harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\int_R^\infty |u|^{(p-1+j)p'}r^\theta\mathrm dr \leq C^{(p-1+j)p'}\|u\|_{X^{1,p}_\infty}^{(p-1+j)p'}R^{(j-2)\frac{\alpha_0+1}{p}}\int_R^{\infty}r^{\theta-(p+1)\frac{\alpha_0+1}p}\mathrm dr &=\dfrac{pC^{(p-1+j)p'}R^{(j-p-3)\frac{\alpha_0+1}p+\theta+1}}{(p+1)(\alpha_0+1)-p\theta-p}\|u\|_{X^{1,p}_\infty}^{(p-1+j)p'}.
[\lambda_a + 2\rho_a, \lambda_b + 2 \rho_b] = \lambda_{[a,b]}- 4 \rho_{[a,b]}- 6 [\lambda_a, \rho_b] \cr = \lambda_{[a,b]}+ 2 \rho_{[a,b]}- 6(\rho_{[a,b]}+ [\lambda_a,\rho_b])
\sqrt{1 - 2\varrho_{k}}\frac{1}{\varrho_{k}}e^{-\frac{\eta_{-}}{\varrho_{k}}v}= \frac{-1}{2\pi i}\int_{\mathcal{C}_{1 - \varrho_{k}}}dz \frac{z - \varrho_{k}}{1 - z - \varrho_{k}}\frac{1}{\sqrt{2z - 1}}\frac{1}{1 - z}e^{- \frac{\eta_{-}}{1 - z}v},
\Psi_{c}(x,t)= \left[ \frac{\Delta \hat{q}\Gamma(1/4)}{2}\right]^{-\frac{1}{4}}\exp\left\{-\frac{(x -\langle \hat{q}\rangle )^{4}}{8(\Delta \hat{q})^{4}}+ \frac{i}{\hbar}\left( \langle \hat{p}\rangle x -\frac{1}{2}\langle \hat{q}\rangle \langle \hat{p}\rangle \right) \right\}\; .
\langle\partial_\gamma v, (\partial_{\delta}h_{\alpha})v\rangle= -\overline{(h_{\alpha}-\lambda I)_{00}^{+}\left((\partial_\gamma h_{\alpha})v\right)_{0}}\left((\partial_\delta h_{\alpha})v\right)_{0}= c({h_{\alpha}}) \overline{(i\gamma\cdot\textbf{x}-\gamma\cdot\textbf{y})}(i\delta\cdot\textbf{x}-\delta\cdot\textbf{y}),
\left( \begin{array}{l}{{u_{A}}}\\{{\overline{{{v}}}^{\dot{B}}}}\\ \end{array}\right) \longrightarrow e^{i \chi}\left( \begin{array}{l}{{u_{A}}}\\{{\overline{{{v}}}^{\dot{B}}}}\\ \end{array}\right) \ \ \ \mathrm{w e ~ h a v e}
\delta_\mathrm{RBA}(u) =~ \big( \delta_\mathrm{Hoch}(u), -u \big), \delta_\mathrm{RBA}(f, g) =~ \big( \delta_\mathrm{Hoch}(f), ~ - \overline{\delta}_\mathrm{Hoch}(g) - \Phi^n (f) \big),
\frac{m(I_k \cap H)}{m(I_k)}= \frac{m(I_k \cap P)}{m(I_k)}+\frac{m(I_k \cap (H \setminus P))}{m(I_k)}< \frac{m(I_k \cap P)}{m(I_k)}+\frac{n_k+1}{2^{n_k}}
G^{n_{1}\ldots n_{p}, a_{1}\ldots a_{r}}( x_{1}\ldots x_{p}, y_{1}\ldots y_{r}) = \langle T \left( \psi^{n_{1}}( x_{1}) \ldots \psi^{n_{p}}( x_{p}) \phi_{a_{1}}( y_{1}) \ldots \phi_{a_{r}}( y_{r}) \right) \rangle ,
\lambda^2_{2N+1,2}(z) \le\frac{1}{2}\lambda^2_{N+1,2}(z^2)+\vert l_{N,N+1}(z^2)\vert^2\lambda^2_{N,2}(e_N)+\lambda^2_{N,2}(z^2)+\frac{1}{2}\vert l_{N,N+1}(z^2)\vert^2.
F_n^{\prime}(t) =\frac{d}{dx}\,\big\{(x-\sqrt{x^2-1})^nP_n(x)\big\}\,x^{\prime}(t) &=2(x-\sqrt{x^2-1})^{n-1}\big(\sqrt{x^2-1}\,P_n^{\prime}(x)-nP_n(x)\big)
\delta \Gamma_{e f f}= t r \int d^{d}x ( D_{\mu}\epsilon^{a}) \frac{\delta \Gamma_{e f f}}{\delta A_{\mu}^{a}}= - t r \int d^{d}x \epsilon^{a}D_{\mu}\left( \frac{\delta \Gamma_{e f f}}{\delta A_{\mu}^{a}}\right) .
\alpha=\frac{\sum_{i=1}^{N}C_{i}[\alpha]_{i}}{\sum_{i=1}^{N}C_{i}}=\sum_{i=1}^{N}(\frac{C_{i}}{\sum_{i=1}^{N}C_{i}})[\alpha]_{i}=\sum_{i=1}^{N}\chi_{i}[\alpha]_{i}
\langle{\cal C}^{1 1}( x_{1}){\cal C}_{1 1}^{\dagger}( x_{3}){\cal K}_{{\bf 1}}( x_{2}) \rangle_{_{( 0 )}}={\frac{N^{2}- 1}{3 ( 2 \pi )^{6}( x_{1 3})^{2}( x_{1 2})^{2}( x_{2 3})^{2}}}\; .
