harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
A= \begin{bmatrix}1-\beta 0.3 \beta 0.4 \beta 0.3 \beta 0.4 \beta 1-\beta 0.3 \beta 0.3 \beta 0.5\beta 0.4\beta 1-\beta 0.1 \beta 0.1\beta 0.2\beta 0.7\beta 1-\beta \end{bmatrix}.
\int f^2 \log \tfrac{f^2}{\int f^2 d\phi_\#m}d\phi_\#m = \int \tilde{f}^2 \log \tfrac{\tilde{f}^2}{\int \tilde{f}^2 dm}d m \leq \frac{1}{2\rho}\int |D\tilde{f}|^2 d m & \leq \frac{\sup |D\phi|_{\ell^2}^2}{2\rho}\int |Df\circ \phi|^2 d m = \frac{\sup |D\phi|_{\ell^2}^2}{2\rho}\int |Df|^2 d \phi_{\#}m.
A_4 = 2Jq^4-3Jq^3Jq^1+Jq^2Jq^2+Jq^1Jq^3=0, A_5 = 5Jq^5-5Jq^4Jq^1+Jq^2Jq^3-2Jq^1Jq^4=0, A_6 = 9Jq^6-7Jq^5Jq^1+Jq^2Jq^4+3Jq^1Jq^5=0.
\omega_{\hat{\alpha}}^{\alpha\rho}= f', \quad \omega_{\hat{\theta}}^{\theta\rho}=g'e^{g-f}, \quad \omega_{\hat{\varphi}}^{\varphi\rho}=g'e^{g-f}\sin\theta, \quad \omega_{\hat{\varphi}}^{\varphi\theta}=\cos\theta,
L_{i j}= \left( \begin{array}{c c c c c}{{0}}&{{k_{1}- k_{3}}}&{{- k_{1}+ k_{2}}}&{{k_{3}- k_{2}}}&{{0}}\\{{- k_{1}+ k_{3}}}&{{0}}&{{k_{1}}}&{{- k_{3}}}&{{0}}\\{{k_{1}- k_{2}}}&{{- k_{1}}}&{{0}}&{{k_{2}}}&{{0}}\\{{- k_{3}+ k_{2}}}&{{k_{3}}}&{{- k_{2}}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{0}}&{{0}}\\ \end{array}\right) .
\begin{aligned}T_{1}\leq\frac{C}{(R-\rho)^{2}}\int_{B_{R}^{+}\backslash B_{\rho}^{+}}\omega^{2}dxdy\leq\frac{C}{(R-\rho)^{2}}\int_{B_{R}^{+}\backslash B_{\rho}^{+}}|\nabla\boldsymbol{w}|^{2}dxdy\leq\frac{C}{(R-\rho)^{2}}.\end{aligned}
G^{\lambda}(x,y) = y_n^{2\mu+1}\int_0^\infty \frac{\exp({-\frac{|\tilde{x}-\tilde{y}|^2}{2w}})}{(2\pi w)^{\frac{n-1}{2}}}{\mathcal{G}}_{\mu}^{\lambda}(x_n,y_n;w)\,dw\/,
\mathcal W_2(\mu_0,\mu_1) =\inf \left\{\int_0^1 \int_{\mathbb{R}^d}|\nu_s|^2 m_s(dx)ds :\, \,\,\partial_s m_s + \div (\nu_s m_s) =0\,,\, m_{i}=\mu_i\,,\,\, i=0,1 \right\},
P(z)=[\begin{matrix}1&a(1+z^{-1})\\ 0&1\end{matrix}][\begin{matrix}1&0\\b(1+z)&1\end{matrix}][\begin{matrix}1&c(1+z^{-1})\\ 0&1\end{matrix}][\begin{matrix}1&0\\d(1+z)&1\end{matrix}][\begin{matrix}k_{1}&0\\ 0&k_{2}\end{matrix}].
u =d_{12}d_{21}+ d_{13}d_{21}+ d_{13}d_{22}+(d_{12}+d_{22}) d_{31}+(d_{13}+ d_{23}) d_{31}+(d_{13}+ d_{23}) d_{32}= (d_{12}d_{21}+ d_{13}d_{21}+ d_{13}d_{22}) +(d_{12}d_{31}+ d_{13}d_{31}+ d_{13}d_{32}) +(d_{22}d_{31}+ d_{23}d_{31}+ d_{23}d_{32}) .
\limsup_{i\in I}\frac{H_\mu(\alpha_{F_i})}{\theta(A_i)}+\mu(f) =\lim_{i\in I}\frac{H_\mu(\alpha_{F_i})+\mu(f_{A_i})}{\theta(A_i)}&\leq \limsup_{i\in I}\frac{\operatorname{P}_{f_{A_i}}(\mathcal{U}_{F_i})}{\theta(A_i)}+\limsup_{i\in I}\frac{|F_i|}{\theta(A_i)}\theta(V)\epsilon &\leq \operatorname{p}_f^{(OW)}(\pi)+\epsilon.
p_c = \frac{1}{\ln V}[C \ln^2\frac{1}{p_c}+C' \ln \frac{1}{p_c}] & = \frac{1}{\ln V}[C (\ln \ln V - 2 \ln \ln \ln V - \ln C +O(\varepsilon))^2 +C' \ln \ln V + O(\varepsilon)] & = \frac{1}{\ln V}[C (\ln^2 \ln V - 4 \ln \ln \ln V \ln \ln V- 2\ln C \ln \ln V) + +C' \ln \ln V + O(\ln^2 \ln \ln V)].
