harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\mathrm{RW}_{C}(d) = \sum_{\ell=1}^L \mathbb{E}_{(Y_\ell, X) \sim \mu_{\ell, L+1}}\left[ (Y_\ell - \delta_0) I(D(X) = \ell) \right] & = \mathbb{E}_{X}\left[ \sum_{\ell=1}^L \left(\mathbb{E}[Y_\ell \mid X] - \delta_0 \right) I(D(X) = \ell) \right].
\frac{|1-a\overline{\alpha}|^2 D^{(a)}(\underline{z})}{|u^{(a)}_{\underline{z}}|}=(1-a^2)(a\overline{\alpha}+a\alpha-2|a\alpha|^2) + a (2a - a^2(\alpha+\overline{\alpha}) - (\alpha+\overline{\alpha}) + 2a|\alpha|^2 ) &= 2a^2(1-a\alpha-a\overline{\alpha}+|a\alpha|^2) = 2a^2|1-a\alpha|^2.
S_{\pm}( x , y ) = \langle q_{\alpha}^{a}( x ){\bar{q}}_{\beta}^{b}( y ) \rangle = \int d^{4}z \, \frac{\left[ \Psi_{z}^{\pm}( x ){\bar{\Psi}}_{z}^{\pm}( y ) \right]_{\alpha \beta}^{a b}}{m^{*}}
d s^{2}= - \left( 1 - \frac{r_{+}^{2}}{r^{2}}\right) d{\hat{t}}^{2}+ \left( 1 - \frac{r_{-}^{2}}{r^{2}}\right) d{\hat{x}}^{2}+{\left( 1 - \frac{r_{+}^{2}}{r^{2}}\right)}^{- 1}{\left( 1 - \frac{r_{-}^{2}}{r^{2}}\right)}^{- 1}\frac{l^{2}d r^{2}}{r^{2}}
-\frac{1}{4n}(h+2n) (r(h-2)-r(h+2)) &=-\frac{1}{4n}((h+2n)r(h-2)-(h+2n)r(h+2)) &\overset{by (\ref{rh-2 rh+2})}{{=\joinrel=\joinrel=}}-\frac{1}{4n}((h-2n)r(h+2)-(h+2n)r(h+2)) &=r(h+2).
\langle \, : ( \Delta \, P_{+})^{2}: \, \rangle_{\beta , \gamma}\simeq \langle \, : ( \Delta \, P_{-})^{2}: \, \rangle_{\beta , \gamma}\simeq 4 \, \langle N \rangle_{\beta , \gamma}\simeq 4 \, \alpha^{\prime}\, M^{2}\, .
\int_{|y|<\frac{1}{2|x_i-x_k|}}Q'(\tau_i^{-1}\tau_kQ)^{p-1}A_{ij,0}=\int_{|y|<\frac{1}{2|x_i-x_k|}}\bigg(\frac{\kappa_0}{(x_k-x_i)^2}\bigg)^{p-1}+O\bigg(\frac{1}{d^3}\bigg) &=\int Q'A_{ij}\bigg(\frac{c_0}{(x_k-x_i)^2}\bigg)^{p-1}+O\bigg(\frac{1}{d^3}\bigg)=O\bigg(\frac{1}{d^3}\bigg),
\frac{\partial^{2}C}{\partial x^{2}}\Rightarrow\frac{1}{2(\Delta x)^{2}}((C_{i+1}^{j+1}-2C_{i}^{j+1}+C_{i-1}^{j+1})+(C_{i+1}^{j}-2C_{i}^{j}+C_{i-1}^{j})),
\Bigg\vert \begin{gathered}\begin{pmatrix}x^{(k+1)}- x^* x^{(k)}- x^*\end{pmatrix}\end{gathered}\Bigg\vert\leq \begin{gathered}\begin{pmatrix}F G I 0 \end{pmatrix}\end{gathered}\Bigg\vert \begin{gathered}\begin{pmatrix}x^{(k)}- x^* x^{(k-1)}- x^*\end{pmatrix}\end{gathered}\Bigg\vert.
Q = \frac{1}{2 \pi}\left( \lim_{r{\rightarrow}{\infty}}\overline{{\cal{N}}}( r ) - \overline{{\cal{N}}}( 0 ) \right) = \frac{1}{2 \pi}\lim_{r{\rightarrow}{\infty}}\overline{{\cal{N}}}( r ) = \frac{m}{2},
\mu-{x}\mu\overline{x}+c_{1}{\bf i}+c_{2}{\bf j}+c_{3}{\bf k}+d_{1}{\bf i}+d_{2}{\bf j}+d_{3}{\bf k}=0, &-x\mu\overline{w}+c_{0}{\bf i}+c_{2}{\bf k}-c_{3}{\bf j}+d_{1}=0, &\mu -{w}\mu \overline{w}-c_{1}{\bf i}-c_{2}{\bf j}-c_{3}{\bf k}+d_{1}{\bf i}+d_{2}{\bf j}+d_{3}{\bf k}=0.
A_0 = \{\omega:\|u_1-u_1'\|>q \|u_0-u_0'\|\}, B_i = \{\omega:\|u_i-u_i'\|\leq q \|u_{i-1}-u_{i-1}'\|\}, A_k = \big(\cap_{i=1}^kB_i\big)\cap B_{k+1}^c, \forall 1\leq k\leq n-1, A_{n}= \cap_{i=1}^{n}B_i.