R(z)=\frac{1}{\kappa_2(z^2)}\frac{(p^2;p^2)_{\infty}}{(p;p)^2_{\infty}}\begin{pmatrix}a(z) 0 0 d(z) 0 b(z) c(z) 0 0 c(z) b(z) 0 d(z) 0 0 a(z)\end{pmatrix}
\begin{aligned}A(t,\rho) \leq S(t,\rho)+\int_0^t\Big[S(\tau,\rho)+\int_0^\tau A(\tau',\rho)\,d\tau'\Big]\,d\tau &\leq S(t,\rho)+ \int_0^t S(\tau,\rho)\,d\tau+ \int_0^t(t-\tau)A(\tau,\rho)\,d\tau.\end{aligned}
\begin{aligned}0 =8 u^3 +\left( 24 \gamma + 8 \kappa \right) u^2+\left( 16 \gamma \kappa + 4 C_\mathfrak{b}\gamma^2 + 18 \gamma^2 + 2 \kappa^2 \right) u & +2 \gamma^2 C_\mathfrak{b}\left( 3 \gamma - \kappa \right) + 2 \gamma \kappa^2 + 10 \gamma^2 \kappa .\end{aligned}
\lim_{N\to\infty}\P\left(\bigcap_{\ell=1}^k \{X_N^{R,{\rm resc}}(T_\ell)\leq U_\ell\}\right)=\frac{\P\left(\bigcap_{\ell=1}^k\{{\mathcal A}_2(T_\ell)-T_\ell^2\leq U_\ell\}\cap \{{\mathcal A}_2(0)\leq R)\}\right)}{\P({\mathcal A}_2(0)\leq R)}
Q(M^{k},\tilde M^{k+1})-Q( M^{k+1},\tilde M^{k+1}) &\hskip -2 cm=\frac{1}{2}\Vert (M^{k}-\tilde M^{k+1})_{\bar \Omega}\Vert^{2}_{2}-\frac{1}{2}\Vert (M^{k+1}-\tilde M^{k+1})_{\bar \Omega}\Vert^{2}_{2}\rightarrow 0.
N \int_I \tilde{G}_{\mu \nu}(z)\mathcal{X}(E) dE=N \int_{I}\tilde{G}_{\mu \nu}(z)\mathbf{1}(\mathcal{N}(E^{-}, E_U)=l)d E =N \int_{I}\tilde{G}_{\mu \nu}(z)q[\operatorname{Tr}\mathcal{X}_{E}(Q_1)]dE,
\frac{N-2\alpha}2\int_{\partial \mathcal M}y^{1-2\alpha}v\frac{\partial v}{\partial\nu}\,dS &=\frac 12 \int_{\partial \mathcal M}y^{1-2\alpha}|\nabla v|^2(X-z_0,\nu)\,dS - \int_{\partial \mathcal M}y^{1-2\alpha}(\nabla v, X-z_0)\frac{\partial v}{\partial\nu}\,dS,
\tilde{M}_{0, \varphi}(x) =\left(\begin{array}{cc}\frac{-8 i + 4 i e^{- i \varphi x}+ 4 i e^{i \varphi x}}{x - i}\frac{12 i - 4 i e^{- i \varphi x}- 8 i e^{i \varphi x}}{x - i}& \frac{- 6 i + 4 i e^{- i \varphi x}+ 2 i e^{i \varphi x}}{x - i}\frac{8 i - 4 i e^{- i \varphi x}- 4 i e^{i \varphi x}}{x - i}\end{array}\right).
\int_{\{-t'_3\le\Psi<-t'_4\}}|F_{t_0,\tilde{c}}|^2e^{-\varphi}\le \int_{\{z\in M:-\Psi(z)\in (t'_4,t'_3]\}}|\tilde F|^2_h+\int_{\{z\in M:-\Psi(z)\in (t'_4,t'_3]\backslash N\}}|F_{t_0,\tilde{c}}-\tilde F|^2_h &+\int_{\{z\in M: -\Psi(z)\in(t'_4,t'_3]\cap N\}}|F_{t_0,\tilde{c}}|^2_h.
\Psi_{l \zeta}^{{\vec{k}}{\vec{m}}}= \exp \left[ i \frac{N_{A}X}{L_{1}}\left( \theta + \frac{2 \pi N_{A}}{\kappa L_{2}}Y \right) \right] \prod_{\alpha < \beta}\left( \frac{\theta_{1}^{\ast}\left( z_{\alpha}- z_{\beta}\mid \tau \right)}{\theta_{1}\left( z_{\alpha}- z_{\beta}\mid \tau \right)}\right)^{\frac{1}{2 \kappa}}\tilde{\Psi}_{l \zeta 0}^{{\vec{k}}}
C_{k+1}:= \inf \{n > C_k: X_{C_k}^{(0)}+ (\xi_{C_k + 1}+ \xi_{C_k + 2}+ \dotsi +\xi_{n}) \geq 0 X_{C_k}^{(0)}\leq -1 X_{C_k}^{(0)}+ (\xi'_{C_k + 1}+ \xi'_{C_k + 2}+ \dotsi + \xi'_{n}) < 0 X_{C_k}^{(0)}\geq 0 \}
\overline{\zeta}_q(-m)= \sum_{j=0}^{2m+1}(-1)^{j+1}\frac{(n+1-q)^j}{m}\binom{m+1}{j}\xi_q(-2m+j-1) &+(-1)^m m!\sum_{j=2m+2}^{2n+2m}(n+1-q)^j \frac{\Gamma(-m+j-1)(-2m-2+j)}{\Gamma(j+1)}\gamma_{n,q}(-2m+j-1)
( g )^{\mu \nu}= \left( \begin{array}{c c c}{{0}}&{{e^{- \psi / 2}}}&{{0}}\\{{e^{- \psi / 2}}}&{{( f^{2}x_{I}^{2}- h_{+ +}) e^{- \psi}+ g_{+ I}\gamma^{I J}g_{J +}}}&{{- e^{- \frac{3}{2}\psi}g_{+ I}\gamma^{I J}}}\\{{0}}&{{- e^{- \frac{3}{2}\psi}g_{+ J}\gamma^{J I}}}&{{e^{- \psi}\gamma^{I J}}}\\ \end{array}\right)
A_{0}( \vec{x}, t ) \rightarrow A_{0}^{\prime}={\frac{1}{\beta}}\int_{0}^{\beta}d t \, A_{0}( \vec{x}, t ) , \qquad A_{i}( \vec{x}, t ) \rightarrow A_{i}^{\prime}= \bar{A}_{i}( \vec{x})
\frac{\partial^3 \mathcal{F}}{\partial t_2^3}+ \frac{\partial^3 \mathcal{F}}{\partial t_1^3}\frac{\partial^3 \mathcal{F}}{\partial t_1 \partial t_2^2}= \left( \frac{\partial^3 \mathcal{F}}{\partial t_1^2 \partial t_2}\right)^2
\Phi_{b}^{( 1 0 )}= - 2 ~ \mathrm{l n}\Big ({\frac{( p^{2}+ q^{2}) \Delta^{\frac{1}{2}}}{p^{2}+ q^{2}\Delta}}\Big ) , ~ ~ \chi ={\frac{p q ( \Delta - 1 )}{p^{2}+ q^{2}\Delta}}.
