harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\begin{aligned}E(f(x_{l+1})-f(\omega_*)) \leq \xi E(f(x_l)-f(\omega^*)) \xi = \frac{(1-\sigma_2)(1-\beta)+2L\alpha_2\sigma_1\sigma_2\beta(1-\beta^m)}{2\mu\sigma_1\alpha_1 m(1-\sigma_2)(1-\beta)}\end{aligned}
|ry-\theta|^2 = |(r-1)y+y-\theta|^2= (1-r)^2|y|^2-2(1-r)\langle y,y-\theta\rangle+|y-\theta|^2 &\geq -2(1-r)\langle \theta,y-\theta\rangle-2(1-r)|y-\theta|^2+|y-\theta|^2 &\geq -2(1-r)|y|+2(1-r)-2(1-r)|y-\theta|^2+|y-\theta|^2 &\geq -2(1-r)|y-\theta|^2+|y-\theta|^2= \frac{|y-\theta|^2}{2},
(\nabla_\xi\Phi)(e_j,e_k) =g(e_j,\nabla_\xi(\xi\times e_k))+g(\nabla_\xi e_k,\xi\times e_j) &=g(e_j,\nabla_\xi\xi\times e_k)+g(e_j,\xi\times\nabla_\xi e_k)+g(\nabla_\xi e_k,\xi\times e_j) &=-g(\nabla_\xi\xi,e_j\times e_k).
[\Sigma_{\lambda\alpha}]_{ab}= i\int \frac{d^4 S}{(2\pi)^4}(-ie\gamma^\mu_{\beta\alpha})(-i g_{\mu\nu}[\tau_3 D(S)\tau_3]_{ba})(-ie \gamma^\nu_{\lambda \gamma}) i (P+S)_\tau \gamma^\tau_{\gamma\beta}[D(P+S)]_{ab}.
\lambda_{\tau, k}=\frac{1}{\sqrt{\tau_{1}(1-\tau_{1})\tau_{2}(1-\tau_{2})}}\left[\begin{array}[c]{c}1 -\nabla G^{(k)}(\tau)[D_{1}^{-1}(\tau_{1}),D_{2}^{-1}(\tau_{2})]^{\top}\end{array}\right] .
d{\tilde{s}}^{2}= - \left( M - \frac{r^{2}}{l^{2}}\right) d t^{2}+ J d t d \varphi + r^{2}d \varphi^{2}+ \left( \frac{r^{2}}{l^{2}}- M + \frac{J^{2}}{4 r^{2}}\right)^{- 1}d r^{2}
I_n(\bar{\epsilon})\cap \left[y_n(\bar{\epsilon}),y_n(\bar{\epsilon})+\frac{\Psi(n)}{\beta^n}\right) =\left\{x\in [0,1]: 0\leqslant x-y_n(\bar{\epsilon})< r_n(\bar{\epsilon})\right\}=[y_n(\bar{\epsilon}), y_n(\bar{\epsilon})+r_n(\bar{\epsilon}))
J^N\cdot \left(\begin{array}{c}\mathcal{R}_0(x) \mathcal{R}_1(x) \cdots \cdots \mathcal{R}_{N-1}(x) \end{array}\right) + \left(\begin{array}{c}0 0 \cdots 0{\cal{R}}_N(x) \end{array}\right)=x\left(\begin{array}{c}\mathcal{R}_0(x) \mathcal{R}_1(x) \cdots \cdots \mathcal{R}_{N-1}(x) \end{array}\right),
\mathrm{int}(F_0(\mathcal G))=\mathrm{int}(O_S[g_1])=\mathrm{int}(\partial(O_S[g_1])+\sum_{s\in S}T_{\tilde s}(\mathrm{Span}_{\mathbb C}\{g_n\,|\,n\geq0\})) =\mathrm{int}(\partial(O_S[g_1]))+\sum_{s\in S,n\geq0}\mathbb C\cdot \mathrm{int}(T_{\tilde s}(g_n))\subset F_0(\mathcal G)+\sum_{s\in S,n\geq0}F_1(\mathcal G)=F_1(\mathcal G)
\tilde A_{\pm}(x) := A_{\pm}(\rho^k x), \ \tilde \lambda_{\pm}(x) := \rho^{2k(1-\alpha)}\lambda_{\pm}(x) \ \mbox{and}\ \tilde f_{\pm}(x) : = \rho^{k(2-\alpha)}f_{\pm}(x).
I_f(p,q) = \sum_{i \in A_{1+\kappa,\infty}}q_i f\left( \frac{p_i}{q_i}\right) + \sum_{i \in A_{1, 1+ \kappa}}q_if\left( \frac{p_i}{q_i}\right) &+ \sum_{i \in A_{1/(1+\kappa),1}}q_i f\left( \frac{p_i}{q_i}\right) + \sum_{i \in A_{0,1/(1+\kappa)}}q_if\left( \frac{p_i}{q_i}\right),
\begin{array}{c c c}{{\it S}}\\{{\downarrow}}\\{{{\it S}^{*}}}\\ \end{array}\begin{array}{c c c}{{}}&{{\longrightarrow}}&{{}}\\{{}}&{{}}&{{}}\\{{}}&{{\longrightarrow}}&{{}}\\ \end{array}\begin{array}{c c c}{{{\bar{\it S}}}}\\{{\downarrow}}\\{{{\bar{\it S}}^{*}}}\\ \end{array}\begin{array}{c c c}{{}}&{{}}&{{}}\\{{}}&{{}}&{{}}\\{{.}}\\ \end{array}
\mathbb{P}(X_{r}>T_{(k)}) =\mathbb{E}\left[\mathbb{P}(X_{r}>T_{(k)})|T_{(k)}\right] &=\sum_{j=0}^{r-1}\frac{s^{j}}{j!}\mathbb{E}\left(e^{-sT_{(k)}}T_{(k)}^{j}\right).
