harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
(\varphi, \tilde g - g_h)_{\Omega_h}= a_h(\tilde w - w_h, \tilde g - g_h) - \operatorname{Res}_w(\tilde g - g_h) - \operatorname{Res}_g(\tilde w - w_h) + \operatorname{Res}_g(\tilde w) &=: I_1 + I_2 + I_3 + I_4,
\psi(u) =\mathbb{P}\left(\sup_{n\geqslant 1}\left\{\sum_{k=1}^nX_k-n\right\}>u\right)=\mathbb{P}\left(\bigcup_{n=1}^\infty\left\{\sum_{k=1}^n X_k>n+u\right\}\right) &\geqslant\mathbb{P}\left(\bigcup_{n=1}^\infty\left\{\sum_{k=1}^n X_k^*>n+u\right\}\right)= \mathbb{P}\left(\sup_{n\geqslant 1}\left\{\sum_{k=1}^nX^*_k-n\right\}>u\right) &=:\psi_{\varepsilon}^*(u),
\mathcal{G}_{\delta}=\left\{\phi \in \mathcal{D}\ \big| \ \phi\phi^*=\phi^*\phi=I, \det \phi = 1\right\}, \mathcal{G}_{0,\delta}=\left\{\phi \in \mathcal{G}_{\delta}\ \big|\ \sigma (\phi)=I\right\},
-Ric(\nabla_{\nu}e_2,e_2)+Ric(\nabla_{\nu}e_1,e_1) =h_{22}R_{22}-\frac{(e_2(R))^2}{4|\nabla f|^3}-h_{11}R_{11}+\frac{(e_1(R))^2}{4|\nabla f|^3}.
R + \frac{\gamma}{4}\left[ \frac{( D \Omega )^{2}}{\Omega^{2}}- 2 \frac{D \cdot D \Omega}{\Omega}\right] + ( 1 - \lambda ) \mu \Omega^{- \lambda}+ \frac{1}{4}( 1 - \frac{\epsilon}{2}) \Omega^{- \epsilon / 2}F^{2}- \frac{\delta}{2}\Omega^{\delta - 1}( D f )^{2}= 0 ,
H^{(2)}= \int dx\left[\frac{1}{2}e^4 (\Phi^1 )^2 +e^2 \Phi^1 \Phi^3+e^2(\partial_1 \Phi^1 )^2 -e^2 \Phi^1 \partial_1 \Phi^2-\Phi^2 \Phi^4 + \frac{1}{2}(\Phi^3 )^2 \right].
E_{k,\ell,j,m,n_1,n_2}= 2 \omega \left(1 + 2 k + \sqrt{\mu + \left( \ell + 2 j + 2 m + 3 n_1 + 3 n_2 + d + 3 a_1 + 3 a_2 ++ \frac{11}{2}\right)^2}\right)
\mathcal{C}_{J,K}^{\exists}:=\left\{\eta \in \mathcal{C}_{J,K}:\: \mbox{there exists a BBS($J$,$K$) carrier for $\eta$}\right\}; \mathcal{C}_{J,K}^{\exists !}:=\left\{\eta \in \mathcal{C}_{J,K}:\: \mbox{there exists a unique BBS($J$,$K$) carrier for $\eta$}\right\}.
f_n(b_n-1) =3(b_n-1)^2-(1+4n)(b_n-1)+(n-2)b_n+2(n+1)+u_n &\le3(b_n-1)^2-(1+4n)(b_n-1)+(n-2)b_n+2(n+1)+b_n(n-2)\;\;\mbox{(by (\ref{ae6}))}&=(b_n-3)\left[3b_n-(2n+2)\right]<0\;\;\mbox{(by (\ref{ae8})).}
\mathbf{y}(t)=\Big(\|x_\nu^{(1)}(t)-x_\nu^{(2)}(t)\|^2,\|x_\nu^{(3)}(t)-x_\nu^{(N)}(t)\|^2,\dots,\|x_\nu^{(\frac{N}{2}+1)}(t)-x_\nu^{(\frac{N}{2}+2)}(t)\|^2\Big)^{\top}, t\in\mathbb{R},
A(k,\theta) = \sum_{m=1}^{2g}m^k \sin(4 \pi m \theta) \sum_{j=0}^{\alpha-1}\left( \frac{1}{2m+j}+ \frac{1}{2m-j}\right) &+ 2 \pi \theta \sum_{m=1}^{2g}m^k \cos ( 4 \pi m \theta) \sum_{j=0}^{\alpha-1}\left( \frac{j}{2m+j}- \frac{j}{2m-j}\right) + O \Big( g^{k-1}\theta \alpha^3 \Big).
\alpha\lambda_{\lambda_{\rho_{x}(z)}\alpha(y)}^{-1}\lambda_{\lambda_{z}(x)}^{-1}(z)=\lambda_{\lambda_{\rho_{\lambda_{x}(y)}\alpha(z)}\alpha\lambda_{\lambda_{x}(y)}^{-1}(x)}^{-1}\lambda_{\lambda_{\alpha(z)}\lambda_{x}(y)}^{-1}\alpha(z).
\int_{- \infty}^{\infty}d z_{1}\int_{- \infty}^{\infty}d z_{2}\psi_{z_{1}, z_{2}}^{\ast \nu , \mu}( u^{0}, x^{\prime}, y^{\prime}) \psi_{z_{1}, z_{2}}^{\nu , \mu}( u^{0}, x , y ) = \delta \left( x - x^{\prime}\right) \delta \left( y - y^{\prime}\right) .
\mathbf{E}\left[|\zeta(s,\mathbb{X}_\alpha)|\right]^2\leq \mathbf{E}\left[|\zeta(s,\mathbb{X}_\alpha)|^2\right]= \sum_{m=0}^{\infty}\sum_{n=0}^{\infty}\frac{\mathbf{E}[\mathbb{X}_\alpha(m) \overline{\mathbb{X}_\alpha(n)}]}{(m+\alpha)^s (n+\alpha)^{\overline{s}}}.