\left(\mathbf{G}\mathbf{O}_2\mathbf{F}^{\dagger}\right)_{ij}= \left(\mathbf{G}\mathbf{O}_2\mathbf{F}^{\dagger}\right)_{ji}= \sum_{k=1}^{n}\mathbf{G}_{jk}\mathbf{O}_{2,kk}\mathbf{F}^{\dagger}_{ki}.
b^{(k)}= (b_{11}^{(k)}-b_{33}^{(k)}) - (b_{22}^{(k)}-b_{44}^{(k)}) = 2b_{13}^{(k)}\cot2\phi_{k+1}- 2b_{24}^{(k)}\cot2\psi_{k+1}& = (b_{13}^{(k)}+ b_{24}^{(k)})(\cot2\phi_{k+1}-\cot2\psi_{k+1}) + (b_{13}^{(k)}-b_{24}^{(k)})(\cot2\phi_{k+1}+\cot2\psi_{k+1}).
0\leq \mathbb{P}(\xi^{(n)}(I_{n,k})=\xi^{(n)}(J_{n,k,j})=0)=\mathbb{P}(\xi^{(n)}(I_{n,k}\cup J_{n,k,j})=0) \leq \mathbb{P}([z_j',z_j'+d_0]\cup \left(\cup_{i=1}^k[z_i,z_i+d_0]\right))=0) \leq \mathbb{P}(\xi^{(n)}([z_j',z_j'+d_0])=0)\prod_{i=1}^k\mathbb{P}(\xi^{(n)}([z_i,z_i+d_0])=0)\leq p_{n,k}^{k+1},
\overline{G_N}( [0, x] \times I \left. \right| \overline{a}_0, \overline{a}_{-1}, \ldots ) =\lim_{n \rightarrow \infty}\overline{G_N}( [0, x] \times I \left. \right| \overline{a}_0, \ldots, \overline{a}_{-n}) \overline{G_N}\mbox{-a.s.}
(a,{\boldsymbol{v}}) \mapsto (a-2,{\boldsymbol{v}}) \mbox{withprobability}\frac{a-1}{\sum_{i=1}^\infty i v_i + a-1}, (a,{\boldsymbol{v}}) \mapsto (a+k-2,{\boldsymbol{v}}-{\boldsymbol{e}}_k) \mbox{with probability}\frac{kv_k}{\sum_{i=1}^\infty i v_i + a-1}, \:k \in \mathbb{N}.
\overline{{{v}}}^{s , \pm}( \mathbf{p}) v^{r , \mp}( \mathbf{p}) = \varepsilon_{s}\delta_{s r}\frac m{p_{0}}, \hspace{0 . 3 i n}\left( \overline{{{v}}}^{s , \pm}( \mathbf{p}) = v^{* s , \pm}( \mathbf{p}) \Gamma_{4}\overline{{{\Gamma}}}_{4}\right) ,
W^2_{\nu_1,...\nu_k}(\mu_0,\mu_1)=\inf_{\gamma_{y_1...y_kx_i}\in \Pi^k_{opt}(\nu_1,...,\nu_k, \mu_i), i=0,1}\int_{X^{k+2}}|x_0-x_1|^2d\gamma(y_1,....,y_k,x_0,x_1).
\dfrac{\partial}{\partial\rho}f_{Y|X}(\rho x'+z'|x) =\frac{1}{\sqrt{2\pi}}\dfrac{\partial}{\partial\rho}\left[e^{-(\rho x'- \rho x+z')^2/2}\right] &=-\frac{1}{\sqrt{2\pi}}e^{-(\rho x'- \rho x+z')^2/2}(\rho x' - \rho x + z') (x'-x),
\begin{aligned}\frac{\partial}{\partial{F_k}}\left\{\sum_{l_k \in \Pi}{\frac{d_{S_k, D_k}^2}{F_k}}+ \theta_1 \left( \sum_{l_k \in \Pi}F_k- \epsilon_{so}\right) \right\}\Bigg{|}_{F_k^*}=0, l_k \in \Pi,\end{aligned}
{\phi}= \left(\psi^{{}^\parallel}\;,\;P_k+i\psi_k\right)\;,\;\;T_0 = \epsilon_{jkl}\pi_j\psi^{k\perp}\psi^{l\perp}-\frac{i\alpha}{2}\zeta\omega\;,\;\;T_k = -\zeta\omega\epsilon_{kjl}\psi^{j \perp}\psi^{l \perp}+\frac{i\alpha}{2}\pi_k\;.
\mathcal{C}_d=\left\{(x_1,\dots,x_{d+1},y_1,\dots,y_{d+1}) \in (0,1)^{d+1}\times \mathbb{R}^{d+1}:\begin{aligned}x_1 y_1 + \cdots + x_{d+1}y_{d+1}= 0 x_1 + \cdots + x_{d+1}= 1 y_i \ne y_{i+1}\textrm{for all}1 \le i \le d\end{aligned}\right\}
DS_{N}(u, v) = \int_{\mathcal{C}_{\{\tau/\sigma_{l}\}}}\frac{dz}{2\pi i}\int_{\mathcal{C}_{\{\sigma_{l}/\tau\}}}\frac{dw}{2\pi i}\frac{\eta_{-}^2}{4\pi}\frac{e^{-\eta_{-}v(1+\frac{1}{z})}}{z\sqrt{1 - z^{2}}}\frac{e^{-\eta_{-}u(w+1)}}{\sqrt{w^{2}-1}}\psi_{N, \alpha, \tau}(z, w),
\int | x | \y^{2}\ \Gamma_{x , y , 0}^{\prime}\ \mathrm{d}^{3}x \mathrm{d}^{3}y - 2 \int | x | \y \cdot z \ \Gamma_{x , y , z , 0}^{\prime}\ \mathrm{d}^{3}x \mathrm{d}^{3}y \mathrm{d}
\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):(i,i,t,p)\in\mathcal{I}'_{k-1}\}|+|\mathcal{I}'_{k-1}|.