T_{\mu \nu}= \frac{{\rm e}^{-\phi}}{2\,\,3!}\left[ 3\,H_{\mu\sigma\rho}H_{\nu}^{\,\,\,\sigma\rho}- \frac{G_{\mu\nu}}{4}H^2\right]+ \frac{{\rm e}^{\phi}}{2\,\,3!}\left[ 3 \,(\tilde{F}_3)_{\mu\sigma\rho}(\tilde{F}_3)_{\nu}^{\,\,\,\sigma\rho}-\frac{G_{\mu\nu}}{4}F_3^2\right]
\langle\bar{\psi}(x)\psi(x)\,\bar{\psi}(\tilde{x})\psi(\tilde{x})\rangle_w \cong -{\langle1\rangle_w \over(4\pi)^2}{\vert\tilde{x}-x_o \vert^4 \over r^4_o}{2\over \vert x-x_o\vert^4}
\delta_m := \frac{T}{m}, m \in \mathbb{N}, [t]_m^- := k \delta_m, k \delta_m \leq t < (k+1)\delta_m, k = 0,\ldots,m-1, [t]_m^+ := (k+1) \delta_m, k \delta_m \leq t < (k+1)\delta_m, k = 0,\ldots,m-1,
\frac{\partial b}{\partial \overline{k}}= \frac{\partial b}{\partial \overline{\alpha}}+ab_0, \frac{\partial a}{\partial{k}}= b\overline{b_0}, b_0={b}(0,k,t), ~~|\alpha|\leq A, ~~k\in \mathbb C,~~0\leq t\leq T.
W ( x_{B}, Q ;{\mathcal T}) = \left( \begin{array}{c c c c}{{W^{0 0}}}&{{0}}&{{W^{0 2}}}&{{0}}\\{{0}}&{{W^{1 1}}}&{{0}}&{{0}}\\{{W^{2 0}}}&{{0}}&{{W^{2 2}}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{0}}\\ \end{array}\right)
\|\varphi\|^p_{1,2,p; W^{2, p}(\mathbb{R}^d)}\ = \ \| \varphi \|^p_{p; W^{2,p}(\mathbb{R}^d)}+ \| \frac{\partial \varphi}{\partial t}\|^p_{p; L^p(\mathbb{R}^d)}, \ 1 \leq p < \infty.
\overline{C}_n(\varepsilon) = \prod_{t=1}^{\eta}\left(1-\frac{v_n^2}{s_{nt^2}}\right)\prod_{t=\eta+1}^{(F_{n}-1)/2}\left(1-\frac{v_n^2}{s_{nt^2}}\right) &\geq \prod_{t=1}^{\infty}\left(1-\frac{(1-2\varepsilon (-\varphi)^{-n})^2}{u_t^2}\right) - \mathcal{O}(\varphi^{n/5}).
\|\hat A h \|_2 = \|\hat A_{T_{01}}h_{T_{01}}+ \sum\limits_{j\geq 2}\hat A_{T_j}h_{T_j}\|_2 & \geq \|\hat A_{T_{01}}h_{T_{01}}\|_2 - \sum\limits_{j\geq 2}\|\hat A_{T_j}h_{T_j}\|_2 & \geq \sqrt{1-\delta}\|h_{T_{01}}\|_2 - \sqrt{1+\delta}\sum\limits_{j\geq 2}\|h_{T_j}\|_2.
K_1 = \Big|f_\epsilon(V+\phi_2 )-f_\epsilon(V+\phi_1) -f^{'}_\epsilon(V) (\phi_2-\phi_1 ) \Big|_{\frac{2n}{n+2}}= \Big|\Big(f_\epsilon^{'}(V+\phi_\varrho )- f^{'}_\epsilon(V) \Big)(\phi_2-\phi_1) \Big|_{\frac{2n}{n+2}}.
L= y^2-(4q-3)y-4(q-1)(q-2) = y^2-(4q-3)y+\Bigl(\frac{4q-3}{2}\Bigr)^2-\Bigl(\frac{4q-3}{2}\Bigr)^2-4(q-1)(q-2) = \Bigl(y-\frac{4q-3}{2}\Bigr)^2-\frac{(4q-3)^2+16(q-1)(q-2)}{4}= \Bigl(y-\frac{4q-3}{2}\Bigr)^2-\frac{32q^2-72q+41}{4}.
v=\frac{x_{A}}{t_{A}}=\frac{\frac{1}{2}c(k^{2}-1)T_{1}}{\frac{1}{2}(k^{2}+1)T_{1}}=c\frac{k^{2}-1}{k^{2}+1}=c\frac{k-k^{-1}}{k+k^{-1}}.
{T_{i_{1}\cdot\cdot\cdot i_{p}}}^{j_{1}\cdot\cdot\cdot j_{q}}\mapsto A_{i_{1}}^{i_{1}^{\prime}}\cdot\cdot\cdot A_{i_{p}}^{i_{p}^{\prime}}{T_{i_{1}^{\prime}\cdot\cdot\cdot i_{p}^{\prime}}}^{j_{1}^{\prime}\cdot\cdot\cdot j_{q}^{\prime}}(A^{-1})_{j_{1}^{\prime}}^{j_{1}}\cdot\cdot\cdot(A^{-1})_{j_{q}^{\prime}}^{j_{q}}.