\mathbb{P}\left(\left|\int f\,d\mu_N-\int f\, d\widetilde\mu_N\right|\geq \frac{\epsilon}{3}\right) \leq \mathbb{P}\left(\frac{\|f\|_{\mathrm{Lip}}}{\sqrt{N}}\left(\mathrm{Tr}[(X_N-\widetilde{X}_N)^2]\right)^{1/2}\geq \frac{\epsilon}{3}\right) &=\mathbb{P}\left(\mathrm{Tr}[(X_N-\widetilde{X}_N)^2]\geq \frac{\epsilon^2 N}{9 \|f\|_{\mathrm{Lip}}^2}\right).
|Tw_1 - Tw_2| \le \frac{p c_p^{1-p}}{(c_p-M)^{1-p}}|w_1 - w_2|_\infty + C_1|d||w_1 - w_2|_\infty = (\alpha + C_1 |d|) |w_1 - w_2|_\infty \le \frac{\alpha +1}2 |w_1 - w_2|_\infty\,.
u(X_{t})=\int_{t}^{\zeta}f_{u}(X_{r})\,dr+\int_{t}^{\zeta}dA_{r}^{\mu}+\int_{t}^{\zeta}dA_{r}^{\nu}+\int_{t}^{\zeta}dM_{r},0\le t\le\zeta,P_x\mbox{-a.s.}
L(x) = \sum_{\substack{d<x^{1/3}(d,hn)=1}}\frac{\gamma_d}{\varphi\bigl([d,q]\bigr)}= \frac{1}{\varphi(q)}\sum_{c|q^{\infty}}\gamma_c\frac{(c,q)}{c}\sum_{\substack{d<x^{1/3}/c (d,hnq)=1}}\frac{\gamma(d)}{\varphi(d)}\ .
A_{\mu}^{\prime}( \tau , \vec{x}) = g^{- 1}( \tau , \vec{x}) A_{\mu}( \tau , \vec{x}) g ( \tau , \vec{x}) + i g^{- 1}( \tau , \vec{x}) \nabla_{\mu}( \tau , \vec{x}) g ( \tau , \vec{x})
(1+\epsilon)F(v)-F((1+\epsilon)v-\rho) = (1+\epsilon)F(v)-F((1+\epsilon)v)+F((1+\epsilon)v)-F((1+\epsilon)v-\rho) \succeq C_l\epsilon \mathbf{1}-C_f\rho\succeq C_l\epsilon \mathbf{1}-C_fK_4\epsilon \hat V^*
G^{n_{1}\ldots n_{p}, a_{1}\ldots a_{r}}( x_{1}, \ldots , x_{p}, y_{1}, \ldots y_{r}) = \int{\cal D}[ U ] \phi_{a_{1}}( y_{1}) \ldots \phi_{a_{r}}( y_{r}) \exp ( - \int_{M}{\cal L}[ U ] ) ,
\zeta_k(-1, \mathfrak{A}) =\frac{\textsl{sgn}\delta(\mathfrak{a})~r_2r_2'}{360N(\mathfrak{a})c^3}\big\{(a+d)^3-6(a+d)N(\varepsilon)-240c^3(\textsl{sgn}c) &\times S^3(a,c)+180ac^3(\textsl{sgn}c)S^2(a,c)-240c^3(\textsl{sgn}c)S^3(d,c) & +180dc^3(\textsl{sgn}c)S^2(d,c) \big\},
(div_{\sigma}\eta)^{(0)}= \frac{1}{3}\tilde{\nabla}^a(\beta_a)=\rho (div_{\sigma}\eta)^{(1)}= \frac{1}{4}\tilde{\nabla}^a D \beta_a = D \rho (div_{\sigma}\eta)^{(2)}= \frac{1}{10}\tilde{\nabla}^a (D^2 \beta_a ) -\frac{2}{15}\tilde{\nabla}^a ( \alpha_{ab}\beta^b) = \frac{1}{2}D^2\rho+\frac{1}{15}( |\alpha|^2- 8 |\beta|^2).
\gamma_{0}\mathcal{D}_{\partial \Omega}[\phi]=- \tfrac{1}{2}\phi + \mathcal{K}_{\partial \Omega}[\phi]\, , \gamma_{0}^\complement\mathcal{D}_{\partial\Omega}[\phi]= \tfrac{1}{2}\phi + \mathcal{K}_{\partial \Omega}[\phi]\, , \forall \phi \in H^{1/2+\tau}(\partial \Omega)\, .
\sup_{x \in \mathbb{R}}\left\|\mathcal{P}^+\left(z^{-i}\bar{r}_1(z)e^{-2iz x}\right)\right\|_{L_z^\infty}\leq\frac{1}{\sqrt{2}}\left\|z^{-i}r_1(z)\right\|_{H^1},i=0,1, &\sup_{x \in \mathbb{R}}\left\|\mathcal{P}^-\left(r_2(z) \mathrm{e}^{2iz x}\right)\right\|_{L_{z}^{\infty}}\leq\frac{1}{\sqrt{2}}\left\|r_2(z)\right\|_{H^{1}}.
\begin{array}{rcl}tt^{-1}=1=t^{-1}t~, begin{align*}2mm] tet^{-1}=q^{2}e~,~~~~tft^{-1}=q^{-2}f~, begin{align*}2mm] [e,f]=\displaystyle\frac{t-t^{-1}}{q-q^{-1}}~.\end{array}
S = \frac{1}{1 6 \pi}\int d^{4}x \sqrt{- g}\left[ R - 2 ( \nabla \phi )^{2}- \frac{2}{3}( \nabla \sigma )^{2}\right. \left. - e^{- 2 \phi}F^{2}- e^{- 2 q \sigma / 3}F^{2}\right] .