\left(\frac{(r^{2}-1-(s-t)^{2})}{\sqrt{\beta}(1+r^{2}+(s-t)^{2}+\sqrt{\beta})}\right) =\frac{-\left(1+\frac{1-r^{2}}{(s-t)^{2}}\right)}{(s-t)^{2}}\left(\frac{1}{\sqrt{1+q}}\cdot \frac{1}{1+y+\sqrt{1+q}}\right)
a_i = \left\langle d - \left\langle d , x^{\rho_1 + \cdots + \rho_{n-1}}) \right\rangle \beta(\rho_1, \dots , \rho_n), x^{\rho_1 + \cdots + \rho_{i-1}}\right\rangle & = \begin{cases}\left\langle d , x^{\rho_1 + \cdots + \rho_{i-1}}\right \rangle i < n 0 i \geq n.\end{cases}
\mu_2(\rho_2) \ge \nu\bigl(\partial (\rho_2B)\cap \left(B+{\bf v}_1\right)\bigr) &=2\pi \sqrt{1+{{4\sqrt{830}}\over{21}}}\left( \sqrt{1+{{4\sqrt{830}}\over{21}}}-{{\sqrt{830}}\over{21}}-1\right).
\int_\Omega \big(\lambda \frac{|\phi_\rho|^{2^*(s)}}{|x|^s}+\mu \frac{|\psi_\rho|^{2^*(s)}}{|x|^s}+2^*(s)\kappa \frac{|\phi_\rho|^\alpha |\psi_\rho|^\beta}{|x|^s}\big)dx \rightarrow \int_\Omega \big(\lambda \frac{|u|^{2^*(s)}}{|x|^s}+\mu \frac{|v|^{2^*(s)}}{|x|^s}+2^*(s)\kappa \frac{|u|^\alpha |v|^\beta}{|x|^s}\big)dx = 1
\forall \,n\geq0,F_n\mathcal O_{mod}(\tilde C):=\{f\in\mathcal O_{mod}(\tilde C)\,|\,\forall\,\gamma_1,\ldots,\gamma_{n+1}\in\Gamma_C,\,\, f_{|(\gamma_1-1)\cdots(\gamma_{n+1}-1)}=0\},
f_{0}(t) =1, f_{1}(t) =-t, f_{2}(t) =\frac{1}{2}(t-3) t, f_{3}(t) =-\frac{1}{6}t \left(t^2-9 t+2\right), f_{4}(t) =\frac{1}{24}t \left(t^3-18 t^2+35 t-42\right), f_{5}(t) =-\frac{1}{120}t \left(t^4-30 t^3+155 t^2-270 t+24\right).
{\cal A}^{\mu}( \hat{p}, \hat{q}) = \int_{- \infty}^{+ \infty}d t \int_{0}^{1}d y \, \varepsilon_{\eta \nu}^{d i l}\, \varepsilon_{\lambda \sigma}^{d i l}\, \langle \, V_{- 1}^{\eta}( i y , - \hat{p}- \hat{q}) \, V_{- 1}^{\nu}( - i y , - \hat{p}- \hat{q}) \, V_{0}^{\mu}( t , 2 \hat{q}) \, V_{0}^{\lambda}( i , \hat{p}) \, V_{0}^{\sigma}( - i , \hat{p}) \, \rangle \quad .
\tilde{g}^{ij}(\Gamma^{k}_{ij}(\tilde{g}) - \Gamma^{k}_{ij}(g))\nabla_ku =O_3(r^{-2\delta-2}), (\tilde{g}^{ij}-{g}^{ij})\nabla_j\nabla_iu =O_2(r^{-2\delta-2}).
\tilde{\beta}^{2}= \frac{\lambda 2^{n}\rho ( \rho - 1 )}{( n + 1 ) ( 1 + \rho )^{n}}\mathrm{{A p p e l l}}F_{1}[ n + 1 , 2 , - n , n + 2 , 1 - \rho , \frac{1 - \rho}{2}] .
\varepsilon_n^2 =\Vert y(x,t)-y_n(x,t)\Vert^2 &=\Vert\sum_{i=n+1}^{\infty}\langle y(x,t),\psi_i(x,t)\rangle\psi_i(x,t)\Vert^2 &=\sum_{i=n+1}^{\infty}\langle y(x,t),\psi_i(x,t)\rangle^2M^2_i ,
\frac{\Gamma(t - 2\eta; \tau, 8\eta)}{\Gamma(t + 2 \eta; \tau, 8\eta)}\frac{1}{\theta_0(t + 2\eta; \tau)}\frac{1}{\theta_0(t + 2\eta; 8\eta)}= -e^{-2\pi i t - 4 \pi i \eta}\Gamma(t - 2\eta; \tau, 8\eta)\Gamma(-t - 2\eta; \tau, 8\eta).