(A \bar\varphi) (\xi) \ = \ \int\limits_{\mathbb R^d}a (\xi-q) \mu (\xi, q) (\bar\varphi (q) - \bar\varphi (\xi) ) \dq \ = \ \int\limits_{\mathbb T^d}\hat a (\xi-\eta) \mu (\xi, \eta) (\bar\varphi (\eta) - \bar\varphi (\xi) ) \d\eta,
\sum_{x \in V_1}\left(d(y,x)-1\right) \ge \sum_{i=2}^r (d(y,w_i)-1)+\sum_{i=2}^r (d(y,u_i)-1) &\ge \sum_{i=2}^r (i-1) + \sum_{i=2}^r (i-2) &=(r-1)^2.
g^{S_0 \oplus S_1}_{(0, i)}= 1 + \beta \sum_{j \in S_1}p({i, j})g^{S_0 \oplus S_1}_{(1, j)}+ \beta \sum_{j \in N \setminus S_1}p_{ij}g^{S_0 \oplus S_1}_{(1, j)}= g^{S_0 \oplus S_1}_{(1, i)}= g^{S_1}_i,
\left.+\frac{\partial}{\partial x_\nu}((x-x')_\rho\delta_{\mu\lambda}-(x-x')_\lambda\delta_{\mu\rho})\right]D_1\left((x-x')^2\right)=\oint\limits_C^{}dx_\mu\oint\limits_C^{}dy_\mu G\left((x-y)^2\right),
\partial_x\bigl[\eqref{va1}\bigr] = \sqrt{\tfrac{it}{2\pi}}e^{-\frac{it}{2}x^2}T(0)\psi(0) & + \sqrt{\tfrac{it}{2\pi}}\int_+ e^{-\frac{it}{2}(\xi-x)^2}\partial_\xi\bigl[T(\xi)\psi(\xi)\bigr]\,d\xi.
4e^{-2 \cdot 2^n}-8e^{-3 \cdot 2^n}-16e^{-4\cdot 2^n}=4e^{-2 \cdot 2^n}\left[ 1-2e^{- 2^n}-4e^{-2 \cdot 2^n}\right]\ge 2e^{-2 \cdot 2^n}>e^{-2^{n+1}},
\cdots \to H^{p}(X, K^{M}_{p}(O_{X_{j+1}}))_{\mathbb{Q}}\to H^{p}(X, K^{M}_{p}(O_{X_{j}}))_{\mathbb{Q}}\xrightarrow{\delta =0}H^{p+1}(X, \Omega^{p-1}_{X/ \mathbb{Q}})
\|(f\circ\phi)'\|^2 = \int |(f\circ \phi)'(x)|^2 dx = \int |f'\circ \phi(x)|^2 |\phi'(x)|^2 dx \leq |\phi'|_{C^0}\int |f'\circ \phi(x)|^2 |\phi'(x)| dx = |\phi'|_{C^0}\int |f'(y)|^2dy \leq C^2 \|f'\|^2,
\begin{aligned}F(a,b,c,d,e) = (a \wedge b \wedge c \wedge bc)(a \wedge d \wedge e \wedge de) \quad{}+ (a \wedge b \wedge d \wedge bd)(a \wedge e \wedge c \wedge ec) + (a \wedge b \wedge e \wedge be)(a \wedge c \wedge d \wedge cd)\end{aligned}
\dfrac{d^{k}}{dx^{k}}\phi_{3\lambda}\left( x\right) =\dfrac{s^{2}\beta_{2}}{\delta \gamma}\cos \left( s\left( c_{1}-a\right) \right) \sin \left(s\left( c_{2}-c_{1}\right) \right) \dfrac{d^{k}}{dx^{k}}\left( \sin s\left(x-c_{2}\right) \right) +O\left( se^{t\left( x-a\right)}\right) ,
\hat{x}_{i+1}\coloneqq x_{i-1}+ \frac{y_{i+1}-y_{i-1}}{a_{i+1}}= x_{i-1}+ \frac{y_i - y_{i-1}}{a_{i+1}}+ \frac{y_{i+1}-y_i}{a_{i+1}}= x_{i-1}+ \frac{a_i}{a_{i+1}}(x_i-x_{i-1}) + (x_{i+1}-x_i).
\P_{w}\big(Y^{D}_{\sigma_{i, 1}}\in \widetilde{J}_i\big) \le 1-\P_{w}( \sigma_{i, 1}=\zeta ) \le 1-\P_{w}\big(\tau^{Y^D}_{D\cap B(w,\rho_Q(w)/2)}=\zeta \big) <k_1,\ \ \w\in \widetilde{J}_i.
\frac{A_{+}e^{5 i \alpha_{+}}+ A_{-}e^{5 i \alpha_{-}}}{A_{+}e^{i \alpha_{+}}+ A_{-}e^{i \alpha_{-}}}= \frac{e^{6 i \alpha_{+}}- e^{6 i \alpha_{-}}}{e^{2 i \alpha_{+}}- e^{2 i \alpha_{-}}}= - \frac{\sin 3 \pi j / 2}{\sin \pi j / 2}= - ( 1 + 2 \cos \pi j ) .
E_{\xi_{i}}^{*}E_{\xi_{j}}\sim q E_{\xi_{j}}( - E_{\xi_{i + 1}}^{*}E_{i}^{*}+ q^{-1}E_{i}^{*}E_{\xi_{i + 1}}^{*}) = q E_{\xi_{j}}E_{\xi_{i}}^{*}.
d_{c}^{2}(f)(x,y,z) =\theta\left(x,[\alpha(y),z] \right)+\theta\left(y,\delta(z,\alpha(x) \right)+\beta\Big(\theta\left(z,[x,y]\right)\Big) + \rho\left(x \right) \theta(\alpha(y),z)+\rho\left(y \right) \theta(z,\alpha(y))+\beta\Big(\rho\left(z \right) \theta(x,y) \Big).
\{e_{1}, e_{2}, e_{3}, 1 | e_{1}, e_{1}| e_{1}, e_{2}| e_{1}, e_{3}| e_{1}, 1 | e_{2}, e_{1}| e_{2}, e_{2}| e_{2}, e_{3}| e_{2}, 1 | e_{3}, e_{1}| e_{3}, e_{2}| e_{3}, e_{3}| e_{3}\}.