\begin{aligned}v^{'}_{(\{l_i\}_{i=1}^{q-1})}:=x_{1}x_{-1}^{k^{'}_1-1}\left\{\prod_{i=2}^{q}x_{c^{'}_{i-1}}x_{1}^{l_{i-1}}x_{-1}^{k^{'}_{i}-1}\right\}&=z_{1}(k^{'}_1)\left\{\prod_{i=2}^{q}z_{c^{'}_{i-1}}(1)z_{1}(1)^{l_{i-1}-1}z_{1}(k^{'}_{i})\right\},\end{aligned}
f(x) =\int\limits_{D\setminus D_n}P_{D_n}(x,y)f(y)dy &=\int\limits_{D_n}M_{D_n}(x,v)\left(G_{D_n}(x_0,v)\int\limits_{D\setminus D_n}j(|v-y|)f(y)dy\right)dv &=\int\limits_{D_n}M_{D_n}(x,v)\eta_{D_n}f(dv),
[W_t(h,a),y_i] =(1+\frac{y_i}2)\cdot W_t(h,a)+(1-\frac{y_i}2)\cdot W_t(h,a) &=W_t\Big((1+\frac{y_i}2)\cdot h,(1+\frac{y_i}2)\cdot a\Big)+ W_t\Big((1-\frac{y_i}2)\cdot h,(1-\frac{y_i}2)\cdot a\Big).
\iota^\bullet \colon \begin{cases}\mathcal F_v (R) \to \mathcal F_v (H) \setminus \{\emptyset \}I \mapsto I^{\bullet}\end{cases}\iota^\circ \colon \begin{cases}\mathcal F_v (H) \setminus \{\emptyset \}\to \mathcal F_v (R) \mathfrak a \mapsto \mathfrak a_{v_R}\end{cases}
x^{\boldsymbol{\omega},-}_i =\tilde{z}_j+k i =2i_kN+j, \: k\in \mathbb{Z}, \: j=1,...,2(i_{k+1}-i_k)N, x^{\boldsymbol{\omega},+}_i =\tilde{z}_{-j+1}+k i =2i_kN-j, \: k\in \mathbb{Z}, \: j=0,...,2(i_k-i_{k-1})N-1.
g_{n}=\frac{\left(-1\right)^{n+1}}{\left(n+1\right)!}\left(x^{n+1}-\left(x-1\right)^{n+1}\right)=\frac{\left(-1\right)^{n+1}}{n!}\int_{0}^{1}\left(x+u-1\right)^{n}du,
a_2=2a_1-a_0,\; a_3=(\nu a_0-1)(2a_1-a_0), \; a_4=\nu a_0 (2a_1-a_0)-a_1,\; d_1=(\nu a_0-1)(2a_1-a_0)+a_1, d_2=\nu a_0 (2a_1-a_0),\; \mbox{where}\; \gcd(\nu a_0-1,a_1-a_0)=1.
x_{U,\epsilon}(t) \geq x_{U,\epsilon}(T_2(\epsilon))=x_{U,\epsilon}(T_1(\epsilon)/(1-q-\epsilon)) &=G_0^{-1}(\lambda(\epsilon)T_1(\epsilon)/(1-q-\epsilon)) &=G_0^{-1}(\eta(\epsilon) G_0(\delta(\epsilon))).
f(\mathbf{x})=\sum_{p\in (n)_m}\left(\sum_{p_0\in [n]\setminus\{p_1\}}\mathbf{x}_{p_0,p_1}\right)\left(\prod_{i=1}^{m-1}\mathbf{x}_{p_i,p_{i+1}}\right)\left(\sum_{p_{m+1}\in [n]\setminus \{p_m\}}\mathbf{x}_{p_m,p_{m+1}}\right)\;.
2 - 2g_{C}= 2m - \gcd(m,\deg(f))\left(\frac{m}{\gcd(m,\deg(f)}- 1\right) - \sum_{j=1}^{n-1}\deg(f_{j})\gcd(j,m)\left(\frac{m}{\gcd(j,m)}- 1\right) &= 2m - (m - \gcd(m,\deg(f))) - \sum_{j=1}^{n-1}\deg(f_{j})(m - \gcd(j,m)) &= m + \gcd(m,\deg(f)) - \sum_{j=1}^{n-1}\deg(f_{j})(m - \gcd(j,m)),
||x^*||_1\,\varphi(t^*,x^*) - \mathbb{E}\left[L(\rho)X(\rho)\varphi \big(t^*-\rho,\mathfrak{X}(\rho)\big)\right] =\, \mathbb{E}\left[\int_0^{\rho}L(s)X(s)\, g\big(t^*-s,s,\mathfrak{X}\big)\, \mathrm{d}s \right] >0\,,
\int\limits_{0}^{\infty}\frac{x^{\alpha - 1}\cos{bx}}{x\sinh{c x}}dx = \frac{\Gamma (\alpha )}{(2c)^{\alpha}}\left\{\zeta \left[\alpha ,\frac{1}{2}\left( 1+\frac{ib}{c}\right) \right] + \zeta \left[\alpha ,\frac{1}{2}\left( 1-\frac{ib}{c}\right) \right] \right\}\,,
m\lfloor\frac{m-1}{k-1}\rfloor-\frac{k-1}{2}\lfloor\frac{m-1}{k-1}\rfloor(\lfloor\frac{m-1}{k-1}\rfloor+1)>\frac{m^{2}}{2(k-1)}-\frac{m}{2}-\frac{k}{2(k-1)}
u\triangleq[\begin{matrix}u_{1}^{A}\\u_{2}^{A}\\u_{2}^{B}\\u_{3}^{B}\\ \end{matrix}]f\triangleq[\begin{matrix}f_{1}^{A}\\f_{2}^{A}\\f_{2}^{B}\\f_{3}^{B}\\ \end{matrix}]g\triangleq[\begin{matrix}0\\g_{2}^{A}\\g_{2}^{B}\\ 0\\ \end{matrix}]u,f,g\in\mathbb{R}^{n}
U_{h,\theta}|_{\Omega_{h,k}^{(i)}}\in O_{\epsilon}({U}_i^{(0)}),\,\, i=1,2,\sigma_{j}^{(k)}\in O_{\hat{\epsilon}}(\sigma_{j0}),\,\, j=2,3,\gamma_{4}^{(k)}\in O_{\hat{\epsilon}}(0).
