harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\mathcal{D}(x,b) = \left\{F \in \mathcal{F}:~ \frac{-\log \overline F(x)}{\log x}\geq b\right\}= \left\{F\in \mathcal{F}:~ \overline F(x)\leq x^{-b}\right\}&= \left\{F\in \mathcal{F}:~ F(x) \geq 1- x^{-b}\right\}= \left\{F\in \mathcal{F}:~ F(x) < 1- x^{-b}\right\}^c.
\begin{cases}\eta(0,t)=0,\,\,\eta(L,t)=0,\,\,\eta_{x}(0,t)=f(t), (0,T), w(0,t)=0,\,\,w(L,t)=0,\,\,w_{x}(L,t)=g(t). (0,T),\end{cases}
S_{\mathrm{e x t}}( \Phi , V ) = S_{\mathrm{F P}}( \Phi ) - \int d^{4}x ( \alpha \partial )^{\mu}\bar{c}({V_{\kappa}}^{\lambda}x^{\kappa}\partial_{\lambda}A_{\mu}+{V_{\mu}}^{\nu}A_{\nu}) .
1 ={\frac{\int d A \exp \left\{- N \beta \mathrm{T r}\left[ \left( A - \sum_{i = 1}^{d}U_{i}\right) \left( A^{\dagger}- \sum_{i = 1}^{d}U_{i}^{\dagger}\right) \right] \right\}}{\int d A \exp \left\{- N \beta \mathrm{T r}\left[ A A^{\dagger}\right] \right\}}}.
b_{n,(j+1)}= \min \Big(c_2 4^{-2(n+j)}b_{n,j}\cdot(b_{n,j}+e^{-{\frac{1}{2}}4^{(n+j)}})\cdot c_6 4^{3n},c_2 4^{-(n+j+1)}\Big)\mbox{for}j\ge 0.
u \ge\frac{M_j+m_j}{2}+\frac{L}{2\cdot 4^{\alpha j}}\delta-\frac{L}{2\cdot 4^{\alpha j}}&=M_j-\frac{M_j-m_j}{2}-\frac{L}{2\cdot 4^{\alpha j}}(1-\delta) &=M_j-\frac{L}{2\cdot 4^{\alpha j}}(2-\delta) B_{4^{-j}R}.
\beta_i'w_j \,\,=\,\,\frac{w_i'(I_d-P_{B,\beta_1,\ldots,\beta_{i-1}})w_j}{\|(I_d-P_{B,\beta_1,\ldots,\beta_{i-1}})w_i\|}\,\,=\,\, \frac{w_i'(I_d-P_{B,w_1,\ldots,w_{i-1}})w_j}{\|(I_d-P_{B,\beta_1,\ldots,\beta_{i-1}})w_i\|}\,\,= \,\,0,
T(x_1,\dots,x_{2n})=\frac{\prod_{i,j=1}^nG(x_i,x_{n+j})}{\Delta(x_1,\dots,x_n)\Delta(x_{n+1},\dots,x_{2n})}\,\det_{1\leq i,j\leq n}\left(\frac{1}{G(x_i,x_{n+j})}\right),
R_{\Lambda^{(n)}\times\Delta^{(n)}}^{(k)}\big(M(\Lambda^{(n)}\times \Delta^{(n)};f_1),\dots,M(\Lambda^{(n)}\times\Delta^{(n)};f_k)\big) &\quad=\frac1{V^{(n)}}\int_{\Lambda^{(n)}}f_1(x)\dotsm f_k(x)\,d\sigma(x)\int_{\Delta^{(n)}}s^k\,d\nu(s)+\sum_{j=2}^k\frac{c_j^{(n)}}{(V^{(n)})^j},
d{{s}^{2}}\,=\,{{r}^{2({{\alpha}_{1}}+1)}}d{{t}^{2}}-{{r}^{m}}{{({{\alpha}_{1}}t+{{\alpha}_{2}})}^{{m}/{{{\alpha}_{1}}}\;}}d{{r}^{2}}-{{r}^{2}}d{{\Omega}^{2}}.