4A(h)C(h)-B^{2}(h)=16u\left( s-v\right)^{2}R(h)\allowbreak{\large ,}{\large (}C(h)D^{2}(h)+A(h)E^{2}(h)-B(h)D(h)E(h){\large )}-F(h){\large (}4A(h)C(h)-B^{2}(h){\large )}= =\allowbreak 16\left( s-v\right)^{2}u^{2}R^{2}(h)
\widehat{\Phi}_{e}* \widehat{\Phi}_{e}= \lim_{\epsilon \to 0}a_{+}K^{( 3 )}| f_{1}^{( 3 ) \prime}( i + \epsilon ) |^{\frac{1}{4}}\left( e^{- \frac{1}{2}\rho ( i )}e^{- \frac{1}{2}\bar{\rho}( i )}\cdot e^{\frac{1}{2}\rho ( i - \epsilon )}e^{\frac{1}{2}\bar{\rho}( i - \epsilon )}\right) \widehat{\Phi}_{e}.
\mathfrak{S}_{n;p,q}^{z,w}=(2\pi^{n})^{-1}\Gamma(n)d(n,p,q)(|z||w|)^{p+q}R_{p,q}^{n-2}\left( \left\langle \frac{z}{|z|},\frac{w}{|w|}\right\rangle \right) .
d X^{0}= \sum_{i = 1}^{g}\left( \lambda_{i}\omega_{i}+ \bar{\lambda}_{i}\bar{\omega}_{i}\right) + \mathrm{e x a c t}, d X^{9}= \sum_{i = 1}^{g}\left( \gamma_{i}\omega_{i}+ \bar{\gamma}_{i}\bar{\omega}_{i}\right) + \mathrm{e x a c t}
f(x_1-x_2) =\ \sum_{r\ge 0}\lambda_r ((x_1-x_2)^{-r-1}-(-x_2+x_1)^{-r-1})+f(-x_2+x_1) =\ f(-x_2+x_1)+\sum_{r\ge 0}\lambda_{r}\frac{1}{r!}\left(\frac{\partial}{\partial x_2}\right)^rx_1^{-1}\delta\left(\frac{x_2}{x_1}\right).
\sum_{k=0}^{m}[3k+1]\frac{(q;q^2)_k^3 q^{-{k+1\choose 2}}}{(q;q)_k^2 (q^2;q^2)_k}\equiv q^{(1-n)/2}[n] \pmod{[n]\Phi_n(q)^2}, \sum_{k=0}^{n-1}(-1)^k [3k+1]\frac{(q;q^2)_k^3}{(q;q)_k^3}\equiv q^{(n-1)^2/4}[n](-1)^{(n-1)/2}\pmod{[n]\Phi_n(q)^2}.
[f(v_{n_{j}}+h_{j})-f(v_{n_{j}})]w =g(v_{n_{j}}+h_{j}) [|v_{n_{j}}+h_{j}|^{p-1}-|v_{n_{j}}|^{p-1}] w & +[g(v_{n_{j}}+h_{j})-g(v_{n_{j}})] |v_{n_{j}}|^{p-1}w.
\alpha_1 = \frac{q^{- l_1 + 1/2}}{t_1}, alpha_2 = \frac{q^{-l_2 +3/2}}{t_2}, alpha_3 = \frac{q^{- l_3 + 1/2}}{t_3}, beta_1 = \frac{q^{(-h_1+h_2+h_3-l_1-l_2-l_3)/2 +1}}{t_1}, & \beta_2 = \frac{q^{(h_1-h_2+h_3-l_1-l_2-l_3)/2 +1}}{t_2}, \beta_3 = \frac{q^{(h_1+h_2-h_3-l_1-l_2-l_3)/2 +1}}{t_3}.
\lim_{t \to 0^+}\det \Big(Q + c_i B_i^\ast A^{(t)}_i B_i + \sum_{\stackrel{1 \le k \le m}{k \neq i}}c_k B_k^\ast A_k B_k \Big) = \det\Big(Q + c_i \tilde{B}_i^\ast \tilde{A}_i \tilde{B}_i + \sum_{\stackrel{1 \le k \le m}{k \neq i}}c_k B_k^\ast A_k B_k \Big) > 0
F_{11}= -\frac{\phi^{(4)}(Y)}{\phi''(Y)^2}Y_1^2+ 2\frac{\phi^{(3)}(Y)^2}{\phi''(Y)^3}Y_1^2- \frac{\phi^{(3)}(Y)}{\phi''(Y)^2}Y_{11}= \bigg(\frac{1}{\phi''}\bigg)''(Y)Y_1^2- \frac{\phi^{(3)}(Y)}{\phi''(Y)^2}Y_{11}\le 0, F_{22}= \bigg(\frac{1}{\phi''}\bigg)''(Y)Y_2^2- \frac{\phi^{(3)}(Y)}{\phi''(Y)^2}Y_{22}\le 0.
\begin{array}{r c l}{{}}&{{}}&{{m ( S \otimes \mathrm{i d}) \Delta ( H ) = S ( H ) + H = 0 ~ ,}}\\{{}}&{{}}&{{m ( S \otimes \mathrm{i d}) \Delta ( X^{+}) = S ( X^{+}) q^{- H}+ S ( q^{H}) X^{+}= 0 ~ .}}\\ \end{array}
\boldsymbol{\phi}_n(f) ``:=" (-1)^n\int f(x_n) R_V(x_n,x_{n-1}) R_V(x_{n-1},x_{n-2})\dots R_V(x_1,x_{0}) \; \left( \lambda(x_{n-1})\dots \lambda(x_0)\right) \cdot \; \cdot\left(\dot\varphi^2(x_{n-1}) \star_\omega \dots \star_\omega \dot\varphi^2(x_0) \right) \phi_0(x_0) \; dx_0\dots dx_n
\mathfrak{E}_{q_1,q_2,q_2'}\ll \begin{cases}\sqrt{QX}Q^3+Q^{5/2+\varepsilon}Q_2^{3/2}\;\;q_2=q_2'; Q^{5/2+\varepsilon}\sqrt{Q_2}\;\;q_2\neq q_2'.\end{cases}
\Phi^{[B;m]}_{1}\left(\frac{\tau}{M},\frac{z_1}{M},\frac{z_2}{M}\right)=\left(\sum\limits_{j,k\geq0 \atop k\,odd}-\sum\limits_{j,k<0 \atop k\,odd}\right)(-1)^j e^{\frac{\pi i(2m+1)}{M}j(z_1+z_2)+\frac{\pi i}{M}k z_1}q^{\frac{1}{M}\left(\left(m+\frac{1}{2}\right)j^2+\frac{jk}{2}\right)}.
