harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
Z [ \eta , \overline{{{\eta}}}] : = \int [ d \psi ] [ d \overline{{{\psi}}}] [ d A ] e^{- S [ \psi , \overline{{{\psi}}}, A ] + \sum_{a}\Big \{( \overline{{{\eta}}}^{( a )}, \psi^{( a )}) + ( \overline{{{\psi}}}^{( a )}, \eta^{( a )}) \Big \}}\; .
\phi(f(X)) = p(f(X)) q(f(X)) &= m(f(X))q(f(X)) + (d^{n-1})^2 r(f(X)) q(f(X)) &= m(f(X))q(f(X)) + d^{n-1}r(f(X)) \cdot d^{n-1}q(f(X))
f''(g_1)(\eta) &=\sum_jf'(h_j)(L(\eta)|_{Y_j})~(\mbox{def. of $T_1$}) &=\sum_j\sum_if(g_i)(\pi(L(\eta)|_{Y_j})|_{X_{g_i, h_j}})~(\mbox{def. of $S_1$}) &=\sum_{j, i}f(g_i)(\eta|_{X_{g_i, h_j}\cap \psi(Y_j)}).~(\mbox{Lemma \ref{lem: composition of pi and L under restrictions}})
\int_{[0< z_1< \dots< z_{n}< t]}\int_{\mathbb R^{nd}}\prod_{j=1}^n|z_j-z_{j-1}|^{-\beta_0}e^{-(z_j-z_{j-1})\Psi(\xi_j)}\mu(d\xi)dz = \int_{\Omega_{t}^{n}}\int_{\mathbb R^{nd}}\prod_{j=1}^ns_j^{-\beta_0}e^{-s_j\Psi(\xi_j)}\mu(d\xi)ds,
\frac{1}{r!}\Big(\frac{d}{dx}\Big)^{r}\bigg[\frac{x^{r}}{(1-x)^{2}}F_{p,\lambda}\Big(\frac{x}{1-x}\Big)\bigg]-\Big(x\frac{d}{dx}\Big)_{p,\lambda}\Big(\frac{1}{1-x}\Big)^{r+1}&=\sum_{k=1}^{\infty}(k)_{p,\lambda}\bigg(\frac{1}{(1-x)^{r+1}}-\sum_{j=0}^{k}\binom{r+j}{j}x^{j}\bigg).
\mathcal{S}_0(sV_A^{\perp})=\frac{2s^2(1+5\cos(2\theta_1+2\theta_2))-19s^3\cos(3\theta_1+3\theta_2)}{8}(JX_1+JX_2)+\frac{s^3}{8}V_A^{\perp}+O(s^4).
\psi(x)=\frac{1}{4\pi}\frac{1}{\left(R^2+r_0^2\right)^{1/2}}\sum\limits_{l=0}^{+\infty}b_l \left(\frac{2r_0 R}{R^2+r_0^2}\right)^l\left[ \sum\limits_{j=1}^{2m}(-1)^j (\nu\omega_j)^l \right], \ \R\rightarrow\infty, & \nu\omega_j = \sin\theta \cos\left(\phi-\frac{\pi(j-1)}{m}\right).
\hat{x}_\mu{\blacktriangleright}\big(\hat{f}\hat{g}\big)=\big(Z^{-1}{\blacktriangleright}\hat{f}\big)(\hat{x}_\mu{\blacktriangleright}\hat{g})-a_\mu\big(p_\alpha^{\rm L}{\blacktriangleright}\hat{f}\big)(\hat{x}_\alpha{\blacktriangleright}\hat{g})
\tilde t = c_1t+c_0, \tilde x=\varphi(x), \tilde u = c_2u+c_4t^2+c_3t+\psi(x), \tilde u_{\tilde x}=\frac{c_2u_x+\psi_x}{\varphi_x}, \tilde f = \frac{\varphi_x^2}{c_1^2}f, \tilde g = \frac{1}{c_1^2}\left(c_2g+\frac{c_2u_x+\psi_x}{\varphi_x}\varphi_{xx}f-\psi_{xx}f+2c_4\right),
W_{\pi}(g_{t,l,v}) = \zeta_F(1)^{-1}q^{-\frac{a_1+t}{2}}q^{s(l_1-l_2)}\epsilon(\frac{1}{2},\chi_1^{-1}) G(\varpi^{-t-a_1}-\varpi^{-l}vb_1, \chi_2).
P_{mn}^{{\rm T}}=P_{nm}, &P_{mn}^{{\rm T}}P_{mn}=P_{mn}P_{mn}^{{\rm T}}=I_{mn}, &{\rm vec}(A)=P_{mn}{\rm vec}(A^{{\rm T}}), A\in \mathbb R^{m\times n}, &P_{mr}(A\otimes I_r)P_{nr}^{{\rm T}}=I_r\otimes A.
\frac{1}{\phi(R)}\left(\frac{1}{|B|}\int\limits_{B}\prod\limits_{i=1}^m |f_i(x)|^{p}dx \right)^{\frac{1}{p}}\le\frac{1}{\phi(R)}\left(\frac{1}{|B|}\int\limits_{B}\prod\limits_{i=1}^m |f_i(x)|^{p^*}dx \right)^{\frac{1}{p^*}}&\le \prod\limits_{i=1}^m \frac{1}{\phi_i(R)}\left(\frac{1}{|B|}\int\limits_{B}|f_i(x)|^{p_i}dx \right)^{\frac{1}{p_i}}.
\int_{-1}^1p_x'(t)w(t)dt= -w(x)+\left(1-p_x(-1)\right)w(-1)+p_x(1)w(1)- &-\sum_{i=1}^L\left(w(s_i+)-w(s_i-)\right)\left(p_x-\chi_{[-1,x]}\right)(s_i)-\int_{-1}^1\bigl(p_x-\chi_{[-1,x]}\bigr)(t)w'(t)dt.
