harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\int_0^\infty \left| \frac{G(e^{\tau}, \delta)}{e^{\sigma\tau}}\right|^2 d\tau = \int_1^\infty \left| \frac{G(y, \delta)}{y^{\sigma+\frac{1}{2}}}\right|^2 dy = \int_1^\infty \left| \frac{G(y, \delta)}{y^{1+\frac{\alpha}{\log(1/\delta)}}}\right|^2 dy > \frac{1}{e^{\alpha b}}\int_1^{1/\delta^b}\left| \frac{G(y, \delta)}{y}\right|^2 dy
\begin{array}{rcl}M \otimes S^{\otimes n}\to M \otimes S^{\otimes n}(m|s_1|\cdots|s_n) \mapsto (\partial_M(m)|s_1|\cdots |s_n) + \sum\limits_{i=1}^n (m|s_1|\cdots | \partial(s_i)| \cdots |s_n)\,.\end{array}
\sqrt{-g}g^{\mu_{1}\nu_{1}}g^{\mu_{2}\nu_{2}}\cdots g^{\mu_{d-p}\nu_{d-p}}{\tilde F}_{\nu_{1}\nu_{2}\ldots \nu_{d-p}}=\frac{1}{p!}\epsilon^{\mu_{1}\mu_{2}\ldots \mu_{d-p}\nu_{1}\nu_{2}\ldots \nu_{p}}F_{\nu_{1}\nu_{2}\ldots \nu_{p}},
v_{l}(i) = \frac{1}{W^{\tilde{d}}}\sum_{\boldsymbol{w}_{\tilde{d}}\in\mathcal{W}^{\tilde{d}}}\psi(\boldsymbol{u}_{l-\boldsymbol{w}_{\tilde{d}}}(i)), l\in\{W-1,\ldots,L-1\},
\begin{cases}\partial_t\widetilde{S}^\alpha\Gamma^aV-\mu\Delta\sum\limits_{l=0}^\alpha C_\alpha^l (-1)^{\alpha-l}\widetilde{S}^l\Gamma^aV-\nabla\cdot\widetilde{S}^\alpha\Gamma^a H=f^1_{\alpha a}, \partial_t\widetilde{S}^\alpha\Gamma^a H -\nabla\widetilde{S}^\alpha\Gamma^aV=f^2_{\alpha a},\end{cases}
[y_1, y_1]=y_4, [y_1, y_2]=y_4+\theta_1y_5, [y_2, y_1]=\frac{\alpha_5}{\alpha_3}y_4+\theta_2y_5, [y_2, y_2]=y_5, [y_1, y_3]=\frac{\alpha_3\beta_4}{\alpha_1\gamma_2}y_5, [y_2, y_3]=y_5=-[y_3, y_2].
T_1 =\left\{k_1(t)=\begin{bmatrix}1 0 0 0 \cos t -\sin t 0 \sin t \cos t\end{bmatrix}:0\leq t\leq 2\pi\right\}T_2 =\left\{k_2(t)=\begin{bmatrix}\cos t -\sin t 0 \sin t \cos t 0 0 0 1 \end{bmatrix}:0\leq t\leq 2\pi\right\}.
\frac{6 \ddot{a}}{a}+ \dot{\phi}^2 = \frac{2 \ddot{a}}{a}+ \frac{4 (\dot{a}^2 + k)}{a^2}= - \frac{1}{\alpha}\left( \ddot{\phi}+ \frac{3 \dot{a}\dot{\phi}}{a}\right) = \Lambda e^{\alpha \phi}\; .
\hat{\cal A}= \hat{\cal A}_{1}+ \hat{\cal A}_{2}= ( - 1 )^{n + 1}{\frac{1}{2^{2 n - 1}n !}}({\frac{g}{\pi}})^{n}\epsilon_{\mu_{1}\nu_{1}. . . \mu_{n}\nu_{n}}[ \hat{F}_{\mu_{1}\nu_{1}}. . . \hat{F}_{\mu_{n}\nu_{n}}- g \theta^{\alpha \beta}\partial_{\beta}( \hat{A}_{\alpha}\hat{F}_{\mu_{1}\nu_{1}}. . . \hat{F}_{\mu_{n}\nu_{n}}) ] .
\mathrm{Fitt}_{\Lambda_n}(A) = \mathrm{Fitt}_{\Lambda_n}( A_{\mathrm{mft}}) \cdot \mathrm{Fitt}_{\Lambda_n}(A') \mathrm{Fitt}_{\Lambda_n}(B) = \mathrm{Fitt}_{\Lambda_n}( B_{\mathrm{mft}}) \cdot \mathrm{Fitt}_{\Lambda_n}(B') .
-A_p(X) = \overline{\nabla}_X \eta = \sum_{i=1}^{n+1}\overline{\nabla}_X(a_i \widetilde{X}_i) = \sum_{i=1}^{n+1}\left[ a_i(p) \overline{\nabla}_X \widetilde{X}_i + X(a_i) \widetilde{X}_i (p) \right] = \overline{\nabla}_X \widetilde{X}_{n+1}+ \sum_{i=1}^{n+1}X(a_i) X_i = \alpha_p(X) + \Gamma_p^{-1}(D \gamma(p) \cdot X),
\left|\sum\limits_{j=0}^{n-1}g_{\lambda_i}^{-\frac{1}{2}}(\alpha\theta_n^{j})\theta_n^{-jk}\right| =\left|\lambda_i^{-1/2}\sum\limits_{j=0}^{n-1}\theta_n^{-jk}+\sum\limits_{j=0}^{n-1}(g_{\lambda_i}^{-\frac{1}{2}}(\alpha\theta_n^{j})-\lambda_i^{-1/2})\theta_n^{-jk}\right|.