\alpha[\frac{s}{m}]=\epsilon_{r}\epsilon_{0}\cdot\frac{4\pi}{1000}\frac{Oe}{A/m}\cdot100\frac{cm}{m}\cdot\alpha[\frac{V}{cmOe}]=1.113\cdot10^{-11}\epsilon_{r}\cdot\alpha[\frac{V}{cmOe}]
\lim_{k}\frac{\log (sp(B^{m_{k}}T(\eta_{1})A^{n_{k}}T(\eta_{2})))}{L(\theta^{-})m_{k}+L(\varphi^{-})n_{k}}=\lim_{k}\frac{n_{k}\log \alpha+m_{k}\log \beta}{L(\theta^{-})m_{k}+L(\varphi^{-})n_{k}}&=\frac{t\log \alpha}{L(\varphi^{-})}+\frac{(1-t)\log \beta}{L(\theta^{-})}.
W_{i j}^{( p^{\prime}) \i j}( \Omega , \Omega ) = \sum_{{\cal P}k l m}q_{i j}^{( p^{\prime}){\cal P}k l m}( \Omega ) q^{( p^{\prime}){\cal P}l m \i j}( \Omega )^{*}= - 7 2 \alpha ( 1 ) .
\begin{array}{c c c}{{s A_{\mu}= - D_{\mu}c \; ,}}&{{s c = c^{2}\; ,}}&{{s \phi = \left[ c , \phi \right] \; ,}}\\{{s \psi_{\mu}= \left\{c , \psi_{\mu}\right\}\; ,}}&{{s \chi_{\mu \nu}= \left\{c , \chi_{\mu \nu}\right\}\; ,}}&{{s \eta = \left\{c , \eta \right\}\; ,}}\\{{s \overline{{{\phi}}}= \left[ c , \overline{{{\phi}}}\right] \; ,}}&{{}}&{{}}\\ \end{array}
\Pi_{it, f}\varphi (v) = e^{-it \log \langle f, v\rangle}\Pi_{it}\left[ e^{it \log \langle f, \cdot \rangle}\varphi(\cdot) \right](v), N_{it, f}\varphi(v) = e^{-it \log \langle f, v\rangle}N_{it}\left[ e^{it \log \langle f, \cdot \rangle}\varphi(\cdot) \right](v).
+\frac{1}{2}\Phi^{-\frac{D-3}{D-2}}g^{ij}D_i A^\alpha \cdot D_j A_\alpha+q\Phi^{-\frac{1}{2}}(\overline{\psi}_+ A\psi_+ +\overline{\psi}_-A\psi_-)-M\Phi^{-\frac{1}{2(D-2)}}(\psi_+^+\psi_- +\psi_-^+\psi_+)
J_n^1 =\mathsf E\sum_{j=0}^{n-1}\frac{\sigma_{j+1}^2}{s_n^2}\left(v_t + \frac{1}{2}\left( \overline\lambda^2 v_{xx}^+ -\underline\lambda^2 v_{xx}^-\right)\right)(t_j,X_j), J_n^2 =\frac{1}{2}\mathsf E\sum_{j=0}^{n-1}\frac{\sigma_{j+1}^2}{s_n^2}\left((\lambda_j^2-\overline\lambda^2)v_{xx}^+ +(\underline\lambda^2-\lambda_j^2) v_{xx}^- \right)(t_j,X_j).
\sum_{i = 1}^{l(v)}\left(\prod_{j = 1}^i\beta(f_j^v)\right)y_{f^v_i, n}= \sum_{i = 1}^{l(v)}\left(\prod_{j = 1}^i\beta(f_j^v)\right)(x_{v_2(f_i^v), n}- \beta(f_i^v) x_{v_1(f_i^v), n}) = \left(\prod_{j = 1}^{l(v)}\beta(f_j^v)\right)x_{v, n}- x_{v_0, n}
\frac{\chi}{4(2\pi)^3}\sum_{(m,n)\in \mathbb{Z}^2-(0,0)}\frac{1}{|m\tau+n|^3}>\frac{\chi}{4(2\pi)^3}\sum_{n\in \mathbb{Z}-\{0\}}\frac{1}{|n|^3}=\frac{\chi\zeta(3)}{2(2\pi)^3}\;.
_r\phi_s\left(\genfrac{}{}{0pt}{}{a_1,a_2,\dots,a_r}{b_1,b_2,\dots,b_s};q,z\right):=\sum_{n=0}^\infty \frac{(a_1;q)_n(a_2;q)_n\dots (a_r;q)_n}{(q;q)_n(b_1;q)_n\dots(b_s;q)_n}\left[(-1)^nq^{n\choose 2}\right]^{1-r+s}z^n.
\sum_{\begin{subarray}{c}0\leq k\leq n+1 k\neq i\end{subarray}}\frac{\prod_{p=0}^{d-1}\big(p\alpha_i+(d-p) \alpha_k\big)}{(\alpha_k- \alpha_i)\prod_{q\neq i,k}(\alpha_k- \alpha_q)}=-\frac{(d \alpha_i)^{d}}{\prod_{q\neq i}(\alpha_i- \alpha_q)}.