- \frac{1}{2}\mathrm{T r}[ \frac{M^{2}}{- \partial^{2}+ M^{2}{\tilde{\omega}}}\alpha \frac{M^{2}}{- \partial^{2}+ M^{2}{\tilde{\omega}}}\alpha ] = -{\frac{1}{4 8 \pi}}\int d^{2}x{\frac{1}{\omega^{2}}}\partial_{\mu}\omega \partial^{\mu}\omega .
d s_{\mathrm{a s y m}}^{2}= \frac{1}{2}M_{i j}d{\bf x}_{i}\cdot d{\bf x}_{j}+ \frac{g^{4}}{2 ( 4 \pi )^{2}}( M^{- 1})_{i j}( d \xi_{i}+{\bf W}_{i k}\cdot d{\bf x}_{k}) ( d \xi_{j}+{\bf W}_{j l}\cdot d{\bf x}_{l}) \, \, .
\Psi_I^{(n)}(\hat\alpha_{a_0}, \ldots, \hat\alpha_{a_n}) = (-1)^{n+\sum_{i>0}|\alpha_{a_0}||\alpha_{a_i}|}\Psi_I^{(n)}(\hat\alpha_{a_1}, \ldots, \hat\alpha_{a_n},\hat\alpha_{a_0}).
S_1(...) = \frac{N^{2/3+i\nu}}{p_1qr^{2/3}}\sum_{\pm}\sum_{n_{1}|p_1qr}\,n^{1/3}_1 \sum_{n_{2}\ll N_0/n^2_1}\frac{A_{\pi}(n_{2},n_{1})}{n^{1/3}_{2}}S\left( r\overline{(a+bq)}, \pm n_{2}; p_1qr/n_{1}\right) & \times \eth_1^{\pm}(n^2_1n_2,u,q,p_1),
\sum_{j=0}^\ell\binom{\ell}{j}\psi_0^{(\ell-j)}(w)f^{(j)}(w)+\sum_{j=n}^{\ell+n}\binom{\ell}{j-n}\psi_n^{(\ell+n-j)}(w)f^{(j)}(w)=\lambda f^{(\ell)}(w).
D_{21}= \frac{1}{2 \pi i \Gamma(s_3)}\int_{at_3}^\infty \int_v^\infty \int_{(c)}\frac{\Gamma(s_3+z) \Gamma(-z)}{v^{s_2-z}u^{s_1+s_3+z}}dzdudv &= \frac{(at_3)^{2-s_1-s_2-s_3}}{2 \pi i (s_1+s_2+s_3-2) \Gamma(s_3)}\int_{(c)}\frac{\Gamma(s_3+z) \Gamma(-z)}{s_1+s_3-1+z}dz.
\dot{\xi}_{\underline{{{c}}}}-{h}_{\ \underline{{{c}}}}^{i}\,{h}_{\ \underline{{{b}}}}^{j}\, T_{i j}^{\underline{{{a}}}}\, \xi_{\underline{{{a}}}}\, \xi^{\underline{{{b}}}}-{h}_{\ \underline{{{c}}}}^{i}\, T_{i 0}^{\underline{{{a}}}}\, \xi_{{a}}= 0 ,
\frac{b_1^2}{b_0b_2}=\frac{n(n-1)(\alpha+3)_2(n+\alpha+\beta+1)_2}{(n-2)(n-3)(\alpha+1)_2(n+\alpha+\beta+3)_2}\xrightarrow{n\to\infty}\frac{(\alpha+3)(\alpha+4)}{(\alpha+1)(\alpha+2)},
\frac{\partial}{\partial \upsilon_j}f(\upsilon) = -2\pi i \cdot \left\langle A^{-T}(\tau) \langle x\rangle , \frac{\partial}{\partial \upsilon_j}\Phi^{-1}(\upsilon+\tau) \right\rangle \cdot e_{\tau}(x,\upsilon) = -2\pi i \cdot (\phi_{\tau}(\upsilon)\cdot x)_j \cdot e_{\tau}(x,\upsilon) .
\Psi(\vec{x}; \vec{z}; \vec{\xi}) := \frac{1}{2^\mu \cdot \mu!}\Bigl(H_{(x_1, x_2)}-z_1^2 - z_2^2\Bigr)^\mu \diamond \bigl(\delta(x_1 -\xi_1, x_2-\xi_2) \cdot \exp(x_1 z_1 + x_2 z_2)\bigr).
s_0(x_0,x_1,x_2) = (x_1+x_2-x_0,x_1,x_2), s_1(x_0,x_1,x_2) = (x_0,x_0+x_2-x_1,x_2), s_2(x_0,x_1,x_2) = (x_0,x_1,x_0+x_1-x_2).
I_{X 2}\equiv \int \frac{d^{4}q}{( 2 \pi )^{4}}\, \Bigg \{\frac{2 q_{\mu}( q_{\mu}+ k_{\mu})}{\Big ( ( k + q )^{2}+ M^{2}\Big )^{2}\Big ( q^{2}+ M^{2}\Big )^{2}}- \frac{4 M^{2}}{\Big ( ( k + q )^{2}+ M^{2}\Big ) \Big ( q^{2}+ M^{2}\Big )^{3}}\Bigg \}. \qquad
\lbrack \mathbf{\Xi}_{\mathrm{p}}(t)]_{+}=-\frac{t^{2}}{2}\mathbf{\mu}\left( \left\{0\right\}\right) -\int\nolimits_{0}^{t}\mathrm{d}s\int\nolimits_{0}^{s}\mathrm{d}\alpha \int\nolimits_{\mathbb{R}\backslash\left\{0\right\}}\mathbf{\mu}\left( \mathrm{d}\nu \right) \cos \left(\alpha \nu \right) \ .