N_2 =\#\Bigl\{(a_2,a_2',a_1)\in\mathcal{C}(A,M)^2\times\mathcal{C}:\,a_1=\frac{v_2a_2}{v_1X}+\frac{v_3}{v_1}+\frac{v_4}{v_1X}=\frac{v_2'a_2'}{v_1'X}+\frac{v_3'}{v_1'}+\frac{v_4'}{v_1'X}, & 0<|v_1|,|v_1'|,|v_2|,|v_2'|,|v_3|,|v_3'|,|v_4|,|v_4'|\le V,\,a_1a_2a_2'\ne 0\Bigr\}.
\sharp F(a,b,\varepsilon,B,r) =\sum_{e_1f_1|b,e_2f_2|a,e_3f_3|b-a}\left(\prod_{i=1}^3\mu(e_i)\right)\sharp F(e_1,e_2,e_3,f_1,f_2,f_3,a,b,\varepsilon,B,r),
\int_{\Omega}[\rho_h^n(\psi_h^{n+1}-\psi_h^n) - ( \rho_{h}^{n+1}-\rho^n) \psi_h^n ]dx + \int_{\partial \Omega}\sigma ( \psi_h^{n+1}-\psi_h^n) ds =A_1(\psi_h^n, \psi_h^{n+1}-\psi_h^n )- A_1(\psi_h^{n+1}-\psi_h^n, \psi_h^n)=0.
\sigma_{i}\cdot\mathbf{x}= t_{1}t_{2}\cdots t_{i-1}t_{i+1}(t_{i+1}^{-1}t_{i}t_{i+1})t_{i+2}\cdots t_{k}, \sigma_{i}^{-1}\cdot\mathbf{x}= t_{1}t_{2}\cdots t_{i-1}(t_{i}t_{i+1}t_{i}^{-1})t_{i}t_{i+2}\cdots t_{k}.
A_1=\frac{1}{4}\left(\alpha +\frac{|\mathbf{c}|^2}{8\omega}\right),\ \A_2=\frac{1}{4}\left(\beta +\frac{|\mathbf{c}|^2}{4\omega}\right),\ \A_3=\frac{1}{4}\left(\gamma +\frac{|\mathbf{c}|^2}{4\omega}\right),
\sup_{n}\mathbb{E}_{\mathbb{P}}\left[\exp \left\{\kappa \sum_{i=1}^{n}\int_{T_{0}}^{T_{0}+\delta}\triangle b_{t}^{i, n,\infty}\cdot d W_{t}^{i}-\frac{\kappa}{2}\sum_{i=1}^n \int_{T_{0}}^{T_{0}+\delta}\left|\triangle b_{t}^{i, n,\infty}\right|^{2}d t\right\}\right]
F_{n}\left({\vec{z}}\right) = \sum_{{\vec{c}}}D_{{\vec{c}}, n}\theta \left[ \begin{array}{c}{{\vec{0}}}\\{\frac{{\vec{c}}}{\ell}}\\ \end{array}\right] \left({\vec{z}}- \frac{\tau}{\omega \kappa}\left( n + \frac{\beta}{2 \pi}\right){\hat{e}}+ \frac{\alpha}{2 \pi \omega \kappa}{\hat{e}}, \Omega \right)
N_1^{1/3}a_{0,1}(\ell,k)=\frac{\alpha}{(2\pi)^4}\int_{\mathbb{R}}dt_1\int_{I_1}ds_1\int_{\mathbb{R}}dt_2\int_{I_2}ds_2J(t_1,s_1,t_2,s_2)\frac{e^{f_1(z(t_1);0)+f_2(w(t_2);0)}}{e^{f_1(\zeta(s_1);y)+f_2(\omega(s_2);x)}}.
\mathcal{B}_1(s_1,s_2) = \prod_{p}\Big( 1-\frac{k \ell}{p^{s_1+s_2+2}}+ O_{k,\ell}(p^{-2.7}) \Big ) \ll_{k,\ell}1 \sigma_1, \sigma_2 \ge -10^{-1}.
