harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\mathbf{J}:\ker(\mathfrak{L}_p|_{\delta=0}) \rightarrow \ker(\mathcal{T}_{a^+,a^-}\circ B); (\eta,\phi) \mapsto \Big(\left(\begin{array}{c}\frac{a^+\eta}{2\sqrt{z}}\frac{a^-\bar{\eta}}{2\sqrt{\bar{z}}}\end{array}\right)+O_{L^2_1}(1)+\phi\Big)
\frac{\delta_{\rm II}\gamma a_i\rho^k\alpha^{k^2/2}}{16^{k^2}}\overset{\eqref{eq:dgd}}{=}\frac{\gamma^5\rho^k\alpha^{k^2/2}}{9\cdot 2^{8k^2}}a_i \overset{(\ref{eq:dgdr})}{\ge}\frac{a_i^{18k^4+24k^2}}{2^{55k^6}}\overset{\eqref{rec}}{\ge}a_{i+1}\,.
b \rightarrow b B ( r ) , \qquad B ( r ) \equiv \left[ 1 +{\frac{\varepsilon_{2}- r}{r - \varepsilon_{1}}}\left({\frac{\varepsilon_{1}}{R_{0}}}\right)^{2}\mathrm{e}^{\mp 2 a \varepsilon_{2}}\right]^{- 1}.
P^-=\frac{1}{2p^+}(p^i)^2+\frac{1}{2p^+}\sum_{n=1}^\infty\left(\alpha^i_{-n}\alpha^i_{n}+\tilde{\alpha}^i_{-n}\tilde{\alpha}^i_{n}+ n \,S^a_{-n}S^a_{n}+ n \,\tilde{S}^{\dot{a}}_{-n}\tilde{S}^{\dot{a}}_{-n}\right)
\sum_{Q\in H(D)}\pi(Q,w,X)^2 \ll \pi(X) + \sum_{1\leq v_0\leq 4X/\sqrt{|D|}}\frac{XW(\frac{Dv_0^2}{X^2})}{\log(|D|)^2L(1,\chi_{D})}\big(\frac{v_0}{\varphi(v_0)}\big)^3 &\ll \pi(X)+ \frac{X}{\log(|D|)^2L(1,\chi_D)}\sum_{1\leq v_0\leq 4X/\sqrt{|D|}}W(\frac{Dv_0^2}{X^2})\big(\frac{v_0}{\varphi(v_0)}\big)^3.
w[\mathbf{u},\mathbf{u}] = \frac{1}{2}\int_{\mathbb{R}^d}\left\langle \sigma_*(\mathbf{u}), e_* (\mathbf{u}) \right\rangle_{\mathbb{C}^m}d \mathbf{x}= \frac{1}{2}\int_{\mathbb{R}^d}\left\langle g(\mathbf{x}) b(\mathbf{D}) \mathbf{u}, b(\mathbf{D}) \mathbf{u}\right\rangle_{\mathbb{C}^m}d \mathbf{x},\mathbf{u}\in H^1 (\mathbb{R}^d; \mathbb{C}^d).
\frac{\alpha d|\sinh \theta - \gamma^{1/2}\cosh \theta ||\cosh \theta - \gamma^{1/2}\sinh \theta|}{\gamma^{1/2}\big( \frac{d}{\gamma}(\cosh \theta -\gamma^{1/2}\sinh \theta)^{2}+W^{2}\big)^{3/2}}begin{align*}3mm] \leq \frac{\alpha d(\sinh M+\gamma^{1/2}\cosh M)(\cosh M+\gamma^{1/2}\sinh M)}{\gamma^{1/2}c_{0}^{3/2}}=:M_{1}.
\mathbb{P}_1=\mathbb{P}\left[^{+}_{\omega \ge \omega^{*}}S(\omega)+r^{h}(\omega) \in d\omega_2, \max\limits_{\omega \ge \omega^{*}}S(\omega)+r^h(\omega) \in dx^{*}\right. \left. \vert S(\omega^{*})=y-x^{*}\right]
a= (\frac{q-3}{2})q^{m-1}+(\frac{q-1}{2})q^{m-2}+\cdots+(\frac{q-1}{2})q^{m-j_0-1}&-(\frac{q-3}{2})q^{m-j_0-2}-i_{m-1}q^{m-j_0-3}-\cdots-i_{j_0+2}q-i_{j_0+1}.
L ( x ) = \tilde{W}^{\prime}( x ) \left[ 1 + \tilde{W}( x ) M ( \tilde{W}{}^{2}( x ) ) \right] , \quad M ( \tilde{W}{}^{2}( x ) ) = \left( \tilde{W}^{2}( x ) + 4 \gamma \right)^{- 1 / 2}.
\Delta \left((\mathbf{1})^n\right) =\left(\Delta (\mathbf{1})\right)^n &=\left(1\otimes(\mathbf{1})+(\mathbf{1})\otimes 1\right)^n &=\sum_{k=0}^n C^n_k (\mathbf{1})^k\otimes(\mathbf{1})^{n-k}
D_{pert, s\eta^i}(\psi_i+s\mathfrak{c}_{i+1}-\mathfrak{h}_{i+1})=s^2\mathcal{A}+s^2\mathcal{B}+s^2\mathcal{C}+s\sum_{j=1}^2\mathcal{Q}_j &-D_{pert}(s\mathfrak{k}^b_{i+1})\mbox{}mod(\ker(D|_{L^2_1}))
- i \left( ( S_{2}\widetilde{F})^{a}\bar{G}^{b}- ( S_{2}\widetilde{F})^{a}( \bar{F}S_{1})^{b}- \widetilde{G}^{a}( \bar{F}S_{2})^{b}+ ( S_{1}\widetilde{F})^{a}( \bar{F}S_{2})^{b}\right) .