(x-i\omega)\sqrt{d_1}u_1^{\mathcal{R}}(x)+\rho_0(x-\alpha_1\rho_0^{-1}-c_0)u_0^{\mathcal{R}}(x)=0, &(x-i\omega)\sqrt{d_{k+1}}u_{k+1}^{\mathcal{R}}(x)+\rho_{k}(x+f_{k}^{-1}\rho_{k}^{-1}s_{k})u_{k}^{\mathcal{R}}(x)+(x+i\omega)\sqrt{d_{k}}u_{k-1}^{\mathcal{R}}(x)=0,
\mathcal{S}\simeq \sum\limits_{p=0}^{n-1}\gamma_{p}^{(\nu ,n)}\left[p!t^{-1-p}+\sum\limits_{s=0}^{+\infty}\frac{(-1)^{s+1}B_{2(p+s+1)}\left(\nu+\frac{1}{2}\right)}{(p+s+1)s!}t^{s}\right] .
\stackrel{(0)}{\omega}_{1}=-W_{ab}\left( H_{0}^{a}\eta^{(2)b}+C_{i}^{(2)a}A^{bi}\right) -M_{bc}^{a}\left( B_{a}^{0i}\eta^{(2)b}A_{i}^{c}+\frac{1}{2}\eta_{a}^{(2)ij}A_{i}^{b}A_{j}^{c}\right) .
A^{ijkl,\varepsilon}:= \lambda g^{ij,\varepsilon}g^{kl,\varepsilon}+ \mu(g^{ik,\varepsilon}g^{jl,\varepsilon}+ g^{il,\varepsilon}g^{jk,\varepsilon}), B^{ijkl,\varepsilon}:= \theta g^{ij,\varepsilon}g^{kl,\varepsilon}+ \frac{\rho}{2}(g^{ik,\varepsilon}g^{jl,\varepsilon}+ g^{il,\varepsilon}g^{jk,\varepsilon}),
[\begin{matrix}1&3&1\\ 1&0&0\end{matrix}]+[\begin{matrix}0&0&5\\ 7&5&0\end{matrix}]=[\begin{matrix}1+0&3+0&1+5\\ 1+7&0+5&0+0\end{matrix}]=[\begin{matrix}1&3&6\\ 8&5&0\end{matrix}]
\Delta_{s,\tau}=\sum_{j=1}^{d}\left[ \left( s-\frac{t}{2}\right)\frac{\partial^{2}}{\partial x_{j}^{2}}+\frac{t}{2}\frac{\partial^{2}}{\partial y_{j}^{2}}-u\frac{\partial^{2}}{\partial x_{j}\partial y_{j}}\right] .
\stackrel{\wedge}{h}= \frac{1}{2}\{A^{-}, A^{+}\}_{q}= \frac{1}{2}\left( q A^{-}A^{+}+ q^{- 1}A^{+}A^{-}\right) = 2 \left( \stackrel{\wedge}{T}^{2}- \cosh \frac{\omega}{4}\right)
\frac{(2k+1)}{2\mu}\frac{\left( -1\right)^k}{\Gamma \left( \frac 12\right) \Gamma \left( k+1\right)}\left(\frac{t}{2\mu}\right)^{2k}a_{d+2k+1}\left[ 2\ln( \frac{t}{2\mu})-2+\Psi(1) +\sum\limits_{l=0}^{k-1}\frac 1{k-l}\right]
\sum g_i(x,x') W_{\vec{\eta}}(x|x')\frac{1}{m+1}= 0 \hbox{for}i= 1, \ldots, m, \sum g_\sigma(x,x') W_{\vec{\eta}}(x|x')\frac{1}{m+1}= \eta_\sigma \hbox{for}\sigma \in T \cup H,
\begin{array}{rll}{\rm SU}(n+1): x+y - P_{n+1}(v)=0, 2k= 0{\rm SO}(2n): x+y - P_{2n}(v^2)=0, 2k=4{\rm SO}(2n+1): x+y - P_{2n}(v^2) =0, 2k=2{\rm Sp}(2n): (x+y)^2 - P_{2n}(v^2)=0, 2k=-2 \end{array}
\lambda_p^{k,v,*}= \lim_{n\rightarrow \infty}n \cdot Cov \left(\frac{\hat{\gamma}_p(k/n;B^H)}{\gamma_p(k/n;B^H)},\frac{\hat{\gamma}_p(v/n;B^H)}{\gamma_p(v/n;B^H)}\right), k,v \in \mathcal{M},
\vert g(f)(x)\vert\leq\sum\limits_{j=1}^{\infty}\lambda_{j}\vert g(a_{j})(x)\vert+\sum\limits_{j=1}^{\infty}\mu_{j}\vert g(b_{j})(x)\vert=\uppercase\expandafter{\romannumeral1}+\uppercase\expandafter{\romannumeral2}.
E_{2}= \frac{v^{2}}{2}\int \frac{d^{3}q}{( 2 \pi )^{2}}\left\{G_{h}^{- 1}( - \vec{q}^{2}) h ( \vec{q}) h ( - \vec{q}) + G_{p}^{- 1}( - \vec{q}^{2}) \vec{p}( \vec{q}) \cdot \vec{p}( - \vec{q}) \right\}
\theta'(s) =\frac{2n-1}{r^2(s)}\left(r(s)\cos{\left(\theta(s)\right)}-c r^3(s)\right) &=\frac{2n-1}{r^2(s)}\left(-\phi_0 r^{2-2n}(s) -\frac{2c}{2n+1}r^3(s)\right).
U ( L_{1}, L_{2}) = \frac{\pi}{4 8}\biggl ( \frac{1}{L_{1}}+ \frac{1}{L_{2}}\biggr ) - \frac{L_{1}L_{2}}{3 2 \pi}\sum_{p , q = - \infty}^{\infty}\hspace{- . 1 5 i n}~^{\prime}\hspace{. 1 i n}( p^{2}L_{1}^{2}+ q^{2}L_{2}^{2})^{- \frac{3}{2}}.