0 =(\tfrac{Z_{22}}{Z_{12}})^2-2b\tfrac{Z_{22}}{Z_{12}}+(b^2-a)=(\tfrac{Z_{22}}{Z_{12}})^2-\left(\tfrac{Z'_{21}}{Z'_{11}}+ \tfrac{Z'_{22}}{Z'_{12}}\right)\tfrac{Z_{22}}{Z_{12}}-\tfrac{Z'_{21}}{Z'_{11}}\cdot \tfrac{Z'_{22}}{Z'_{12}}= &=\left(\tfrac{Z_{22}}{Z_{12}}-\tfrac{Z'_{21}}{Z'_{11}}\right)\left(\tfrac{Z_{22}}{Z_{12}}-\tfrac{Z'_{22}}{Z'_{12}}\right).
\psi^{c}=(\begin{matrix}\psi_{L}^{c}\\ \psi_{R}^{c}\end{matrix})=\eta_{c}C\overline{\psi}^{T}=\eta_{c}(\begin{matrix}-i\sigma^{2}&0\\ 0&i\sigma^{2}\end{matrix})(\begin{matrix}\psi_{R}^{*}\\ \psi_{L}^{*}\end{matrix})=\eta_{c}(\begin{matrix}-i\sigma^{2}\psi_{R}^{*}\\i\sigma^{2}\psi_{L}^{*}\end{matrix})
W :=\left\langle\iota_i(\underline{\lambda}^{(d_i)})\mid 1\leq i\leq d \right\rangle, V :=\left\langle\iota_i(e_{\pi_1(v)}) -\iota_j(e_{\pi_2(v)})\mid v\in V_{ij}, 1\leq i < j\leq t\right\rangle.
\sum\limits_{\substack{i=0 \mbox{\scriptsize $m+i$ even}}}^m (-1)^i{m \choose i}\sum\limits_j{\frac{m+i}{2}\brack j}x^{2j-1}+ \sum\limits_{\substack{i=0 \mbox{\scriptsize $m+i$ odd}}}^m (-1)^i{m \choose i}x^{m+i}= 0,
\big\Vert \hat{\Pi}_{W_1}(\hat{\Pi}_V\boldsymbol{\epsilon})_1-\hat{\Pi}_{W_1}\big(\hat{\Pi}_{V_1}-\sum_{j=1}^k(\hat{\Pi}_{V_1}\hat{\Pi}_{V_2})^j(1-\hat{\Pi}_{V_1})\big)\boldsymbol{\epsilon}\big\Vert_n &\leq\frac{\rho_{0,\delta}^{2k+1}}{1-\rho_{0,\delta}^2}\Vert \boldsymbol{\epsilon}\Vert_n.
\begin{array}{l c l}{{\zeta_{+ 1}= \beta_{+ 1}^{- 1},}}&{{\qquad}}&{{\zeta_{- 1}= \beta_{- 1},}}\\{{\zeta_{+ 2}= - \beta_{+ 1}^{- 1}\mu_{+ 1 2},}}&{{\qquad}}&{{\zeta_{- 2}= \mu_{- 2 1}\beta_{- 1},}}\\{{\zeta_{+ 3}= - \beta_{+ 1}^{- 1}( \mu_{+ 1 3}- \mu_{+ 1 2}\mu_{+ 2 3}) ,}}&{{\qquad}}&{{\zeta_{- 3}= \mu_{- 3 1}\beta_{- 1}.}}\\ \end{array}
\epsilon_{V}( t , x ) = \sum_{n}\left[ \frac{e^{- i \omega_{k}t}}{\sqrt{2 \omega_{k}}}a ( k ) e^{i k x}+ \frac{e^{i \omega_{k}t}}{\sqrt{2 \omega_{k}}}a^{\dagger}( k ) e^{- i k x}\right]
N_{0 , 0}= i g_{0}^{2}\{\frac{m_{H}^{2}}{1 6 m^{2}}[ ( \sum_{a = 1}^{3}\phi_{a}^{2})^{2}+ 2 H^{2}\sum_{a = 1}^{3}\phi_{a}^{2}] + \lambda H^{4}\}\delta ( x_{1}- x_{2}) ,
S_{u^j}(-z) = \left( S_{u^j}(\chi)+\displaystyle\frac{\partial}{\partial z}S_{u^j}(\chi)(-z-\chi)+\dotsc+\displaystyle \frac{1}{K!}\frac{\partial^K}{\partial z^K}S_{u^j}(\chi)(-z-\chi)^K \right) + (-z-\chi)^{K+1}R(z^{-1})
\langle f, C_\theta\, k_\lambda^\theta\rangle= \overline{(C_\theta f)(\lambda)}\quad\ \ \quad\qquad\langle f, C_{\alpha,\frac{\theta}{\alpha}}\, k_\lambda^\theta\rangle= \overline{(C_{\alpha,\frac{\theta}{\alpha}}f)(\lambda)}.
\frac{r_{n}^{2}}{h_{n}}\leq \frac{(f(c_{n}))^{2}}{h_{n}t_{n-1}^{2}(c_{n}- c_{n-1})^{2}}\leq \frac{(c_{n}^{3/2})^{2}}{h_{n}c_{n}^{2}}\Big{(}\frac{c_{n}}{c_{n}- c_{n-1}}\Big{)}^{2}\frac{1}{t_{n-1}^{2}}= \frac{c_{n}}{h_{n}}\Big{(}\frac{1}{1 - \frac{c_{n-1}}{c_{n}}}\Big{)}^{2}\frac{1}{t_{n-1}^{2}}\to 0.