P(W^k_{\ell_2\Delta,t}(j)=1,A|Z_0(\Delta\ell_2)=m,Z_0(\Delta \ell_1)=n,X_{\ell_1,\Delta}=1,B_{\ell_1\Delta}(\ell_2\Delta)=i) = i \nu \Delta p^k_{j,+}(t-\Delta\ell_2).
=((0,1,1,1)[\begin{matrix}0&1\\ 1&0\end{matrix}]^{\otimes2})^{\otimes|U|}(([\begin{matrix}0&1\\ 1&0\end{matrix}][\begin{matrix}1\\ 0\end{matrix}])^{\otimes3}+([\begin{matrix}0&1\\ 1&0\end{matrix}][\begin{matrix}0\\ 1\end{matrix}])^{\otimes3})^{\otimes|V|}
\tilde{\delta}_{k,L_n}:= \tilde{\eta}_{k,L_n}+\frac{\beta_0}{2}\sum_{j=0}^{k-1}\left( \sum_{i=0}^{k+L_n-j-2}\varrho_0^i \right)^2 +\frac{\beta_0}{2}\sum_{j=0}^{k-1}\sum_{i=1}^{2(k+L_n-j-1)}i \varrho_0^{i-1}.
c = \sqrt{|\eta| + (kr_{1})^{2}}- \sqrt{|\eta|}\ln \left(\sqrt{1 + \frac{|\eta|}{(kr_{1})^{2}}}+\frac{\sqrt{|\eta|}}{kr_{1}}\right), \;\;\;\;\; \left({\rm for}\; \eta = \alpha^{2}- \frac{1}{4}< 0 \right).
M_H(Z_0,J,\rho)\ge \sum_{1\le j\le n}M_{H,j}(Z_0,J,\rho) = \sum_{1\le j\le n}M_{\partial D_j}\times M_{M_j}= \sum_{1\le j\le m}M_{\partial D_j}\times \prod_{1\le l\le n,l\not=j}M_{D_l}\ge \pi^{n-1}\sum_{1\le j\le m}M_{\partial D_j}\times \prod_{1\le l\le n,l\not=j}M_{\partial D_l}= n\pi^{n-1}M_S(Z_0,J,\lambda).
\Phi^\omega_m = \epsilon(\omega) \left(\sqrt{\omega^2-m^2}\:u \:+\: \frac{M m^2}{\sqrt{\omega^2-m^2}}\:\log u \right) A = \left( \begin{array}{cc}\cosh \Theta \sinh \Theta \sinh \Theta \cosh \Theta \end{array}\right) \;,\Theta = \frac{1}{4}\: \log \left| \frac{\omega-m}{\omega+m}\right| |R_\infty| \leq \frac{C}{u}\:,
(a^2-(a+1)^2) (1-(a+1)^{p-3}) - (1-(a+1)^2)(a^{p-3}-(a+1)^{p-3}) &= a^{-2}(a+1)^{-2}(a^2(a^2-(a+1)^2)((a+1)^2-1) - (1-(a+1)^2)((a+1)^2-a^2)) &= -a^{-1}(a+1)^{-1}(a+2)(2a+1)(a-1)
e^{-b^{2}/2}\le Q_{n}(a,b)\le\frac{1}{2}\sqrt{\frac{2n-1}{2}+\frac{\zeta^{2(1-n)}}{2(1-\zeta^{2})}}[e^{-(b-a)^{2}/2}+e^{-(b+a)^{2}/2}],\zeta<1
f \in I_m(W^c_m)I_{m-1}(W^c_m) &\subseteq I_t(W^c_m)^{(m-t+1)}I_{t}(W^c_m)^{(m-t)}\hbox{\cite[Theorem 3.16 (a)]{conca98}}&\subseteq I_{t}(W^c_m)^{(2m-2t+1)}= I_t(W^c_m)^{(h)}.
\xi = ([\xi_0^i: \xi_1^i])_{i=1}^N{\xi}'' = ([{\xi''}_0^i :{\xi''}_1^i])_{i=1}^N \omega = ([\omega_0^{(k,l)}: \omega_1^{(k,l)}])_{1 \leq k<l\leq N}{\omega}'' = ([{\omega''}_0^{(k,l)}:{\omega''}_1^{(k,l)}])_{1 \leq k<l\leq N}
\mathbb{E}\left[P_{\rm{u}}^\frac{2}{\eta}\right]= \frac{\rho^{\frac{2}{\eta}}\gamma\left(2,\pi \lambda\left(\frac{P^{({\rm{M}})}_{\rm{u}}}{\rho}\right)^{\frac{2}{\eta}}\right)}{\pi \lambda}+ \left(P^{({\rm{M}})}_{\rm{u}}\right)^{\frac{2}{\eta}}e^{-\pi \lambda\left(\frac{P^{({\rm{M}})}_{\rm{u}}}{\rho}\right)^{\frac{2}{\eta}}},
\left|b_{u}^{'}-b_{v}^{'}\right|=\left|a_{i}\gamma_{1}-a_{j}\gamma_{1}\right|\leq \left|a_{i}-a_{j}\right|~for~all~a_{i}^{'}\in A_{i}^{'}~~and~a_{j}^{'}\in A_{j}^{'}~for~i,j\in\{1,\ldots,s\}
\Delta\tilde{u}_m^i(y)=(\mu_m^i)^{\frac{n+2}{2}}\bigg[\Big(\Delta u_m(x)\Big)\chi_m^i(x)+2 \nabla u_m(x)\nabla\chi_m^i(x)+ u_m(x)\Big(\Delta\chi_m^i(x)\Big)\bigg].
\Theta_\nu\left(A,h,N;\tau\right)=(-i)^\nu (-i\tau)^{-\frac12-\nu}A^{-\frac12}\sum_{\substack{k\pmod{N}Ak\equiv 0 \pmod{N}}}e\left(\frac{Akh}{N^2}\right)\Theta_\nu \left(A,k,N;-\frac{1}{\tau}\right).
