1}_{{\sf R}_{\sf mid}<|\xi|<2{\sf R}_{\sf mid}}.
f_i(x)= \frac{2\chi(\frac{\gamma_i-Nx}{\alpha_i-x})+1}{\chi(\frac{\gamma_i-Nx}{\alpha_i-x})\left[\chi(\frac{\gamma_i-Nx}{\alpha_i-x})+1\right]}(\alpha_i-x) &-\frac{1}{\chi(\frac{\gamma_i-Nx}{\alpha_i-x})\left[\chi(\frac{\gamma_i-Nx}{\alpha_i-x})+1\right]}(\gamma_i-Nx)+\frac{x}{N},
\int_{0}^{\infty}\left( \frac{1}{t + M^{2}n^{2}}\right)^{\beta}t^{\alpha}d t = ( M^{2}n^{2})^{\alpha + 1 - \beta}\frac{\Gamma ( \alpha + 1 ) \Gamma ( \beta - \alpha - 1 )}{\Gamma ( \beta )},