E^{2}= \frac{1}{p^{2}+ \left( \frac{4 \pi^{2}\alpha^{\prime}}{L_{1}L_{2}}\right)^{2}q^{2}}\left( \left( \frac{2 \pi}{L_{1}}n_{1}\right)^{2}+ \left( \frac{2 \pi}{L_{2}}n_{2}\right)^{2}\right)
(g,h)_{f_{u}}:=\int_{\mathbb{R}^{|u|}}g(\mathbf{x}_{u})h(\mathbf{x}_{u})f_{u}(\mathbf{x}_{u})d\mathbf{x}_{u}=:\mathbb{E}\left[g(\mathbf{X}_{u})h(\mathbf{X}_{u})\right]
w_{sustr}={1\over 2(2\pi)^{D-1}}\ \sum_{states \S}{(e_1+e_2)E\over \epsilon}\sum_{k=1}^{\infty}(-)^{(k+1)(a_{S}+1)}\Bigl({\vert\epsilon\vert\over k}\Bigr)^{D/2}\exp{\Bigl(-{\pi k m_S^{\ 2}\over \vert\epsilon\vert}\Bigr)},
[a^{\dagger}b,b^{\dagger}a] = a^{\dagger}bb^{\dagger}a - b^{\dagger}aa^{\dagger}b = a^{\dagger}(aa^{\dagger}-\delta)a - b^{\dagger}(bb^{\dagger}+\delta)b = (a^{\dagger}a)^2 - (b^{\dagger}b)^2 - \delta(a^{\dagger}a + b^{\dagger}b) = 2(\gamma-\delta)\left(a^{\dagger}a - \frac{\gamma}{2}\right).
\begin{aligned}N_{r,i}^{n^{-1}}=\frac{\Phi_{i}}{(1+\lambda_{i})^{2}}\binom{n}{r}\left[\frac{2^{2i}(i!)^{2}}{(2i)!}\Psi_{k,i}^{n,r}+\left(\frac{2^{2i}(i!)^{2}}{(2i)!}\right)^{2}\sum_{d=0}^{i}\frac{(2d)!\lambda_{d}}{2^{2d}(d!)^{2}}\Psi_{j,d}^{n,r}\right],\end{aligned}
b_{(i,\alpha, r_1, r_2), (j, \beta, r'_1, r'_2)}= a_{(i, \alpha, r_1, r_2), (j, \beta, r'_1, r'_2)}- \underset{\substack{d^{r''_1, r''_2}_{l, \gamma}> d^{i, \alpha}_{r_1, r_2}d^{r''_1, r''_2}_{l, \gamma}> d^{r'_1, r'_2}_{j, \beta}}}{\sum}b_{(i, \alpha, r_1, r_2), (l, \gamma, r''_1, r''_2)}
\Psi'_{u}(t)=s_1t^{2s_1-1}|\nabla_{s_1}u|_{2}^{2}+s_2t^{2s_2-1}|\nabla_{s_2}u|_{2}^{2}-\frac{1}{t}\displaystyle\int_{\mathbb{R}^{d}}W(\frac{x}{t})u^{2}dx-\frac{d}{t^{d+1}}\displaystyle\int_{\mathbb{R}^{d}}\widetilde{G}(t^{\frac{d}{2}}u(x))dx.
\gamma \alpha_{0}=\partial_{\mu}\left( \varepsilon^{\mu \nu \lambda \rho}f_{\;\;bc}^{a}\left( C_{\rho a}A_{\nu}^{b}A_{\lambda}^{c}+C_{a}\partial_{\rho}\left( A_{\nu}^{b}A_{\lambda}^{c}\right) \right) \right) .
\int\limits_0^\pi Cf(0,\beta(\phi),\psi)d\psi =\int\limits_0^\pi \int\limits_0^\infty f(r\sin(\psi+\phi),r\cos(\psi+\phi))dr d\psi + \int\limits_0^\pi \int\limits_0^\infty f(-r\sin(\psi-\phi),r\cos(\psi-\phi))dr d\psi.
P^i_{\mathbb{R}}=\sum_{k=1}^n\left(\frac{\partial P_\mathbb{R}^{ik}}{\partial x^k}+\frac{\partial P_\mathbb{R}^{in+k}}{\partial x^{n+k}}\right)=\frac{1}{4}\sum_{k=1}^n\left(\frac{\partial P^{ik}}{\partial z^k}+\frac{\partial \overline{P^{ik}}}{\partial \overline{z}^k}\right)
h_{\mu}\left(G/Q,\nu_{Q}\right)=\sum_{\ell=1}^{k}\sum_{r_{\ell-1}<i\le r_{\ell},j>r_{\ell}}\lambda_{i}-\lambda_{j}=h_{\mu}\left(G/P,\nu_{P}\right)-\sum_{\ell=1}^{k}\sum_{r_{\ell-1}\le i<j\le r_{\ell}}\lambda_{i}-\lambda_{j}.
I(P,V) = O\left( m^{2/3}\left(m + \sum_{x \in P}i(x) (i(x)-1) (i(x)-2) \right)^{1/3}\right) & = O\left( m + m^{2/3}\left(\sum_{x \in P}i(x) (i(x)-1) (i(x)-2) \right)^{1/3}\right).
\begin{array}{c}\frac{2Li_{(k_1,...,k_r)}(1-e^{-t})}{(1+e^t)^r}e^{rxt}=\sum\limits_{n=0}^{\infty}\mathbf{E}_{n}^{(k_1,...,k_r)}(x){t^{n}\over n!}\end{array}.
p_{i_j}=\langle i_j,\, i_{j-1}\ldots i_2 i_{1}\rangle,p'_{i_j}=\langle i_j,\, i_{j}\ldots i_2 i_{1}\rangle,p''_{i_j}=\langle i_j,\, i_{j+1}\ldots i_2 i_{1}\rangle, q_{i_j}=\langle i_1i_2\ldots i_{j-1},\, i_j\rangle^{-1},q'_{i_j}=\langle i_1i_2\ldots i_{j},\, i_j\rangle^{-1},q''_{i_j}=\langle i_1i_2\ldots i_{j+1},\, i_j\rangle^{-1}.