\delta_Kj_T^{0M_1\cdots M_p}=\sum_{k=1}^p\partial_{\mu_k}\left[ \frac 1{p!}\epsilon^{0\mu_1\cdots \mu_k\cdots \mu_p}\cdot \phi_{,\mu_1}^{M_1}\cdots \delta_K\phi^{M_k}\cdots \phi_{,\mu_p}^{M_p}\right] \quad .
x^\omega(x^{\omega+p}y)^{\omega+q}=(x^{\omega+p}y)^{\omega+q},\quad\quad (x^{\omega+p}y)^{\omega+q}x^\omega=(x^{\omega+p}yx^\omega)^{\omega+q}, begin{align*}1mm](yx^{\omega+p})^{\omega+q}x^\omega=(yx^{\omega+p})^{\omega+q},\quad x^\omega(yx^{\omega+p})^{\omega+q}=(x^\omega yx^{\omega+p})^{\omega+q}.
\log_t (x, w) \geq - (s^\lambda, y^\lambda) + \biggl(\lambda - \log_t \Bigl(\frac{N}{1-\theta}\Bigr) - \delta(t)\biggr)e = (x^\lambda, w^\lambda) - \biggl(\log_t \Bigl(\frac{N}{1-\theta}\Bigr) + \delta(t)\biggr)e \, ,
{_aI_x^{1-\alpha}}u'(x_{n+1}) =\sum_{j=1}^{n+1}\int_{I_j}\omega_{1-\alpha}(x_{n+1}-s)u'(s)\,ds &=\int_{I_1}\omega_{1-\alpha}(x_{n+1}-s)u'(s)\,ds+\sum_{j=1}^{n}\int_{I_j}\omega_{1-\alpha}(x_{n}-q)u'(q+h)\,dq.
((M^\dagger)_{\subset t-1})_p \cong((M^\dagger)_p)_{\subset t-1}&\simeq(\textstyle\sum^{\dim R\slash p}M_p^{\dagger p})_{\subset t-1}&=\textstyle\sum^{\dim R\slash p}((M_p^{\dagger p})_{\subset t-\dim R\slash p-1}) &=\textstyle\sum^{\dim R\slash p}((M_p^{\dagger p})_{\subset \dim_{R_p}M_p-1}).
v_i = \frac{u_i^0}{\nu_{m}}, \dot{v}_i = \frac{\dot{u}_i^0}{\nu_{m}}, \omega = \frac{\tau^0}{\nu_{m}}, \dot{\omega}= \frac{\dot{\tau}^0}{\nu_{m}}, r_i = \frac{q^0_i}{\nu_{m}}\Omega \times \{0\}.
{\rho}_{1}({x_1}) = \int\limits_{\mathbb{R}}{\rho}(x_1, x_2)~dF({x_{2}}), \: \: h^{(1)}({x_1}) ={\rho}_1({x_1}) - \theta, & \: h^{(2)}({x_1},{x_2}) ={\rho}({x_1},{x_2}) -{\rho}_1({x_1}) -{\rho}_1({x_2}) + \theta.
\tilde{D}_{da, bc}^{li,jk}\equiv{1\over N}\sum_{\tilde{m}=0}^{P-1}\tilde{B}(l-j-Q\tilde{m}){\rm e}^{i{2\pi\over P}(d-a)\tilde{m}}\delta_{ab}\delta_{cd}\delta_{li}\delta_{jk},
\mathrm{Hom}(\mathrm{Coker}\,\phi,\mathbb F_p) \simeq \mathrm{Ker}\,\phi^{\vee}\simeq \mathrm{Ker}\,\xi' \simeq \mathrm{Ker}\,\xi &\simeq (\widetilde{\mathcal{U}}_S \cap \overline{k^{\times}}\mathcal{J}_k^p)\mathcal{J}_k^p/\mathcal{J}_k^p=\widetilde{V}_S\mathcal{J}_k^p/\mathcal{J}_k^p\simeq \widetilde{V}_S/(\overline{k^{\times}})^p .