J_1\leq 3^4 \int_{|z'|\leq\left(\frac{1}{3M_0}\right)^{\frac{1}{1+\alpha}}}2M_0\left(\frac{h}{\rho_0}\right)^\alpha|z'|^{1+\alpha}dz' = C \left(\frac{h}{\rho_0}\right)^\alpha,
T^{\mbox{\scriptsize{reg}}\:(1)}(s,l) \;=\; -\frac{1}{32 \pi^3}\:\lim_{a \rightarrow 0}\int_{-\infty}^\infty \left[ \frac{1}{u}\;e^{-\frac{i a l}{u}}\:\Theta(u) \:-\: (1+\log a) \:\delta(u) \right]e^{-i u s}\:du \;.
I_{\Phi}^{(N)}(t,\varphi) := -\frac1{N\sigma_N}\sum_{i=1}^{N}\varphi(t, x_i(t)) \sum_{\substack{j=1 j \neq i}}^{N}\Phi'\left(\frac{| x_i (t)- x_j(t)|}{\sigma_N}\right) \frac{x_i(t) - x_j(t)}{| x_i(t) - x_j(t)|},
A=\Omega_1:= \left(\begin{array}{ccccccc}E_1(L_1) \hdots E_{s_1}(L_{1}) E_{s_1+1}(L_{1}) \hdots E_N(L_1) \vdots \hdots \vdots \hdots \hdots \vdots E_1(L_{s_1}) \hdots E_{s_1}(L_{s_1}) E_{s_1+1}(L_{s_1}) \hdots E_N(L_{s_1}) 0 \hdots 0 \hdots 0 0 \vdots \hdots \vdots \hdots \hdots \vdots 0 \hdots 0 \hdots 0 0\end{array}\right),
\sigma_j.((m_1,i_1),\ldots, (m_\ell,i_\ell))=((m_1,i_1),\ldots, (m_{j+1},i_{j+1}),(m_j,i_j),\ldots, (m_\ell,i_\ell)), 1<j<\ell, &\sigma_\ell.((m_1,i_1),\ldots, (m_\ell,i_\ell))=((m_\ell+e,i_\ell),(m_2,i_2),\ldots, (m_1-e,i_1)).
P(\vert\boldsymbol{Y}_{\mathcal{I}}\vert=n) = \sum_{k\geq n}\frac{a_{\vert\boldsymbol{\theta}_{\mathcal{I}}\vert}(n) a_{\vert\boldsymbol{\theta}_{-\mathcal{I}}\vert}(k-n)}{a_{\vert\boldsymbol{\theta}\vert}(k)}P(\vert\boldsymbol{Y}\vert=k)
( X , Y ) = \int d^{n}x \left( \frac{\delta^{R}X}{\delta \Phi^{A}( x )}\frac{\delta^{L}Y}{\delta \Phi_{A}^{*}( x )}- \frac{\delta^{R}X}{\delta \Phi_{A}^{*}( x )}\frac{\delta^{L}Y}{\delta \Phi^{A}( x )}\right) \, ,
\left\langle \mathbf{K}\left[1-e^{-h}\right](e_n),e_n \right\rangle =\int_{E\times E}\sqrt{1-e^{-h(x)}}K(x,y) \sqrt{1-e^{-h(y)}}e_n(x)e_n(y)dxdy =\int_{E\times E}\sum_{i=1}^{+\infty}\lambda_i \varphi_i(x)\varphi_i(y)e_n(x)e_n(y)dxdy
\begin{array}{c c c}{K_{a b}^{- 1}=}&{p \left( \begin{array}{c c}{\sqrt{g}}&{{\frac{1}{\sqrt{g}}}}\\{{\frac{1}{\sqrt{g}}}}&{\sqrt{g}}\\ \end{array}\right)}&{+ ( 1 - p ) \left( \begin{array}{c c}{g}&{{\frac{1}{g}}}\\{{\frac{1}{g}}}&{g}\\ \end{array}\right)}\\ \end{array}
\{S_{\alpha}^{a}, S_{\beta}^{b}\}= \epsilon_{\alpha \beta}\; \tilde{\cal Z}^{a b}= \left( \begin{matrix}{{0}}&{{| Z | ( 1 + \sigma_{3})}}\\{{- | Z | ( 1 + \sigma_{3})}}&{{0}}\\ \end{matrix}\right) \ .
\left( \begin{array}{c}{a_{D}}{a}\end{array}\right) \rightarrow \left( \begin{array}{c c}{0}{- 1}{1}{0}\end{array}\right) \left( \begin{array}{c}{a_{D}}{a}\end{array}\right) = \left( \begin{array}{c}{- a}{a_{D}}\end{array}\right) ,
\phi(e_\lambda^3)x_{j,g}=\phi(e_\lambda^2)x_{e_\lambda j,g}=\phi(e_\lambda^2)x_{e_\lambda,g}\alpha_g(\phi(\alpha_{k_0}^{-1}(j))),\ \ \ \ \lambda\in \Lambda,\g\in G,\j\in I_{k_0},
\begin{aligned}X_{{11}}(-j-1)\sp{b_{11}}X_{{12}}(-j)\sp{a_{12}}X_{{11}}(-j)\sp{a_{11}}\,,\ \, b_{11}+a_{12}+a_{11}= k+1, &X_{{11}}(-j-1)\sp{b_{11}}X_{22}(-j)\sp{a_{{22}}}X_{{12}}(-j)\sp{a_{12}}\,,\ \, b_{{11}}+a_{{22}}+a_{{12}}= k+1, \end{aligned}
\exp \left\{-\frac{g^2}2\int d^4xd^4x^{\prime}\left[ \phi (x)\right]N\left( x,x^{\prime}\right) \left[ \phi (x^{\prime})\right] \right\}\equiv \int D \xi \;P \left[ \xi \right] \exp \left\{-ig\int d^4x\;\xi(x)\left[\phi (x)\right] \right\}
\begin{array}{lll}\displaystyle \int_{L_s}(s^2-l^2) \gamma \delta_{N+1}(v^{\gamma+1}\rho) \dA = \displaystyle \int_{L_s}(s^2-l^2) v^\gamma \rho^2 \dA + 2 \int_{L_s}v^{\gamma+1}\rho l \delta_{N+1}l \dA begin{align*}12pt] \leq \displaystyle \int_{L_s}s \left( R^{-1}\|\delta l\|_{\L^{N+1}}^2 + (s + R) \rho^2\right)v^\gamma \dA.\end{array}
v_{ij}v_{ik}= -v_{ik}v_{ij}v_{ji}v_{ki}= -v_{ki}v_{ji}j \neq k v_{ij}v_{kl}= v_{kl}v_{ij}i\neq k j \neq l \sum_{\sigma \in S_3}v_{1 \sigma(1)}v_{2 \sigma(2)}v_{3 \sigma(3)}= 1.