\pi^*\,\tilde{l}_k(\sigma_N^1,\dots,\sigma_N^k) =\varsigma(k)~ \iota_{v_N^k}\dots\iota_{v_N^1}~\omega_N &=\varsigma(k)~ \iota_{v^k|_N}\dots\iota_{v^1|_N}~j^*\omega &=j^*\big(\varsigma(k)\, \iota_{v^k}\dots\iota_{v^1}\omega\big) &=\pi^*\,\tilde{l}_k(\sigma^1,\dots,\sigma^k)_N,
\begin{array}{c | c c c c c c c}{{p}}&{{3}}&{{5}}&{{7}}&{{9}}&{{1 1}}&{{1 3}}&{{\cdots}}\\ \hline{{\Delta}}&{{5}}&{{1 1}}&{{1 7}}&{{2 1}}&{{2 9}}&{{3 5}}&{{\cdots}}\\ \end{array}
\Phi_{xx}^2+\zeta\Phi_{xq}(\Phi_{yq}-\Phi_{yp})+1 = -\Phi_{xy}\Phi_{yx}, \Phi_{qq}^2 +\zeta\Phi_{xq}(\Phi_{yq}-\Phi_{yp})+1 = -\Phi_{qp}\Phi_{pq}.
\|\overline{u}\|_{L^{m\chi(\alpha+1)}(\Omega_{R_1})}\le C^{1/(\alpha+1)}\Bigg[\frac{1}{(R_1-R_2)}+\left(\frac{G(R_1,R_2)}{m\alpha+1-Cm\epsilon^{m'}}\right)^{1/m}+\left(\frac{\|f\|_q}{(m\alpha+1-Cm\epsilon^{m'})k^{m-1}}\right)^{N/m(mq-N)}\Bigg]^{1/(\alpha+1)}\|\overline{u}\|_{L^{m(\alpha+1)}(\Omega_{R_2})}.
{\hat F[L,~f]}~=~ \alpha^{-\sigma[L,~f]}~\sum_{\{j_i\}}~~\mu_{j_1}~\mu_{j_2}~....~\mu_{j_r}~ V[L;~n_1,~n_2,~...~n_r;~ j_1,~j_2,~....~j_r]
R ~ = ~{\sqrt{\frac{\pi}{2}}}\omega^{3 / 2}\left[{\frac{\alpha}{2}}\frac{({\frac{\rho}{2}})^{- \epsilon}}{\Gamma ( 2 - \epsilon )}~ + ~{\frac{\beta}{2}}\left( \left({\frac{\rho}{2}}\right)^{- \epsilon}\cot \pi ( 1 - \epsilon ) - \frac{\epsilon}{{\mathrm{s i n}}\pi ( 1 - \epsilon )}\left({\frac{\rho}{2}}\right)^{- 2 + \epsilon}\right) \right] ~ .
\int_{|u-\alpha_1| \leqslant\frac{d}{M}}\frac{\,\mathrm{d}u}{\big((u-\alpha_1)^2+(v-\alpha_2)^2\big)^{\frac{p}{2}}}\leqslant{}\int_{|u-\alpha_1| \leqslant\frac{d}{M}}\Big(\frac{d}{\sqrt{1+M^2}}\Big)^{-p}\,\mathrm{d}u ={}2d^{1-p}M^{-1}(1+M^2)^{\frac{p}{2}},
\epsilon\cdot\mathbb{F}_{p}\otimes_{\mathbb{Z}}^{L}\mathbb{F}_{p}=Ker(\mathbb{F}_{p}\otimes_{\mathbb{Z}}^{L}\mathbb{F}_{p}\rightarrow\epsilon\cdot\mathbb{F}_{p}\otimes_{\mathbb{Z}}^{L}\mathbb{F}_{p}).
\begin{aligned}\frac{1}{2}\Big( \psi^{+0}_{n,p}\psi^{+0}_{n-1,p}\rho^{0+}_{n-2,p}-2\psi^{+0}_{n,p}\rho^{0+}_{n-1,p}\psi^{+0}_{n-1,p-1}+\rho^{0+}_{n,p}\psi^{+0}_{n,p-1}\psi^{+0}_{n-1,p-1}\Big) = - \varphi^{++}_{n,p}\psi^{+0}_{n-1,p-1}& = - \psi^{+0}_{n,p}\varphi^{++}_{n-1,p}\,.\end{aligned}
S_{v}p=[\begin{matrix}v_{x}&0&0\\ 0&v_{y}&0\\ 0&0&v_{z}\\ \end{matrix}][\begin{matrix}p_{x}\\p_{y}\\p_{z}\end{matrix}]=[\begin{matrix}v_{x}p_{x}\\v_{y}p_{y}\\v_{z}p_{z}\end{matrix}].
\langle0|\log Z|0\rangle=\lim_{{\alpha_{\Re}}\rightarrow0^+}\frac{\langle0|S_{{\lambda_{vac}}}^{-{\alpha_{\Re}}}|0\rangle-1}{{\alpha_{\Re}}}=\lim_{{\alpha_{\Re}}\rightarrow0^+}\frac{\langle0|Z_{{\lambda_{vac}}}^{{\alpha_{\Re}}}|0\rangle-1}{{\alpha_{\Re}}}
\overline{T}_{2k,2a}(x;q) =(1+xq)\sum_{i=1}^a (xq)^{2i}\left[\sum_{h=1}^{k-i}(xq^2)^h\overline{T}_{2k,2k-2h}(xq^2;q) +\sum_{h=0}^{k-i-1}(xq^2)^h\overline{T}_{2k,2k-2h}(xq^2;q)\right] &\quad+ (1+xq) \sum_{i=0}^{a-1}(xq)^{2i}\left[\sum_{h=1}^{k-i}(xq^2)^h\overline{T}_{2k,2k-2h}(xq^2;q)+\sum_{h=0}^{k-i-1}(xq^2)^h\overline{T}_{2k,2k-2h}(xq^2;q)\right].