\biggl\{-4z^2(1-z) \frac{d^2}{dz^2}+ 2[(4a+3)z^2 - (4a-4b_1-2l-d+3)z - 4b_1] \frac{d}{dz}+ [2a(2a+1) - A]z - 4a^2 + 8ab_1 + 2a(2l+d-1) - 2b_1 - \epsilon + \frac{-8(a-1)b_1 + 4b_1^2 + 2b_1(2l+d-3) + B_1}{z}+ \frac{-4b_1^2 + B_2}{z^2}\biggr\}\phi(z) = 0.
\begin{cases}\|\sum_{l\le j\epsilon}T_a^{j, l}\|_{L^{1}\to L^{1}}\lesssim_{\epsilon}2^{j(m+n(1-\rho)/2 + \varepsilon)}, \|T_a^{j, l}\|_{L^{1}\to L^{1}}\lesssim_{\epsilon}2^{10n(m-n)(j+l)}, l>j\epsilon.\end{cases}
({\cal B}_V)_{cl}= \sum_{{\vec n},{\vec m}}\sum_{s,s'}\left[(a^{s\dagger}_{\vec n}\bar{u}^s_{\vec n}+b^{s}_{-{\vec n}}\bar{v}^s_{-{\vec n}})(a^{s'}_{\vec m}\gamma^0 u^{s'}_{\vec m}+b^{s'\dagger}_{-{\vec m}}\gamma^0 v^{s'}_{-{\vec m}})\right]_{cl}F({\vec n}-{\vec m},X).
\int_{\Delta_{\theta,t}^{n+1}}f_1(s_1) \int_{\Delta_{\theta,s_1}^p}g_1(r_1) \dots g_p(r_p) dr_p \dots dr_1 f_2(s_2) \dots f_{n+1}(s_{n+1}) ds_{n+1}\dots ds_1 &= \int_{\theta}^t f_1(s_1) \left( \int_{\Delta_{\theta,s_1}^n}\int_{\Delta_{\theta,s_1}^p}g_1(r_1) \dots g_p(r_p) dr_p \dots dr_1 f_2(s_2) \dots f_{n+1}(s_{n+1}) ds_{n+1}\dots ds_2 \right) ds_1 .
\underline\dim_{{\rm loc}}(\mu,x)=\liminf_{r\rightarrow 0^{+}}\frac{\log(\mu(B(x,r)))}{\log(r)}\ \mbox{and}\ \overline\dim_{{\rm loc}}(\mu,x)=\limsup_{r\rightarrow 0^{+}}\frac{\log (\mu(B(x,r)))}{\log(r)}.
S_k^{m,1}(t)=\frac{\displaystyle\sum_{l=0}^{\infty}\frac{S_{k+(r+1)l}^{m,1}(0)t^l}{l!}}{\displaystyle\sum_{l=0}^{\infty}\frac{S_{(r+1)l}^{1,1}(0)t^l}{l!}};S_k^{1,n}(t)=\frac{\displaystyle\sum_{l=0}^{\infty}\frac{S_{k+(q+1)l}^{1,n}(0)t^l}{l!}}{\displaystyle\sum_{l=0}^{\infty}\frac{S_{n-1+(q+1)l}^{1,n}(0)t^l}{l!}};k\in \mathbb{Z}_+.
\begin{pmatrix}\alpha' \beta'\end{pmatrix}=\begin{pmatrix}1 1 1 . -1 -1 . . 1 -1 . .\end{pmatrix}\begin{pmatrix}\alpha' \beta'\end{pmatrix}=\begin{pmatrix}1 1 -1 . 1 -1 . . -1 1 . .\end{pmatrix}.
1 0 3 ( \overrightarrow{a}\cdot \overrightarrow{b}) = \frac{\overrightarrow{a}\cdot \overrightarrow{b}}{\vert \overrightarrow{a}/ / \overrightarrow{b}\vert}= \frac{- 1}{1 \cdot 2}= - \frac{1}{2}.
\omega(t)=-\frac{1}{4}(-t)^2 -\frac{1}{8}(-t)^{-1}-\frac{9}{64}(-t)^{-4}-\frac{189}{128}(-t)^{-7}-\frac{21663}{512}(-t)^{-10}\hphantom{\omega(t)=}{}-\frac{4825971}{2048}(-t)^{-13}-\frac{3540311739}{16384}(-t)^{-16}-\frac{241980297111}{8192}(-t)^{-19}+\cdots.