O_{1}^{( k )}= \left< \sum_{i}( q_{i}^{( k )}- 6 )^{2}\right> \; \mathrm{a n d}\; O_{2}^{( k )}= \left< \sum_{< i j >}( q_{i}^{( k )}- 6 ) ( q_{j}^{( k )}- 6 ) \right> ,
\sum_{j = 0}^k{( - 1)^j j^3 G_j}= 81G_0 - 50G_1 - ( - 1)^k (k^3 + 9k^2 + 39k + 81)G_k + ( - 1)^k (k^3 + 6k^2 + 24k + 50)G_{k + 1}\,.
C_{2}[ \omega , \omega_{1}, \omega_{2}, B ] ={\frac{m^{8}}{4 B^{2}\omega \omega_{1}\omega_{2}}}\int_{0}^{\infty}{\frac{d s}{s}}e^{- m^{2}s}J_{2}( s , \omega , \omega_{1}, \omega_{2}, B ) ~ ,
U'(x(T,y))= p(0,y) \exp \big( \int_0^T [\ln(1-\frac{b_0(s,y)}{\lambda\gamma_0(s,y)})-\ln(1+\Psi(s,y))]d\tilde{N}(s) &+\lambda\int_0^T\{[\ln(1-\frac{b_0(s,y)}{\lambda\gamma_0(s,y)})+\frac{b_0(s,y)}{\lambda\gamma_0(s,y)}]-[\ln(1+\Psi(s,y)) &-\Psi(s,y)]\}ds \big)=:p(0,y)\Gamma(T,y),
\dot{x}_i = v_{i}\left(\cos{\psi_i}- \sin{\psi_i}\tan{\beta_i}\right) \dot{y}_i = v_{i}\left(\sin{\psi_i}+ \cos{\psi_i}\tan{\beta_i}\right) \dot{\psi}_i = \frac{v_{i}}{l_r}\tan{\beta_i}\dot{\beta}_i = \omega_i \dot{v}_{i}= a_{i},
\sum_{n\geq 0}\sum_{j= 0}^{\lfloor{n\over 2}\rfloor}\sum_{m= 0}^{\lfloor{n-2j\over 2}\rfloor}A(n,j,m)=\sum_{l\geq 0}\sum_{j\geq 0}\sum_{m\geq 0}A(l+2j+2m,j,m).
A_k(\xi^m) = \sum_{i=0}^{k-1}a_{k-i}Jq^{k-i}Jq^i(\xi^m) = \sum_{i=0}^{k-1}a_{k-i}\binom{m}{i}\binom{m+i}{k-i}= m(m-1) (b_{k-2}m^{k-2}+b_{k-3}m^{k-3}+ \cdots +b_0),
p \leq \sum_{t=1}^{\lfloor 2\ln(e|J|)\rfloor}\exp\left(-\frac{C_{\ref{l: pl on sqrt d scale}}t\lceil |J|(2e)^{-t}\rceil}{4a}\right) &\leq \lfloor 2\ln(e|J|)\rfloor\max_{t=1,\ldots, \lfloor 2\ln(e|J|)\rfloor}\exp\left(-\frac{C_{\ref{l: pl on sqrt d scale}}t |J|(2e)^{-t}}{4a}\right)\ll \frac{1}{n^3},
\limsup_{D\ni x\to \infty}\overline M^D_Y(x,z)\le c \limsup_{D\ni x\to \infty}\frac{u(|x-z|^2)|x_0-z|^{d-2}}{u(|x_0-z|^2)|x-z|^{d-2}}\le c \limsup_{D\ni x\to \infty}\frac{|x_0-z|^{d-2}}{|x-z|^{d-2}}=0.
f_{\alpha}( \phi ) ={\frac{e^{- i \pi / 4}}{\sqrt{2 \pi p}}}\left({\frac{1}{1 + e^{i \phi}}}\right) e^{i \alpha \phi}\left( e^{- i \alpha \pi}- e^{i \alpha \pi}\right) \; , \quad 0 \le \alpha < 1 \; .
\dfrac{M_{2}(\mathcal{C}(G))}{|e(\mathcal{C}(G))|}- \dfrac{M_{1}(\mathcal{C}(G))}{|v(\mathcal{C}(G))|}= \dfrac{32n^{4}(2n-11) + 32n^{2}(21n - 16) + 128n}{8n^{2}(n-2)+16n-5}:= \frac{f(n)}{g(n)}.