\begin{cases}\begin{array}{ll}u(.,t)\in SPSH(\Omega), \dot{u}=\log\det (u_{\alpha\bar{\beta}})-Au+f(z,t)\;\;\; \mbox{on}\;\Omega\times (0,T), u=\varphi \mbox{on}\;\partial\Omega\times [0,T), u(.,t)\stackrel{L^1}{\longrightarrow}u_0. \end{array}\end{cases}
I_2 =\frac{1}{2\pi i}\int_{-T_2}^{+T_2}dt\frac{1}{t-is}\omega_0^{(\Lambda,N)}(a^\ast P_0^{(\Lambda,N)}(\phi)[a,B^{(\Lambda)}(t;\phi)];\phi)e^{-\alpha t^2}&=I_2^{-}+I_2^{0}+I_2^{+}
\bigl\{n-1+i_1,\, n-1-(l+1)+i_2,\, \dots,\, n-1-(p-2)(l+1)+i_{p-1}\bigr\}\cr=\, \bigl\{n-1-l,\, n-1-l-(l+1),\, \dots,\, n-1-l-(p-2)(l+1)\bigr\}.
u\left( x,x_{0},k\right) =A\left( x,y\right) e^{ik\tau \left( x,y\right)}\left( 1+h\left( x,y\left( x_{0}\right) ,k\right) \right) ,k\rightarrow\infty ,\forall x\in \overline{\Omega},\forall x_{0}\in \left[ 0,1\right] ,
\begin{cases}L=\pi \sqrt{\dfrac{(1-a^2b)\alpha(k,l,m,n,s)}{3r}}, \xi_0 = -\dfrac{\pi}{3}(5k+4l+3m+2n+s), p= \sqrt{\dfrac{(1-a^2b)\xi_0\xi_1\xi_2\xi_3\xi_4\xi_5}{c}},\end{cases}
\sum_{d = 1}^{r_i+s_i}\sum_{\substack{\mu_1+\dots+\mu_{d}= r_i+s_i \mu_i \ge 1}}\frac{(-1)^{d}}{\mu_1!\cdots\mu_{d}!}H(r_i + s_i, r_i, \mu_1, \mu_{d}).
\beta_1^{n+1}= \Big|\Big|J_{\mathcal A_{{t_{n+1}},~t_{n+1/2}}}\Big|\Big|_{L_{\infty}(\Omega_{t^{n+1}})}||\nabla \cdot \mathbf{w}_{h}||_{L_{\infty}(\Omega_{t^{n+1/2}})}, \beta_2^n = \Big|\Big| J_{\mathcal A_{{t_{n}},~t_{n+1/2}}}\Big| \Big|_{L_{\infty}(\Omega_{t^{n}})}|| \nabla \cdot \mathbf{w}_{h}||_{L_{\infty}(\Omega_{t^n})},
\psi^{s}= \psi_{l}= \left( \begin{array}{c}{E + m}\\{- i k}\\ \end{array}\right) e^{i k x}+ l ( k ) \left( \begin{array}{c}{E + m}\\{i k}\\ \end{array}\right) e^{- i k x}, \qquad x \leq - \xi
S = - \sum_{I = 1}^{K}\, \int d \tau_{I}\, ( m_{0}^{I}\, \sqrt{v_{\mu}^{I}\, v_{I}^{\mu}}+ \mathrm{t r}\, Z_{I}\lambda_{I}^{- 1}{\dot{\lambda}}_{I}) - \int d^{4}x \, \mathrm{t r}\, \bigl ( j_{\mu}\, A^{\mu}+{\frac{1}{1 6 \pi}}\, F_{\mu \nu}\, F^{\mu \nu}\bigr ) .
\gamma_{\xi}: = \mu \frac \partial{\partial \mu}\xi_{\mathrm{R}}= 2 \lambda_{\mathrm{R}}\xi_{\mathrm{R}}( \xi_{\mathrm{R}}- 1 ) \left( \xi_{\mathrm{R}}-{\frac{1}{2}}\right) \frac{C_{2}( G )}{( 4 \pi )^{2}}g_{\mathrm{R}}^{2},
d^nf(s_1,\ldots,s_{n+1}) :=\kappa_{s_1}f(s_2,\ldots,s_{n+1}) +\sum_{i=1}^n{(-1)^nf(s_1,\ldots,s_i+ s_{i+1},\ldots,s_{n+1})}+(-1)^{n+1}f(s_1,\ldots,s_n)
\left|\frac{1}{q}\sum_{m\not\equiv \frac{\pm ql}{2l+1}}\zeta_q^{-im}\prod_{p=0}^{k-1}\left(1-\zeta_q^{b^pm}\right)\right|=\left|\frac{1}{q}\sum_{m\not\equiv \frac{\pm ql}{2l+1}}\zeta_q^{-im}\left(\prod_{p=0}^{s-1}(1-\zeta_q^{b^pm})\right)^{k_1}\left(\prod_{p=0}^{k_2-1}(1-\zeta_q^{b^pm})\right)\right|.
\begin{aligned}\mathcal{M}_1(q) = -2q^{-10}m(-q^{-7}, q^{48}, q^6) + 2q^{-23}m(-q^{-23}, q^{48}, q^6) + 2m(-q^{25}, q^{48}, q^{-6}) & - 2q^{-10}m(-q^{-7}, q^{48}, q^{-6}) + \frac{2q}{(q)_{\infty}}\Theta_{3,2}(q^6, q^4, q).\end{aligned}
\left(v_{xx}^{[n]}-v^{[n]}\right)_x-u_t=0, v_{xx}^{[j]}-v^{[j]}+u\omega^{[j+1]}=0, j=1\dots n-1, \left(u\omega^{[1]}\right)_x+u_y=0, &\omega_x^{[j]}=uv_x^{[j]}, j=1\dots n.