\left(z;q\right)_{\infty}\sum_{n=0}^{\infty}\frac{q^{\binom{n}{2}}z^{n}}{(q,z;q)_{n}}=\sum_{n=0}^{\infty}\frac{q^{2n^{2}-n}z^{2n}}{(q^{2};q^{2})_{n}}.
\alpha_{\pm}^1:=\frac{1}{\sqrt 2}(\theta^1\wedge\theta^2\pm\theta^3\wedge\theta^4) \alpha_{\pm}^2:=\frac{1}{\sqrt 2}(\theta^1\wedge\theta^3\pm\theta^4\wedge\theta^2) \alpha_{\pm}^3:=\frac{1}{\sqrt 2}(\theta^1\wedge\theta^4\pm\theta^2\wedge\theta^3)
\lambda_{1}=\inf_{v \in S_{0}^{1,p}(\Omega)\setminus\{0\}}\left\{\frac{\int_{\Omega}|X v|^{p}d x}{\int_{\Omega}|v|^{p}d x}\right\}=\frac{\int_{\Omega}|X u_{1}|^{p}d x}{\int_{\Omega}|u_{1}|^{p}d x},
\frac{1}{1-\frac{\alpha_{1}t}{1-\frac{\alpha_{2}t}{1-\ddots}}}=\frac{1}{1-\alpha_{1}t-\frac{\alpha_{1}\alpha_{2}t^{2}}{1-(\alpha_{2}+\alpha_{3})t-\frac{\alpha_{3}\alpha_{4}t^{2}}{1-(\alpha_{4}+\alpha_{5})t-\frac{\alpha_{5}\alpha_{6}t^{2}}{\ddots}}}}
T_{p,s}:=\sup_{u\in W^{s,p}_0(\mathbb{R}^N)}\left\{\left(\int_{\mathbb{R}^N}|u|^\frac{N\,p}{N-s\,p}\,dx\right)^\frac{N-s\,p}{N}\, :\, [u]_{W^{s,p}(\mathbb{R}^N)}=1\right\}.
\mathsf{d}_{\mathrm{com}}(\{X_{ni}\})= 2\sum_{n=1}^{d_1^2}\|X_{n1}\|_1\leq 2d_1^{\frac{3}{2}}\sqrt[4]{\sum_{n=1}^{d_1^2}\|X_{n1}\|_1^4}=2d_1^{\frac{3}{2}}\sqrt{\nu}.
\bar{\gamma}_{ki}=\frac{\rho_{kii}^{2}\bar{\upsilon}_{ki}}{\rho_{kii}{\displaystyle \sum_{j=1}^{L}\alpha_{j}\mathbb{E}\left[\mathsf{\Gamma}_{ji}\right]+\bar{\upsilon}_{ki}\sum_{j\neq i}^{L}\kappa_{j}\mathbb{E}\left[\mathsf{\Gamma}_{ji}^{2}\right]}}
\int\limits_{\gamma(\pi,h)}a_{k}(t)\Psi_{k,t}+a_{-(k+1)}(t)\Psi_{-(k+1),t}dt=\int\limits_{[\pi-h,\pi+h]}a_{k}(t)\Psi_{k,t}+a_{-(k+1)}(t)\Psi_{-(k+1),t}dt.
\int_t^{t+1}|A^{1/2}u(\tau)|_{\alpha,\sigma}^2d\tau \le | u(t)|_{\alpha,\sigma}^2+3c_1^2\int_t^{t+1}(1+\tau)^{-2\lambda}d\tau &\le 2c_*^2 (1+t)^{-2\lambda}+3c_1^2(1+t)^{-2\lambda}&= \Big(2 c_*^2 +\frac{c_*^2}{M_2}\Big)(1+t)^{-2\lambda}.