\phi ( \tau , \sigma ) = q + \frac{p}{2 \pi}\tau + \frac{i}{\sqrt{4 \pi}}\sum_{n \neq 0}\frac{1}{n}\, \left({a_{n}}\, e^{- i n ( \tau + \sigma )}+{\bar{a}_{n}}\, e^{- i n ( \tau - \sigma )}\right) .
\begin{cases}\partial_t \psi(x,t) + d\Delta \psi(x,t) = -\theta(x,t), (x,t)\in \Omega_{\tau, T}, \nabla \psi(x,t) \cdot \nu(x) = 0, (x,t)\in \partial\Omega\times(\tau,T), \psi(x,T) = 0, x\in\Omega.\end{cases}
\partial_{(c,b_{i+1},b_{n})}f(\mu)(v_{1},...,v_{p},v_{p+1}) \cdot e_{1}\otimes ... \otimes e_{i}\otimes e_{i+1}\otimes e_{{n}}&= \partial_{(c,b_{n},b_{i+1})}f(\mu)(v_{1},...,v_{p},v_{p+1}) \cdot e_{1}\otimes ... \otimes e_{i}\otimes e_{{n}}\otimes e_{i+1}.
g(2,3) > g(1,4)\,, g(3,3) > g(1,5)\,, g(3,4) > g(1,6)\,, g(1,2k-1) > g(k,k) k \ge 4\,, g(1,2k) > g(k,k+1) k \ge 4\,, g(1,6,6) > g(4,4,5)\,, g(4,4,4) \ge g(1,5,6) \,.
\tau^b =(d_2-d_1+1-\lambda)!{d_1+1 \choose 2d_1-d_2+\lambda}{d_2+1 \choose d_1+\lambda}{n_1 \choose d_1+1}{n_2 \choose d_2+1}& + \sum_{2r_0+r_1=2d_1-d_2+\lambda}{d_1+1 \choose r_0}{d_1+1-r_0 \choose r_1}\frac{(d_2+2)!}{2^{r_2}\lambda!}{n_1 \choose d_1+1}{n_2 \choose d_2+2},
\begin{array}{r c l}{G}&{=}&{( \varphi ( x ) \varphi ( y ) ) \star i \Delta ( z , m^{2}) + i \Delta ( z , m^{2}) \star ( \varphi ( y ) \varphi ( x ) )}\\{}&{}&{+ \varphi ( x ) \star_{x}i \Delta ( z , m^{2}) \star_{y}\varphi ( y ) + \varphi ( y ) \star_{y}i \Delta ( z , m^{2}) \star_{x}\varphi ( x ) ,}\\ \end{array}
d\left( \tilde{x}_{n+1},\tilde{x}_{n}\right) = d\left( T\tilde{x}_{n},T\tilde{x}_{n-1}\right) \tilde{\preceq}\tilde{t}\left( d\left( T\tilde{x}_{n},\tilde{x}_{n}\right) + d\left( T\tilde{x}_{n-1},\tilde{x}_{n-1}\right) \right) = \tilde{t}\left( d\left( \tilde{x}_{n+1},\tilde{x}_{n}\right) + d\left( \tilde{x}_{n},\tilde{x}_{n-1}\right) \right).
(\psi_j,\psi_{j'})_{\rm QFT}= \delta_{jj'}, (\overline{\psi_j},\overline{\psi_{j'}})_{\rm QFT}= -\delta_{jj'}, \qquad\mbox{and}(\psi_j,\overline{\psi_{j'}})_{\rm QFT}= 0.
\frac{1}{\hat{\Pi \hspace{-8pt}/}- M + i 0}\> = \>\int_0^{\infty}dT \int d\chi \> \exp \left [ \> - \frac{i}{2 \kappa_0}( M^2 T + M \chi) \> \right ]\> \exp \left [ \> \frac{i}{2 \kappa_0}( \hat{\Pi \hspace{-8pt}/}^2 T + \hat{\Pi \hspace{-8pt}/}\chi) \> \right ] \>
\sum_{i=1}^k \sum_{\alpha\in I_{i,k-1}}\biggl(\biggl[\nabla_{v_1}\biggl\{\eqref{aneurysm}\biggr\}\biggr](v_2 \otimes \cdots \otimes v_{k+1}) + \sum_{\ell=2}^{k+1}\biggl\{\eqref{aneurysm}\biggr\}\biggl[v_2 \otimes \cdots\otimes \frac{\nabla \gimel_{s_\ell}}{\partial s_1}\otimes \cdots \otimes v_{k+1}\biggr]\biggr).
H_{L C}= \int_{0}^{l}\mathcal{H}_{L C}= \int_{0}^{l}d \sigma \, \frac{G_{+ -}}{2 \, \pi_{-}}\left( G^{i j}\pi_{i}\pi_{j}+ \frac{1}{( 2 \pi \alpha^{\prime})^{2}}G_{i j}Z^{i \, \prime}Z^{j \, \prime}\right)
\rho^{(1)}_{\Lambda_M}:=\frac{1}{\sqrt 2}(\rho_{\Lambda_M}+\rho_{\Lambda_{M\cup\Psi}}),R^{(1)}_{\Lambda_M}:=\frac{1}{\sqrt 2}(R_{\Lambda_M}+R_{\Lambda_{M\cup\Psi}}),
C_{p,g}^{(n)}= \sum_{j=1}^{p}(-1)^{j-1}\sum_{t=0}^{\lfloor\frac{p-j}{2}\rfloor}\binom{n-p+j+2t}{t}2^{p-j-2t}\binom{n}{p-j-2t}\sum_{i=0}^{j-1}\binom{0}{p+i-j-t-g}&= \sum_{j=1}^{p}(-1)^{j-1}\sum_{t=p-g-j}^{\lfloor\frac{p-j}{2}\rfloor}\binom{n-p+j+2t}{t}2^{p-j-2t}\binom{n}{p-j-2t}
\frac{d}{dt}F_t(x)=\begin{cases}-4ad\lambda F_t(x)+2a\lambda\sum_{y:y\sim x}F_t(y) x\neq O, \big[1-4d\lambda(a+b-1)+2d\lambda(b^2-1)+2da^2\lambda\big]F_t(O)+4abd\lambda F_t(e_1) x=O.\end{cases}
-\tilde c^{m,\ell}_{q,\kappa}\tilde b_{m,\ell}= (q_n +1) (q_n +2)( \tilde b_{q+ \overline{n},\kappa}- b_{q+ \overline{n},\kappa}) & + \sum_{i \neq n}(q_i + 1) (q_i + 2)(\tilde b_{q+ 2 \overline{i}- \overline n,\kappa}- b_{q+ 2 \overline{i}- \overline n,\kappa}) - (\kappa+1)( \tilde b_{q- \overline{n}, \kappa+1}- b_{q- \overline{n}, \kappa+1}).