\alpha_{-1}:= \frac{3}{2^4}(\alpha_0 \cdot \alpha_1 + \beta_0 \cdot \beta_1 - \frac{1}{2^2}(\beta_0, \beta_0)a_1 + \frac{1}{3}(\alpha_0 + \alpha_1)) \in M_0^{(a_1)}\beta_{-1}:= \frac{3}{2^4}(\alpha_0 \cdot \beta_1 + \alpha_1 \cdot \beta_0 - \frac{3}{2^3}(\beta_0 + \beta_0)) \in M_{\frac{1}{2^2}}^{(a_1)}.
\tilde{\mathcal{A}}^{-1/2}\in L(\mathcal{H}, \mathcal{H}) \tilde{\mathcal{A}}^{1/2}\colon D(\tilde{\mathcal{A}}^{1/2}) := \mathrm{im}\, \tilde{\mathcal{A}}^{-1/2}\to \mathcal{H}
C_{X^\infty\rightarrow{Y}^{\infty}}^{FB,A.1}=\lim_{n\longrightarrow\infty}\frac{1}{n+1}C_{X^n\rightarrow{Y}^{n}}^{FB,A.1}=\nu_0\Big(H(\nu_{0|0})-H(\gamma)\Big)+(1-\nu_0)\Big(H(\nu_{0|1})-H(\delta)\Big) &\qquad\qquad+\xi_0\Big(H(\gamma)-H(\alpha)\Big)+\xi_1\Big(H(\delta)-H(\beta)\Big)
d_{s}= \inf_{t \geq 0,x_1,x_2 \in E_{s-t_0}}\frac{\P_{s,v_{s,x_1,x_2}}(\tau_{A}> t+s)}{\sup_{x \in E_{s}}\P_{s,x}(\tau_{A}> t+s)}; &d'_{s}= \inf_{t \geq 0}\frac{\P_{s,v_{s}}(\tau_{A}> s+t)}{\sup_{x \in E_{s}}\P_{s,x}(\tau_{A}> s+t)}.
\sum_{\lambda \vdash d}P(\lambda)\nu(\lambda) = \sum_{\lambda\vdash d}\frac{1}{z_\lambda}\sum_{k=0}^{d-1}\frac{P(\lambda)\psi_d^k(\lambda)}{q^k}= \sum_{k=0}^{d-1}\frac{1}{q^k}\left(\sum_{\lambda \vdash d}\frac{P(\lambda)\psi_d^k(\lambda)}{z_\lambda}\right) = \sum_{k=0}^{d-1}\frac{\langle P, \psi_d^k\rangle}{q^k}.
{\cal L}= \frac{1}{2}( \partial_{\mu}\sigma )^{2}+ \frac{1}{2}( \partial_{\mu}\pi )^{2}\nonumber \, - \frac{1}{2}\mu^{2}( \sigma^{2}+ \pi^{2}) - \frac{\lambda}{4}( \sigma^{2}+ \pi^{2})^{2}+ c \sigma \quad .
\sum_{a\in Q_1}[a, D_a(w)] =\sum_{a\in Q_1}\sum_{\{s|b_s=a\}}\big (ab_{s+1}\ldots b_r\ldots b_{s-1}-b_{s+1}\ldots b_r\ldots b_{s-1}a \big ) &=\sum_{s=1}^r b_sb_{s+1}\ldots b_r\ldots b_{s-1}-\sum_{s=1}^r b_{s+1}\ldots b_r\ldots b_{s-1}b_s &=0.
~f(x_1+2, x_2, \ldots, x_n) - f(x_1, \ldots, x_n) = ~ u(x_1+2, x_2, \ldots, x_n) - u(x_1, \ldots, x_n) - 2 x_1 - 2 = ~(x_1 + 2) (\xi_1 + 2) - x_1 \xi_1 - \psi(\xi_1 + 2, \xi_2, \ldots, \xi_n) + \psi(\xi_1, \ldots, \xi_n) - 2 x_1 - 2 = ~0.
{\frac{1}{\sigma_{R}\sqrt{\pi}}}~ e^{-{\frac{( E - E_{R})^{2}}{\sigma_{R}^{2}}}}= \int d E^{\prime}~ w ( E^{\prime}) ~{\frac{1}{\sigma_{Q}\sqrt{\pi}}}~ e^{-{\frac{( E - E^{\prime}- E_{Q})^{2}}{\sigma_{Q}^{2}}}},
l(y) <f(z) \frac{y-p}{z-p}+ \frac{z-y}{z-p}\left( \frac{y-p}{y-x}f(x) + \frac{p-x}{y-x}f(y) \right) &=(z-\alpha) \frac{y-p}{z-p}+ \frac{z-y}{z-p}\left( \frac{y-p}{y-x}(x-\alpha) + \frac{p-x}{y-x}(y-\alpha) \right) &=y-\alpha=f(y),
v=\frac{1}{5}\left[ \frac{1}{12}v_0'''+\frac{1}{12}u_0''v_0'+\frac{1}{4}u_0'v_0'' +\frac{1}{6}\left( u_0'\right)^2 v_0' -\frac{8}{27}\left( v_0'\right)^3 \right] \quad etc \quad ,
\dim(\Lambda(\lambda_0)\cap W) =\mu_{Mas}(\Lambda,W)=k(1,\varepsilon)-k(0,\varepsilon) &=\sum_{0\leq\theta\leq\varepsilon}\dim\ker(d(1)-e^{i(\pi+\theta)})-\sum_{0\leq\theta\leq\varepsilon}\dim\ker(d(0)-e^{i(\pi+\theta)}).