\left. \hat{F}_{\alpha \beta}^{I}\right|_{\mathrm{r e g}}= - \frac{2 \, e_{\alpha}^{a}\, \hat{\xi}_{a b}^{+}\, e_{\beta}^{b}}{\left( r^{2}+ \rho^{2}\right)^{2}}\qquad \mathrm{o r}\qquad \left. \hat{F}_{a b}^{I}\right|_{\mathrm{r e g}}= - \frac{2 \, \hat{\xi}_{a b}^{+}}{\left( r^{2}+ \rho^{2}\right)^{2}}
a_{n+1}= \frac{c(n + 1)}{2 (n + \alpha + 1)}\sqrt{\frac{h_{n+1}}{h_{n}}}\ ,b_n= \frac{c(n + 2\alpha+1)}{2 (n + \alpha + 1)}\sqrt{\frac{h_{n-1}}{h_{n}}}\ .
\left|\sum_{k = m}^{z_1}|W(k)| - (r(n) - \sqrt{n})n \right| \le \left|\sum_{k = m}^{z_1}\left(|W(k)| - b(k)\right) \right| + \left| \sum_{k= m}^{z_1}b(k) - (z_1 - \lfloor \sqrt{n}\rfloor)n\right| + \big|(z_1 - \lfloor \sqrt{n}\rfloor)n - (r(n) - \sqrt{n})n\big|.
\lambda_{0}=\lim_{n\rightarrow \infty}\lambda_{0}^{(n)}\geqslant\lim_{n\rightarrow \infty}\inf_{x \in B_n}\frac{f(x)}{G_Y^{0}f(x)}\geqslant \inf_{x \in \mathbb{R}\setminus\{0\}}\frac{f(x)}{G_Y^{0}f(x)},
\left\Vert \log \left\vert h(e^{iw};\boldsymbol{\xi})\right\vert^{2}-\log\left\vert h(1;\boldsymbol{\xi})\right\vert^{2}\right\Vert_{L^{2}\left( \mathbb{T}\right)}=\left\Vert \log \left\vert f(e^{iw};\boldsymbol{\xi})\right\vert^{2}-\log \left\vert f(1;\boldsymbol{\xi})\right\vert^{2}\right\Vert_{H^{2}\left( \mathbb{D}\right)}.
\int_{S^{n-1}}g(u)\,d\widetilde{C}_{q}(\varphi K,\varphi Q,u)=\int_{S^{n-1}}g\left(\frac{\varphi^{-t}u}{\|\varphi^{-t}u\|}\right)d\widetilde{C}_{q}( K,Q,u).
\langle\psi({\bf p}_1)\psi^+({\bf p}_2)\rangle=\langle\phi({\bf p}_2)\phi^+({\bf p}_1)\rangle=f_1\!\left(\frac{|{\bf p}_1|}{|{\bf p}_2|}\right)\frac{1+z_1\bar{z}_2}{\sqrt{|{\bf p}_1||{\bf p}_2|}}\,\delta_{z\bar{z}}(z_1-z_2)
\begin{array}{l}{{{\cal W}( x_{1}^{0}- \mathrm{i}x_{1}^{4},{\bf x_{1}}; \dots ; x_{n}^{0}- \mathrm{i}x_{n}^{4},{\bf x_{n}}) =}}\\{{\quad = \langle \, 0 | \phi ( 0 ,{\bf x_{1}}) \mathrm{e}^{- \mathrm{i}( x_{1}^{0}- x_{2}^{0}) H}\mathrm{e}^{- ( x_{1}^{4}- x_{2}^{4}) H}\phi ( 0 ,{\bf x_{2}}) \dots \phi ( 0 ,{\bf x_{n}}) | 0 \, \rangle \, .}}\\ \end{array}
M\left(1+t \:h(x) +yt\: h(x)\right) = 1+12t, M\left(1+x(x^3+1)+t\: h(x) +y\left((1+x^3)+ t\:h(x)\right)\right) = 5+12t, M\left(1+t \; h(x) +y\left((1+x^3)+t\:h(x)\right)\right) =-3^3 (1+4t).