\frac{1}{n}\mathbb L_1\cdots\mathbb L_{n-1}\mathbb L_nF^n = \rho^{G_n}\left(\widetilde{F}^n(W_{(n,1)},\ldots,W_{(n,n)})\right) = \rho^{G_n}\left(F\left(\frac{1}{n}\sum_{k=1}^n\delta_{W_{(n,k)}(1)}\right)\right).
4K_1^2+4M+4M^{-\frac{p-2}{2}}K_2^p+8M^{-\frac{p-2}{2}}K_2^p\frac{\alpha^2}{\varrho^2}&\quad\leq4K_1^2+4M+12M^{-\frac{p-2}{2}}K_2^p\frac{\alpha^2}{\varrho^2}=4K_1^2+16K_2^2\alpha^{\frac{4}{p}}\varrho^{-\frac{4}{p}}\leq16K^2\alpha^{\frac{4}{p}}\varrho^{-\frac{4}{p}}.
f_{Z_{\alpha}}\left ( t \right )=\alpha \frac{\mathbb{E}\left [ \left ( Y-t \right )_{+}^{\alpha -1}\right ]-\mathbb{E}\left [ \left ( X-t \right )_{+}^{\alpha -1}\right ]}{\mathbb{E}\left [ Y^{\alpha}\right ]-\mathbb{E}\left [ X^{\alpha}\right ]},t\geq 0,
\begin{array}{ll}\int_D \lambda \, \tau \, dx + \int_D \nabla s \cdot \nabla \tau \, dx = - \int_D \psi_g \, \tau \, dx \forall \ \tau \in H^1_0(D) [6pt]\int_D \nabla \lambda \cdot \nabla v \, dx [4pt]= - \int_D f \, v \, dx - \int_D \nabla \psi_g \cdot \nabla v \, dx \forall \v \in H^1_0(D) \, .\end{array}
I(D)=\int_0^{\infty}\!\!\! d\hat T_1\, d\hat T_2\int_0^1 \!\!\! da\, db\, dc\, dd\,{\biggl[(\hat T_1+G_{Bab})(\hat T_2+G_{Bcd})-{C^2\over 4}\biggr]}^{-{D\over 2}}
\hat{b}= \prod_{i\in\{0,8,16\}}(i+3,i+8)(i+4,i+7), \hat{c}= (3,6)(4,5)(11,14)(12,13)(17,24)(18,23), \hat{d}= (5,6)\prod_{i=9}^{16}(i,i+8), \hat{e}= (21,22)\prod_{i=1}^8 (i,i+8).
D^2_X.\pi_*([\tilde{E}]^2)=D^2_X.\pi_*(\frac{1}{2}\pi^*E.\tilde{E})=\frac{1}{2}D^2_X.E^2=\frac{1}{2}\pi_*(\tilde{D}^2.\tilde{E}^2)=-\frac{1}{2}\pi_*(\beta^*((D_{|\Sigma})^2)).
\bar{A}^{(2)}_{\beta,1}=\left. A^{(0)}_{\beta,1}\right|_{{\cal C}_\beta}N^{(2)}(d-2)\int_{\Sigma}+\left. A^{(2)}_{\beta,1}\right|_{{\cal C}_\beta}\int_{\Sigma}+ \left. A^{(1)}_{\beta,1}\right|_{{\cal C}_\beta}(d-2)\int_{\Sigma}~~~.
\widetilde{H}_{c}^{\prime}= \frac{1}{4}( B_{i j}- Q_{i j})^{2}- \frac{1}{2}( A_{i}+ \partial_{i}\theta )^{2}+ \frac{1}{4 m^{2}}F_{i j}F^{i j}- \frac{1}{2 \cdot 3 ! m^{2}}H_{i j k}H^{i j k}.
(-1)^{k+1}G^{k+1}_m(x) = \int_{0}^{x}(-1)^{k}zG^{k}_m(z) dm(z) + x\int_{x}^{y}(-1)^{k}G^{k}_m(z) dm(z) + x\int_{y}^{1}(-1)^{k}G^{k}_m(z) dm(z).
\Sigma_a = \sum_{\substack{2 \le k \le \frac{n}{2}k \}}|a_{k,n}^{H^F}| i_{2 \times 2}(C_{H^F}(a_{k,n})) < \sum_{2 \le k \le \frac{n}{2}}q^{\frac{1}{8}n^2+\frac{1}{8}(n-2k)^2+k(n-k)+d'n}.
\Big[-\frac{\hbar^2}{2M}\sum_{i}{\nabla_i}^2-\frac{\hbar^2}{2m}\nabla^2-\lambda\delta(x-x_1)-\lambda\delta(x-x_2)]\Big]\Psi(x;x_1, x_2)=E\Psi(x;x_1,x_2).
\pi_1(M_0)=\left \langle a,b,c,d,e,f\ \left| \begin{array}{l l}adb^{-1}d^{-1}, acb^{-1}c^{-1}, eaf^{-1}a^{-1}, ebf^{-1}b^{-1}, ced^{-1}e^{-1}, cfd^{-1}f^{-1}, a^4,b^4,c^4,d^4,e^4,f^4 \end{array}\right. \right \rangle.