\mathcal{C}_{\mathrm{\tiny ~ F P}}c = \bar{c}\; , \; \; \mathcal{C}_{\mathrm{\tiny ~ F P}}\bar{c}= - c \; , \; \; \mathcal{C}_{\mathrm{\tiny ~ F P}}B = B + \widetilde Z_{1}( 1 - \alpha ) ( \bar{c}\times c ) \; , \; \; \mathcal{C}_{\mathrm{\tiny ~ F P}}A = A \; .
M_{\alpha}(t)=\frac{\sqrt{\pi}e^{t\mu}}{\Gamma(1/\alpha)}{\rm{H}}_{1,2}^{1,1}\left[\frac{-\sigma^2\Gamma(1/\alpha)}{4\Gamma(3/\alpha)}t^2\left|\genfrac{}{}{0pt}{}{(1-\frac{1}{\alpha},\frac{2}{\alpha})}{(0,1),(\frac{1}{2},1)}\right.\right].
\bar{\gamma}_{ki}^{s,p}=\frac{\bar{\upsilon}_{ki}^{s,p}}{{\displaystyle \sum_{j=1}^{L}\alpha_{j}\mathbb{E}\left[\mathsf{\bar{L}}_{ji}\right]+\bar{\upsilon}_{ki}^{s,p}\sum_{j\neq i}^{L}\frac{\ell_{kji}^{2}}{\ell_{kjj}^{2}}}}
V ( x ) = \frac{1}{N}\ln ( x^{2}) + V^{\prime}( x ) , \; \; V^{\prime}( x ) = \frac{2}{N}\left[ \mathrm{K}_{0}\left( e \sqrt{\frac{N}{\pi}}| x | \right) + \ln \left( \frac{e}{2}\sqrt{\frac{N}{\pi}}\right) + \gamma \right] \; .
\sum_{i_0=0}^\infty \sum_{i_1\geq i_0}^\infty \cdots \sum_{i_{d}\geq i_{d-1}}^\infty 2^{-(i_0+i_1+\cdots +i_{d})}|\Delta(s,i_0,\dots,i_{d})| q^{-s(d-k)}\lesssim 1.
\frac{\tilde\lambda_{j-1}}{\tilde\lambda_{j-1}-\tilde\lambda_j}=1+\frac{\tilde\lambda_{j}}{\tilde\lambda_{j-1}-\tilde\lambda_j}\leq 1+2\frac{\lambda_{j-1}}{\lambda_{j-2}-\lambda_{j-1}}\leq 2\frac{\lambda_{j-2}}{\lambda_{j-2}-\lambda_{j-1}}.
a_n(\kappa, t) = \frac{2}{2^{2n}n}\sum_{k=1}^{n}\binom{2n}{n-k}e^{-kt}\left\{\sum_{m= 0}^{k-1}L_{k-m-1}^{(m+1)}(2kt)\,2^{m}P_n^{(m)}(\kappa^2)\right\}.
U^-_1(x,t) = f^-_0\int\limits_{\eta}^{\infty}\! \mathrm{Ai}(\theta)\, d\theta +\sum\limits_{n=1}^{N-1}f^-_n (3t)^{-n/3}\left[ \int\limits_{-\infty}^{-1}\theta^{-n}\mathrm{Ai}(\eta-\theta)d\theta +B^-_n(\eta) \right]+
\det (I_{n+2M}-u{\cal A})=(1+2( \beta +2 \delta )u+4 \delta ( \delta +1) u^2 )^{M-N}&\times \det ((1+ \delta u)(1+2( \beta +2 \delta )u+4 \delta ( \delta +1) u^2 ) I_N &- (1+2 \delta u) \beta u (1+( \beta +2 \delta )u){\bf A}(G) + \beta{}^2 (1+2 \delta u) u^2{\bf D}_G ) .
0 \le \frac{\partial^2}{\partial s^2}G(\rho(s))\Big|_{s=0}=\Bigl(G''(\rho(s))(\rho'(s),\rho'(s))+G'(\rho(s))\rho''(s)\Bigr) \Big|_{s=0}&= G''(u)(v,v)+G'(u)\rho''(0)= G''(u)(v,v),
\frac{1}{(n-1)!}[\begin{matrix}n\\k+1\end{matrix}]=\sum_{i_{1}=1}^{n-1}\sum_{i_{2}=i_{1}+1}^{n-1}\cdot\cdot\cdot\sum_{i_{k}=i_{k-1}+1}^{n-1}\frac{1}{i_{1}i_{2}\cdot\cdot\cdot i_{k}}=\frac{w(n,k)}{k!}
\frac{1}{4\beta}F^\infty_{\nu,\alpha\beta}(x,y)=-\int_1^\infty\frac{d\alpha}{\alpha^2}j^{\infty}_{\nu,\alpha\beta}(\alpha x,y)-\frac{1}{2}\frac{\partial}{\partial x^\nu}\int_1^\infty \frac{d\alpha}{\alpha}\left (1-\frac{1}{\alpha}\right)x^\rho j^{\infty}_{\rho,\alpha\beta}(\alpha x,y)
f(x_{\lfloor\lambda\rfloor+1})\ll C_p(f)^{\frac{1}{p-1}}\left(\frac{1}{\deg(u)^{\frac{1}{p-1}}}+\sum_{n=0}^{\lfloor\log_2(\lambda/\deg(u))\rfloor}\max_{\substack{A\subset\Gamma\setminus U' u\in A A \lambda/2^{n+1}<|A|\le\lambda/2^n}}j_{A,p}\right).