\frac{\partial L_{0}}{\partial\xi}-\frac{\partial}{\partial t}(\frac{\partial L_{0}}{\partial(\partial\xi/\partial t)})-\frac{\partial}{\partial x}(\frac{\partial L_{0}}{\partial(\partial\xi/\partial x)})-\frac{\partial}{\partial y}(\frac{\partial L_{0}}{\partial(\partial\xi/\partial y)})=0.
\Delta_0^+ := \{\varepsilon_j-\varepsilon_k | 1\leq j < k \leq N+1 \}, \Delta_1^+ := \{\varepsilon_j-\delta | 1\leq j \leq N+1 \}, \Delta^+ := \Delta_0^+ \cup \Delta_1^+, \Delta:= \Delta^+\cup\Delta^-.
2\pi \sum_{\substack{\rho' = \beta' + i\gamma', t_q < \pm \gamma' \leq T}}(\beta' - b') = \int_{t_q}^T \log{| G_1(b' \pm it, \chi) |}dt - \int_{t_q}^T \log{| G_1(a_q \pm it, \chi) |}dt &\quad\mp \int_{b'}^{a_q}\arg{G_1(\sigma \pm it_q, \chi)}d\sigma\pm \int_{b'}^{a_q}\arg{G_1(\sigma \pm iT, \chi)}d\sigma.
\bar{\lambda}(g_{a, \vec{b}}) \doteqdot \bar{\lambda}(Y) \doteqdot\left ( \prod_{i=1}^mY_i^{- \frac{2n_i}{n}}\right )\cdot \sum_{i=1}^m (2n_i p_iY_i -\frac{1}{2}n_i q_i^2 Y_i^2 )
\xi_{2+k+1}'(a) = T_a'(\xi_{2+k}(a)) \xi_{2+k}'(a) \biggl(1 + \frac{\partial_a T_a(\xi_{2+k}(a))}{T_a'(\xi_{2+k}(a)) \xi_{2+k}'(a)}\biggr) = (T_a^{k+1})'(\xi_2(a)) \xi_2'(a) \prod_{j=0}^k \biggl(1+\frac{\partial_a T_a(\xi_{2+j}(a))}{T_a'(\xi_{2+j}(a)) \xi_{2+j}'(a)}\biggr)
g(0) = \lim_{\varepsilon \to 0}\frac{1}{\varepsilon}\int_0^\varepsilon g(y)\d y \geq \lim_{\varepsilon \to 0}\frac{1}{\varepsilon}\frac{g(1)}{\lambda}\left(\frac{1}{2}-\frac{1}{2}\sqrt{1-\frac{4 \varepsilon}{b}}\right) = \frac{g(1)}{b\lambda}>0.
\Pi\left( \frac{A_2+V_2}{G(C)}-\frac{A_1+V_1}{nC^{n-1}}\right) = \Pi\left(\frac{V_2 P'-V_1Q'-h+(A_1+V_1)F'}{nC^{n-1}G(C)C'}\right) &=\Pi\left(\frac{\Pi_-(V_2 P'-V_1 Q')+(A_1+V_1)F'}{nC^{n-1}G(C)C'}\right) &=0,
\int [d \delta{\hat g}] e^{-|\delta{\hat g}|^2/2}\equiv \prod_{\sigma}\int [d \delta{\hat g}]_{\sigma}e^{-|\delta{\hat g}|_{\sigma}^2/2}= 1 = J_M({\cal \phi},{\hat g}) \int [d \delta{\cal \phi}]_{e^\phi{\hat g}}\int [d \delta \eta]'_{\hat g}\int_0^{\infty}d l e^{-|\delta{\hat g}|^2/2}\quad ,
e^z = e^{x+\epsilon y + \epsilon^2 z}= e^x e^{\epsilon y}e^{\epsilon^2 z}= e^x\left(1 + \epsilon y + \frac{\epsilon^2 y^2}{2}\right)\left(1 + \epsilon^2 z \right) = e^x\left(1 + \epsilon y + \epsilon^2\left(z + \frac{y^2}{2}\right)\right)
{1\over x_1}={1\over 1+x_2}+{1\over 1+x_3}\ , \{1\over x_2}={1\over 2+x_1}+{1\over 1+x_3}\ , \{1\over x_3}={1\over 2+x_1}+{1\over 1+x_2}\, .
\mathcal{F}(\hat{H}^\sharp) =-\overbrace{\mathrm{sf}(\hat{H}^\sharp|_{\{k^0_x\}\times S^1})}^{0}[\overline{\alpha}_x]+ \overbrace{\mathrm{sf}(\hat{H}^\sharp|_{S^1 \times \{k^0_y\}})}^{0}[\overline{\alpha}_y]+ \overbrace{\mathrm{sf}(\hat{H}^\sharp|_{\partial \overline{D}_1})}^{1}[\overline{\alpha}_1]=[\overline{\alpha}_1].
\left(\begin{matrix}\mathfrak{a}\mathfrak{b}\end{matrix}\right) =S^{-1}A(\varepsilon_1)\cdots A(\varepsilon_{\nu-1})\left(\begin{matrix}\alpha \beta\end{matrix}\right)\ \ (\mathfrak a'\,,\mathfrak b') = (1\,,0) A(\varepsilon_{\nu-1}) \cdots A(\varepsilon_1) S.
\tilde{\cal L}^{( 0 )}= \pi_{a}\dot{\sigma}^{a}+ \Psi \dot{\theta}- \frac{1}{2}\,{\sigma^{\prime}}_{a}^{2}- \frac{1}{2}\, \lambda \, \bigl ( \sigma_{a}^{2}- 1 \bigr ) - \frac{1}{2}\, \pi_{a}^{2}+ \sigma_{a}\pi^{a}\theta - \frac{1}{2}\sigma_{a}^{2}\theta^{2}.