\|\boldsymbol{v}\|_{\boldsymbol{H}^{\emph{\textbf{curl}},\frac{1}{2}}}^2 := T(\|\boldsymbol{v}_0^c\|_{\Omega}^2 + \|\emph{\textbf{curl}}\, \boldsymbol{v}_0^c\|_{\Omega}^2) + \frac{T}{2}\sum_{k=1}^{\infty}\Big( (1+k\omega) \|\boldsymbol{v}_k\|_{\Omega}^2 + \|\emph{\textbf{curl}}\, \boldsymbol{v}_k\|_{\Omega}^2] \Big).
P(z) = V\frac{(y-1)^2}{h^2}-D\frac{y-1}{h}+ Q = \frac{1}{h^2}Vy^2 - \left(2\frac{1}{h^2}V + \frac{1}{h}D\right)y + \left(\frac{1}{h^2}V+\frac{1}{h}D+Q\right),
\frac{1}{\mathbb{E}_{\alpha}[T_{A}]}=\lambda=\frac{\langle (I-P) f_{1},f_{1}\rangle_{\pi}}{\|f_1\|_2^2}=\mathrm{Var}_{\pi}f_1 \frac{\langle (I-P) f_{1},f_{1}\rangle_{\pi}}{\mathrm{Var}_{\pi}f_1}.
b_p^{\mathcal{D}_{\mathbb{F}G}}(G) = \frac{b_p^{\mathcal{D}_{\mathbb{F}N}}(N)}{[G:N]}= \frac{[H:N] b_p^{\mathcal{D}_{\mathbb{F}H}}(H)}{[G:N]}= \frac{b_p^{\mathcal{D}_{\mathbb{F}H}}(H)}{[G:H]},
\mathbb{E}\left[\left\|\overline{\mathcal{M}}\left(\frac{1}{n}\sum_{i=1}^{n}G_{i}^{\prime}\mathcal{X}_{i}\right)\right\|_{}\right]\leq C_{9}K\left(\sqrt{\frac{p_{2}\lceil \sqrt{p_3}\rceil}{n}}+\sqrt{\frac{p_{1}\lceil \sqrt{p_3}\rceil}{n}}+\sqrt[4]{\frac{p_{1}\lceil \sqrt{p_3}\rceil^2 p_{3}}{n^2}}\right).
K_{\widetilde{f}}^2= \; -n\sum_{k\ge 1}\alpha_k+2n(n-1)\left(-\frac{2M}{n}+\sum_{k\ge 1}k\alpha_k\right)+n(n-1)^2\left(\frac{2rM}{n^2}-\sum_{k\ge 1}k^2\alpha_k\right) = \;\frac{2(n-1)((n-1)r-2n)}{n}M-n\sum_{k\ge 1}((n-1)k-1)^2\alpha_k
J_{3,2}=6T_{x^2y}T_{xy}-3T_{xy^2}T_{x^2}-2T_{x^2y}T_{x}T_{y}+T_{xy^2}T_{x}^2-4T_{xy}^2T_{x}&+2T_{xy}T_{x}^2T_{y}-3T_{x^3}T_{y^2}+4T_{x^2}T_{x}T_{y^2}-T_{x}^3T_{y^2}+T_{x^3}T_{y}^2-T_{x^2}T_{x}T_{y}^2
\epsilon({AdS})=\left[\sqrt{f^{1/2}+1}-\sqrt{f^{1/2}-1}\gamma_{\overline{r}}\right]e^{-{1\over2}\gamma_{\overline{\theta r}}}e^{-{1\over2}\gamma_{\overline{\phi\theta}}}e^{-{gt\over\sqrt{2}}\gamma_{\overline{0}}}\epsilon_0.
\Omega_{B}^{A}=d\omega_{B}^{A}+\omega_{C}^{A}\omega_{B}^{C}\Omega_{B^{\prime}}^{A^{\prime}}=d\omega_{B^{\prime}}^{A^{\prime}}+\omega_{C^{\prime}}^{A^{\prime}}\omega_{B^{\prime}}^{C^{\prime}},
\sum_{m \geq 0}(-1)^m \frac{(p(j+2\nu)+1)_m}{(b_j+1)_m}\frac{v^{m+p(j+2\nu)-1}}{m!}= v^{p(j+2\nu)-1}{}_1F_1\left(b_j-a_j+1, b_j+1, -v\right) & = e^{-v}v^{p(j+2\nu)-1}{}_1F_1\left(a_j, b_j+1, v\right).
E_k = \frac{1}{4|k|}\int_0^{\pi}\biggl[(f')^2 + k^2\biggl(\frac{\sin f}{\sin\theta}\biggr)^2 + \biggl(\frac{kf'}{L}\frac{\sin f}{\sin\theta}\biggr)^2 + 2L^2{\cal E}^{(0)}(\cos f) \biggr] \sin\theta\,d\theta,
~(T_{\lambda+0i+h}-T_{\lambda+0i})\psi(x) = ~i\int_{-\infty}^xe^{i\lambda(x-y)}[e^{ih(x-y)}-1]u(y)\psi(y)~dy - (\widetilde{G}_{\lambda+h}-\widetilde{G}_{\lambda})*(u\psi)(x)
I(f)(\varpi^n\xi) =\sum_{j=-B}^{\infty}q^{j(\ell-2)}f(\varpi^{n-j}\xi,0,\varpi^j) =\sum_{j=-B}^{A-1}q^{j(\ell-2)}f(0,0,\varpi^j)+\sum_{j=A}^{n-A}q^{j(\ell-2)}f(0)+\sum_{j=n-A+1}^{n+B}q^{j(\ell-2)}f(\varpi^{n-j}\xi,0,0).
\alpha^{-1}\epsilon^{\alpha}(1-\epsilon)^{b_1-1}= \int_0^\epsilon x^{\alpha-1}dx (1-\epsilon)^{b_1-1}\le I_1(\alpha,\beta;\epsilon)\le \int_0^\epsilon x^{\alpha-1}dx (1-\epsilon)^{b_0-1}= \alpha^{-1}\epsilon^{\alpha}(1-\epsilon)^{b_0-1}.