0 = (P_V^{\dagger}U)^a=\sqrt g\left[{V^b\kern 1pt\nabla_bU^a-{\textstyle{1 \over 2}}V_b\kern 1pt\nabla^aU^b+\nabla_bV^b\kern 1ptU^a+{\textstyle{1 \over 2}}\nabla^aV_b\kern 1ptU^b}\right]
\hat{f}(n)=\int_0^1 f(x)e(-nx)\,dx=\int_0^1 f(x)\,d[\frac{1}{-2\pi in}e(-nx)] &=\frac{1}{-2\pi in}[D_\mu(x)-x]e(-nx)|_0^1+\frac{1}{2\pi in}\int_0^1 e(-nx)\,d [D_\mu(x)-x] &=\frac{1}{2\pi in}\int_0^1 e(-nx)\,d\mu(x)=\frac{\hat{\mu}(n)}{2\pi in},
\Vert x_{f(k+1)}-y_{f(k)}\Vert^2 \le\exp\left(-2\theta_{f(k)}\sum_{j=f(k)}^{f(k+1)}\lambda_j\right)\cdot\exp(2\theta_{f(k)}\lambda_{f(k+1)})\cdot\Vert x_{f(k)}-y_{f(k)}\Vert^2 \qquad+2M^2(\theta_{f(k)}-\theta_{f(k+1)})\cdot\sum_{j=f(k)}^{f(k+1)}\lambda_j+4M^2\sum_{j=f(k)}^{f(k+1)}\lambda_j^2.
\min_{\tilde{\beta}_{t,n}, \tilde{\beta}_{r,n}}\mathrm{Re}( \breve{\vartheta}_{t,n}^* \tilde{\beta}_{t,n}) + \mathrm{Re}( \breve{\vartheta}_{r,n}^* \tilde{\beta}_{r,n}) begin{align*}-0.5em] \mathrm{s.t.}\tilde{\beta}_{t,n}^2 + \tilde{\beta}_{r,n}^2 = 1, 0 \le \tilde{\beta}_{t,n}, \tilde{\beta}_{r,n}\le 1.
\kappa^{-\lambda}e^{-\frac{1}{2}S_{\kappa}(t)}= \exp\left(-\frac{1}{2}\sum_{t^{\kappa}< p \leq t}\frac{|g(p)|}{p}+ \frac{\log(1/\kappa)}{\log_2 t}\sum_{1 \leq j \leq m}c_j\sum_{p \leq t \atop p \in E_j}\frac{|g(p)|}{p}\right) \ll_B e^{-\frac{1}{4}S_{\kappa}(t)}.
1 + \sum_{i=1}^{L-k}3 \cdot 4^{i-1}+ \sum_{i=1}^{k}2^i \cdot 4^{L-k}= 1 + (4^{L-k}- 1) + (2^{k+1}-2) 4^{L-k}= (2^{k+1}-1) 4^{L-k}
\Gamma_{\lambda_{1}\mu_{1},\lambda_{2}\mu_{2}}^{\nu\rho}=\sum_{\alpha,\beta,\eta,\theta}(\sum_{\kappa}c_{\kappa,\alpha}^{\lambda_{1}}c_{\kappa,\beta}^{\mu_{2}})(\sum_{\gamma}c_{\gamma,\eta}^{\lambda_{2}}c_{\gamma,\theta}^{\mu_{1}})c_{\alpha,\theta}^{\nu}c_{\beta,\eta}^{\rho},
\left| D^n_s \left( \eta^{\rm slow}\circ \phi \right) \right| \lesssim_{n,k_{0}}\left( \log k \right)^n\sum_{\left\{m_{\ell}\right\}}\left| (D^{m}_s \eta^{\rm slow}) (\phi) \right|k^{m\psi}& \lesssim_{n,k_{0}}\left( \log k \right)^n\sum_{m=0}^{n}\left| (D^{m}_s \eta^{\rm slow}) (\phi) \right|k^{m\psi}.
S_{C S}= \frac k{4 \pi}\int_{M^{3}}\{\epsilon^{i j k}( A_{i}^{a}( \partial_{j}A_{k}^{b}- \partial_{k}A_{j}^{b}) g_{a b}+ \frac 2 3 C_{a b c}A_{i}^{a}A_{j}^{b}A_{k}^{c}) \},
V_x(y) \sim \begin{cases}\displaystyle \frac{E\big(y_{n_0+l-1,x}\big)}{y_{n_0+l-1,x}}, \mbox{if}y_{n_0+l-1,x}\geq t^{\ast}, \displaystyle \frac{E\big(y_{n_0+l-1,x}+ t^{\ast}\big)}{y_{n_0+l-1,x}+ t^{\ast}}+ \frac{t^{\ast}}{\Tilde{t}}, \mbox{if}y_{n_0+l-1,x}< t^{\ast}.\end{cases}
{1\over |R|}\int_R |b(x_1,x_2,x_3)-b_R| dx_1dx_2dx_3 &={1\over a^4}\int_{a}^{a+a^2}\int_{a}^{2a}\int_{a}^{2a}\bigg|x_1 -{3a\over2}\bigg|\,dx_1dx_2dx_3 &={a\over 4}.
\frac{L^\alpha[e^a]}{e^a}=\begin{cases}\frac{L^\alpha[e^g]}{e^{g}}+ \frac{1}{2}\int_{|y|<R}\left( H_\epsilon(a_0,x,y) + H_\epsilon(a_0,x,-y) \right) K dy begin{align*}2.5mm]+\int_{|y| \geq R}G_\epsilon(x,y) \left(1 - \frac{\exp(a_0(\eta^\epsilon(x,y),t)/\epsilon)}{\exp(a_0/\epsilon)}\right) K dy,\end{cases}
N^{-1/2}\mathcal{E}(t)=\mathcal{E}_1(t)+\mathop{\mathcal{E}_1^\ast(t)}^{\mathclap{\substack{\downarrow\,\,\,\,}}}+\mathcal{E}_2(t)+\mathcal{E}_2^\ast(t)+\mathcal{E}_2^{\mathrm{sa}}(t)+\mathcal{E}_3(t)+\mathcal{E}_3^\ast(t)+\mathcal{E}_4(t)+\mathcal{E}_4^\ast(t)+\mathcal{E}_4^{\mathrm{sa}}(t)
c_{n-2,\xi_{\boldsymbol{s}k},\xi_{\boldsymbol{t}l}}^{\ast\ast}=\frac{\Gamma\left((n-2)/2\right)}{(2\pi)^{(n-2)/2}}\left[\int_{0}^{\infty}x^{(n-4)/2}\overline{\mathcal{G}}_{n}\left(\frac{1}{2}\xi_{\boldsymbol{s}k}^{2}+\frac{1}{2}\xi_{\boldsymbol{t}l}^{2}+x\right)\mathrm{d}x\right]^{-1}.