u(t,x,y)=u_0\big(x-tU(y),y\big)+tU'(y)\int_0^y u_{0x}\big(x-tU(z),z\big)dz, &v(t,x,y)=-\int_0^y\Big\{u_{0x}\big(x-tU(z),z\big)+t\big[U(y)-U(z)\big] u_{0xx}\big(x-tU(z),z\big)\Big\}dz,
\sum_{\substack{1\leq e<\infty (e, g)=1}}\frac{\mu(e)}{e}\sum_{\substack{a|g (ea,\frac{mh}{g}\cdot \frac{nk}{g})=1}}\frac{\mu(a)}{\phi(ea)}R(0;ae,mnhk/g^2) = \frac{\phi(mnhk)}{mnhk}.
h(\theta) = \frac{(\theta -\tau) ( \frac{1}{\tau}-\theta)}{\sqrt{(\frac{1}{\tau}-\tau)(\tau +\frac{1}{\tau}-2\theta)}}, x(\theta)= \frac{(\frac{1}{\tau}-\tau)^2}{(\theta -\tau) ( \frac{1}{\tau}-\theta)},
\hat{\chi}= \sum_{M = 1}^{n}\left( \bar{z}_{+}^{M}\bar{z}_{-}^{M}\right) \Big ( \begin{array}{c c}{\mathrm{e}^{V}}\\ \end{array}\Big ) \left( \begin{array}{c}{z_{+}^{M}}\\{z_{-}^{M}}\\ \end{array}\right) \ ,
\frac{1}{4\pi i}\int^{1+i\infty}_{1-i\infty}\frac{\zeta_F'}{\zeta_F}(s,\chi)\hat{g}(s)ds=\sum_{v}\frac{1}{4\pi i}\int^{1+\epsilon+i\infty}_{1+\epsilon-i\infty}\frac{L'_v}{L_v}(s,\chi)\hat{g}(s)ds+\frac{\delta_\chi}{4}g(1)
\left\{\lim_{m \rightarrow \infty}\textrm{diam}\left( \varphi_{t_m}(\cdot,V(\cdot))\right) =0 \right\}& \subset \left\{\lim_{m \rightarrow \infty}\textrm{diam}\left( \varphi_{t_m}\left( \cdot,B \left( x, \frac{1}{k}\right) \right) \right) =0 \textrm{for some}k \in \mathbb{N}\right\}.
\|\sin[\frac{\pi}{2}A]-\sin[\frac{\pi}{2}B]\|_{F}^2 =\sum_{i,j}(\sin(\frac{\pi}{2}A_{ij})-\sin(\frac{\pi}{2}B_{ij}))^2 \le \sum_{i,j}\frac{\pi^2}{4}(A_{ij}-B_{ij})^2=\frac{\pi^2}{4}\|A-B\|_{F}^2.
lim_{t\rightarrow\infty}e^{-t}\sum_{n=0}^{\infty}\frac{1-z^{n+1}}{1-z}\frac{t^{n}}{n!}=lim_{t\rightarrow\infty}\frac{e^{-t}}{1-z}(e^{t}-ze^{tz})=\frac{1}{1-z},
\lim_{t\to \infty}\left( e^{-2^{-1}mt}u(t) -\sum_{i\in I_1}\textup{Re}\left(w_ie^{\mathbf{i}\beta_i t}\right)\varphi_i-\sum_{i\in I_2}c_i \varphi_i\right) =0\ \textup{in}\C^\infty(\Sigma;\mathbf{V}).
\langle{T_{\k}^{j}}\rangle - \langle T_{_{\oslash}k}^{j}\rangle \sim{\frac{\pi M^{2}}{2}}\Phi^{2}\Big ({\frac{1}{c^{2}}}u^{j}u_{k}-{\frac{c_{_\mathrm{S}}^{2}}{c^{2}}}\gamma_{_k}^{j}\Big ) \ln \Big \{{\frac{w_{\!_{\odot}}}{w}}\Big \}.
\Phi_{n + 1}= \frac{- 1}{1 - \Phi_{n}^{2}}\left[ \frac{d \Phi_{n}}{d r}- \frac{n}{r}\Phi_{n}\right] = \Phi_{n - 1}- \frac{2}{1 - \Phi_{n}^{2}}\frac{d \Phi_{n}}{d r}, n \geq 1 .