\frac{2}{\kappa_j}\left[\cos(\kappa_j x) +\left(\frac{\xi}{\kappa_j}\right)\sin(\kappa_j |x|)\right]\sin(\kappa_j t)= 2\left( \frac{\xi L}{2}\right)\left[ \frac{t}{Z_j^2}+2\sum_{n=1}^\infty\frac{1}{k_n}\frac{\cos(k_n x) \sin(k_n t)}{Z_j^2 - (\pi n)^2}\right]
\partial_p c^1(p_1) = \lim\limits_{h\searrow 0}\frac{c^1(p_1 + h)-c^1(p_1)}{h}\geq 0\ \ \partial_p c^1(p_1) = \lim\limits_{h\searrow 0}\frac{c^1(p_1)-c^1(p_1-h)}{h}\leq 0
\mathcal{P}^+\left(f(z) e^{-2iz x}\right) =\int_{2x}^{+\infty}\widehat{f}(\xi)e^{iz(\xi-2x)}\mathrm{d}\xi, &\mathcal{P}^-\left(f(z) e^{2iz x}\right) =-\int^{\infty}_{2x}\widehat{f}(\xi)e^{-iz(\xi-2x )}\mathrm{d}\xi,
S_2v=-K_{y\to x}\left( B[v](|y|)\left[\widetilde a_{ij}\theta_i\theta_j-\overline{\widetilde a_{ij}\theta_i\theta_j}\right]+\widetilde a_{ij}\theta_i\theta_j-\overline{\widetilde a_{ij}\theta_i\theta_j}\right),
{B}_j^d(\xi)=\Big(\frac{x_j-\xi}{x_j-x_{j-d}}\Big){B}_j^{d-1}(\xi)+\Big(\frac{\xi-x_{j-d-1}}{x_{j-1}-x_{j-d-1}}\Big){B}_{j-1}^{d-1}(\xi),\;\;\xi\in \mathbb{R}.
T(r,s,t) = ( - 1)^t \sum_{i = t - s + 2}^t{( - 1)^i \binom{2t-i-1}{t-1}\zeta (s - t + i)\zeta (r + 2t - i)}+ ( - 1)^s \sum_{i = 2}^t{\binom{s+t-i-1}{s-1}\zeta (i)\zeta (r + s + t - i)}\quad- ( - 1)^s \sum_{i = 1}^t{\binom{s+t-i-1}{s-1}\sum_{\mu = 1}^\infty{\frac{{H_{\mu ,i}}}{{\mu^{r + s + t - i}}}}}\,,
d_{i,j}(z) = \sum_{k = 1}^{p+1}(T(-z)^{-1})_{i,k}( \mathcal{A}(z) )_{k,j}= - \sum_{k = 1}^{p+1}(T(z)^{-1})_{i,k}( \mathcal{A}(z) )_{k,j}.
\int_0^1 t^s h^n_{n-i}(t)dt =\sum_{k=0}^n d_{n,k}^i \frac{1}{k+s+1}=c_{n,i}\sum_{k=0}^n\frac{(-n)_k(n-i+1)_k}{(1)_k(s+2)_k}q(k),
\{x^i,x^j\}_\mathbb{I}=\frac{1}{4}{\rm Im}\,\{z^i,z^j\}\,,\,\{x^i,x^{n+j}\}=-\frac{1}{4}{\rm Re}\,\{z^i,z^j\}\,,\,\{x^{n+i},x^{n+j}\}=-\frac{1}{4}{\rm Im}\,\{z^i,z^j\}.
\begin{aligned}V(t_0,x_0,\nu_0) := \frac{1}{2}\mathbb{E}\Big\{\int_{t_0}^T\Big[Q\left(x_t^{*,t_0,x_0,\xi}+l(\nu_t^{*,t_0,\xi})\right)^2+R\left(\alpha_t^{*,t_0,x_0,\xi}+h(\mu_t^{*,t_0,\xi})\right)^2\Big]dt+G\left(x_T^{*,t_0,x_0,\xi}+g(\mu_T^{*,t_0,\xi})\right)^2\Big\}\end{aligned}
\lvert \mathrm{Im}\log\Phi'(z)\rvert \leqslant \frac{1}{\pi}\int_{-\infty}^{+\infty}\frac{y\lvert\mathrm{Im}\log\Phi'(t)\rvert\,\mathrm{d}t}{(t-x)^2+y^2}\leqslant \frac{\theta_0}{\pi}\int_{-\infty}^{+\infty}\frac{y\,\mathrm{d}t}{(t-x)^2+y^2}= \theta_0,
Xe^{-t}- \frac{e^{-\alpha t}Y}{(1-e^{- \alpha t})^{1 - \frac{1}{\alpha}}}=_{\mathcal{L}}\left(e^{- \alpha t}\dfrac{(1-e^{- \alpha t})^{\alpha -1}+e^{-\alpha t}}{(1-e^{-\alpha t})^{\alpha - 1}}\right)^{\frac{1}{\alpha}}X .