\int_{0}^{\infty}dx\, x^{2}\log \left[1-e^{-\left(x^{2}+m_{eff}^{2}\beta^{2}\right)^{1/2}}\right]=-\left(m_{eff}\beta \right)^{2}\, \sum_{n=1}^{\infty}\frac{1}{n^{2}}K_{2}\left(m_{eff}\beta n\right).
P^{(\mathcal{B}_n)}(\mathbf{w})=2^{-n(R_1+R_2)}\sum_{(i,j)\in\mathcal{I}_n\times\mathcal{J}_n}p^n_{W|U,V}\big(\mathbf{w}\big|\mathbf{u}(i),\mathbf{v}\big(i,j)\big).
\sum_{i\in[n]}c(v_i) \ge c^*\big(\tfrac{1}{k}-(k-1)\gamma\big)n+\tfrac{1}{k-1}\cdot k(k-1)\gamma n+(1-c^*) \cdot \big(\tfrac{1}{k}- (k-1)\gamma \big)n &=\big(\tfrac{1}{k}-(k-1)\gamma\big)n + k\gamma n >\tfrac{n}{k},
\begin{array}{r c l}{{Y_{2}A_{1}}}&{{=}}&{{L_{2}^{+}S L_{2}^{-}A_{1}}}\\{{}}&{{=}}&{{L_{2}^{+}A_{1}S L_{2}^{-}R_{2 1}}}\\{{}}&{{=}}&{{A_{1}R_{1 2}L_{2}^{+}S L_{2}^{-}R_{2 1}}}\\{{}}&{{=}}&{{A_{1}R_{1 2}Y_{2}R_{2 1},}}\\ \end{array}
t=\sum_i\frac{h_i(\ell_{i(n_i+1)},\dots,\ell_{ip})}{L_{i1}^{s_{i1}}\dots L_{in_i}^{s_{in_i}}}+\phi_i(L_{i1},\dots,L_{in_i},\ell_{i(n_i+1)},\dots,\ell_{ip})
\omega_A^{2}(T) = \omega_A^{2}(T^{\sharp_A}) =\sup_{\alpha^{2}+\beta^{2}=1}\left\Vert\alpha [\Re_A(T)]^{\sharp_A}+\beta [\Im_A(T)]^{\sharp_A}\right\Vert_A^2\quad( \ref{lm2}) \leq \frac{1}{2}\left\Vert TT^{\sharp_A}+ T^{\sharp_A}T\right\Vert_A.
L\left( X_\kappa ,\Theta_\kappa ,A_\kappa \right) =L\left( X,\Theta,A\right) -\frac 12L_{MN}\left[ \Omega_g^{MN}-\Omega^{MN}\left( \Theta\right) \right] -\frac 12\left( A_\kappa^{ij}-A^{ij}\right) X_i\cdot X_j.
\Lambda_e =\Big\{{\sum}_{i=0}^{e-1}a_i(p^e-p^i)\colon 0\le a_i<p\Big\}&\ \ \ \ -\Big\{1+p^e\delta_{n-r}+{\sum}_{i=1}^rp^{d'_i}\colon (d'_1,\ldots,d'_r)\le\theta_e,\d'_r=0\Big\}.
2\ell\left( -s,a,\chi\right) =-\chi\left( -1\right) \sum\limits_{j=0}^{l}\left( -1\right)^{j}\frac{(s)_{j}}{j!}E_{j,\overline{\chi}}(0)a^{s-j}& \quad+\frac{(s)_{l+1}}{l!}\int\limits_{0}^{\infty}\overline{E}_{l,\overline{\chi}}\left( -t\right) \left( t+a\right)^{s-l-1}dt,
\phantom{=}\int_{\mathbb{R}^N}(K\ast |u_\varepsilon^*|^{p^\sharp}) |u_\varepsilon^*|^{p^\sharp}dx &\geq C_U(S^*)^{\frac{N+\alpha}{ps}}-|O(\varepsilon^{\frac{N+\alpha}{2(p-1)}}) |-|O(\varepsilon^{\frac{N+\alpha}{p-1}})| &= C_U(S^*)^{\frac{N+\alpha}{ps}}-|O(\varepsilon^{\frac{N+\alpha}{2(p-1)}}) |,
v=\dot{z}=\frac{d}{dt}(Re^{i\theta[t]})=R\frac{d}{dt}(e^{i\theta[t]})=Re^{i\theta(t)}\frac{d}{dt}(i\theta[t])=iR\dot{\theta}(t)e^{i\theta(t)}=i\omega Re^{i\theta(t)}=i\omega z
\frac{1-\left|\eta_{\mu_{1}}(z)\right|}{1-\left|z\right|}=\frac{1+2\Re\psi_{\mu_{1}}(z)}{\left(1+\left|\eta_{\mu_{1}}(z)\right|\right)\left|1+\psi_{\mu_{1}}(z)\right|^{2}\left(1-\left|z\right|\right)},z\in\mathbb{D},
V(\varphi_{0},\beta)=\frac{2}{\beta}\sum^{\infty}_{s=1}\sum^{+\infty}_{n=-\infty}\int\frac{d^{d}p}{(2\pi)^{d}}\frac{(-1)^{s}}{s}\frac{(M+g\varphi_{0})^{2s}}{(\omega^{2}_{n}+q^{2})^{s}}.
\beta_{j_0}= \frac{2}{a}\sqrt{\left( 1 - \frac{2(m-1)(1-a)}{N}\right)^2 + \frac{4(1-a)a(m-1)}{N}(N+2-m)}&+ \frac{2}{a}\left( 1 - \frac{2(m-1)(1-a)}{N}\right).