U_k=\left(\begin{array}{ll}g_{k-1}g_k\end{array}\right) \left(\begin{array}{cc}\frac{p^THp}{(p^TBp)^2}-\frac{1}{p^TBp}-\frac{p^THp}{(p^TBp)^2}-\frac{p^THp}{(p^TBp)^2}\frac{p^THp}{(p^TBp)^2}+\frac{\phi_k}{p^TBp}\end{array}\right)\left(\begin{array}{c}g_{k-1}^T g_k^T \end{array}\right),
\sigma_{\!\operatorname{ess}}(B_{{-},0}) = \emptyset \Leftrightarrow \lim_{x\to b_{-}}R_{-}(x) \int_{x}^{b_{-}}w_{-}(\xi) d\xi = 0, 0\not\in\sigma_{\!\operatorname{ess}}(B_{{-},0}) \Leftrightarrow \sup_{x\in(0,b_{-})}R_{-}(x) \int_{x}^{b_{-}}w_{-}(\xi) d\xi < +\infty.
\dim_{\mathrm{k}}(AB)=\dim_{\mathrm{k}(t)}(A^{\prime}B^{\prime}),\dim_{\mathrm{k}}(A)=\dim_{\mathrm{k}(t)}(A^{\prime})\quad\quad\dim_{\mathrm{k}}(B)=\dim_{\mathrm{k}(t)}(B^{\prime})
\big|\beta_k V^{\beta_k}(x,i) - \lambda_j^{\beta_k}\big| \ = \ \beta_k\big|V^{\beta_k}(x,i) - V^{\beta_k}(0,j)\big| \ \leq \ \beta_k\frac{L_2}{\gamma}|x| + \beta_k\hat c \ \overset{k\rightarrow\infty}{\longrightarrow}\ 0,
\left(\int_{\mathbb R^{n}}|f(x)|^{2}dx\right)^{2}\leq \left(\frac{2}{n}\right)^{2}\int_{\mathbb R^{n}}\left| \nabla f(x) \right|^{2}dx \int_{\mathbb R^{n}}\|x\|^{2}|f(x)|^{2}dx, n\geq2,
w^{( a )}= w_{\mathrm{f l a t}}^{( a )}\equiv - \sum_{A}\left( \frac{1}{2 r_{A}}\epsilon^{a b c}d r_{A}^{b}\wedge d r_{A}^{c}+ d r_{A}^{a}\wedge ( d \psi_{A}+ \cos \theta_{A}d \phi_{A}) \right) ,
+ \frac{q^{2}}{6 4 \pi^{2}f_{a}}\sqrt{- g}\epsilon^{\mu \nu \rho \sigma}\! \Big ( V_{\mu \nu}{}^{j}+ \sqrt{2 \kappa}( a_{\nu}D_{\mu}A^{j}- a_{\mu}D_{\nu}A^{j}+ A^{j}f_{\mu \nu}) \Big )
c(d+1,d+1) = \frac{2}{d+1}c(d,d) = \frac{2}{d+1}\frac{2}{d}c(d-1,d-1) = \cdots = \frac{2}{d+1}\frac{2}{d}\cdots \frac{2}{2}c(1,1) = \frac{2^d}{(d+1)!}
\frac{\partial^{k+1}F}{\partial v^{k+1}}(a,v) =\int_a^v\frac{\partial^{k+1}f}{\partial u^{k+1}}(v,v')dv'+\frac{\partial^kf}{\partial u^k}(v,v) &+\sum_{l=0}^{k-1}\binom{k}{l+1}\frac{\partial^{k}f}{\partial u^{k-l}\partial v^l}(v,v)+\sum_{l=0}^{k-1}\binom{k}{l+1}\frac{\partial^{k}f}{\partial u^{k-1-l}\partial v^{l+1}}(v,v).
\frac{(-a)_k^2}{(1)_{k}^2}\cdot(H_{a-k}-H_k)=\frac{d}{d x}\bigg(\frac{(-a)_{a-k}(-a-x)_{a-k}}{(1)_{a-k}^2}-\frac{(-a)_k(-a-x)_{k}}{(1)_k^2}\bigg)\bigg|_{x=0}\pmod{p}.
\delta(f)(g_1, g_2, \ldots, g_{n+1}):= g_1f(g_2, \ldots, g_{n+1})+\sum_{i=1}^{n}(-1)^{i}f(g_1,\ldots,g_ig_{i+1},\ldots,g_{n+1}) +(-1)^{n+1}f(g_1,\ldots,g_n).
I=2\pi i\frac{e^{\frac{1}{4}\pi i}}{-1+i}(\frac{17}{4}-5^{\frac{3}{4}}2^{\frac{1}{4}})=2\pi2^{-\frac{1}{2}}(\frac{17}{4}-5^{\frac{3}{4}}2^{\frac{1}{4}})
X(t) <Z^{+}(t;T_{n-1}^{+})\,\,\,\,t\in[T_{n-1}^{+},T_{n}^{+}), X(T_{n}^{+}) \in[Z^{+}(T_{n}^{+};T_{n-1}^{+}),Z^{+}(T_{n}^{+};T_{n-1}^{+})+c_{2}T_{2}),
\Omega_1(x_2) = x_2, \Omega_2(x_2, x_4) = x_4 - 5x_2(x_2-1), \Omega_3(x_2, x_4, x_6) = x_6 - 7x_4(x_2-1) + \frac{35}{3}x_2(x_2 - 1)(x_2 - 2) + \frac{14}{3}x_2,
\sum\limits_{i=1}^kI(U_{\mathcal{A}_i};\hat{S}_{a(i)}|U_{\cup_{j=1}^{i-1}\mathcal{A}_j})\leq\rho\sum\limits_{i=1}^kI(V_{\mathcal{A}_i};Y_{a(i)}|V_{\cup_{j=1}^{i-1}\mathcal{A}_j})
\sum_{n=1}^{\infty}\frac{p^{2}+\lambda_{n}^{2}}{p\left(p+\frac{1}{\pi}\right)+\lambda_{n}^{2}}\,\frac{e^{2\pi\lambda_{n}}}{\left(\sigma(\lambda_{n})e^{2\pi\lambda_{n}}-1\right)^{2}}\cdot\left\{\pi\,\sigma\left(\lambda_{n}\right)+\frac{p}{\left(p-\lambda_{n}\right)^{2}}\right\}=\frac{\pi}{24}\cdot\frac{1+\frac{3}{\pi p}(1+\frac{1}{\pi p})}{\left(1+\frac{1}{\pi p}\right)^{3}}-\frac{1}{8}.