1_{N_{B}}( x_{\eta}^{- 1}x_{\zeta}) \beta_{i , j}( x_{\eta}^{- 1}x_{\zeta}) \frac{1}{| K |}\int_{K}\mathrm{d}k \, \phi ( k x_{\eta}g_{B}^{- 1}x_{\eta}^{- 1}x ) = 1_{N_{B}}( x_{\eta}^{- 1}x_{\zeta}) \beta_{i j}( x_{\eta}^{- 1}x_{\zeta}) \frac{1}{| K |}\int_{K}\mathrm{d}k \, \phi ( k x )
|z_k - p| = |z_0 + k + \zeta_k - p| \geq |\Re(z_0 + k + \zeta_k - p)| = |\Re(z_0) + k + \Re(\zeta_k) - p| \geq |k - p + k_0| - |\Re(z_0) - k_0| - |\Re(\zeta_k)| &\geq |k - p + k_0| - 2 \geq \frac{1}{3}| k - p + k_0|,
hg (hge_nhg) \cdots (hg^{\frac{n}{2}-1}e_nhg^{\frac{n}{2}-1}) (hg^{\frac{n}{2}+1}e_nhg^{\frac{n}{2}+1})\cdots (hg^{n-1}e_nhg^{n-1}) = (hge_nhg) \cdots (hg^{\frac{n}{2}-1}e_nhg^{\frac{n}{2}-1}) (hg^{\frac{n}{2}+1}e_nhg^{\frac{n}{2}+1})\cdots (hg^{n-1}e_nhg^{n-1}),
\begin{aligned}(\lambda_{m+1}-\lambda_m)\frac{c'}{r_m}\leq\sup_{\{f=r_{m}\}}(\lambda_{m+1}-\lambda_m)|\operatorname{Ric}(g)|\leq \frac{C'}{r_m^l}(\lambda_m+\lambda_{m+1}) \leq\frac{C'}{r_m^l}(\lambda_{m+1}-\lambda_m)+\frac{C''}{r_m^{l-\frac{1}{2}}}. \end{aligned}
u_{\lambda}\left( x\right) =\lambda^{\alpha_{1}}\ln^{\beta_{1}}\lambda\,R_{\left( \alpha_{1},\beta_{1}\right)}\left( x\right)+\cdots+\lambda^{\alpha_{k}}\ln^{\beta_{k}}\lambda\,R_{\left( \alpha_{k},\beta_{k}\right)}\left( x\right) \,,
\sum_{k=1}^{\infty}e^{-t\tau_k}= \int_{\partial \Omega}\bigg[ \frac{1}{(2\pi)^{n-1}}\int_{\mathbb{R}^{n-1}}\bigg( \frac{i}{2\pi}\int_{\mathcal{C}}e^{-t\tau}\sum_{j \leqslant -1}\operatorname{Tr}\phi_{j}\,d\tau \bigg) \,d\xi \bigg] \,dS.
y\left( x\right) =y_{a}\frac{\left( \psi \left( x\right) -\psi \left( a\right) \right)^{\gamma -1}}{\Gamma \left( \gamma \right)}+\frac{1}{\Gamma \left( \alpha \right)}\int_{a}^{x}\psi^{\prime}\left( t\right) \left( \psi \left( x\right) -\psi \left( t\right) \right)^{\alpha-1}f\left( t,y\left( t\right) \right) dt.
{{\cal G}^{-1}}^{\nu b}_{\mu a}\eta^{\rm T,L}_{\nu b}=\frac{1}{2}\left(k^{2}-\Lambda_{l}+\frac{1}{2r^{2}}(N-1)(N-2)-\frac{N+d}{N+d-2}\Lambda\right)\eta^{\rm T,L}_{\mu a},
M_i - (\times_\dagger -\overline{\lambda}_i-1) = M_i - (k_i-\overline{\lambda}_i-1) + k_i - \times_\dagger \le M_{i'}- (k_{i'}-\overline{\lambda}_{i'}-1) + k_{i'}- \times_\dagger &= M_{i'}- (\times_\dagger -\overline{\lambda}_{i'}-1)
H_{N}(x^{N}(\eta))-H_{N}(x^{N}(0))-2a\left[\int_{0}^{\eta}\sum_{i=1}^{N}\rho_{\epsilon,i}\left(x^{N}(t)\right)dt-NA^{N}(\eta)\right] =\frac{4a}{N}\sum_{i=1}^{N}\int_{0}^{\eta}\left[\sum_{j:j\neq i}p_{\epsilon}'(x_{i}^{N}(s)-x_{j}^{N}(s))\right]dx_{i}^{N}(s).
\Lambda_{0}^{1}= \frac{1}{\tilde{\Delta}_{( m , n )}^{1 / 2}}\left( \begin{array}{c c}{{m \qquad}}&{{- n \, e^{- \phi_{0}}+ \chi_{0}\, ( m - \chi_{0}n ) \, e^{\phi_{0}}}}\\{{n \qquad}}&{{( m - \chi_{0}n ) \, e^{\phi_{0}}}}\\ \end{array}\right) ,
\frac{1}{r_1^2}\int_{\{\Psi=-\infty\}}|f|^2_h =\lim_{p\rightarrow0+0}\frac{1}{r_1^2}\int_{\{p\Psi<2\log r_1\}}|f|^2_h &\ge G(0;c\equiv1,\Psi,h,I_+(h,2a_{z_0}^f(\Psi;h)\Psi)_{z_0},f).
j_{i_{1}\ldots i_{s}}= \sum_{k = 0}^{s}{\frac{( - 1 )^{k}}{k ! \left( k +{\frac{D - 4}{2}}\right) ! ( s - k ) ! \left( s - k +{\frac{D - 4}{2}}\right) !}}\partial_{i_{1}}\cdots \partial_{i_{k}}\phi^{*}\; \partial_{i_{k + 1}}\cdots \partial_{i_{s}}\phi - \mathrm{t r a c e s}
V^{1-1-3,l}(\psi):=\sum_{n=2}^{\infty}V^{1-1-3,l,(n)}(\psi),V^{1-1-4,l}(\psi):=\sum_{n=2}^{\infty}V^{1-1-4,l,(n)}(\psi), &V^{1-2-2,l}(\psi):=\sum_{n=2}^{\infty}V^{1-2-2,l,(n)}(\psi),V^{2,l}(\psi):=\sum_{n=1}^{\infty}V^{2,l,(n)}(\psi), &V^{1-1,l}(\psi):=\sum_{j=1}^4V^{1-1-j,l}(\psi),V^{1-2,l}(\psi):=\sum_{j=1}^2V^{1-2-j,l}(\psi).
p_1(x, \xi)= q_0(x, \xi) \, q_1(x, \xi)+ q_1(x, \xi) \, q_0(x, \xi) + \sum_{j=1}^4 -i \frac{\partial q_1}{\partial \xi_j}(x, \xi) \, \frac{\partial q_1}{\partial x_j}(x, \xi),
|A_\omega(y)-A_\omega(y')| \le \sum_{j=0}^{p-1}\sum_{\ell=0}^{r(F^jy)-1}|v|_\theta d_\theta(F^jy,F^jy')= |v|_\theta\sum_{j=0}^{p-1}r(F^jy)\theta^{-j}d_\theta(y,y') \le |v|_\theta \,\theta^{-p}(1-\theta)^{-1}r_p(y)d_\theta(y,y'),
q(x|z_i=1) = \sum_{k=1}^{H_0}b^0_{kj}\prod_{i=1}^M (a^0_{ik})^{x_{i}}, p(x|z_i=1,A,B) = \sum_{k=1}^{H}b_{kj}\prod_{i=1}^M (a_{ik})^{x_{i}}.