\kappa_n := \{p_1,\dots, p_i\}\cdot \{q_1, \dots, q_j\}- \{p_1, \cdots p_{i-1}\}\cdot \{q_1, \cdots q_{j}, p_{i}\}- \{p_1, \cdots p_{i}, q_j\}\cdot \{q_1, \cdots q_{j-1}\}&+ \{p_1, \cdots p_{i-1}, q_j\}\cdot \{q_1, \cdots q_{j-1}, p_{i}\}+ \{p_1, \cdots p_{i-1}, q_1, \cdots q_{j-1}\}\cdot \{p_i, q_j\}
\left[\epsilon^0 q^{d}\right] \frac{(-1)^{(N-1)d}}{\det(z_i^{N-j+1})}\,\begin{vmatrix}z_1^{d+N}z_1^{d+N-1}\cdots z_1^{d+1}z_2^{d+N}z_2^{d+N-1}\cdots z_2^{d+1}\vdots \vdots \cdots \vdots z_N^{d+N}z_N^{d+N-1}\cdots z_N^{d+1}\end{vmatrix}f(z_{N+1}).
\widehat{R}^{\mu}_{i\: \nu \rho \sigma}=\kappa^{\mu}_{i\: \rho}\kappa^{i}_{\nu \sigma}-\kappa^{\mu}_{i\: \sigma}\kappa^{i}_{\nu \rho}+h^{\alpha}_{i\: \nu}h^{\beta}_{i\: \rho}h^{\gamma}_{i\: \sigma}R^{\mu}_{\:\: \alpha \beta \gamma}
\delta S\vert_{\hat{E}^I=0=\tilde{E}^i}={1 \over 2}\int_{{\cal M}^{1+1}}\left( \hat{M}_2^I U^I_{\underline{m}}+ j_1 \wedge \left( \hat{E}^{++}U^{--}_{\underline{m}}- \hat{E}^{--}U^{++}_{\underline{m}}- 2 m f^iu^{i}_{\underline{m}}\right) \right) \delta \hat{X}^{\underline{m}}
\int_t^u \dfrac{\mu'(\tau)}{\mu(\tau)}\left(\dfrac{\mu(\tau)}{\mu(t)}\right)^{p\gamma}\|U(\tau,t) P(t) x\|^p d\tau = \mu(t)^{-p \gamma}\int_t^u \|H(\tau) U(\tau,t) P(t) x\|^p d\tau \leq \mu(t)^{-p \gamma}\left[L(t,P(t)x) - L(u,U(u,t) P(t) x)\right] \leq \mu(t)^{-p \gamma}L(t,P(t)x),
g\left(\sum\alpha_{i,,j}\left(\hat{w}_{i,j}+\frac{Q_{i}}{-Q_{j}}\hat{w}_{-j,-i}\right)\right)=\sum\left(g(\alpha_{i,,j})\hat{w}_{g(i),g(j)}+\frac{Q_{g(i)}}{-Q_{g(j)}}\hat{w}_{g(-j),g(-i)}\right).
[a_{(1\,l)},a_p]_c = -\lambda_{ppp+1}(1-\xi^2)\chi_p(g_{(p+2\,l)})\big(a_{(1\,p-1)}g_{ppp+1}a_{(p+2\,l)}&-\chi_{(p+2\,l)}(g_p)\chi_{(p+2\,l)}(g_{(1\,p+1)})a_{(p+2\,l)}a_{(1\,p-1)}g_{ppp+1}\big) &=-\lambda_{ppp+1}(1-\xi^2)\chi_p(g_{(p+2\,l)})\chi_{(p+2\,l)}(g_{ppp+1}) &\qquad\qquad [a_{(1\,p-1)},a_{(p+2\,l)}]_cg_{ppp+1}=0.
{\cal L}={\frac{1}{2 g \sqrt{\alpha^{\prime}}}}\mathrm{T r}\left( \left({\dot{\Phi}}_{i}-{\frac{i \lambda}{2 ( 2 \pi \alpha^{\prime})}}\epsilon_{i j k}\left[ \Phi_{i}, \Phi_{j}\right] \right)^{2}+{\frac{1 + \lambda^{2}}{2 ( 2 \pi \alpha^{\prime})^{2}}}\left[ \Phi_{i}, \Phi_{j}\right]^{2}\right) .
P_j(u,v)-P_j(w,v) &=\sum_{|\alpha|\geq 1,|\beta|=l}c_{\alpha,\beta}[(x_Q+2^jru)^{\alpha}-(x_Q+2^jrw)^{\alpha}](x_Q+d_Qv)^{\beta}+\tilde{Q}(u,w,v).
R = \sum_{k=1}^K \log_2 \left( \left| \mathbf{I}_{N_S}+ \frac{P}{KN_S}\mathbf{R}_{i}^{-1}\mathbf{M}_k^H \tilde{\mathbf{H}}_k \mathbf{B}_k \mathbf{B}_k^H \tilde{\mathbf{H}}_k^H \mathbf{M}_k\right| \right),
\mathbf{e}^{z_1+z_2}=e^{(x_1+x_2)+i(y_1+y_2)}=(e^{x_1+x_2},y_1+y_2) =(e^{x_1}e^{x_2},y_1+y_2)=(e^{x_1},y_1)(e^{x_2},y_2)=\mathbf{e}^{z_1}\mathbf{e}^{z_2}.