\frac{\delta\Gamma[A_{k}]}{\delta \omega (x)}= \int dy\,\frac{\delta A_{i}(y)}{\delta \omega (x)}\frac{\delta\Gamma [A_{k}]}{\delta A_{i}(y)}= D_{i}\,\frac{\delta \Gamma[A_{k}]}{\delta A_{i}(x)}= 0
\begin{pmatrix}(a_n+ a_n')u(n) u(n-1)\end{pmatrix}= \frac{1}{2i}\left(Z(n)\begin{pmatrix}a_n\varphi(n) \varphi(n-1)\end{pmatrix}-\overline{Z(n)}\begin{pmatrix}a_n\overline{\varphi(n)}\overline{\varphi(n-1)}\end{pmatrix}\right) = \mathrm{Im}\left[Z(n)\begin{pmatrix}a_n\varphi(n) \varphi(n-1)\end{pmatrix}\right].
\left\|{\boldsymbol\tau}\left(\boldsymbol f\right)-{\boldsymbol\tau}\left(\boldsymbol g\right)\right\|_{p',B}\leq 2C_{p}\left\|\boldsymbol f-\boldsymbol g\right\|_{p,B}^{p-1}\mbox{with}\C_{p}=1+2^{\frac{2-p}{2}},
\psi^*\alpha-\alpha =\int_0^1 \left(\frac{d}{ds}\phi_s^*\alpha\right) \, ds &=\int_0^1 \phi_s^* (L_{X_s}\alpha) \, ds &=\int_0^1 \phi_s^* (d\iota_{X_s}\alpha+\iota_{X_s}d\alpha) \, ds &=d \int_0^1 \phi_s^* (\alpha(X_s)-H_s)\, ds=-du.
g_1^*(b) = \begin{cases}f^*(b), b \neq b_{k+1}f^*(b_k), b=b_{k+1}\end{cases}\mbox{or}g_2^*(b) = \begin{cases}f^*(b_{k+1}), b \leq b_{k}f^*(b), b \geq b_{k+1}. \end{cases}.
Z_{S^{3}}= \int [ d x_{\mu}] [ d p_{\mu}] \mathrm{T r}\exp \left[ i \int \left( p_{\nu}\dot{x}_{\nu}- H_{S^{3}}^{0}\right) d t \right] \mathrm{T r}\exp \left[ \int A_{S^{3}}^{0}d t \right] \ ,
\,\,\, (2\pi)^{-1}\int_{-\infty}^{\infty}e^{-ix}(A - ix)^a ( B - ix)^b (C-ix)^c dx = \,\,\, \frac{1}{\Gamma(a) \Gamma(b) \Gamma(c)}\int_0^1 \int_0^{1-t}(1 - t -u)^{a-1}t^{b-1}u^{c-1}e^{-(A -(A -B)t - (A - C)u )}dudt.
\E \left [ P^{\ast}\left ( \max_{x}\lvert S_{n,l}^{\ast}(x_{i_k(x)}(k),x_{i_{k+1}(x)}(k+1)) \rvert > \epsilon /(k+3)^2 \right ) \right ] = \C \left ( l^{-\eta}(k+3)^4 \epsilon^{-2}+ 2^{-(k+1)}l^2/d_l^2 E[\tilde \mu_{n,l}(H_m)]^2 \right ).
f ( S , T_{i}, A )_{a}= S -{\frac{1}{8 \pi^{2}}}( b_{1}- b_{0}) \log \lbrack h ( T_{i}) A \rbrack -{\frac{1}{8 \pi^{2}}}\sum_{i = 1}^{3}({b^{\prime}}_{a}^{i}- \delta_{G S}^{i}) \log \eta ( T_{i})^{2}{}.
+ C_{\xi}^{* k}C_{\eta}^{k}e_{n + 1 + k , M}\otimes e_{k N}( \frac{q^{n + 1}}{1 - q^{2}})^{k}q^{k^{2}}\sqrt{\frac{[ n + 1 ]_{q^{2}}!}{[ n + 1 + k ]_{q^{2}}! [ k ]_{q^{2}}!}}( 1 - q^{2 ( n + k + 1 )}) )
q_1:=-8x_2^4+\lambda^2 x_2^2x_3^2-8\lambda x_2x_3u^2-8x_3^4+8u^4, q_2:=-8x_2^3x_3+\lambda^2x_2^2x_3^2-8x_2x_3^3-8\lambda x_2x_3u^2+8u^4, q_3:=8x_2^3x_3+\lambda^2x_2^2x_3^2t+8x_2x_3^3+2\lambda x_2x_3u^2+u^4, q_6:=-8x_2^4+\lambda^2x_2^2x_3^2-8x_2^3x_3-8\lambda x_2x_3u^2+8u^4, q_7:=8x_2^4+\lambda^2x_2^2x_3^2+8x_2^3x_3+2\lambda x_2x_3u^2+u^4.