P\left(\frac{n-A_n^*}{f_1(n)}\in C\cap (0,\ell_1^{-1})\right) \leq P\left(\kappa_1<\frac{n-A_n^*}{f_1(n)}\leq\kappa_2\right) &=P\left(\frac{n-A_n^*}{f_1(n)}>\kappa_1\right)-P\left(\frac{n-A_n^*}{f_1(n)}>\kappa_2\right).
\widetilde{\psi}_{k}^{(0^{2})}\big(x;\omega ,N,- \Vert \psi_{0}\Vert^{2}\big)\sim \widetilde{L}_{k,N}^{(0^{2})}( z ) \frac{z^{(N+1/2) /2}\exp ( -z/2 )}{\sum\limits_{l=0}^{N}z^{l}/l!}, k\geq 1.
S\frac{\operatorname{d}\!\widetilde{u}}{\operatorname{d}\!\widetilde{t}}=\frac{\operatorname{d}(S\widetilde{u})}{\operatorname{d}(S\widetilde{t})}\neq \frac{\operatorname{d}(S\widetilde{u})}{\operatorname{d}\!\widetilde{t}}.
\tilde{H}_1(\beta)\begin{cases}\infty, \beta\in[0,\beta_P(d)], \in(0,\infty), \beta\in(\beta_P(d),\beta_P(d)+\epsilon_d], \in[0,\infty), \beta \in (\beta_P(d)+\epsilon_d, \beta_c(d)), \end{cases}
v_{\uparrow}^{^{M R}}( p^{\mu}) = - i [ u_{\downarrow}^{^{M R}}( p^{\mu}) ]^{\ast}\quad , \quad v_{\downarrow}^{^{M R}}( p^{\mu}) = + i [ u_{\uparrow}^{^{M R}}( p^{\mu}) ]^{\ast}\quad ,
\chi : \{(V_{i_0},E_{i_1}, F_{i_2})\, |\, i_k=1,\cdots,M_k (k=0,1,2),\, E_{i_1}\cap F_{i_2}=E_{i_1},\,V_{i_0}\cap E_{i_1}=V_{i_0}\}\longrightarrow X
\mathrm{depth}\,(R/J) =1+\mathrm{depth}\,(K[x_{k+1},\ldots,x_{n}]/I(D_{F})) &=1+\mathrm{dim}\,(K[x_{k+1},\ldots,x_{n}]/I(D_{F})) &=1+(n-k)-(g-k) &=1+n-g.
d^2_{R_i}(p_1,p_2) = g_{xx}(p_1) \; (x_1 + x_2)^2 + g_{yy}(p_1) \; (y_2-y_1)^2 + O(x_1^2) + O( | p_1 - p_2|^2 ) d^2_{R_{ij}}(p_1,p_2) = g_{xx}(p_1) \; (x_1 + x_2)^2 + g_{yy}(p_1) \;(y_1+y_2)^2 + O(x_1^2 + y_1^2) + O( |p_1-p_2|^2)
m! \left(1-\frac{|E(H)|}{m-q}\right) \geq m! \left(1- \frac{|E(H)|}{m}\cdot\left(1+\frac{2q}{m}\right)\right) &= m!\left(1-\frac{|E(H)|}{m}\right)\left(1-\frac{2q|E(H)|}{m(m-|E(H)|)}\right) &\geq m!\left(1-\frac{|E(H)|}{m}\right)\left(1-3\frac{|E(H)|^2}{m^2}\right),
e^{X}\left(ku^k\frac{\partial}{\partial q_{k}}\right)e^{-X}= ku^k\frac{\partial}{\partial q_{k}}+\sum_{n=1}^{\infty}\frac{(-1)^n}{n!}\sum_{m_1,\dots,m_n>0}ka_{m_1}(k+m_1)\dots &\quad \dots a_{m_n}(k+\sum_{j=1}^n m_j)u^{k+\sum_{j=1}^n m_j}\frac{\partial}{\partial q_{k+\sum_{j=1}^n m_j}}
S_1=\frac{X}{V}\sum_{u_1,\ldots,u_k}\frac{y_{u_1,\ldots,u_k}^2}{\prod_{i=1}^k\varphi(u_i)}+O\left(\frac{y_{max}^2\varphi(VP_f)^kX(\log X)^k}{V(VP_f)^kD_0}\right).
\kappa^{2}g^{2}\alpha^{\prime}\left( - 2 \cdot ( \frac{( 3 2 - 5 ) \cdot 2^{7 / 2}}{3^{5}}) ( - \frac{5 \cdot 2^{7 / 2}}{3^{5}}) \right) \left( A_{i}^{2}A_{j}^{2}\right) = \kappa^{2}g^{2}\alpha^{\prime}\left( \frac{1 7 2 8 0}{5 9 0 4 9}\right) \left( 2 A_{i}A_{i}A_{j}A_{j}\right) .
\langle S_\lambda^* (f \sqrt{d\mu}), g \sqrt{d\nu}\rangle = \int_{Z(s(\lambda))}\overline{f \circ \sigma_\lambda(x)}g(x) \sqrt{\frac{d(\mu \circ \sigma_\lambda)}{d((\mu \circ \sigma_\lambda) + \nu)}}\sqrt{\frac{d\nu}{d((\mu \circ \sigma_\lambda)+ \nu)}}d((\mu \circ \sigma_\lambda) + \nu).
{\bf\Psi}_n(t_0)=\left[\Psi_{j\ell}\right]_{j,\ell=0}^n, \Psi_{j\ell}=(-1)^{\ell}\left(\begin{array}{c}\ell j \end{array}\right)t_0^{\ell+j+1},0\le j\leq\ell\le n.