A_{11}=\frac{1}{2}(R_{1212}+2R_{1234}+R_{3434}), A_{12}=A_{21}=\frac{1}{2}(R_{1213}+R_{1334}+R_{1242}+R_{3442}), A_{13}=A_{31}=\frac{1}{2}(R_{1214}+R_{1434}+R_{1223}+R_{2334}), A_{22}=\frac{1}{2}(R_{1313}+2R_{1342}+R_{4242}), A_{23}=A_{32}=\frac{1}{2}(R_{1314}+R_{1442}+R_{1323}+R_{2342}), A_{33}=\frac{1}{2}(R_{1414}+2R_{1423}+R_{2323});
G(x, y| \xi) = \frac{1}{\Gamma(1 + x\partial_{x})}\sum_{n=0}^{\infty}\xi^n (y - x)^n & = \frac{1}{\Gamma(1 + x\partial_{x})}\left[\frac{1}{(1 - y \xi) + \xi x}\right] & = \frac{1}{(1 - y \xi) + \xi c_{z}x}\, \frac{1}{\Gamma(1+z)}\Big\vert_{z=0}& = \frac{1}{1 - y \xi}\exp\left(- \frac{x\xi}{1-y\xi}\right).
\mathbf f_{\mathbf D_u}^i(x)= \underbrace{(f_{k}*\cdots*f_{l})}_{|\mathbf D_u|}(x),\,\mathbf g_{\mathbf D_u}^i(x)= \underbrace{(g_{k}*\cdots*g_{l})}_{|\mathbf D_u|}(x),
(a^4,a^3b,ab^3,b^4)(x_1,\ldots,x_n)^2 \cap a^2b^2(x_1^2,\ldots,x_n^2) \subseteq (a^3,b^3)\cap (a^2b^2)\cap (x_1^2,\ldots,x_n^2) =a^2b^2(a,b) \cap (x_1^2,\ldots,x_n^2)=a^2b^2(a,b)(x_1^2,\ldots,x_n^2),
\begin{aligned}\sum\limits_{i=1}^{n}{{{w}_{i}}f\left( M+m-\left( \left( 1-t \right)\overline{a}+t{{a}_{i}}\right) \right)}\ge f\left( \sum\limits_{i=1}^{n}{{{w}_{i}}\left( M+m-\left( \left( 1-t \right)\overline{a}+t{{a}_{i}}\right) \right)}\right) =f\left( M+m-\overline{a}\right). \end{aligned}
- 2 g(V,W) \psi = V \cdot_g W \cdot_g \psi + W \cdot_g V \cdot_g \psi, \nabla^g_V g(\psi, \psi') = g(\nabla^g_V \psi, \psi') + g(\psi, \nabla^g_V \psi'), \nabla^g_V(W \cdot_g \psi) = \nabla^g_V W \cdot_g \psi + W \cdot_g \nabla^g_V \psi, g(V \cdot_g \psi, \psi') = (-1)^{s+1}g(\psi, V \cdot_g \psi'),
A_{1}^{\alpha , i}\equiv \frac{1 + \sigma_{3}}{2}\otimes \psi_{- 1 / 2}^{\alpha , i}| 0 , k \rangle , A_{2}^{\alpha , i}\equiv \frac{1 - \sigma_{3}}{2}\otimes \psi_{- 1 / 2}^{\alpha , i}| 0 , k \rangle
1-2\beta re^{if(r)}\psi'_{\mu_{1}}(re^{if(r)}) =\frac{2\beta}{f(r)}\Im\frac{1}{1-\eta_{\mu_{1}}(re^{if(r)})}-2\beta re^{if(r)}\psi'_{\mu_{1}}(re^{if(r)}) =2\beta\int_{\mathbb{R}_{+}}\left[\frac{1}{f(r)}\frac{tr\sin(f(r))}{|1-tre^{if(r)}|^{2}}-\frac{tre^{if(r)}}{(1-tre^{if(r)})^{2}}\right]\,d\mu_{1}(t).
A(s_{\alpha_k},\chi) =c_{\alpha_k}(\chi)-q^{-1}+(-\varrho,\varpi)_2(\varpi,\varpi)_2^{k-1}q^{-1/2}\chi^{-1/2}(a_{\alpha_k}) &=-\chi^{-1}(a_{\alpha_k})\frac{(1+q^{-1/2}\epsilon_{\varrho,k}\chi^{-1/2}(a_{\alpha_k}))(1-q^{-1/2}\epsilon_{\varrho,k}\chi^{1/2}(a_{\alpha_k}))}{1-\chi^{-1}(a_{\alpha_k})}=y_{\alpha_k}(\chi^{-1}).
{\frac{\displaystyle \prod_{j = 1 , j \neq i}^{N - 1}( y_{j}^{2}- y_{i}^{2})}{\displaystyle \prod_{j = 1}^{N - 1}( v_{j}- y_{i}^{2})}}d y_{i}= - d x , \quad 1 \leq i \leq N - 1 ,
\begin{array}{lcl}\lambda^{(0)}= \gamma_{1,6'}\lambda^{(0)}\gamma_{1,6'}^{-1}, \lambda^{(0)}=-\gamma_{\Omega R,6'}\lambda^{(0)^{T}}\gamma_{\Omega R,6'}^{-1}\lambda^{(3)}= e^{2\pi iv_i}\gamma_{1,6'}\lambda^{(3)}\gamma_{1,6'}^{-1}, \lambda^{(3)}= -\gamma_{\Omega R,6'}\lambda^{(3)^T}\gamma_{\Omega R,6'}^{-1}\end{array}
S_1 =\{v_1,v_2,v_5,v_6,\ldots,v_{4k-3},v_{4k-2}\}, S_2 =\{v_2,v_3,v_6,v_7,\ldots,v_{4k-2},v_{4k-1}\}, S_3 =\{v_3,v_4,v_7,v_8,\ldots,v_{4k-1},v_{4k}\}, S_4 =\{v_1,v_4,v_5,v_8,\ldots,v_{4k-3},v_{4k}\}.
ds^2_{\rm target}= \frac{1}{2}d\varphi^2 + \frac{1}{2}d\beta^2 +\frac{1}{2}e^{2\varphi}\left( d\sigma_1 -\chi d\sigma_2 \right)^2 +\frac{1}{2}e^{\varphi}\left[ e^{\sqrt{3}\beta}d\chi^2 +e^{-\sqrt{3}\beta}d\sigma_2^2 \right] .