\sum_{k=0}^{n-1}\frac{(aq^r;q^d)_k (bq^{d-r};q^d)_k (x;q^{d})_k q^{dk}}{(q^d;q^d)_k (abq^{2d};q^{2d})_k}\equiv (-1)^{\langle -r/d\rangle_n}\sum_{k=0}^{n-1}\frac{(aq^r;q^d)_k (bq^{d-r};q^d)_k (-x;q^{d})_k q^{dk}}{(q^d;q^d)_k (abq^{2d};q^{2d})_k},
\mathcal{X}= \mathcal{M}(D_2, \cdots, D_{5}, D_7, D_8, D_{9}, D_{11}, D_{12}, D_{13}, D_{16}, D_{19}, D_{20}, D_{23}) \mathcal{Y}= \mathcal{M}(D_6, D_{10}, D_{14}, D_{15}, D_{17}, D_{18}, D_{21}, D_{22}, D_{24}).
\lim_{r\to\infty}q^{-m\binom{r}{2}}P_{(m^r)}(1,q,q^2,\dots;q^n) =\frac{(q^{\kappa};q^{\kappa})_{\infty}^{n-1}}{(q)_{\infty}^n}\cdot \prod_{1\leq i<j\leq n}\theta(q^{j-i};q^{\kappa}) &=\frac{(q^{\kappa};q^{\kappa})_{\infty}^{m-1}}{(q)_{\infty}^m}\cdot \prod_{1\leq i<j\leq m}\theta(q^{j-i};q^{\kappa}).
\left|f\left(\sum_{j=1}^mx_j\right)\right| \leqslant \left(\sum_{j\in B}C_j|g_j(x_j)|\right) + \theta\left(\sum_{q\in A_2}|c_q|\cdot\left|f_q\left(\sum_{j\in E_q}x_j\right)\right|\right).
F_{\lambda}= \sum_{\lambda_{0}, . . . , \lambda_{k - 1}}d \rho_{\lambda_{0}}\wedge d \rho_{\lambda_{1}}\wedge . . . \wedge d \rho_{\lambda_{k - 1}}\wedge d \log g_{\lambda \lambda_{0}\lambda_{1}. . . \lambda_{k - 1}}
\sum_{k=0}^{n-1}\frac{(q^2;q^4)_k(-q;q^2)_k^2}{(q^4;q^4)_k(-q^4;q^4)_k^2}[6k+1]q^{k^2}\equiv0\pmod{[n]}, \sum_{k=0}^{(n-1)/2}\frac{(q^2;q^4)_k(-q;q^2)_k^2}{(q^4;q^4)_k(-q^4;q^4)_k^2}[6k+1]q^{k^2}\equiv0\pmod{[n]}.
f_{k+1}= f_kf_{k-1}= \varphi^{(k+1)/2}(y/u)\varphi^{(k-1)/2}(x) = \varphi^{(k-1)/2}(x\cdot y/u)\varphi^{(k-1)/2}(x) = \varphi^{(k-1)/2}((x\cdot y/u)x) = \varphi^{(k-1)/2}(x(y/u\cdot x)) = \varphi^{(k+1)/2}(x),
\lim_{n\to \infty}\mathbb P \left[ T_N(n)\ge - x \right]=\lim_{n\to \infty}\left\{1-\mathbb P \left[T_N(n)\le -x \right]\right\}=1-\mathbb P \left[T_N\le -x\right]=\mathbb P \left[ T_N \ge -x\right].
- ((u-w_n)(x_j - t_j)) w_n'(x_j - t_j) = \lim_{t \uparrow x_j}\left( ((u-w_n)(t)) w_n'(t)\right) & \phantom{=}{}- ((u-w_n)(x_j - t_j)) w_n'(x_j - t_j) &=\lim_{t \uparrow x_j}\int_{x_j - t_j}^t \left((u-w_n) w_n'\right)'(s) \, ds & = \int_{x_j - t_j}^{x_j}\left((u-w_n) w_n'\right)'(s) \, ds,
-\int_\Sigma \varphi\Delta_f \varphi+(\|A\|^2+1)\varphi^2e^{-f}d\mathrm{vol}_\Sigma =-\int_\Sigma \varphi^2 +\|A\|^2a^2+a\phi\|A\|e^{-f}d\mathrm{vol}_\Sigma &= -\int_\Sigma \varphi^2 +\|A\|^2a^2 e^{-f}d\mathrm{vol}_\Sigma &< 0.
\mathcal{G}^{4s}_{\log}(\vec{a};\vec{b}) =2\pi \times (i)^{4s+1}\frac{\log(\langle \vec{a},\vec{b}\rangle)}{||\vec{a}||^{4s+1}+||\vec{b}||^{4s+1}}\int \limits_{|a_n|}^{|b_n|}\int \limits_{|a_{n-1}|}^{|b_{n-1}|}\cdots \int \limits_{|a_1|}^{|b_1|}\sqrt[4s]{\sum \limits_{j=1}^{n}x^{4s}_j}dx_1dx_2\cdots dx_n
\{a, b_{1}b_{2}\}= [\tilde{r}_{a}, \tilde{\tau}_{b_1 b_2}]= [\tilde{r}_a, \tilde{\tau}_{b_1}\tilde{\tau}_{b_2}] = [\tilde{r}_{a}, \tilde{\tau}_{b_1}] \cdot [\tilde{r}_{a}, \tilde{\tau}_{b_2}] \cdot [\tilde{\tau}_{b_1}, [ \tilde{\tau}_{b_2}, \tilde{r}_{a}] ]^{-1}= \{a, b_1\}\{a, b_2\}.