(D_t - Op_h^w(x\xi +p(\xi))v^{\Sigma}_{\Lambda}= h\, Op_h^w(\Gamma)Op_h^w(\Sigma) \Big[\sum_{|I|=3}Op_h(m_I)(v_I) + \sum_{|I|=3}Op_h(\widetilde{m}_I)(v_I)\Big] + h R(v) \, ,
R_{2,H,f}(V_k)-R_{2,H,f}(W) &=f(H+V_k)-f(H)-T^{H,H}_{f^{[1]}}(V_k)-(f(H+W)-f(H)-T^{H,H}_{f^{[1]}}(W)) &=f(H+V_k)-f(H+W)-T^{H,H}_{f^{[1]}}(V_k-W) &=\int_0^1T^{H_t,H_t}_{f^{[1]}}(V_k-W)dt-\int_0^1 T^{H,H}_{f^{[1]}}(V_k-W)dt.
\left(\delta_{e_1/a_1}-\delta_0\right)*\sigma_j &=\delta_{c_j}\ast (\delta_{e_1\,N_1/a_1}-\delta_0)\ast \frac{1}{\,R_j}\,\sum_{0\le k_2\le a_1 R_j-1}\,\dots \sum_{0\le k_s\le a_s R_j-1}\,\delta_{(0,-k_2/a_2,\dots,-k_1/a_s)}\ast \nu_\mu,
\int_0^Td\tau \psi_\mu\psi_\nu\omega_{\mu\nu}=-T\int_0^Td\tau(\psi\cdot\dot{x})^\cdot(\psi\cdot\dot{x})+T\int_0^Td\tau(\psi\cdot\dot{x})(\dot{\psi}\cdot\dot{x})+{T\over 4}\int_0^Td\tau\dot{x}\cdot\ddot{x}
\int_{-\pi}^{\pi}e^{-i\theta k}\left( \left(\sum_{j = 0}^{\infty}a_j e^{i j \theta}\right) - \left(\sum_{j = 0}^{\infty}\overline{c}_j e^{-i j \theta}\right)^{<\frac{1}{\alpha - 1}>}\left( f(\theta) \right)^{\frac{-1}{\alpha - 1}}\right) d\theta = 0, k = 0, 1,\dots
\begin{array}{l}{{\nu_{r}^{\left( l \right)}\mid \Phi \rangle = 0 \qquad r > 0}}\\{{J_{n}^{\left( l \right)}\mid \Phi \rangle = 0 \qquad n > 0}}\\ \end{array}\begin{array}{c}{{l = 1 , 2 , \ldots N}}\\ \end{array}
m(y+b_i) =p_im(y)+q_i u(e^ye^{b_i}) =p_iu(e^y) \textrm{, where}m(y)=u(e^y)+\frac{q_i}{p_i-1}u(e^y) =\alpha e^{\beta y}\textrm{(by Lemma~\ref{lem:srnie})}m(y) =\alpha e^{\beta y}+\frac{q}{p-1}, g(x) =\alpha e^{\beta f(x)}+\frac{q}{p-1}, g(x) \simeq e^{\beta f(x)}.
\lambda_{\pm}( x_{0}, r ; z ; q ) = \frac{1}{2}\left\{\sum_{n = 0}^{\infty}\frac{( \frac{\rho_{+}}{\rho_{-}}; q )_{n}}{( q ; q )_{n}}( i \rho_{-}( 1 - q ) z )^{n}\pm \sum_{n = 0}^{\infty}\frac{( \frac{\rho_{-}}{\rho_{+}}; q )_{n}}{( q ; q )_{n}}( i \rho_{+}( 1 - q ) z )^{n}\right\}.
\hat x_{12}= x_{1_A}- x_{2_A}+{4i\over (12)}[(1^-2) \theta^+_1 \bar\theta^+_1 + (2^-1) \theta^+_2 \bar\theta^+_2 + \theta^+_1 \bar\theta^+_2 + \theta^+_2 \bar\theta^+_1]
\left(m I - \frac{\partial P}{\partial u}\right)^{-1}= \begin{bmatrix}\frac{1}{m}-\frac{n (n+1)}{a_0^4 m (m+1)}-\frac{n (n+1)}{a_0^2 m (m+1)}-\frac{2 n}{m^2+m}0 \frac{1}{m}0 0 0 0 \frac{1}{m}0 0 -\frac{n+1}{2 a_0^4 m (m+1)}-\frac{n+1}{2 a_0^2 m (m+1)}\frac{1}{m+1}\end{bmatrix}
\gamma_{n}=lim_{m\rightarrow\infty}\{\sum_{k=1}^{m}\frac{(lnk)^{n}}{k}-\int_{1}^{m}\frac{(lnx)^{n}}{x}dx\}=lim_{m\rightarrow\infty}\{\sum_{k=1}^{m}\frac{(lnk)^{n}}{k}-\frac{(lnm)^{n+1}}{n+1}\}.
\mathbb{X}_t= 1+\int_{0}^{t}\mathbb{X}_{s^-}\otimes dX_s+\frac{1}{2}\int_{0}^{t}\mathbb{X}_{s^-}\otimes d[X,X]^c_s &+\sum_{0<s\leq t}\mathbb{X}_{s^-}\otimes \{\exp(\Delta X_s)-\Delta X_s -1\},
Q_{0}^{(N,M)}(z) =-\frac{1}{2}\int_{-1}^{z}d\zeta (1-\zeta)^{N}(1+\zeta)^{M}=- ( z+1 )^{M+1}\sum_{k=0}^{N}\frac{2^{N-k-1}(-1)^{k}}{M+k+1}\binom{N}{k}( z+1 )^{k}
\langle (\pi^* \alpha)^{n}\rangle - n\langle (\pi^* \alpha)^{n-1}\rangle \cdot \widehat{\omega}= \langle (\pi^* \alpha)^{n}\rangle - n\langle (\pi^* \alpha)^{n-1}\cdot \widehat{\omega}\rangle \geq \langle (\pi^* \alpha)^{n}\rangle -n\langle (\pi^* \alpha)^{n-1}\cdot \pi^* \beta\rangle >0,
q = -{\frac{a_{0}\ddot{a}_{0}}{\dot{a}_{0}^{2}}}={\frac{\dot{I}}{2 H I}}+{\frac{1}{2}}( I^{- 3 / 2}\Omega_{\varrho}+ I^{- 1}\Omega_{\Phi}) +{\frac{1 6 \pi^{2}G_{5}^{2}\sigma}{3 c^{6}H^{2}}}( I^{- 1 / 2}\wp + I^{- 1}\wp_{\Phi}+ 2 I^{- 1 / 2}\wp_{\Lambda_{4}}) + I^{- 1}\Omega_{\cal C},
r\circ\, s\,\circ\, S_0\,\circ\, \overline{\Omega}= S_2\circ R^{NC}_{2}\circ p\circ \Omega_Y =R^{PC}_{2}\circ T_2\circ p\circ \Omega_Y \stackrel{R^{PC}_{2}\circ T_2(\gamma)}{\leadsto}0.