{\cal F}(\Lambda_{24})=\wedge_{i=1}^{3}d\lambda_{i}\frac{a_{12}a_{15}^{2}a_{25}^{2}a_{35}^{2}a_{36}a_{46}}{a_{45}a_{56}}\prod_{k<l}a_{kl}^{-3}\Theta(\Lambda_{24}),
\sum_{j = 1}^{p - 1}\frac{\partial}{\partial x_{j}}\frac{\partial}{\partial x_{2}}\frac{\partial^{2}}{\partial x_{3}^{2}}\dots \frac{\partial^{p - 2}}{\partial x_{p - 1}^{p - 2}}f \biggr |_{x_{1}= x_{2}= \dots = x_{p - 1}}= 0 .
D_{a,b}= \sum_{w=0}^{p}(-1)^w \tbinom p{a-b}\tbinom{p-(a-b)}{b}\tbinom{a-b}{a-(p-w)}\tbinom{a-b}{2(a-b)-r-1}= \tbinom p{a-b}\tbinom{p-(a-b)}{b}\tbinom{a-b}{2(a-b)-r-1}\sum_{z=a-p}^{a}(-1)^{p - (a - z)}\tbinom{a-b}{z},
\mathbb{P}\left( \sup_{t\leq\delta}\frac{W_{t}^{n}}{h\left( t\right)}\geq\sqrt{1+\varepsilon}\right) \leq\sum_{k=0}^{\left[ 2^{n+1}\delta\right]}\mathbb{P}\left( \sup_{t\in I_{n+1,k}}\frac{W_{t}^{n}}{h\left( t\right)}\geq\sqrt{1+\varepsilon}\right) .
2a^{\beta\alpha}_{rs}\Big(Q^{(r)}_{\gamma\beta}\Big)_t(x')= \, 2a^{\beta\alpha}_{rs}t^{1-n}(\partial_r E_{\gamma\beta})(x'/t,1) begin{align*}4pt]= \,2t^{1-n}(x'/t,1)_s \,k_{\gamma\alpha}(x'/t,1) begin{align*}4pt]= \,(x'/t,1)_s\,\big(P_{\gamma\alpha}\big)_t(x')=\frac{x_s}{t}\big(P_{\gamma\alpha}\big)_t(x').
x_i^{\pm}(z) = \sum_{k \in \mathbb{Z}}x_i^{\pm}(k) z^{-k}, \phi_i(z) = \sum_{m \in \mathbb{Z}_+}\phi_i(m) z^{-m}, \varphi_i(z) = \sum_{m \in \mathbb{Z}_+}\varphi_i(-m) z^{m}.
\gamma_z((s_\lambda s_\mu^*)\cdot_\sigma(s_\nu s_\tau^*)) =\sigma(g,h)\gamma_z((s_\lambda s_\mu^*)(s_\nu s_\tau^*)) &=\sigma(g,h)\gamma_z\Big(\sum_{(\alpha,\beta)\in \Lambda^{\min}(\mu,\nu)}s_{\lambda\alpha}s_{\tau\beta}^*\Big) &=\sigma(g,h)\sum_{(\alpha,\beta)\in \Lambda^{\min}(\mu,\nu)}z^{d(\lambda)+d(\alpha)-d(\tau)-d(\beta)}s_{\lambda\alpha}s_{\tau\beta}^*.
\frac{(\mathcal X_s)}{(\mathbb B_0^{(d)}(r))}\le \frac{(\mathbb B_0^{(d-1)}(r)) \frac{\delta r}{\sqrt{d}}}{(\mathbb B_0^{(d)}(r))}= \frac{\delta}{\sqrt{\pi d}}\frac{\Gamma(\frac{d}{2}+1)}{\Gamma(\frac{d}{2}+\frac{1}{2})}\le \frac{\delta}{\sqrt{\pi d}}\cdot \sqrt{\frac{d}{2}+\frac{1}{2}}\le \delta,
\begin{array}{c c c c}{{w ( t ) ~ = ~ 1 / \, t}}&{{\quad}}&{{\mathrm{w i t h}}}&{{k = 0 ~ ,}}\\{{w ( t ) ~ = ~ 1 / \sin t}}&{{\quad}}&{{\mathrm{w i t h}}}&{{k = + 1 ~ ,}}\\{{w ( t ) ~ = ~ 1 / \sinh t}}&{{\quad}}&{{\mathrm{w i t h}}}&{{k = - 1 ~ ,}}\\ \end{array}
e_Q(\sigma_{\ell_Q - \ell_P})_{\ell - \ell_Q}(T_w^{Q*}) = (-1)^{\ell(w) - \ell_Q(w)}e_Q(\sigma_{\ell_Q - \ell_P})(T_w^{Q*}) & = (-1)^{\ell(w) - \ell_Q(w)}\sigma_{\ell_Q - \ell_P}(T_w^{P*}) & = (-1)^{\ell(w) -\ell_Q(w)}(-1)^{\ell_Q(w) - \ell_P(w)}\sigma(T_w^{P*}) & = (-1)^{\ell(w) - \ell_P(w)}\sigma(T_w^{P*}) = \sigma_{\ell - \ell_P}(T_w^{P*}).
G_{\beta}^{B}( \mathrm{\bf r}_{2}, \mathrm{\bf r}_{1}) ={\frac{m \omega_{c}}{\pi}}e^{- \beta \omega_{c}}\exp \left[ -{\frac{m \omega_{c}}{2}}\left\{r_{2 1}^{2}+ 2 i \epsilon \hat{\mathrm{\bf k}}\cdot ( \mathrm{\bf r}_{2}\times \mathrm{\bf r}_{1}) \right\}\right] ,
\exp \left[ g\left( xp+px\right) \right] p^{2}\exp \left[-g\left(xp+px\right) \right] =p^{2}\sum\limits_{j=0}^{\infty}\left( 4i\right)^{j}\frac{g^{j}}{j!}=p^{2}\exp \left( 4ig\right).