{\cal M}(\nu(p_1)\to \nu(p_2)\gamma(q))=-{G_F\over \sqrt{2}e}Z\epsilon^{\alpha}\bar{u}(p_2)\gamma^{\beta}(1-\gamma_5)u(p_1) \left(g_V\Pi_{\alpha\beta}(-q)-g_A\Pi^5_{\alpha\beta}(-q)\right),
\mathrm{1}_{h'=h}+ \frac{1}{n}\sum_{w^n = (-1)^{\theta_2}}\frac{w^{h-h'}e^{- (\lambda+\theta_1 \pi i+Tw)}}{1 - e^{- (\lambda+\theta_1 \pi i +Tw)}}= \mathrm{1}_{h'=h}- \frac{1}{n}\sum_{w^n = (-1)^{\theta_2}}w^{h-h'}+ \frac{1}{n}\sum_{w^n = (-1)^{\theta_2}}\frac{w^{h-h'}}{1 - e^{- (\lambda+\theta_1 \pi i +Tw)}}.
\limsup_{n \to \infty}\frac{1}{n}\log \int_{A_{n,\eta}}d \theta P_n(\theta) e^{\sum_j h(\theta_j)}= \lim_{k \to \infty}\frac{1}{n_k}\log \int_{A_{n_k,\eta}}d \theta P_{n_k}(\theta) e^{\sum_j h(\theta_j)}.
\widetilde{\Gamma}_{k p}\ = \ \frac{1}{\mu_{k}{-}\bar{\mu}_{p}}\, T_{k}^{\dagger}\, T_{p}\quad , \qquad \textrm{i . e .}\qquad \sum_{k = 1}^{m}\, \Gamma^{\ell k}\, \widetilde{\Gamma}_{k p}\ = \ \delta_{\p}^{\ell}\quad .
e_\mu^a = e^{\frac{\rho(x)}{2}}\delta^a_\mu, \quad e_a^\mu = e^{- \frac{\rho(x)}{2}}\delta^\mu_a,\quad \omega_{m, ab}= \frac{1}{2}(\eta_{a m}\partial_b \rho(x)- \eta_{b m}\partial_a \rho(x)), \quad \omega_{c, ab}= e^m_c\omega_{m, ab}= e^{-\frac{\rho(x)}{2}}\omega_{c, ab}.
{\mathbf T}_j(E,\ell)= \xi^{-(j+1)}\mathbf{D}^{+}(E \Omega^{j+1},\ell) \mathbf{D}^{-}(E \Omega^{-(j+1)},\ell) &-\xi^{(j+1)}\mathbf{D}^{-}(E \Omega^{j+1},\ell) \mathbf{D}^{+}(E \Omega^{-(j+1)},\ell) .
\begin{cases}\left(1-\hbar\right)\left(\gamma\times\dot{\gamma}\right) \hbox{or}\dot{\gamma}\times\ddot{\gamma}\le 0 \quad\hbox{on}S^1_+\cap[T^2 - \Delta(\varepsilon)]_+\,, &\left(1-\hbar\right)\left(\gamma\times\dot{\gamma}\right) \hbox{or}\dot{\gamma}\times\ddot{\gamma}\ge 0 \quad\hbox{on}S^1_-\cap[T^2 - \Delta(\varepsilon)]_+\,. \end{cases}
\lvert \{P \in S \mid \deg(P) > c'\deg(D), f \in W_P \cap \mathcal{L}(D)_I^{d+1}\}\rvert \leq \lvert \left\{P \in S \mid \deg(P) > c'\deg(D), f \in V_P\cap \mathcal{L}(D)_I^{d+1}\right\}\rvert + \lvert \{P \in S \mid \deg(P) > c'\deg(D), x^d f(1/x) \in U_P\cap \mathcal{L}(D)_I^{d+1}\}\rvert =: \I + II.
I_1(r)=C_{\nu}e^{i(\frac{\nu\pi}2+\frac{\pi}4)}\int_0^\infty(r\rho)^{-\frac{n-1}2}e^{i(-r\rho+t\rho^2)}\chi_{N}(\rho)\rho^{n-1}\mathrm{d}\rho,
\tilde{U}(v_0 \otimes v_0) = 0 \tilde{U}(v_1 \otimes v_1) = 0 \tilde{U}(v_0 \otimes v_1) = v_1 \otimes v_0 - q v_0 \otimes v_1 \tilde{U}(v_1 \otimes v_0) = -q^{-1}(v_1 \otimes v_0 - q v_0 \otimes v_1).