|\int_s^t \int_{B^+_\rho(x_0)}\eta(x_0-y)u_t(\tau,y)\,dy\,d\tau| &=|\int_s^t \int_{B^+_\rho(x_0)}\eta(x_0-y) (\sum_{|\alpha|\le m, |\beta|\le m}D^\alpha ((-1)^{m+1}A^{\alpha\beta}D^\beta u(\tau,y)) &\quad+\sum_{|\alpha|\le m}D^\alpha f_\alpha)\,dy\,d\tau|.
e(S_j',T_j) \geq|S_j'|\cdot(\frac{n}{2(2l+1)}-2)-\left(\sum\limits_{q=1}^{2l+1}e(S_j',S_q-S_j')+2e(S_j')\right) &\geq|S_j'|\cdot(\frac{n}{2(2l+1)}-2)-(2l+3)kn &\geq|S_j'|\cdot\frac{n}{2(2l+1)}-(2|S_j'|+(2l+3)kn) &=|S_j'|\cdot\frac{n}{4(2l+1)}+|S_j'|\cdot\frac{n}{4(2l+1)}-(2|S_j'|+(2l+3)kn) &>|S_j'|\cdot \frac{n}{4(2l+1)}
\frac{x+2x^{2}+3x^{3}+\cdot\cdot\cdot}{1+x+x^{2}+x^{3}+\cdot\cdot\cdot}=\frac{x(1-x)}{(1-x)^{2}}=\frac{x}{1-x}=\frac{1}{x^{-1}-1}
\left\{\prod_{i = 0}^{n}( \gamma_{\mu_{i}})_{2 1}\frac{1}{\tilde{x}_{0}- \tilde{x}_{\pi ( 1 )}}\frac{1}{\tilde{x}_{\pi ( 1 )}- \tilde{x}_{\pi ( 2 )}}. . . . . \frac{1}{\tilde{x}_{\pi ( n )}- \tilde{x}_{0}}\; + \; c . c . \right\}\; ,
M_{\gamma \to 2 \gamma}= - \frac{4}{1 5 \sqrt{\pi}}\biggl ( \frac{e^{4}}{3 m^{4}}\sin \theta \left[ \frac{7}{4}( \phi_{1}^{2}+ \phi_{2}^{2}) - \phi_{1 2}^{2}\right] - \frac{Q g ( Q^{2}- g^{2})}{8 M^{4}}\cos \theta \, \phi_{1 2}^{2}\biggr ) \omega \omega_{1}\omega_{2}H \cos 2 \beta
2 \mu{\frac{\sqrt{J_{1}J_{2}J_{3}}}{N}}\big ( \bar{O}^{\prime J_{3}}\Pi^{\prime J_{1}}\bar{\Pi}^{\prime J_{2}}\delta_{J_{3}+ J_{1}, J_{2}}+ O^{J_{3}}\Pi^{\prime J_{1}}\bar{\Pi}^{\prime J_{2}}\delta_{J_{3}+ J_{2}, J_{1}}\big ) .
S\begin{bmatrix}d_1 0 \cdots{}0 0 d_2 \ddots{}\vdots{}\vdots{}\ddots{}\ddots{}0 0 \cdots{}0 d_n\end{bmatrix}S^\mathsf{T}=\begin{bmatrix}d_n 0 \cdots{}0 0 d_1 \ddots{}\vdots{}\vdots{}\ddots{}\ddots{}0 0 \cdots{}0 d_{n-1}\end{bmatrix},
\omega_n^{(\alpha,\sigma)}\approx\widetilde{\omega}_n^{(\alpha,\sigma)}= 2\tau^{1+\alpha}e^{-n\sigma\tau}\mathbf{Im}\left(\sum_{j=0}^{N-1}w_j^{(\ell)}(\lambda_j^{(\ell)})^{\alpha}(1-\lambda^{(\ell)}_j\tau)^{-1-n}F_{\omega}(\lambda^{(\ell)}_j) \right),
A_a \ = \ \begin{psmallmatrix}1 0 0 1 0 0 \end{psmallmatrix}, A_b \ = \ \begin{psmallmatrix}0 0 1 0 0 1 \end{psmallmatrix}, A_c \ = \ \begin{psmallmatrix}1 0 0 0 0 1 \end{psmallmatrix}.
\Lambda_{H W}\! = \! 7 . 6 \! \times \! 1 0^{1 6}\mathrm{G e V}\! \left( 4 M_{G U T}a \right)^{- 3 / 2}\! \left( \frac{\alpha_{Y M}}{( 1 / 2 5 )}\right)^{- 1 / 4}.
G_{\varphi}(x) := R_{\varphi}(x) F(x) R_{\varphi}^{-1}(x) =\left(\begin{array}{cc}\frac{x - 3 i}{x + i}\frac{- 2 i e^{i \varphi x}}{x + i}& \frac{4 i e^{- i \varphi x}}{x + i}\frac{x + 3 i}{x + i}\end{array}\right).