\| N(Z,W)X \|^2 =\big{\langle}(M(Z,W)+ \langle Z,W \rangle (\nabla_ZJ)_W)^2X, X \big{\rangle}&=\big{\langle}M(Z,W)^2X,X \big{\rangle}+ \langle Z,W \rangle^2 \| (\nabla_ZJ)_W X \|^2.
\|w\|_{\infty,\mathcal{T}_h}= \sup_{R \in \mathcal{T}_h}\|w\|_{L^{\infty}(R)},\|w\|_{L^p(\mathcal{T}_h)}= \left(\sum_{R \in \mathcal{T}_h}\|w\|^p_{L^p(R)}\right)^\frac{1}{p},\,\,\forall\,w \in L^p(\mathcal{T}_h),
0 \geq \widetilde{P}_\nu (\widetilde{x_1}, \widetilde{t_1}) = [u_{\nu \nu}(\widetilde{x_1}, \widetilde{t_1}) - \sum\limits_j{u_j d_{j\nu}}- D_{\nu}\varphi] - (A + \frac{1}{2}M) \geq \max_{\partial \Omega}u_{\nu \nu}- C - A - \frac{1}{2}M,
w_F = \frac{1}{2}\Big{(}\nabla^F(\vec H_F) + \,\langle (A_F), \vec H_{F}\rangle_{g_{\textnormal{euc}}}^{\sharp_{g_F}}+ \,\langle (A_F^0), \vec H_{F}\rangle_{g_{\textnormal{euc}}}^{\sharp_{g_F}}\Big{)},
\lim_{n \to \infty}P_{0,\pi}^V(n - \sigma_{i_n}> k) = \lim_{n \to \infty}P_{0,\pi}^V(\mbox{no renewals times in}\{n-k, n-k+1,\ldots, n\}) & = \lim_{n \to \infty}P_{0,\pi}^V(A_{n-k, k+1}) \leq \lim_{n \to \infty}P_{0,\pi}^V(A_{n-k, k}) \leq c_{14}\cdot k^{1-\delta}.
|a_{u}(n)| \ll n^{1/2}\sum_{lm=n}l^{-1}m^{1/2}|\rho(m)| \ll n^{1+\epsilon}\max_{1\leq m\leq n}|\rho(m)| \ll n^{1+\epsilon}k^{\epsilon}r^{5/8}e^{\pi r/2}.
h(p) = (e^{K\varphi(p)}-1) W(p) = (e^{\frac{3}{4}K}-1)W(p) \le h(q) &\le (e^K - 1) \max\left\{\frac{2}{\sqrt{3}}, \frac{32 u(p)}{R}\right\}&\le (e^K-1)\left(\frac{2}{\sqrt{3}}+\frac{32 u(p)}{R}\right)
C'_1 = \frac{1}{2}\log\Bigl(1+\frac{aP_1+P_2}{N}\Bigr) - C_2 &= \frac{1}{2}\log\Bigl(1+\frac{a P_1}{P_2+N}\Bigr) C'_2 = \frac{1}{2}\log\Bigl(1+\frac{aP_1+P_2}{N}\Bigr) - C_1 &= \frac{1}{2}\log\Bigl(1+\frac{(a-1)P_1+P_2}{P_1+N}\Bigr).
A_k'(n) = \{(\lambda = (\lambda_1^{m_1}, \lambda_2^{m_2}, \ldots, \lambda_\ell^{m_\ell}), \lambda_{i_0}, m)\mid & \lambda \in \mathcal{P}(n), \,1 \leq i_0 \leq \ell, \,m_i < k \, \forall i \not = i_0, \,m_{i_0}\ge k, \,k \mid \lambda_{i_0}, \, 0\le m < k \}.
\Phi(\rho_k(x)) =\Phi([(\rho_{m,k}(x))_{m>k}]) &=[(\varphi_m(\rho_{m,k}(x)))_{m>k}] &=[((\Psi^{-1}\circ\rho_m)(\rho_{m,k}(x))_{m>k}] &=[(\Psi^{-1}\circ\rho_k(x))_{m>k}] &=\iota\circ\Psi^{-1}\circ\rho_k(x)
\begin{array}{l l l}{{L_{I J}^{( - )}L_{J K}^{( - )}}}&{{-}}&{{( - 1 )^{P_{J}( P_{I}+ P_{K})}L_{J K}^{( - )}L_{I J}^{( - )}=}}\\{{}}&{{}}&{{( q - q^{- 1}) ( - 1 )^{P_{K}P_{J}}L_{J J}^{( - )}L_{I K}^{( - )}, ( I > J > K )}}\\ \end{array}
\underset{k\rightarrow \infty}{\lim}\min \left( \frac{\mathbf{x}_{k}}{\mathbf{y}_{k+1}}\right) =\underset{k\rightarrow \infty}{\lim}\left( \frac{\overline{\lambda}_{k}-\overline{\lambda}_{k+1}}{1-\gamma_{k}}\right) =0.
I_1=\int_{B}W_{\delta}(x)(U_{\delta,e_n}^p-U_{\delta,-e_n}^p) = \int_{B}U_{\delta,e_n}^{p+1}-U_{\delta,e_n}U_{\delta,-e_n}^p -U_{\delta,-e_n}U_{\delta,e_n}^p +U_{\delta,-e_n}^{p+1}.
\mathcal{N}(\mathcal{F})=\sum_{\substack{\mathbf{x}\in\,\mathcal{X},\,\mathbf{y}\in\, \mathcal{X}\mathbf{x}^{T}C\mathbf{x}=\mathbf{y}^{T}C\mathbf{y}\mathbf{x}^{T}H=\mathbf{y}^{T}H}}w(\mathbf{x})w(\mathbf{y}).