V=\int_{0}^{r_{g}}A\sqrt{g_{rr}}dr=2\pi(\frac{r_{g}^{9/2}arcsin\sqrt{\frac{r_{s}}{r_{g}}}}{r_{s}^{3/2}}-\frac{r_{g}^{4}\sqrt{1-\frac{r_{s}}{r_{g}}}}{r_{s}})
\Psi(\varphi)=\Psi^{\alpha\beta}_{ml}(L,D,R)=\sum\limits^{N(\alpha)}_{n=1}\sum\limits^{N(\beta)}_{k=1}D^{\alpha}_{mn}(L)f^{\alpha\beta}_{nk}(D)D^{\beta}_{kl}\left(R^{-1}\right),
[a_{l-1},a_{(1\,l)}]_c =-\chi_{(1\,l)}(g_{l-1})[a_{(1\,l)},a_{l-1}]_c &=\lambda_{l-1l-1l}(1-\xi^2)\chi_{(1\,l)}(g_{l-1})a_{(1\,l-2)}g_{l-1l-1l}.
\begin{array}{rcl}\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}
\Phi_{0}\vert_{V_{\ell}}( \tau , z , \nu ) = - \sum_{g = 0}^{\infty}x^{2 g - 2}\varphi_{2 g - 2 , m}\vert_{V_{\ell}}( \tau , z ) \, , \quad ( \ell = 0 , 1 , 2 , \ldots ) \, .
<F>=\frac{<p_{g.s.}|F|p_{g.s.}>}{<p_{g.s.}|p_{g.s.}>}=\frac{<g.s.|F|g.s.>+\epsilon^{2}<p_{g.s.}^{1}|F|p_{g.s.}^{1}>}{<g.s.|g.s.>+\epsilon^{2}<p_{g.s.}^{1}|p_{g.s.}^{1}>}
S(t)(u_{0}^{n}, u_{1}^{n}) = \sum_{j = 1}^{N}S(t - t_{n}^{j}) (\lambda_{n}^{j}\phi_{0}^{j}(\lambda_{n}^{j}x), (\lambda_{n}^{j})^{2}\phi_{1}^{j}(\lambda_{n}^{j}x)) + S(t)(R_{0, n}^{N}, R_{1,n}^{N}).
X_s=2ig_s\sum_{\alpha\in\Delta_s}x_{d}^{(1/2)}(\alpha\cdot q, \xi)E_{d}(\alpha),\quad Y_s=ig_s\sum_{\alpha\in\Delta_s}y_{d}^{(1/2)}(\alpha\cdot q, \xi)E_{d}(\alpha),\quad E_{d}(\alpha)_{\Upsilon \Omega}=\delta_{\Upsilon-\Omega,2\alpha}.
A_i^{(N)}(t) = - \sum_{\substack{j=1 j\neq i}}^{N}\frac{1}{\sigma_N}\Phi' \left( \frac{\left|x_i^{(N)}(t) - x_j^{(N)}(t)\right|}{\sigma_N}\right) \frac{x_i^{(N)}(t) - x_j^{(N)}(t)}{\left|x_i^{(N)}(t) - x_j^{(N)}(t)\right|},
\frac{R^{\prime \prime}}{R}= \frac{1}{R}\frac{\partial}{\partial \tilde{U}}\left( \frac{\partial}{\partial \tilde{U}}R \right) = \frac{G^{2}}{R W^{2}}\frac{\partial}{\partial u}\left( \frac{G^{2}}{W^{2}}\frac{\partial}{\partial u}R \right) .
I_{4 d}^{\mathrm{M C S}}= -{\frac{1}{2 \kappa_{( 4 )}^{2}}}\int d^{4}x \; \left({\frac{1}{4}}F^{2}+{\frac{1}{2}}( d a )^{2}+{\frac{k}{2}}\; a \; F \wedge F \right) \ ,
\gamma_{2134}\left( d_{1},d_{2},d_{3},d_{4}\right) & =\left( d_{1},d_{2},d_{3},d_{4},d_{1}d_{2},\underset{6}{0},...,\underset{11}{0},d_{1}d_{2}d_{3},\underset{13}{0},...,\underset{16}{0},d_{1}d_{2}d_{4},\underset{18}{0},...,\underset{35}{0},d_{1}d_{2}d_{3}d_{4},\underset{37}{0},...,\underset{53}{0}\right)
\frac{\mathcal{D}\left((s_{k}')_{k=0}^{2m-3}, (as^{(1)}_{k-1})_{k=0}^{m-1}\right)}{ab\mathcal{D}\left((s_{k}')_{k=0}^{2m-3}, (1,0,\ldots,0) \right)}= \frac{\mathcal{D}\left((s_{k}')_{k=0}^{2m-3}, (s^{(1)}_{k-1})_{k=0}^{m-1}\right)}{b\mathcal{D}\left((s_{k}')_{k=0}^{2m-3}, (1,0,\ldots,0) \right)}.
V(r) \sim -\frac{1}{\pi^2 M_5^3 \lambda r}\left[ \frac{\pi}{2}- \frac{\frac{2R}{\lambda}}{\frac{2R}{\lambda}- \frac{15}{8}}\left\{\frac{\pi}{2}- f\left(\frac{r}{R}\left(\frac{2R}{\lambda}- \frac{15}{8}\right) \right) \right\}\right]
[z_1, \dots, z_N ]^T = [\underbar{z}^{(CR)^T}_1, \dots, \underbar{z}^{(CR)^T}_K ]^T = [\underbar{z}^{(LI)^T}_1, \dots, \underbar{z}^{(LI)^T}_L ]^T = \underbar{z}\in \mathbb{R}^N .
\mathbf{W}_{\rm{s}}=\arg\max\nolimits_{{\mathsf{tr}}\left({\mathbf{W}}{\mathbf{W}}^{\mathsf{H}}\right)\leq p}{\log_2\det\left({\mathbf{I}}_M+L{\mathbf{W}}^{\mathsf{H}}{\mathbf{R}}{\mathbf{W}}\right)}.