\sum_{i,j}b_i^* h(T_i^*T_j)b_j \ =\ \sum_{i,j}b_i^* \left<x,T_i^*T_j x\right> b_j &\ =\ \sum_{i,j}\left<T_i x b_i,T_j x b_j\right> &\ =\ \bigl\langle\sum_i T_i x b_i, \sum_i T_i x b_i\bigr\rangle \ \geq\ 0,
g_{a}(R_{i},\Sigma_{i}) =\int x_{i}q\bigl(x_{i}|R_{i},\Sigma_{i}\bigr)dx_{i}, g_{c}(R_{i},\Sigma_{i}) =\int x_{i}^{2}q\bigl(x_{i}|R_{i},\Sigma_{i}\bigr)dx_{i}-g_{a}^{2}(R_{i},\Sigma_{i}).
g(s) :=sg_1+(1-s)g_2, \Sigma(s) :=h\{(s\eta(t),t)|t\in S^1\}\mbox{such that}\Sigma(0)=\Sigma_1\mbox{and}\Sigma(1)=\Sigma_2, \psi(s) :=s\psi_1+(1-s)\psi_2
\lVert P_A \rVert_{L^p_\alpha}\le \delta^{1/2}\int_{|1-r| < \delta}\int^{1}_{0}|1-r|^{-1/2}\lVert \partial_u F_{u}\big|_{u=u(t)}\rVert_{L^p_{\tau}}\, dt \,dr \le 4\delta M.
\begin{gathered}\int_0^x \rho^{\nu+1}({q^2\rho^2}/{x^2}; q^2)_{-\alpha}J_\nu (u\rho; q^2) \,d_q \rho = x^{\nu+\alpha+1}u^{\alpha-1}(1-q)(1-q^2)^{-\alpha}\Gamma_{q^2}(1-\alpha )J_{\nu-\alpha +1}(u x; q^2). \end{gathered}
\phi_{k}^{\pm,\varepsilon}(I_{\pm}) =\frac{1}{2\pi}\int_{0}^{2\pi}\phi(x\left( I_{\pm},\theta_{\pm}\right) )e^{-ik\theta_{\pm}}d\theta_{\pm}, \Psi_{k}^{\pm,\varepsilon}(I_{\pm}) =\frac{1}{2\pi}\int_{0}^{2\pi}\Psi(x\left( I_{\pm},\theta_{\pm}\right) )e^{-ik\theta_{\pm}}d\theta_{\pm}.
\mathcal{I}_{\nu}(t) = \left(\frac{t}{2}\right)^{\nu}\sum_{k=0}^{\infty}\frac{(t^2/4)^{k}}{k! \Gamma(\nu + k +1)}, K_{\nu}(t) = \frac{\pi}{2}\frac{\mathcal{I}_{-\nu}(t) - \mathcal{I}_{\nu}(t)}{\sin(\pi \nu)}, \nu \notin \mathbb{Z},
\begin{matrix}{{{N_{A}}^{B}={M^{B}}_{A}( - 1 )^{\varepsilon_{A}( \varepsilon_{B}+ 1 )},}}&{{{( N^{- 1})_{A}}^{B}={( M^{- 1})^{B}}_{A}( - 1 )^{\varepsilon_{A}( \varepsilon_{B}+ 1 )}.}}\\ \end{matrix}
\hat{m}= \left( \begin{array}{c c c}{{l_{1}}}&{{l_{2}}}&{{l_{3}}}\\{{m_{1 1}}}&{{m_{1 2}}}&{{m_{1 3}}}\\{{m_{1 2}}}&{{m_{2 2}}}&{{m_{2 3}}}\\{{m_{1 3}}}&{{m_{2 3}}}&{{m_{3 3}}}\\ \end{array}\right)
A(\nu';\nu) = \frac{\zeta\left(\frac{3}{2}+2\nu'+\nu\right)}{\zeta\left(\frac{3}{2}+3\nu'\right)}\prod_p \left(1 + a(p)\frac{1-p^{\nu'-\nu}}{p^{3/2+3\nu'}-1}\right) &= \prod_{p \equiv 1\mod 3}\left(1 + O\big(p^{-5/2 - 2\nu' - \nu}+ p^{-5/2 - 3\nu'}+ p^{-3-4\nu'-2\nu}+p^{-2}\big)\right).
V^{-1}(\rho\circ \varphi) V=V^{-1}\circ\rho\circ V( V^{-1}\circ \varphi \circ V)=0_s\oplus \begin{bmatrix}(V^{-1}\varphi V)^{s+1}(z) \vdots (V^{-1}\varphi V)^n(z) \end{bmatrix}.
\sum_{\mathbf{y}\in{\cal Y}^{n}}\Gamma_{s,1}(\mathbf{y}) =\sum_{\mathbf{y}\in{\cal Y}^{n}}\prod_{i=1}^{n}\left(\sum_{x\in{\cal X}}P_{X}(x)W^{1-s}(y_{i}|x)\right)\left(\sum_{x\in{\cal X}}P_{X}(x)V^{s}(y_{i}|x)\right) =\left[\sum_{y\in{\cal Y}}\left(\sum_{x\in{\cal X}}P_{X}(x)W^{1-s}(y|x)\right)\left(\sum_{x\in{\cal X}}P_{X}(x)V^{s}(y|x)\right)\right]^{n},
\sum_{\lambda\in S_{2}}m_{g}(C,\lambda)+\sum_{\lambda\in S_{2}}d(C,\lambda) =\sum_{\lambda\in S_{2}}m_{a}(C,\lambda) &\leq n-\left(n-|\Lambda(A)|\cdot{\rm rank}(B)-d(A)+\sum_{\lambda\in S_{1}}d(C,\lambda)\right) &=|\Lambda(A)|\cdot{\rm rank}(B)+d(A)-\sum_{\lambda\in S_{1}}d(C,\lambda).