S[A_0 + a_0, a_j,{\bar\psi},\psi]= \int d^3 x \left[ \frac{1}{4}F_{\mu\nu}(A+a)F_{\mu\nu}(A+a)+ \sum_b{\bar\psi}_b (\not \!\partial + i e_N (\not\!\! A + \not \! a) ) \psi_b \right]\;.
\langle \partial X^{\mu}( z ) \bar{\partial}X^{\nu}( \bar{w}) \rangle = - \delta^{\mu \nu}\left[ \pi \delta^{( 2 )}( z - w ) -{\frac{\pi}{2}}\, \omega ( z ) \cdot ( \mathrm{I m}\tau )^{- 1}\cdot \bar{\omega}( \bar{w}) \right] ,
I = 2 \left({\frac{\alpha}{\sinh^{2}A + \sin^{2}{\frac{\gamma}{2}}}}-{\frac{\sinh{\frac{2 A}{\alpha}}}{\sinh{2 A}}}~{\frac{1}{( \sinh^{2}{\frac{A}{\alpha}}+ \sin^{2}{\frac{[ \gamma ]}{2 \alpha}})}}\right) ~ ~ .
\hat{\bf V}_{1}^{\dagger}\hat{\bf V}_{1}= \hat{\bf V}_{2}^{\dagger}\hat{\bf V}_{2}= 1 , \quad \hat{\bf V}_{1}^{n}= \hat{\bf V}_{2}^{n}= 1 , \quad \hat{\bf V}_{1}\hat{\bf V}_{2}= \hat{\bf V}_{2}\hat{\bf V}_{1}e^{2 \pi i \theta_{n}},
\phi(e_1) ={\tau}^{-1}\sum_iE_M(e_1{\lambda_i}^*e_1)f{\lambda_i}= \sum_iE_N({\lambda_i}^*)f \lambda_i~~~~~~~~~(\textrm{since}~E_M(e_1)=\tau) = \sum_i f{\lambda_i}^*f\lambda_i~~~~~~~~~~~~~~(\textrm{by assunption}~(1)) = f~~~~~~~~~~~~~~(\textrm{since}, ~~\sum_i{\lambda_i}^* f{\lambda_i}= 1).
e^{i S_{\Lambda - \delta \Lambda}( \phi_{+}, \phi_{-})}\equiv \int_{\mathrm{C T C}}\prod_{\Lambda_{0}> |{\bf q}| > \Lambda - \delta \Lambda}{\cal D}[ \phi_{+}({\bf q}, t ) ]{\cal D}[ \phi_{-}({\bf q}, t ) ] \; e^{i S_{\mathrm{c l}}[ \phi_{+}, \phi_{-}]}.
knc \le \lambda^2 c = \sum_{u \sim v}\sum_{w \sim u}\mathrm{x}_w = d(v) c + \sum_{u \sim v}\sum_{\substack{w \sim u, w \neq v}}\mathrm{x}_w \le d(v) c + \sum_{u \in S_1}\sum_{\substack{w \sim u w \in L_1 \cup L_2}}\mathrm{x}_w + 2e(S_1)\alpha + 2e(L) + e(L_1, S_1)\alpha + e(N_1, S_2)\alpha.
P_{\lambda}(z_1,\ldots ,z_n,z;q,t,\mathbf{t}) = & \sum_{\substack{\mu\subset m^n,\, \mu\preceq\lambda m-d\leq r\leq m}}(-1)^{m-r+|\lambda|-|\mu|}C^{(n+1)^m-\lambda^\prime,m}_{n^m-\mu^\prime ,r}(t,q,\mathbf{t}) P_{\mu}(z_1,\ldots ,z_n;q,t,\mathbf{t}) \langle z;t_0\rangle_{q,m-r}
A=(\begin{matrix}0&1&1&1&1&1\\ 1&0&1&-1&-1&1\\ 1&1&0&1&-1&-1\\ 1&-1&1&0&1&-1\\ 1&-1&-1&1&0&1\\ 1&1&-1&-1&1&0\end{matrix}).
\tau\left( (x_g : g \in G_{k+1,p}) \right) :=\bigwedge_{(g_1, \ldots, g_{k+1}) \in R_{k+1}}\rho_{\leq r}(x_{g_1}, \ldots, x_{g_{k+1}}) \land \bigwedge_{(g_1, \ldots, g_{k+1}) \in \prod_{i \in [k+1]}P_i \setminus R_{k+1}}\rho_{\geq s}(g_{g_1}, \ldots, x_{g_{k+1}}).