\hat{Y}_{\ell m}\hat{Y}_{\ell^{\prime}m^{\prime}}= \sum_{\ell^{\prime \prime}m^{\prime \prime}, ( \ell^{\prime \prime}\leq \mathrm{m i n}( \ell + \ell^{\prime}, 2 j ) )}C_{\ell m , \ell^{\prime}m^{\prime}}^{\ell^{\prime \prime}m^{\prime \prime}}\hat{Y}_{\ell^{\prime \prime}m^{\prime \prime}}
\begin{cases}u_{tt}(t) - \mathcal{L}u(t) + bu_t(t)+ mu(t)= f(u), t>0, u(x,0) = u_0(x),\,\,\, u_0 \in H^1_{\mathcal{L}}(\mathbb{H}^n), u_t(x,0) = u_1(x),\,\,\, u_1 \in L^2(\mathbb{H}^n),\end{cases}
\eta_k \coloneqq\begin{cases}1 \overline{B}_{R_k}(\mathbf{0}) \; , 1 - \bigl( |\mathbf{x}| - R_k)^2 B_{R_k+1}(\mathbf{0}) \setminus B_{R_k}(\mathbf{0}) , 0 .\end{cases}
(\hat{\nu}_{\epsilon},a) = \sum_{i,k}\hat{\nu}_{\epsilon\,i}(\delta_{ij}-\frac{1}{2}{\cal C}_{ij})\,a_j = \sum_{i\in\Gamma_{\epsilon}^{(0)}}\hat{\nu}_{\epsilon\,i}\,a_i - cos\frac{\pi}{h}\sum_{i\in\Gamma_{\epsilon+1}^{(0)}}\hat{\nu}_{\epsilon+1\,i}\,a_i
S=\frac{1}{4g_{YM}^{2}}{\rm Tr}\left( V\sum_{\alpha \beta}(F_{\alpha \beta}+ \phi_{\alpha\beta}\cdot{\bf 1})g^{\alpha \mu}g^{\beta \nu}(F_{\mu \nu}+ \phi_{\mu\nu}\cdot{\bf 1}) \right) \, .
J_2(X;z^{\prime},\bar{z}^{\prime}) =(n+m+1)\big((-r)^{n+m+1}L_{\Omega}\big)\frac{D(-r)}{-r}&=(n+m+1)\big(\Psi+\Phi(-r)^{n+m+1}\log(-r)\big)\frac{D(-r)}{-r}.
\underset{s\in S}{\sum}|C_K(s)|= \underset{s\in X_S}{\sum}|C_K(s)|+\underset{s\in Z(S,K)}{\sum}|C_K(s)|+\underset{s\in S\setminus (X_S\cup Z(S,K))}{\sum}|C_K(s)| = |X_S| + |K| |Z(S, K)| + \underset{s\in S\setminus (X_S\cup Z(S,K))}{\sum}|C_K(s)|.
\sum_{j=1}^3\sum_k\int_{\mathbb{R}}e^{(|k|+|\eta|)/2}\big|\widehat{g_j'}(t,k,\xi)\big|^2\,\mathrm{d}\xi\lesssim t^2M_0(t)\lesssim e^{CT_0^{21/20}}(Z(t))^2.
\begin{pmatrix}1 0 0 1 0 1 R_{M_1}0 1 0 -1 0 1 \mathbf{0}_m 0 0 1 1 0 0 R_{M_1}0 0 0 0 1 1 \mathbf{0}_m 0 0 0 0 0 2 \mathbf{0}_m 0 0 0 0 0 0 M_{n}- M_{1}\vdots \vdots \vdots \vdots \vdots \vdots \vdots 0 0 0 0 0 0 M_{n}- M_{n-1}\end{pmatrix}
X:=\prod_{v\in V_{\tau}}\mathcal{S}_{F_{\tau}(v)}(\mathbb{P}^{n},d_{\tau}(v))^{\circ}\cong\prod_{v\in V_{\sigma}}\mathcal{S}_{F_{\sigma}(v)}(\mathbb{P}^{n},d_{\sigma}(v))^{\circ}\times\prod_{v\in V_{\sigma'}}\mathcal{S}_{F_{\sigma'}(v)}(\mathbb{P}^{n},d_{\sigma'}(v))^{\circ}.