\int [d \delta \phi ]_{e^\phi{\hat g}}~ e^{-S_L[\phi,{\hat g}] + S_{\rm boundary}[\phi ,{\hat e}]}= \int [d \delta \phi ]^{\prime}_{{\hat g}}\int_{-\infty}^{\infty}d\phi_0 ~ e^{-S[\phi_R,{\hat g}]}\quad ,
W(\rho,\psi) =W(\rho_0\otimes\widetilde{\chi_F\chi},\psi)= W(\chi_F\chi,\psi)^{\rm{dim}(\rho_0)}\cdot\det(\rho_0)(c(\chi_F,\psi)) =W(\chi_F,\psi)^d\cdot\det(\rho_0)(c).
\begin{pmatrix}p_1 (\nu_j) a (\nu_j) p_2 (\nu_j) a (\nu_j) \dots p_N (\nu_j) a (\nu_j) p_1 (\nu_j) d (\nu_j-1) p_2 (\nu_j) d (\nu_j-1) \dots p_N (\nu_j) d (\nu_j-1) \hdotsfor{4}\hdotsfor{4}\end{pmatrix},
E\big[ \sum_{i\in S}d_i E(\hat{y}_i | x_i) \big] = E\big[ \sum_{i\in S}d_i E(y_i | x_i, i\in B) \big] = E\big[ \sum_{i\in S}d_i E(y_i | x_i, i\in U) \big] & = \sum_{i\in U}E(y_i | x_i, i\in U) = E(Y | x_U)
\mathbf{v}_{T}=\sum_{\ell=1}^{\infty}\sum_{m\in\iota_{\ell}}\left( v_{\ell}^{m}\overrightarrow{\operatorname*{curl}\nolimits_{\Gamma}}Y_{\ell}^{m}+V_{\ell}^{m}\nabla_{\Gamma}Y_{\ell}^{m}\right) ,
\sum_{k=3}^\infty \frac{k^{k-2}}{k!}\Bigg[k-1- k\beta\Bigg]\big(\beta e^{-\beta}\big)^{k-2}< \left(\sum_{k=3}^\infty \frac{\frac{3}{5}k-1}{k^{5/2}}\right)\frac{e^2}{\sqrt{2\pi}}\beta e^{1-\beta}&< \frac{3}{5}\frac{e^2}{\sqrt{2\pi}}\beta e^{1-\beta},
[ K , \mathcal{O}_{+}^{i n}( \Omega ) ] = - 2 \overline{{{c}}}\mathcal{O}_{+}^{o u t}( \Omega ) \ , \left[ K , [ K , \mathcal{O}_{+}^{i n}( \Omega ) ] \right] = 4 c \overline{{{c}}}\mathcal{O}_{+}^{i n}( \Omega ) \ .
\begin{aligned}\dfrac{d}{dt}\eta_i^\varepsilon = -\sigma (u_i-\bar u_i)^2 -\varepsilon^2 (u_i-\bar u_i) \dfrac{d}{dt}\bar u_i -\dfrac{1}{\Delta x}(\psi_{i+1/2}- \psi_{i-1/2}) - \dfrac{1}{2 \Delta x}(\bar u_{i+1}-\bar u_{i-1})p (\tau_i|\bar \tau_i) +R_i^u + R_i^\tau. \end{aligned}
N_1(c) = c^2 \mathcal{Q}(G_0) +c^3 \mathcal C (G_0) , N_2(c,H) = \mathcal{Q}(H) + 2 c \mathcal{Q}(G_0,H) + 3 c^2 \mathcal{C}(G_0,G_0,H) + 3 c \mathcal{C}(G_0,H,H).
\dot x_\text s(t) =A_Lx_\text s(t)-Ly(t)+(B_u+\gamma^{-2}YC_z'D_{zu})u(t), \dot x_\text a(t) =A_Fx_\text a(t),x_\text a(t_i)=(I-\gamma^{-2}YX)^{-1}x_\text s(t_i) u(t) =Fx_\text a(t).
S=\int d^4x \sqrt{-g}\left[\frac{R}{16\pi}-\frac{1}{2}g_{\mu\nu}\partial^\mu \Phi \partial^\nu \Phi-\frac{V(\Phi)}{16 \pi}\right]+ S(\psi_m, \mathcal{A}(\Phi)^2g_{\mu\nu})\,.
\begin{aligned}E \leq F^{ii}(\nabla_i \phi)^2 \leq \, C \delta^2 F^{ii}U_{ii}^2 + \frac{C b^2}{\zeta^2}F^{ii}(\nabla_i \zeta)^2 + C b^2 \sum F^{ii}\leq \, C \delta^2 F^{ii}U_{ii}^2 + \frac{C b^2}{\zeta}\sum F^{ii}.\end{aligned}
\sup\limits_{B_r^+(x)}|w| \leq \begin{cases}Cr^{2-\frac{n+1}{q}}(\ln r)^2 q\in (n+1,2(n+1)) Cr^{\frac{3}{2}}(\ln r)^2 q \in [2(n+1),\infty]\end{cases},r\in (0,1/2),
\begin{array}{r c l}{d s^{2}}&{=}&{\left( \sigma^{- 1}\eta_{i j}- \sigma^{- 2}\eta_{i k}\eta_{j l}x^{k}x^{l}\right) d x^{i}d x^{j}}\\{}&{=}&{- \displaystyle \frac{d x J ( I - x^{\prime}x J )^{- 1}d x^{\prime}}{1 - x J x^{\prime}},}\\ \end{array}
\bigl\langle W(\xi\otimes T\zeta'),L\xi'\otimes T\zeta\bigr\rangle =\bigl\langle(\omega_{\xi,L\xi'}\otimes\operatorname{id})(W)T\zeta',T\zeta\bigr\rangle &=\bigl\langle T\zeta',(\omega_{L\xi',\xi}\otimes\operatorname{id})(W^*)T\zeta\bigr\rangle &=\bigl\langle T\zeta',T(\omega_{\xi,L\xi'}\otimes\operatorname{id})(W^*)\zeta\bigr\rangle,
\left|\Psi\left(\frac{z+z^{-1}}{2}; \omega\right)\right| = |e^{\frac{\gamma}{2}(z + z^{-1}- 2)}|\cdot\prod_{i=1}^{\infty}{\frac{|1 + \frac{\beta_i}{2}(z-1)||1 + \frac{\beta_i}{2}(z^{-1}- 1)|}{|1 - \frac{\alpha_i}{2}(z-1)||1 - \frac{\alpha_i}{2}(z^{-1}-1)|}}.