Y(t_{n,i+1})-Y^{(n)}(t_{n, i+1})=Y(t_{n, i})-Y^{(n)}(t_{n, i})+\int_{t_{n, i}}^{t_{n, i+1}}(g(s, W_0^s, Y_0^s)-g(s, W_0^{t_{n, i}}, Y_0^{(n), t_{n, i}})) ds.
d s^{2}/ \alpha^{\prime}= \frac{U}{R_{5}f_{1}^{1 / 2}}( - d t^{2}+ d x_{1}^{2}) + \frac{U f_{1}^{1 / 2}}{R_{5}}( d x_{2}^{2}+ \cdots + d x_{5}^{2}) + U R_{5}f_{1}^{1 / 2}\left( \frac{d U^{2}}{U^{2}}+ d \Omega_{3}^{2}\right) .
\begin{array}{l}c_{2a}f_{-}(u_{+}){\bf D}_{a+1}{\bf D}_{a+1}^{'}+c_{1a}f_{+}(u_{-}){\bf D}_{a+1}{\bf U}_{a-1}^{'}+f_{-}{\bf X}^{a+1}_{a,a+2}{\bf X}^{'a+1}_{a+2,a}\mbox{}=c_{1a}f_{-}(u_{+}){\bf U}_{a-1}^{'}{\bf U}_{a-1}+c_{2a}f_{+}(u_{-}){\bf D}_{a+1}^{'}{\bf U}_{a+1}+f_{-}{\bf X}^{'a-1}_{a,a-2}{\bf X}^{a-1}_{a-2,a}\end{array}
\int_{\Omega}\left\langle A(x)Xu(x),Xu(x)\right\rangle^{\frac{p(x)-2}{2}}\left\langle A(x)Xu(x),Xv(x)\right\rangle \,dx & +\int_{\Omega}|u(x)|^{p(x)-2}u(x)v(x)\omega(x)^{p(x)}\,dx=\int_{\Omega}f(x)v(x)\,dx,
(2d-1)\sum_{1\leq i<j\leq M}\lambda_i\lambda_j\|AD(\beta_i-\beta_j)\|_2^2=\sum_{i=1}^{M}\lambda_i\|AD(\sum_{j=1}^{M}\lambda_j\beta_j-d\beta_i)\|_2^2-\sum_{i=1}^{M}\lambda_i(1-d)^2\|AD\beta_i\|_2^2.
{\frac{m_{i}^{( d )}}{m_{j}^{( d )}}}={\frac{\mu_{i}^{( d )}}{\mu_{j}^{( u )}}}\left[{\frac{\lambda_{j}^{( d )}\lambda_{j}^{( u )}}{\lambda_{i}^{( d )}\lambda_{i}^{( u )}}}\right]^{1 / 2}
{\bf D}^{(z)}(t,x)~=~z_x(t,0-)\cdot (t-x)\cdot \phi_0'(x-t)+(z_1(t,0)-z_1(t,x))\cdot \phi'_0(x)+\left[a^{(z)}_1(t)-a^{(z)}_2(t)\right]\cdot \phi_0'(t),
as_{\lambda}(F_1, F_2, \psi) = \int_{X_\psi}\Big(F_1 (\psi(x))\Big)^{1-\lambda}\Big(F_2(\langle x, \nabla \psi(x)\rangle - \psi(x))\Big)^{\lambda}\Big(\det \, \nabla^2 \psi(x)\Big)^{\lambda}dx.
\frac{\partial^2 A_\varphi}{\partial \rho^2}+ \frac{1}{\rho}\frac{\partial A_\varphi}{\partial \rho}- \frac{A_\varphi}{\rho^2}- \gamma C_2\bigg(A_\varphi+ \frac{1}{\rho^2}\int^{\rho}_{0}A_\varphi(\rho^{\prime}) \rho^{\prime}d\rho^{\prime}\bigg)= m_{CS}\gamma e\lambda K_1(\lambda \rho).
e^{i \Gamma_C [\vec{\varphi}]}= \langle T_C \mbox{exp}\left( i \int_C \left[{\cal L}' (\vec{\phi}(x)) - \frac{\delta \Gamma_C [ \vec{\varphi}]}{\delta \vec{\varphi}(x)}\cdot \{\vec{\phi}(x) - \vec{\varphi}(x) \}\right] \right) \rangle .
(x-1)^{\kappa(\Gamma)}{\rm Tutte}(\Gamma,x,y):= T_n(\{q_{ij}\}, x,y) \bigg |_{q_{ij}=0 \, \, (i,j) \notin \Gamma \atop q_{ij}={[m_{ij}]}_{y}\, \, (i,j) \in \Gamma}.
\partial_{y}^{2}\Big [ ( 1 + c \delta_{0}) f_{n}^{S}\Big ] = \frac{a \delta_{0}^{\prime}}{1 + a \delta_{0}}\partial_{y}\Big [ ( 1 + c \delta_{0}) f_{n}^{S}\Big ] - m_{n}^{2}( 1 + a \delta_{0}) f_{n}^{S},
e^{-\mathcal{A}_{D,\varepsilon}t}-e^{-\mathcal{A}_D^0t}-\varepsilon \mathcal{K}_D(t;\varepsilon) &=-\frac{1}{2 \pi i}\int_\gamma e^{-\zeta t}\left( (\mathcal{A}_{D,\varepsilon}-\zeta I)^{-1}-(\mathcal{A}_D^0-\zeta I)^{-1}-\varepsilon K_D(\varepsilon;\zeta )\right) \,d\zeta ,
\Phi_k^-(x^{(k)})\star1 (\mathfrak{X}_k^{-}(u, \lambda), \mathfrak{X}_k^{+}(v, \lambda))+ \mathfrak{X}_k^{-}(u, \lambda)\star \mathfrak{X}_k^{+}(v, \lambda) (1, \Phi_k(x^{(k)})) = \mathfrak{X}_k^{+}(v, \lambda)\star \mathfrak{X}_k^{-}(u, \lambda) (1, \Phi_k(x^{(k)}))+\Phi_k^+(x^{(k)})\star1 (\mathfrak{X}_k^{+}(v, \lambda), \mathfrak{X}_k^{-}(u, \lambda) ).