2 ) - 3 \sqrt{3}\sin 2 x + 3 \cos 2 x = - 6 ( \frac{\sqrt{3}}{2}\sin 2 x - \frac{1}{2}\cos 2 x ) = - 6 \sin ( 2 x - \frac{\pi}{6}) = 6 \sin [ 2 x + \arctan ( \frac{3}{3}) ] = 6 \sin ( 2 x + \frac{7 \pi}{6})
\hat{\Theta}_{\Delta}^{i}[N]=-\frac{2}{3!}\frac{1}{i\hbar}\;N(v(\Delta))\;\epsilon^{JKL}{\hat{\Phi}}^{E}(F_{JK})\,tr\left( \tau^{i}\;h_{s_{L}(\Delta )}\left[h_{s_{L}(\Delta )}^{-1},\;\sqrt{\hat{V}(v(\Delta ),\epsilon )}\right] \right)
\hat{\phi}( x^{0}, x^{1}) = \frac{1}{\sqrt{2 \pi}}\int_{0}^{\infty}d k^{1}\frac{1}{\sqrt{k^{1}}}\left[ \hat{\Lambda}( k^{1}) \, e^{- i k^{1}( x^{0}+ x^{1})}\, \, + \, \, \hat{\Lambda}^{\dagger}( k^{1}) \, e^{i k^{1}( x^{0}+ x^{1})}\right] ,
\begin{array}{ccccccc}\beta_1\, = \,\alpha_2+2\alpha_3+3\alpha_4+2\alpha_5+2 \alpha_6+\alpha_7 ; \beta_2 = \alpha_1 \cr\beta_3\, = \,\alpha_2 ; \beta_4 = \alpha_3 \cr\beta_5\, = \,\alpha_4 ; \beta_6 = \alpha_6 \cr\beta_7\, = \,\alpha_7 \null \null \null \null \cr\end{array}
\begin{cases}\alpha_1 \displaystyle \frac{r(T_1^p, S_j)}{o(S_j)}+ \cdots +\alpha_u \displaystyle \frac{r(T_u^p, S_j)}{o(S_j)}= 0 \ \ \j=1,\dots,v, &\alpha_1 \epsilon_1 + \cdots + \alpha_u \epsilon_u = 0. \end{cases}
\begin{cases}0<\underbar r_1^{n-1}\leq \underbar r_1^n\leq \bar r_1^n\leq \bar r^{n-1}_1\le \bar A_1\cr0<\underbar r_2^{n-1}\leq \underbar r_2^n\leq \bar r_2^n\leq \bar r_2^{n-1}\le \bar A_2\end{cases}
\Psi^{-1}(\{P\in \mathrm{Spec}(k[X_{1}, ..., X_{n}]):f_{1}, ...., f_{m}\in P\})=\left\{p\in S_{n}^{K}(k):\bigwedge_{i=1}^{m}f_{i}(\overline{v})=0\in p\right\}
S^{( D 9 )}= \mathrm{T r}\, \int_{P P_{6}\times R^{4}}\frac{1}{2}F \wedge * F + \frac{1}{4 !}C^{( 2 )}\wedge F \wedge F \wedge F \wedge F + \frac{1}{2 !}C^{( 6 )}\wedge F \wedge F
\int d^4v\,V_\rho(y,v,0)=\frac{ig}{16\pi^4}\,(f^2-\frac{1}{2}g^2)\,\gamma_5\gamma_\sigma\left(\frac{y_\sigma y_\rho}{y^6}-\frac{\delta_{\sigma\rho}}{4y^4}\right) \, .
\left(\langle \frac{1}{i}\cdot \left(\alpha \cdot \iota(\psi) \right) + \gamma \cdot (\iota(l(\psi))) , l(\psi) \rangle_{\mathcal{S}}\right) = \left(\langle d \cdot \left(-a \cdot \psi_- + b \cdot \psi_+ \right) + \gamma \cdot (a \psi_- - b \psi_+), a\psi_+ - b \psi_- \rangle_{\mathcal{S}}\right) = 0.
\|h_{1}\|_{L^\infty}= \sup_{x\in (-\infty, x_0)}\bigg| \frac{1}{2i}\int_{-\infty}^x(|u_y|^2 f_1+u_yf_2)dy\bigg| \leq \frac{1}{2}( \|f_1\|_{L^\infty}\|u_x\|_{L^2}^2+\|f_2\|_{L^\infty}\|u_x\|_{L^1})dy \leq\frac{1}{2}(\|u_x\|^2_{L^2}+\|u_x\|_{L^1})\|f\|_{L^{\infty}}.
x_{i}=r_{i}cos\phi_{i}+\sqrt{l_{i}^{2}(r_{i}sin\phi_{i})^{2}}=l_{1}+r_{i}cos\phi_{i}-\frac{r_{i}^{2}}{l_{i}}(1-cos(2\phi_{i}))/2+O(r_{i}^{3}/l^{2})
e_k (id:\ell^n_p(\mathbb{R})\rightarrow \ell^n_q(\mathbb{R}))\sim \begin{cases}1 1\leq k\leq \log_2n, \displaystyle\biggl(\frac{\log_2(1+n/k)}{k}\biggr)^{\frac{1}{p}-\frac{1}{q}}\log_2n\leq k\leq n, begin{align*}10pt]\displaystyle2^{-\frac{k-1}{n}}n^{\frac{1}{q}-\frac{1}{p}}n\leq k.\end{cases}
\epsilon ( \omega^{( 0 )},{\bf k}) = 1 +{\frac{3 \omega_{p}^{2}}{K^{2}}}[ 1 -{\frac{\omega^{( 0 )}}{2 K}}\bigl ({\ln}|{\frac{K + \omega^{( 0 )}}{K - \omega^{( 0 )}}}| - i \pi \Theta ( K - \omega^{( 0 )}) \bigr ) ] .
d\boldsymbol{X}_t=d\boldsymbol{W}_t\sqrt{\frac{I+\boldsymbol{X}_t^2}{2}}+\sqrt{\frac{I+\boldsymbol{X}_t^2}{2}}d\boldsymbol{W}_t^*+\left[(-N-2\Re(s))\boldsymbol{X}_t+2\Im(s)\boldsymbol{I}+ Tr\left(\boldsymbol{X}_t\right)\boldsymbol{I}\right]dt.
\frac{R''}{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).
\kappa_n ( \xi \eta ) = \sum_{\begin{array}{c}{\scriptstyle \pi , \rho \in NC(n) \such}{\scriptstyle that \ \pi \vee \rho = 1_n}\end{array}}\ \Bigl( \, \prod_{V \in \pi}\kappa_{|V|}( \xi ) \, \Bigr) \,\Bigl( \, \prod_{W \in \rho}\kappa_{|W|}( \eta ) \, \Bigr) .
{\cal W}= \epsilon_{\alpha \beta \gamma}\Phi_{\alpha}\chi_{\beta}\chi_{\gamma}+ \chi_{1}P^{1}{\widehat R}^{1}+{\widetilde \chi}_{1}P^{1}R^{1}+{\widehat \chi}_{1}{\widehat P}^{1}R^{1}+ \dots ~ .