\Big|\frac{1}{N}\sum_{i=1}^N d_i Q_i \Big|\leq N^{\frac{\varepsilon}{3}}\hat{\Pi}\,, \Theta\Big(z,{\frac{N^{3\varepsilon}}{(N\eta)^{\frac13}}}\,,{\frac{N^{3\varepsilon}}{\sqrt{N\eta}}}\Big)\cap \Omega_2(z)\,.
\partial_t c_{\min \, |t}= \partial^\square c_{|(t,p)}= \Delta_\Gamma G'(c) + cVH_{\, |(t,p)}&= G''(c) \Delta_\Gamma c + G'''(c) \big| \nabla_\Gamma c \big|^2 + g(c)H^2c_{\, |(t,p)}\geq g(c)H^2c_{\, |(t,p)}.
R(\Gamma,q,T)-R(\Gamma\backslash e,q,T)-R(\Gamma/e,q,T) =\frac{((q + 2)T^2 - 1)T}{(1-T)^3}&= -T + (2q + 1)T^3 + (2q^2 + 4q + 2)T^4 + O(T^5).
\lim_{q\to 1}<(a^{*})^{k_{1}}(a)^{k_{2}}(c^{*})^{k_{3}}(c)^{k_{4}}>_{q}=\int (\alpha^{*})^{k_{1}}(\alpha )^{k_{2}}(\gamma^{*})^{k_{3}}(\gamma)^{k_{4}}du ,
[ \alpha^{I}\, \underline{{{\alpha}}}_{I}+ \beta_{I}\, \underline{{{\beta}}}^{I}, \hat{\alpha}^{I}\, \underline{{{\hat{\alpha}}}}_{I}+ \hat{\beta}_{I}\, \underline{{{\hat{\beta}}}}{}^{I}] = ( \alpha^{I}\hat{\beta}_{I}- \hat{\alpha}^{I}\beta_{I}) \, \underline{{{\epsilon}}}_{0}+ \omega^{i}( \alpha , \beta , \hat{\alpha}, \hat{\beta}) \, \underline{{{\omega}}}_{i}\ .
\overline\zeta_{3}:=\frac{(d-3)k^{4}}{6(g(k^2)\pi^{d/2})^2}\int\frac{dP_1dP_2dP_3~\delta(k-P_1)}{P_1^2P_2^2P_3^2(P_1-P_2)^2(P_2-P_3)^2(P_3-P_1)^2}
\Pr(Y \geq X) = \Pr(Y = X = 0) + \sum_{k=1}^{10 c \log n}\Pr(Z = k) \Pr(Y \geq X \mid Z = k) +O(n^{-2}) & \asymp n^{-c}\left( 1+ \sum_{k=1}^{10 c \log n}\frac{(2 c e \theta \log n)^k}{k^{k+1}}\right),
s' : =s-\big(b_kp^k+(b_{k-1}-b_k)p^{k-1}+\cdots+(p-1-b_{k-m})p^{k-m-1}-(p-1)p^{k-m-2}\big) &= \lfloor b_k \cdots b_{k-m}\underbracket[0.1ex]{(p-1)\; 0 \cdots 0}_{\ell-1}\, b_{k-m-\ell-1}\cdots b_0\rfloor_p.
Ht_-(\tau)=\frac{1}{2}\log\mid\frac{\sin(\frac{\pi(\mu\tau-x)}{2K})}{\sin(\frac{\pi(\mu\tau+x)}{2K})}\mid+\frac{\mu\tau\pi}{2K}\frac{\vartheta'_1}{\vartheta_1}(\frac{\pi x}{2K})-2\sum_{n=1}^\infty\frac{q^{2n}}{n(1-q^{2n})}\sin(\frac{n\pi\mu\tau}{K})\sin(\frac{n\pi x}{K}),
\mathbf{\mathcal{W}}(u) = (\pm 2^{\frac{n+1}{2}}H^{(r)}_{2^{p-2}},\textbf{0}_{2^{p-2}})\quad\mbox{or}\mathbf{\mathcal{W}}(u) = (\textbf{0}_{2^{p-2}},\pm 2^{\frac{n+1}{2}}H^{(r)}_{2^{p-2}})
\langle f(t,\cdot), \varphi(t,v) \rangle - \langle f_0 , \varphi(0,\cdot)\rangle - \int_0^t \langle f(\tau, \cdot) \partial_{\tau}\varphi(\tau,\cdot)\rangle \, \mathrm{d}\tau &= \int_0^t \langle Q(f,f)(\tau, \cdot), \varphi(\tau,\cdot) \rangle \, \mathrm{d}\tau, t\geq 0,
\frac{\gamma\left( a,a\right)}{\Gamma\left( a\right)}=\frac{1}{2}+\frac{1}{\sqrt{2\pi}}\left\{\frac{1}{3a^{1/2}}+\frac{1}{540a^{3/2}}+O\left( a^{-5/2}\right) \right\}a\rightarrow\infty.
\left\{\sup_{0\le t\le\eta}\left|\frac{1}{\sqrt{N}}\sum_{j=1}^{N}\int_{0}^{t}\phi_{\epsilon}^{'}\left(x_{j}^{N}(s)-x_{i}^{N}(0)\right)d\beta_{j}(s)\right|>\frac{\gamma\epsilon}{2}\sqrt{N}\right\}
1- \overline{\Psi(w,x_{1i},x_{2i},x_{3i})}\Psi(z,x_{1j},x_{2j},x_{3j}) &=1-\overline{\Psi(w,x(\lambda_i))}\Psi(z,x(\lambda_j)) &=(1-\overline{w}z)\overline{\gamma(\lambda_i,w)}\gamma(\lambda_j,z)+(1-\overline{\lambda_i}\lambda_j)\eta(\lambda_i,w)^*\frac{I-F(\lambda_i)^*F(\lambda_j)}{1-\overline{\lambda_i}\lambda_j}\eta(\lambda_j,z).