P^+ = \{\lambda\in P\mid \forall i\in I,\langle h_i,\lambda\rangle\geq0\}, K(\lambda) = \sum_{i\in I}U_qe_i+\sum_{i\in I}U_q(q^h-q^{\langle h_i,\lambda\rangle}\cdot1)+\sum_{i\in I}f_i^{1+\langle h_i,\lambda\rangle}.
A =\chi \sum_{j=2}^{K}d_{j}(e_bq_+)^j S =\chi \sum_{j=2}^K \sum_{\ell=0}^{j-1}\tilde d_{j,\ell}e_{b}^j(q_+)^\ell (q_-)^{j-\ell}, d_j, \tilde d_{j,\ell}\in \mathbb{R}.
(\nabla_{Y}X)^{k}=D_{(Y^{1},Y^{2})}X^{k}|_{f^{-1}(p)}+\sum_{i=1}^{2}\sum_{j=1}^{2}(\Gamma_{ij}^{k}X^{j})|_{f^{-1}(p)}Y^{i},(k=1,2)
\lambda(f) :=(\log q)^{-1}\frac{d}{ds}\left(-\zeta(s)^{-1}Z_{r_2}(f,s)+\zeta(s+1)^{-1}Z_{r_1}(d_2(f), s+1)\right)\Big|_{s=0}=-\sum_{j=-A}^{A-1}f(0,0,\varpi^j) +2Af(0)-\sum_{j=-A}^{A-1}f(0,0,0,\varpi^j,0,0) =(n+1)\tilde{c}_2(I(f))-I(f)(0,0,0,\varpi^n)
|F_{2j,k}^{(c)}|\le \| \bold{F}^{(i)}\|_2(\sum_{p=1, p\neq k}^{2j-1}|F_{2j,k}^{j,p}|^2 )^{\frac{1}{2}}=\frac{j^{\frac{1}{2}}}{M}\frac{j^{\frac{3}{2}}}{M^2}+\frac{j^{\frac{1}{2}}}{M^2}\le \frac{K_f j}{M^2},
\mathbb{L}_{BRST}^{\mathrm{I}}= \frac12 \left(AdA + dc A+ cdc\right) + \frac16 A[A,A] - \frac{1}{12}c[c,c] \mathbb{L}_{BRST}^{\mathrm{II}}= \frac12 \left(AdA + c dA\right) + \frac16 A[A,A] - \frac14A[c,c] - \frac{1}{12}c[c,c]
d s_{1 1}^{2}= H^{- 2 / 3}( r ) \left( - f ( r ) d t^{2}+ d x_{1}^{2}+ d x_{2}^{2}\right) + H^{1 / 3}( r ) \left( \frac{d r^{2}}{f ( r )}+ r^{2}d \Omega_{7}^{2}\right) ,
(Ju,u) =\sum_{n=1}^\infty\left( \begin{pmatrix}0 1 1 0 \end{pmatrix}v_{n-1}+\begin{pmatrix}s 0 0 t\end{pmatrix}v_n+ \begin{pmatrix}0 1 1 0 \end{pmatrix}v_{n+1}, v_n\right)_{\mathcal{K}}&+\sum_{n=1}^\infty\left(O(n^{\alpha-1})u_{n-1}+O(n^{\alpha-1})u_{n+1}, u_n\right)_{\mathcal{K}}\,.
f^L(\vec x)= \sum_{i=0}^{n-1}(m(i+1)x_{\sigma(i+1)}-m(i)x_{\sigma(i)})f(V_{x_{\sigma(i)}}^+,V_{x_{\sigma(i)}}^-) = \sum_{i=1}^{n-1}x_{\sigma(i)}m(i)(f(V_{x_{\sigma(i-1)}}^+,V_{x_{\sigma(i-1)}}^-)-f(V_{x_{\sigma(i)}}^+,V_{x_{\sigma(i)}}^-)) &+x_{\sigma(n)}m(n)f(V_{x_{\sigma(n-1)}}^+,V_{x_{\sigma(n-1)}}^-).
Lu(t,x) = \int_{\mathbb{R}^{n}}[\tilde{u}(s,y)-\tilde{u}(s,z)]\tilde{a}(y,z)r^{-n-2\beta}\tilde{k}_{0}(y,z)r^{n}dz & = r^{-2\beta}L\tilde{u}(s,y).
c_{\Omega_0}^{\Omega_0}= q^{-1/6}\varphi(q)^{-4}\dfrac{1}{2}q^{-1/20}(\varphi(-q^{1/2})^2\varphi(q)^{-2}G(-q^{1/2})+\varphi(q^{1/2})^2\varphi(q)^{-2}G(q^{1/2})) = q^{-13/60}\dfrac{1}{2}\varphi(q)^{-6}(\varphi(-q^{1/2})^2G(-q^{1/2})+\varphi(q^{1/2})^2G(q^{1/2})).
h^{(2m)}_n := \frac{1}{s^{(2m)}_{n+1}}\left\{1+ \,\sum^m_{\ell=1}\left\{\binom{2m}{2\ell}+\frac{\alpha}{n}\binom{2m}{2\ell-1}\right\}s^{(2\ell)}_n\right\}-1,
\left|A_{\bar{\Gamma}_{\tau}}\right|\,=\,\frac{\left|\partial_{zz}^{2}\hat{w}\left(z,\,\tau\right)\right|}{\left(1+\left(\partial_{z}\hat{w}\left(z,\,\tau\right)\right)^{2}\right)^{\frac{3}{2}}}\,\leq\,C\left(n\right)
H(x,\xi,t)=\left(4\pi A\right)^{-\frac{d}{2}}\left(\det Q_t\right)^{-\frac{1}{2}}\exp\left(-Bt-\left(4 A\right)^{-1}\left\langle Q_t^{-1}(e^{tS}x-\psi),(e^{tS}x-\psi)\right\rangle\right)
U\epsilon(f)=U_2U_1\epsilon(f_A,f_{A^c})=U_2\epsilon(f_A)\otimes \epsilon(f_{A^c})=\epsilon(f_A)\oplus \bigoplus_{n=1}^\infty \epsilon(f_A)\otimes \frac{1}{\sqrt{n!}}f_{A^c}^{\otimes n}
\varrho ={\frac{1 7^{1 / 2}}{4 \pi G}}{\frac{1}{\bar{\cal E}\bar{\cal P}_{\phi}}}, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \wp ={\frac{1 7^{3 / 2}}{6 4 \pi G}}{\frac{1}{\bar{\cal E}\bar{\cal P}_{\phi}}}.