H_{m, n}^{+, 1}(x) = \frac{4 M T}{\pi}\int_{u = -\infty}^\infty \int_{\zeta = -\infty}^\infty e^{-u^2}V(m^2 n, M u + T) \cos(x \cosh \zeta) e\Bigl( \frac{u M \zeta}{\pi}\Bigr) e\Bigl( \frac{T \zeta}{\pi}\Bigr) \, d u \, d \zeta,
\| u \|_{C^{\mathbf{s}(0,\lambda,\delta)}_T}= \sup_{t \in [0,T]}\| u (\cdot, t) \|_{C^{0,\lambda,\delta}}+ \sup_{t', t'' \in [0,T] \atop t' \neq t''}\frac{\|u (\cdot,t') - u (\cdot,t'') \|_{C^{0,0,\delta}}}{|t'-t''|^{\lambda/2}}.
\langle \mathfrak{e}_{x},\Delta_{\omega ,\vartheta}^{(\mathbf{A})}\mathfrak{e}_{y}\rangle =\exp \left( i\int\nolimits_{0}^{1}\left[ \mathbf{A}(t,\alpha y+(1-\alpha )x)\right] (y-x)\mathrm{d}\alpha \right) \langle\mathfrak{e}_{x},\Delta_{\omega ,\vartheta}\mathfrak{e}_{y}\rangle
B_1=\begin{pmatrix}a_{1,5}a_{1,6}a_{1,7}a_{1,8}\cdots a_{1,n}a_{2,5}a_{2,6}a_{2,7}a_{2,8}\cdots a_{2,n}a_{3,5}a_{3,6}a_{3,7}a_{3,8}\cdots a_{3,n}\end{pmatrix},
\begin{multlined}J_\varepsilon(u_0-W_{\delta_\varepsilon (t),\xi})= (\dfrac{1}{2}-\dfrac{1}{2^\ast})\int_M u_0^{2^\ast}dV + \dfrac{\varepsilon}{2^\ast}\int_M u_0^{2^\ast}(\ln u_0 - \dfrac{1}{2^\ast})dV + I_{1,\varepsilon,t,\xi}+I_{2,\varepsilon,t,\xi}+I_{3,\varepsilon,t,\xi}+O(\varepsilon^2),\end{multlined}
z_{1,n}' =\tau(z_{1,n}-z_{2,n})-\gamma z_{1,n}, z_{2,n}' =2\tau(z_{2,n}-z_{2,n}^{1.5})-2\gamma z_{2,n}+(1/n)(\tau+\gamma), &z_{1,n}(0)=u,\z_{2,n}(0)=u^2.
(\rho^{*}_{(1)}\otimes \rho^{*}_{(1)}) E_{\Sigma_0}= (\rho^{*}_{(1)}\otimes \rho^{*}_{(1)}) (\partial_{\Sigma_0}\otimes \partial_{\Sigma_0}) E_{\mathcal{M}_0}= (\partial_{\Sigma}\otimes \partial_{\Sigma}) (\chi^{*}\otimes \chi^{*}) E_{\mathcal{M}_0}=: E_{\Sigma}.
Q_t^{(\lambda)}(x,y):=\left( \Big|\frac{\kappa_t(y)-x}{t^{1/\alpha}}\Big|^\lambda \wedge t^{-\lambda/\alpha}\right)\frac{1}{t^{d/\alpha}}G^{(\alpha)}\left({\kappa_t(y)-x\over t^{1/\alpha}}\right), \lambda\in [0,\alpha).
\mathfrak{J}(\{\pmb{F}_k\}_{k=1}^K, \{\pmb{U}_k\}_{k=1}^{K+1}) = \mathfrak{J}_1(\{\pmb{F}_k\}_{k=1}^K, \{\pmb{U}_k\}_{k=1}^K)+\mathfrak{J}_2(\{\pmb{F}_k\}_{k=1}^K, \pmb{U}_{K+1})
\left(\frac{1}{4}(2-k)(1-r)\sin((k-2)x)-\frac{k}{2}(1+r)\sin(kx)-\frac{1}{4}(2+k)(1-r)\sin((k+2)x)\right)\frac{r}{(1+r)^2}+ \frac{1}{4\pi}\sum_{i=1}^{5}I_{i},
\nabla_{j}= \nabla_{j}^{0}+ i \left( \begin{array}{c c}{{( A_{j})_{i}^{k}}}&{{\langle ( b_{j})_{i}^{\beta}|}}\\{{| ( b_{j})_{\alpha}^{k}\rangle}}&{{( \hat{D}_{j})_{\alpha}^{\beta}}}\\ \end{array}\right) \, .
(\theta^{(j)}- \theta)_t - \kappa \Delta (\theta^{(j)}- \theta) = \tau \sum_{i=1}^N (\nabla^{\delta_0}u_i^{(j)}\cdot \nabla \theta^{(j)}- \nabla^{\delta_0}u_i \cdot \nabla \theta) \mbox{in}Q(T) \mbox{for}j.
\mathbb{E}^{0}\left( r_{t}\mathbb{E}^{0}\left[\int_{0}^{t}U_{s}dQ_{s}|\mathcal{F}_t^{W}\otimes\mathcal{F}_t^M\right]\right) = \mathbb{E}^{0}\left[r_{t}\int_{0}^{t}U_{s}dQ_{s}\right] = \mathbb{E}^{0}\left[\int_{0}^{t}r_{s}\gamma_{s}U_{s}d<W,Q>_{s}\right] + \mathbb{E}^{0}\left[\int_{0}^{t}U_{s}\int_{A}r_{s}(e^{\beta_{s}(x)}-1)d<\tilde{M},Q>_{s}\right]= 0 .