h \overbrace{f_{s_1}(w_1)\cdots f_{s_{i_{l(s)}-1}}(w_{i_{l(s)}-1})}^{b_1}f_{s_{i_{l(s)}}}(w_{i_{l(s)}}) \overbrace{f_{s_{i_{l(s)}+1}}(w_{i_{l(s)}+1})\cdots f_{s_m}(w_m)}^{b_2}= \ \underbrace{f_{t_1}(v_1)\cdots f_{t_{j_{l(t)}-1}}(v_{j_{l(t)}-1})}_{c_1}f_{t_{j_{l(t)}}}(v_{j_{l(t)}}) \underbrace{f_{t_{j_{l(t)}+1}}(v_{j_{l(t)}+1})\cdots f_{t_m}(v_m)}_{c_2}.
\mathcal Q(\ell, \omega) = \frac1\ell \mathcal Q_0 - \frac{2 \varkappa \cdot |\partial_3 \psi_{(\varkappa,0),3}(0)|^2}{\sqrt{\Lambda -\omega}\cdot \sqrt{2 \zeta_1''(\varkappa)}}\cdot \int_{\{|\theta| = 1\}}T_\ell^* P_{\varkappa \theta}T_\ell \; \mathrm{d}\theta + R(\ell,\omega) .
\int_{0}^{\infty}\frac{(\lambda'(t))^{2}}{r \lambda(t)}\left(\mathcal{F}^{-1}(y_{0}(t,\frac{\cdot}{\lambda(t)^{2}}))(\frac{r}{\lambda(t)})\right)^{2}r dr = (\lambda'(t))^{2}\lambda(t)^{2}\int_{0}^{\infty}|y_{0}(t,\omega)|^{2}\rho(\omega \lambda(t)^{2}) d\omega &\leq C \frac{\lambda'(t)^{2}\lambda(t)^{2}}{t^{4}\log^{\epsilon}(t)}
0 = \int_\Sigma \Big \langle Q(h, v), \big( k^\intercal, H'(k) \big) \Big\rangle d\sigma = \int_\Sigma \Big \langle \big( vA + \bar u A'(h) - \nu (v) g^\intercal, 2v \big), \big( -2\zeta A'(h),\zeta A\cdot A'(h) \big) \Big\rangle\, d\sigma = -\int_\Sigma 2\bar u\zeta |A'(h)|^2 \, d\sigma
\P(E_{n, M})=[z^n][u_{s_1}^0]\cdots [u_{s_{n-q}}^0]\sum_{m_{i_1}, \ldots, m_{i_q}\ge 0}[u_{i_1}^{m_{i_1}}]\cdots [u_{i_q}^{m_{i_q}}]F(u_1,\ldots, u_r; z).
\lambda_p^{k,v}= \lim_{n\rightarrow \infty}n \cdot Cov \left(\frac{\hat{\gamma}_p(k/n;B^H)}{\gamma_p(k/n;B^H)},\frac{\hat{\gamma}_p(v/n;B^H)}{\gamma_p(v/n;B^H)}\right), k,v = 1, 2, \ldots, m,
m\left(B_r(x)\setminus B_r(y)\right) =m(B_r(x))-m(B_r(x)\cap B_r(y)) \leq m(B_r(x))-m(B_{r-d/2}(z)) \leq\left\{1-\frac{F(r-d/2)}{F(r+d/2)}\right\}m(B_r(x)) =\left\{\frac{F'(r-d/2)}{F(r+d/2)}d+o(d)\right\}m(B_r(x)) ,
P_{S,S_1|V,Y}(s,s_1|v,y)=P_{S|V,Y}(s|v,y)P_{S_1|V,Y}(s_1|v,y),\quad\forall(v,y,s,s_1)\in\mathcal{V}\times\mathcal{Y}\times\mathcal{S}\times\mathcal{S}_1.
d s_{4}^{2}= - \frac{1}{\sqrt{-{\frac{1}{6}}H^{0}C_{A B C}H^{A}H^{B}H^{C}}}d t^{2}+ \sqrt{-{\textstyle{\frac{1}{6}}}H_{0}C_{A B C}H^{A}H^{B}H^{C}}d \vec{x}^{2}.
\prod_{j=0}^{n-1}(1+t\omega^{ij}_n) \cdot \sum_{j=0}^{n-1}\frac{j\omega_n^{i(j-1)}t}{1+t\omega_n^{ij}}= \sum_{r=0}^{n}\omega_n^{i(\binom{r}{2}-1)}t^r \Big( \binom{r}{2}{n \brack r}_{\omega^i_n}+\omega_n^i{n \brack r}^{'}_{\omega_n^i}\Big).
\Gamma_{\Lambda R_{1}}= \Gamma_{\Lambda R_{1}}^{0}+ \hbar \sum_{n = 0}\Lambda^{- n}\Gamma_{\Lambda R_{1}}^{\left( 1 \right) n}+ \hbar^{2}\sum_{n = - 2}\Lambda^{- n}\Gamma_{\Lambda R_{1}}^{\left( 2 \right) n}+ O \left( \hbar^{3}\right)
\Psi_4 =\{[1,5,3,4,2] \}, \Psi_5 =\{[1,5,7,3,2,6,4],~[1,5,4,7,2,6,3] \}, \Psi_6 = \big\{w:=[1,3,2,8,6,9,4,5,7], w^{-1}, \dots \big\}, \Psi_7 = \{??? \},
\tilde{d}_\ell(x) := \sum_{1\le i\le \ell}\bigg( \frac{A_{3,4+p_i}(x)}{A_{0,1+p_i}(x)}-3\frac{A_{1,2+p_i}(x)A_{2,3+p_i}(x)}{A_{0,1+p_i}(x)^2}+ 2\frac{A_{1,2+p_i}(x)^3}{A_{0,1+p_i}(x)^3}\bigg).
|\lambda + M(\eta)|^2 \geq M(\eta)^2\left( \left( \frac{\cot(\omega)^2}{1 + \cot(\omega)^2}\right)^2 + \frac{\cot(\omega)^2}{(1+\cot(\omega)^2)^2}\right) = M(\eta)^2 \frac{\cot(\omega)^2}{1+\cot(\omega)^2}= M(\eta)^2 \cos(\omega)^2.