\nabla_{\!y}p_a(y)=\nabla_{\!y}\left[ p_a(x)+p_{a-1}(x)\,x_{n+1}\right]=\nabla_{\!x}p_a(x)+x_{n+1}\,\nabla_{\!x}p_{a-1}(x)+p_{a-1}(x)\,e_{n+1}\,,
a(g)( f_1\otimes_s\cdots\otimes_s f_n ) =\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\langle g,f_i \rangle f_1\otimes_s\cdots\otimes_s \widehat{f}_i\otimes_s\cdots\otimes_s f_n a^\dagger(g)( f_1\otimes_s\cdots\otimes_s f_n ) =\sqrt{n+1}g\otimes_s f_1\otimes_s\cdots\otimes_s f_n
\mathbf c(g,h)^{-1}=\det\big(A(gh)A(h)^{-1}A(g)^{-1}\big)=\det\big(A(g)^{-1}A(gh)A(h)^{-1}\big) \hphantom{\mathbf c(g,h)^{-1}}{}=\det\big(1+A(g)^{-1}B(g)C(h)A(h)^{-1}\big).
H(x,y,h_1,h_2,h_3,z):=\big( \Delta_{-h_1,-2h_2}B(y+2z)\Delta_{-h_2,-h_3}B(y+z)\Delta_{-h_1,-h_2}C(x+y+2z) \Delta_{-h_3}C(x+y+z)\Delta_{-h_1}D(2x+y+2z)\Delta_{h_2,-h_3}D(2x+y+z)\big)
\int_{x}^y \frac{(qz/x, qz/y, abcz; q)_\infty}{(az/y, bz/x, cz; q)_\infty}d_q z =\frac{(1-q)y(q, x/y, qy/x, ab, acx, bcy; q)_\infty}{(ax/y, by/x, a, b, cx, cy; q)_\infty}.
(\partial_\nu^A \psi)^2 -(A\nabla_\Gamma \psi\cdot \nu)^2=\left|A^{\frac{1}{2}}\nu\right|^2 \left(\left|A^{\frac{1}{2}}\nabla \psi\right|^2-\left|A^{\frac{1}{2}}\nabla_\Gamma \psi\right|^2 \right).
a=(\psi_1,\psi_2,\psi_0): V_1\oplus V_2\oplus V_0 \to V_1\oplus V_2\oplus V_0 (v_1,v_2,v_0) \mapsto (\psi_1(v_1,v_2,v_0),\psi_2(v_1,v_2,v_0),\psi_0(v_1,v_2,v_0)).
\|\mathcal{T}_{1,l}^{(a_1,a_2)}-\mathcal{T}_{2,l}^{(a_1,a_2)}\|_{\rm F}^2 =\sum_{k_1,k_2}s_{k_1,k_1}^{r}s_{k_2,k_2}^{r}(\mu_{k_1,l}^{(1)}-\mu_{k_1,l}^{(2)})(\mu_{k_2,l}^{(1)}-\mu_{k_2,l}^{(2)}) &=\bigg\{\sum_{k}s_{k,k}^{r}(\mu_{k,l}^{(1)}-\mu_{k,l}^{(2)})\bigg\}^{2}=O(p^2).
\oint_{\infty}d\Sigma^{r}\epsilon_{0}^{\ast}\frac{\tilde{\epsilon}}{r^2}= \oint_{\infty}d\Sigma^{r}\lbrace \epsilon_{0}^{\ast}\gamma^{r}\gamma^{m}\Gamma_{m}\epsilon_{0}+ \epsilon_{0}^{\ast}\frac{\gamma^{r}}{r^3}(\gamma^{\alpha}\partial_{\alpha})\tilde{\epsilon}\rbrace
U_n := (t_{B_{0,2}^g}t_{B_1^g}t_{B_2^g}\cdots t_{B_g^g}V_1) (t_{B_{0,2}^g}t_{B_1^g}t_{B_2^g}\cdots t_{B_g^g}V_1^{\phi_n}) t_{a_{g+1}}^{-1}t_{a_{g+1}^\prime}^{-1},
|T_n|^r= \psi_r(T_n) = \sum_{i=1}^n\psi_r(T_i)-\psi_r(T_{i-1})= \sum_{i=1}^n Y_i\int_0^1 \psi'_r(T_{i-1}+t Y_i) \, dt &= \sum_{i=1}^n Y_i\int_0^1 \left (\psi'_r(T_{i-1}+t Y_i)-\psi'_r(T_{i-1})\right ) \, dt+ \sum_{i=1}^n Y_i \psi'_r(T_{i-1}) \, .
\Phi_{t}= - \Phi_{x x x}+ 3 \left( \Phi{\cal D}_{1}{\cal D}_{2}\Phi \right)_{x}+ \frac{a - 1}{2}\left({\cal D}_{1}{\cal D}_{2}\Phi^{2}\right)_{x}+ 3 a \Phi^{2}\Phi_{x}\quad .
B_{( m , \psi_{m}) [ \alpha , \psi_{\alpha}]}={\frac{\sqrt{| \cal G |}}{\sqrt{|{\cal S}_{m}|}|{\cal S}_{\alpha}|}}{\frac{\sum_{J}\psi_{m}( J ) \psi_{\alpha}( J ){S}_{m \alpha}^{J}}{\sqrt{{S}_{0 m}}}},