\sum_i \bar{F}\left( \frac{2\alpha}{\theta_b a_i}\right) \leq (1+\epsilon)\sum_i \left(\frac{a_i}{2\alpha}\right)^{\alpha-\epsilon}\bar{F}\left( \frac{1}{\theta_b}\right) = o( \theta_b), \theta_b^2 = o \left( \theta_b \right), \bar{F}\left( \frac{b}{k}\right) \exp \left(\theta_b \frac{b}{k}\right) \sum_i a_i^\alpha = k,
\begin{array}{rcl}x_1+ix_2 = y(\partial_{\sigma}\bar{y})^{1/3}/\mbox{Re}\;(\partial_{\sigma}y)^{1/3}begin{align*}5pt]x_- = -\mbox{Im}\;(\partial_{\sigma}y)^{1/3}/ \mbox{Re}\;(\partial_{\sigma}y)^{1/3}begin{align*}5pt]x_+ = x_-|y|^2+i\int d\sigma\;(y\partial_{\sigma}\bar{y}-\bar{y}\partial_{\sigma}y)+C\end{array}
\displaystyle \epsilon^2 \partial_t \bigg(||\partial^m g||_{L^2_{x,v}}^2 + \sum_{j=0}^{m-1}C_{jm}\, ||\partial^j g||_{L^2_{x,v}}^2\bigg) \leq \begin{cases}-2\lambda\, ||g^{\perp}||_{\Lambda}^2\,,m=0, begin{align*}4pt] - \lambda\, ||\partial^m g^{\perp}||_{\Lambda}^2 \,,m\geq 1. \end{cases}
\phi_\lambda(x,t):=\int_0^t\tfrac p\lambda \tau^{p-1}+ \min\{\phi_-'(x,\tau), p\lambda \tau^{p-1}\}\, d\tau=\tfrac 1\lambda t^p + \int_0^t\min\{\phi_-'(x,\tau), p\lambda \tau^{p-1}\}\, d\tau.
L^{a}{}_{b}= \left( \begin{array}{c c}{{\cosh v}}&{{\sinh v}}\\{{\sinh v}}&{{\cosh v}}\\ \end{array}\right)_{\, \; b}^{\! \! a}\; = \; \delta_{b}^{a}+ \epsilon^{a}{}_{b}\, v + \mathcal{O}( v^{2}) \, ,
L ( \phi , \Lambda ) = \sum_{m = 1}^{\infty}{\frac{1}{( 2 m ) !}}\int{\frac{d p_{1}. . . d p_{2 m}}{( 2 \pi )^{8 m - 4}}}L_{2 m}( p_{1}, . . . , p_{2 m}; \Lambda ) \delta^{4}( p_{1}+ . . . + p_{2 m}) \phi ( p_{1}) . . . \phi ( p_{2 m})
I_{jk}(x;z)=\int_{-\infty}^{+\infty}\mathrm{d}\theta\,\int_0^{+\infty}\mathrm{d}u\left[e^{\imath\,\sqrt{k}\,\Psi_x(u,\theta)}\cdot\mathcal{A}_{jk}(x;u,\theta,z)\right],
g'_0(x,a,b)=a\bigl(2F^{0,2}+4a^2F^{1,3}\bigr)\quad\mbox{and}&g''_0(x,a,b)=a\bigl(2F^{0,3}+8a^2F^{1,4}+8a^4F^{2,5}\bigr),\qquad\qquad\qquad
\Phi_{K\to L}(\xi,v,w) =\int_\xi^{+\infty}a_{K\to L}(\zeta,v,w)d\zeta \Phi^n_{K\to L}(\xi) =\Phi_{K\to L}(\xi,v^n_K,v^n_L)=\int_\xi^{+\infty}a^n_{K\to L}(\zeta)d\zeta.
\sum_{m}\frac{u_{m}( z ) u_{m}( z^{\prime})}{m^{2}+ p^{2}}\longrightarrow \int_{0}^{\infty}d m ~ \lim_{z_{c}\to \infty}\frac{1}{\pi k e^{- k z_{c}}}\left( \frac{u_{m}( z ) u_{m}( z^{\prime})}{m^{2}+ p^{2}}\right)
\inf\left\{||b_0-\tilde{b}||_{L^{2}(\mathbb{R}^{3})}\ |\ \tilde{b}\in S_{h_C,W,0}\ \right\}=\inf\left\{||b_0-\tilde{b}||_{L^{2}(\mathbb{R}^{3})}\ \middle|\ \tilde{b}=U_C(\cdot +ze_z)-B_{\infty},\z\in \mathbb{R}\right\}&=\inf\left\{||b_0+B_{\infty}-U_C(\cdot +ze_z)||_{L^{2}(\mathbb{R}^{3})}\ \middle|\z\in \mathbb{R}\right\}.
G_{n,k}(z)=A z\psi_{n,k}(z)+B\psi_{n+1,k}(z)=z\int_{a_{k-1}}^{b_{k-1}}\frac{A\tau\psi_{n,k-1}(\tau)+B\psi_{n+1,k-1}(\tau)}{z-\tau}\,d\sigma_{k-1}^{*}(\tau)
- 7 2 5 3 1 8 4 a^{9}N + 9 0 6 6 4 8 a^{9}N^{2}- 1 6 2 7 1 7 8 8 3 2 3 a N^{2}- 6 0 9 7 8 7 5 9 9 1 a N^{4}+ 5 2 1 2 6 7 5 3 5 a N^{6}
(I): =\sum_{n=0}^{n_0}Z_{x|n}(r_N)\mu([x|n]) &\leq (n_0+1) Z_{x|n_0}(r_N)\leq (n_0+1) \frac{r_N^d}{\phi^d(T_{x|n_0})}\leq \frac{(n_0+1)}{\phi^d(T_{x|n_0})}\cdot r_N^{s+\epsilon}.
\sum_{j \in J}|a_i c_j - b_i d_j| \leq \sum_{j \in J}( |a_i c_j - a_i d_j| + |d_j a_i + d_j b_i|) = \sum_{j \in J}(a_i |c_j - d_j| + d_j |a_i - b_i| ) \leq \delta a_i + |a_i - b_i|.
{\cal O}\equiv\int d^4x\,{\rm tr}\, \phi^2, \qquad\bar{Q}_\mu{\cal O},\qquad\bar{Q}_\mu \bar{Q}_\nu{\cal O},\qquad\bar{Q}_\mu \bar{Q}_\nu \bar{Q}_\rho{\cal O},\qquad\bar{Q}_\mu \bar{Q}_\nu \bar{Q}_\rho \bar{Q}_\sigma{\cal O}.