A_1=1+\sum_{1\leq j,k\leq N}\frac{\lambda_j\lambda_k}{(2\pi)^2}\frac{1}{(\alpha-z_j)\overline{(\alpha-z_k)}}\frac{i}{\overline{z_k}-z_j}-\sum_{j=1}^N \frac{\lambda_j}{\pi}Re\Big\{\frac{D_tZ-\dot{z}_j}{(\alpha-z_j)^2}\Big\}.
\sum\limits_{n=1}^{\infty}e^{-\frac{c}{2}n}\left\|r_4^n \right\| \leq \|\beta_1\|_{\mathrm{L_1}(0,\infty)}\sum\limits_{n=1}^{\infty}e^{-\frac{c}{2}n}\Biggr( 2\int_{0}^{k}\left\|\mathbf{B}_1u_{t}(\zeta)\right\| d\zeta + k \int_{k}^{t_n}\left\|\mathbf{B}_1u_{tt}(\zeta)\right\| d\zeta \Biggr).
W_{ij}'' =\sum_{k=1}^N \phi_{ikj}(V'_{ik}\otimes H_{kj}) V_{ij}'' =\left\{v\in V_{ij}\mid \phi_{ijk}(v\otimes h)\in W_{ik}'' \mathrm{~for~all~}k \mathrm{~and~for~all~}h\in H_{jk}\right\}.
\phi^{\, p}(a) \le \phi(c^n) &\le \frac{C_1^p}{c^p}\left( \rho^{\, p}(c)+\rho^{\, p}(c^2) +K^p \sum_{k=2}^n \psi^{\, p}(c^{k-2})\right) &\le \frac{C_1^p}{c^p}\left( \rho^{\, p}(c)+\rho^{\, p}(c^2) +K^p C^p \psi^{\, p}(c^{n-1})\right) &\le \frac{C_1^p}{c^p}\left( \rho^{\, p}(c)+\rho^{\, p}(c^2) +K^p C^p \psi^{\, p}(a)\right).
\theta_1 = \x_1 dx^2 - x_2 dx^1 + x_3 dx^4 - x_4 dx^3 \theta_2 = \x_1 dx^3 - x_3 dx^1 + x_4 dx^2 - x_2 dx^4 \theta_3 = \x_1 dx^4 - x_4 dx^1 + x_2 dx^3 - x_3 dx^2.
J_1(N; \theta) = \frac{N}{1 - \theta}\int_0^1 \frac{1-y^N}{1-y}\, dy = \frac{N}{1 - \theta}\int_0^1 \left(\sum_{j=0}^{N-1}y^j\right)dy= \frac{N}{1 - \theta}\sum_{j=1}^N \frac{1}{j},
dL_{x}^{-1}B_{H}(dL_{x}(X_{\alpha ,i}),dL_{x}(Y))=\begin{cases}0 \left( Y \in V_{2}\right) \displaystyle -\frac{1}{2}[X_{\alpha ,i},Y]^{\perp}\left(Y\in V_{1}\right).\end{cases}
[(b_1-b_2)^2-2]R_{1212}+[(b_2-b_3)^2-2]R_{2323}+[(b_1-b_3)^2-2]R_{1313}&=\frac{2}{9}-\frac{10}{3}tr(A)+3[b_1^2a_1+b_2^2a_2+b_3^2a_3].
\prod_{j=1}^m(q;q)_{x_2-x_1+l_j}\mapsto\prod_{j=1}^l(q;q)_{a+b-C+m+x_j}\prod_{j=1}^m(q,q)_{a+b-C+j-1}\prod_{j=l+1}^{b}(q;q)_{a+b-C-x_j-1}&\sim(q;q)_m^l\tilde H_q(a+b-C+m), \prod_{j=1}^m(q;q)_{y_2-y_1-l_j}\mapsto\prod_{j=1}^l(q;q)_{b+c-A-x_j-1}\prod_{j=1}^m(q,q)_{b+c-A+j-1}\prod_{j=l+1}^{b}(q;q)_{b+c-A+x_j+m}&\sim(q;q)_m^{b-l}\tilde H_q(b+c-A+m).
\left[ \widetilde{I}_{\mu}, I_{\nu}\right] = 2 \widetilde{I}\delta_{\mu \nu}, \hspace{0 . 3 i n}\left[ I_{\mu}, I_{\nu}\right] = 4 I_{\mu \nu}, \hspace{0 . 3 i n}\left[ \widetilde{I}_{\mu}, \widetilde{I}_{\nu}\right] = - 4 I_{\mu \nu},
\begin{aligned}\left\| g_i^{k+1}- (1-\alpha^k)g_i^k\right\| &\leq \left\| g_i^{k+1}- g_i^k\right\|+\|\alpha^kg_i^k\|\leq L\left\| x_i^{k+1}- x_i^k\right\|+\alpha^kC,\end{aligned}
(z^t,z^x,z^y,z^p,z^u,z^v,z^\omega,z^\phi,z^T)=(t,x,y,p,u,v,\omega,\phi,T), (\tilde z^t,\tilde z^x,\tilde z^y,\tilde z^p,\tilde z^u,\tilde z^v,\tilde z^\omega,\tilde z^\phi,\tilde z^T)=(\tilde t,\tilde x,\tilde y,\tilde p,\tilde u,\tilde v,\tilde \omega,\tilde \phi,\tilde T).