R_v=\sum_{i=0}^{k-1}\frac{i!(k-1-i)!}{k}\sum_{<r_1,\ldots,r_i>}{a_1-1\choose r_1}{a_2\choose r_2}\cdots{a_i\choose r_i}(-1)^{r_2+r_4+r_6+\cdots},
V ( \Phi ) ={\frac{1}{2}}\Omega^{2}\Phi^{2}+{\frac{1}{1 2 8 \pi^{2}}}[ 2 \Omega^{4}l n{\frac{\Omega^{2}}{m^{2}}}- ( 3 - 2 k ) ( \Omega^{2}- m^{2})^{2}- 2 m^{2}( \Omega^{2}- m^{2}) ] +{\cal D}_{1}
- f^{a}{}_{c b}C^{c}{}_{d}{}^{\nu}A^{\prime b}{}_{\mu}A^{\prime d \mu}+ \left[ - \partial_{\mu}C^{a}{}_{d}{}^{\mu}+ C^{a}{}_{c \mu}C^{c}{}_{d}{}^{\mu}\right] A^{\prime d \nu}.
\begin{aligned}x_{21}x_{22}\det(A_1A_0)\cdot (A_1A_0)_{2,1}= x_{21}x_{22}a_{11}a_{22}(x_{11}x_{22}-x_{12}x_{21})a_{22}x_{21}= x_{21}^2 x_{22}a_{11}a_{22}^2 (x_{11}x_{22}- x_{12}x_{21}). \end{aligned}
\qquad\widehat{V}_j^n = \frac{\pi}{2}\frac{n}{k_n-1}\sum_{i=(j-1)k_n+2}^{jk_n}\bigl|\Delta_{i-1}^nX\bigr|\bigl|\Delta_i^nX\bigr|,j=1,\ldots,\lfloor n/k_n\rfloor.
\int\limits_{M_\delta}dp\; p^\alpha \ \leq \ \frac{1}{\delta^2}\int\limits_0^\infty dp\;p^\alpha \left| \chi \left( p \right) - \phi_0 \right|^2 \ < \ \frac{q_0^2 \varepsilon \left(q_0\right)}{\delta^2}\ .
c(p):=\inf_{\left(s,t\right) \in\mathbb{R}^{2}\setminus\left\{\left( 0,0\right) \right\}}\frac{\left[ t^{2}+s^{2}+2s+1\right]^{\frac{p}{2}}-1-ps}{\left[t^{2}+s^{2}\right]^{\frac{p}{2}}}\in\left( 0,1\right]
\rho_\mathfrak{a}^k(0, w) = (*) \frac{L^k_\mathfrak{a}(2w-1)}{L^k_\mathfrak{a}(2w)}\frac{\Gamma(2w-1)}{\Gamma(w + \frac{k}{2}) \Gamma(w - \frac{k}{2})}\rho_\mathfrak{a}^k(h, w) = (*) \frac{\lvert h \rvert^{w-1}}{L^k_\mathfrak{a}(2w)}\frac{1}{\Gamma(w + \frac{\lvert h \rvert}{h}\frac{k}{2})}D_\mathfrak{a}^k(h,w)
a_{n+1}=\frac{||Q_{2^s}\left(\cdot;\mu_{K(\gamma)}\right)||_{L^{2}\left(\mu_{K(\gamma)}\right)}}{||Q_{2^{s-1}}\left(\cdot;\mu_{K(\gamma)}\right)||_{L^{2}\left(\mu_{K(\gamma)}\right)}\cdot a_{2^{s-1}+1}\cdot a_{2^{s-1}+2}\cdots a_{2^{s}-1}}.
{\rm Fb}\circ \varphi (a) = \left( a^{p^i - 1}- a^{p^{n-i}-1}\right)^{p^i}= \left( a^{p^i - 1}\right)^{p^i}- \frac{1}{a^{p^{i}-1}}= \psi_2 \circ \psi_1 (a),
\mu^2\frac{\partial_t{g}^{-1}(t)}{v\left(g^{-1}(t)\right)}=\mu^2\partial_t\ln\left({v\left(g^{-1}(t)\right)}\right)=\partial_t\ln\bigr({v\left(g(t)\right)}\bigr)=\frac{\partial_t{g}(t)}{v\left(g(t)\right)}
\begin{array}{cclccl}\omega{^a}_b = \gamma{^a}_b - \frac{1}{2}\textsf{k}_{ij}\eta^{ac}\mathbf{F}{^j}_{cb}e^i \omega{^a}_i = \frac{1}{2}\textsf{k}_{ij}\eta^{ac}\mathbf{F}{^j}_{bc}e^b \omega{^i}_a = \frac{1}{2}\mathbf{F}{^i}_{ab}e^b \omega{^i}_j = \frac{1}{2}c^i_{jk}(e^k-2\mathbf{A}^k) \end{array}
\begin{cases}w_t-w_{rr}-\frac{N-1}{r}w_r\leq f(w) \mbox{for}r\in [c_{k}t-L\log t, \tilde c_k t], w(c_{k}t-L\log t,t)\leq u(c_{k}t-L\log t, t) \mbox{for}t\geq T, w(\tilde c_k t,t)\leq u(\tilde c_k t, t) \mbox{for}t\geq T, w(r, T)\leq u(r, T) \mbox{for}r\in [c_{k}T-L\log T, \tilde c_kT].\end{cases}
\begin{array}{lll}\epsilon(\widehat{\lambda}{^{\mu}}_{\nu})=\delta{^{\mu}}_{\nu}\,\,\,\,\,\,\,\, S(\widehat{\lambda}{^{\mu}}_{\nu})= g^{\mu\rho}\widehat{\lambda}_{\ \rho}^{\sigma}g_{\nu\sigma}= \widehat{\lambda}_{\nu}{^{\mu}}\epsilon(\widehat{x}^{\mu}) =0 S(\widehat{x}^{\mu})=-\widehat{\lambda}_{\nu}{^{\mu}}\widehat{x}^{\nu}\end{array}