f_{Y}(x;\mu,\sigma^{2})=\frac{1}{\sqrt{2\pi\sigma^{2}}}e^{-\frac{(x-\mu)^{2}}{2\sigma^{2}}}+\frac{1}{\sqrt{2\pi\sigma^{2}}}e^{-\frac{(x+\mu)^{2}}{2\sigma^{2}}}
\tilde{\psi}^{\alpha}( \tilde{x},{\tilde{\tau}},{\tilde{\omega}}) = \psi^{\alpha}( \tilde{x},{\tilde{\tau}}) + \frac{{\tilde{\omega}}^{2}}{4}\tilde{\cal D}^{2}\psi^{\alpha}( \tilde{x},{\tilde{\tau}}) ,
\mathcal{H}^2(\Omega_b) = \left\{F \in\mathcal{O}(\Omega_b)\,:\, \sup_{\epsilon>0}\int_{\mathbb{C}^n\times\mathbb{R}}|F_{\epsilon}(\vec{z},t)|^2\,d\vec{z}\,dt \equiv ||F||_{\mathcal{H}^2}^2 <\infty\right\}.
\hat{\exp}(\xi_1,h_2) =\int_{-\infty}^{\infty}\exp \Big(-4\pi\big((\sqrt{ky}tu+ \frac{ix\sqrt{k}}{t\sqrt{y}}h_2)^2+\frac{k|z|^2h_2^2}{yt^2}\big)-2\pi i u \xi_1 \Big) du = \frac{1}{2t\sqrt{ky}}\exp\big( -\frac{4\pi}{yt^2}\big| \sqrt{k}zh_2 +\frac{\xi_1}{4\sqrt{k}}\big|^2\big).
D(1^{n_1}2^{n_2}\ldots l^{n_1}) = D(\overbrace{j_1, \ldots, j_1}^{n_1}, \overbrace{j_2, \ldots, j_2}^{n_2}, \ldots, \overbrace{j_l, \ldots, j_l}^{n_l}),
r_{n+1}=m^{2}r_{n},\ \ \nu_{n+1}=u_{s_k}, \delta_{n+1}={b}^{-1}R_n^{-6m}(\nu_{n+1}\tilde M_n^{-1})^{\kappa}\delta_{n}, \tilde{M}_{n+1}=b(\nu_{n+1}^{-1}\tilde{M}_{n})^{\kappa}, c_{n+1}=({b}^{-1}R_{n}^{-1}r_{n}^{-1}\delta_{n}\nu_{n+1}\tilde M_n^{-1}c_{n})^{\kappa^{r_{n}}}- r_{n}^{\kappa r_{n}}\epsilon_{n}.
Z = \int [ d \Phi^{A}] [ \det \omega ]^{-{\frac{1}{2}}}[ \det f ]^{-{\frac{1}{2}}}\exp \{\frac{i}{\hbar}S_{n m}[ \Phi^{A}, \Phi_{A}^{*}-{\frac{\partial \Psi}{\partial \Phi^{A}}}] \}
\partial_{\alpha}\,(\,\Gamma^{\,\alpha,\,\alpha_{0}}(\Phi_{c})-\frac{1}{2}<\Phi_{c},\{\tilde C_{\alpha}^{\,\alpha_{0}}\}^{\,-1}\Phi_{c}> )=[\partial_{\alpha}\,\tilde\Delta(\alpha,\alpha_{0})\,]\L^{\,\alpha,\,\alpha_{0}}(\Phi)\,\vert_{\,\Phi=\tilde C_{\alpha}^{\,\alpha_{0}}\!J(\Phi_{c})}\quad .
\varphi_{C}= \frac{1}{g r}\left[{\cal{N}}{\sin}( \overline{{{\cal{N}}}}+{\cal{N}}) +{\cos}( \overline{{{\cal{N}}}}+{\cal{N}}) - 1 \right] +{\cal T}_{C}\left[{\cos}( \overline{{{\cal{N}}}}+{\cal{N}}) - 1 \right] +{\cal T}_{A}\,{\sin}( \overline{{{\cal{N}}}}+{\cal{N}}) .
q(k,1)^{2^{k-1}}=d_1^{-2^{k-1}}d_{2^{n-1-k}-1}^{2^{k-1}}\equiv d_{2^{k-1}}^{-1}\left(d_{2^{k-1}}+r_{2^{k-1}}\right)= &=1+d_{2^{k-1}}^{-1}r_{2^{k-1}}\pmod{2}.
=\frac{1}{\omega}G_{p+\tau,q+\sigma}^{m+\nu,n+\mu}(\begin{matrix}a_{1},...,a_{n},-d_{\tau},a_{n+1},...,a_{p}\\b_{1},...,b_{m},-c_{\sigma},b_{m+1},...,b_{q}\end{matrix}|\frac{\eta}{\omega}).
S(x) \ll{\sum_n}^* \Biggl| \sum_{j=1}^m e(\lambda_j n)\Delta_j(\log(n+\alpha)) \Biggr|\sum_{j=1}^m \bigl| \Delta_j(\log(n+\alpha)) \bigr| &\ll{\sum_n}^* \Biggl| \sum_{j=1}^m e(\lambda_j n) \Delta_j(\log(n+\alpha)) \Biggr|\sum_{j=1}^m \bigl|\Delta_j(x^{(j)}_0)\bigr|+O(e^{-(A+\sigma_1-1)x}).
\kappa (s)=\frac{\sqrt{\left\vert \ddot{y}(s)^{2}-\ddot{z}(s)^{2}\right\vert}}{(\dot{x}(s))^{2}},\tau (s)=\frac{\ddot{y}(s)\dddot{z}(s)-\dddot{y}(s)\ddot{z}(s)}{\left\vert \dot{x}(s)\right\vert^{5}\cdot \kappa^{2}(s)},