\bigg[\partial_{n+1}x_{n+1}^{1-2s}\partial_{n+1}+x_{n+1}^{1-2s}\sum_{j,k=1}^{n}a_{jk}\partial_{j}\partial_{k}\bigg]\tilde{u}=f\quad\mathbb{R}_{+}^{n+1}, \lim_{x_{n+1}\rightarrow0}x_{n+1}^{1-2s}\partial_{n+1}\tilde{u}=V\tilde{u}\quad\mathbb{R}^{n}\times\{0\},
W(0,z) =0, z\in\left[ 0,\infty\right), \frac{\partial W}{\partial r}(\gamma(\Omega),z) =0,z\in\left[ 0,\infty\right), \frac{\partial W}{\partial z}(r,0) \geq-d_s\int_{0}^{r}f^{\circledast}(\sigma)\,d\sigma,r\in(0,\gamma(\Omega)).
f_{F}^{( 1 )}( x ; \omega ) = \frac{1 + i \gamma^{1}}{2}f_{F}( x ; \omega ) = \frac{1}{2}\left( \begin{array}{c}{{f_{F}^{+}( x ; \omega ) - f_{F}^{-}( x ; \omega )}}\\{{- f_{F}^{+}( x ; \omega ) + f_{F}^{-}( x ; \omega )}}\\ \end{array}\right)
S_{1}:=\sum\limits_{k=1}^{k_{0}}\sum\limits_{m \in \mathbb{Z}^{n}}\gamma_{k,m}\delta^{l}(Q_{k,m})\varphi \le C(n,l,k_{0},C_{\gamma})\sum\limits_{m \in \mathbb{Z}^{n}}\gamma_{0,m}\|\varphi|L_{1}(Q_{0,m})\| \le \le C(n,l,k_{0},C_{\gamma}) \|f|W^{l}_{1}(\mathbb{R}^{n+1},\gamma)\|.
\zeta_{f}( s ) = - ( \varepsilon - 1 )^{2}\, \frac{\sin ( \pi s )}{\pi s}a^{2 s}\, 2^{4 s - 7}\, \frac{( s^{2}- 3 s + 4 ) \Gamma ( - 2 s + 2 )}{( s - 1 )},
\int x^{n-2}(q^2x^2;q^2)_\infty h_n(x;q) d_qx=\frac{x^{n}(x^2;q^2)_\infty}{[n-1]_q}\left(\frac{h_n(\frac{x}{q};q)}{x}- \frac{1}{q}h_{n-1}(\frac{x}{q};q) \right).
R_{\alpha,0}=\rho,R_{\alpha, \beta}=0, |\beta|=1, R_{\alpha,\beta}=-\frac{1}{\beta_1}\sum_{k=1}^D\theta_{1k}\left( R_{\alpha,\beta-e_1-e_k}-G(\beta-e_1-e_k) \right) + G(\beta), |\beta|>1,
x_{1,2j}+ x_{1,2j+1}+ x_{m,(n+1)-2j}+ x_{m,(n+1)-(2j+1)}&= x_{1,2j-1}+ x_{1,2j}+ \bigl( \tfrac{1}{2}S - x_{m,2j}\bigr) + \bigl( \tfrac{1}{2}S - x_{m,2j+1}\bigr) &= \bigl( (n_1 +1) m +1 \bigr) + S - \bigl( (n_1 +1) m +1 \bigr) =S.
b'_n = - \sum_{k = 2}^{n}B_{n,k}(b_1', b_2', \ldots) \frac{(k-1)!}{2}\sum_{j=0}^{k-1}a_j a_{k-1-j}\left(\frac{2n(j+1)(k-j) - k(k+1)}{n-1}\right)
-\left(\sum_{r=0}^l (-1)^l \binom{l}{r}t^{b-1,-a+r+1,\hat{J}}\right)(1-t_1) = -t^{b-1,-a+1,\hat{J}}(1-t_2)^l\,(1-t_1) & = t^{b,-(a-1),\hat{J}}(1-t_2)^l - t^{b-1,-(a-1),\hat{J}}(1-t_2)^l
\sum_{n \geq 1}\frac{a_n q^{(m+1)n}}{(1-q^n)^{k+1}}= \frac{1}{(q; q)_{\infty}}\sum_{n \geq 1}\sum_{i=1}^{\left\lfloor \frac{n}{m+1}\right\rfloor}\sum_{j=0}^{\left\lfloor \frac{n-m}{i}\right\rfloor}\binom{k-1+j}{k-1}s_{n-m-ji, i}\cdot a_i \cdot q^n,
\mathsf{B}\mathbb{P}=\{B\in\mathsf{F}\mathbb{P}:\mathbb{P}=B^\lessgtr\}. \mathsf{C}\mathbb{P}=\{C\in\mathsf{P}\mathbb{P}:\exists B\in\mathsf{B}\mathbb{P}\ (B\leq C)\}.
T \equiv \beta_{H}^{- 1}= \left. \frac{\tilde{A}^{\prime}}{4 \pi \sqrt{\tilde{A}\tilde{B}}}\right|_{r_{+}}\nonumber = \frac{\tilde{d}}{4 \pi r_{+}}\left[ 1 - \left( \frac{r_{-}}{r_{+}}\right)^{\tilde{d}}\right]^{( \tilde{d}- 2 ) / 2 \tilde{d}},