{\bf E}\bigl[M^l_{(\tau, \lambda_1, \lambda_2)}\bigr] =\prod_{k=0}^{l-1}\frac{\Gamma(1-(k+1)/\tau)}{\Gamma(1-1/\tau)}\frac{\Gamma(1+\lambda_1-k/\tau)\Gamma(1+\lambda_2-k/\tau)}{\Gamma(2+\lambda_1+\lambda_2-(l+k-1)/\tau)}.
\langle \tilde{\Theta}\tilde{\Theta}U\varphi,\tilde{\Theta}B\varphi\rangle =\langle U\varphi,\tilde{\Theta}B\varphi\rangle &=-\langle U\varphi,B\tilde{\Theta}\varphi\rangle &=\langle B\varphi,U\tilde{\Theta}\varphi\rangle=\langle B\varphi,\tilde{\Theta}U\varphi\rangle.
F_j^{\mu_1,\mu_2}(\phi) = \xi^+_j (-1)^{j/2}\Big[ P_{\frac{j}{2}}^{(\mu_1-1/2,\mu_2-1/2)}(-\cos 2\phi) + \cos(\phi)\sin(\phi)P_{\frac{j}{2}-1}^{(\mu_1+1/2,\mu_2+1/2)}(-\cos 2\phi)\Big].
\langle T_{\tau \tau}\rangle = -{\frac{\kappa \alpha^{2}}{4 [ ( \alpha \tau + \beta )^{2}+ \kappa ]^{3 / 2}}}\left[ 3 ( \alpha \tau + \beta ) + 5 \sqrt{( \alpha \tau + \beta )^{2}+ \kappa}\right] \ \ ,
\int_{\rho B_{1}}\int_{\rho B_{1}}(\psi(x)-\psi(y))^{2}(w^{2}(s,x) + w^{2}(s,y))k(x,y)dxdy & \quad\quad\quad\quad\quad\leq C(\delta,\Lambda)(\rho-\rho')^{-2\beta}\int_{\rho B_{1}}w^{2}(s,x)dx.
C_{\alpha\beta}=[\begin{matrix}C_{11}&C_{11}-2C_{66}&C_{13}&0&0&0\\C_{11}-2C_{66}&C_{11}&C_{13}&0&0&0\\C_{13}&C_{13}&C_{33}&0&0&0\\ 0&0&0&C_{44}&0&0\\ 0&0&0&0&C_{44}&0\\ 0&0&0&0&0&C_{66}\end{matrix}].
\left\langle T_{i i}( x ) \right\rangle = \frac{1}{4}\sum_{n}\omega_{n}\left| \phi_{n}\right|^{2}- \frac{1}{4}\sum_{n}\frac{1}{\omega_{n}}\left| \vec{\nabla}\phi_{n}\right|^{2}+ \frac{1}{2}\sum_{n}\frac{1}{\omega_{n}}\left| \partial_{i}\phi_{n}\right|^{2}
\mathcal I_4(L,r) =\int_S\int_S \frac{1}{|1-r^2\alpha_1|^{2L}-(1-r^2)^{2L}}\left(\sum_{j=1}^n \alpha_j \gamma_j(r\alpha)\right) \left(\sum_{k=1}^n \eta_k \gamma_k(r\eta)\right) =A_n \int_S \frac{1}{|1-r^2\alpha_1|^{2L}-(1-r^2)^{2L}}\left(\sum_{j=1}^n \alpha_j \gamma_j(r\alpha)\right)\ ,
V_p(x):=\left|\frac{u_p(x_p^++\mu_p^+x)}{u_p(x_p^+)}\right|^{p-1}=\left| 1+\frac{v_p^+(x)}{p}\right|^{p-1}\ \ \ \ \ \mbox{($v_p^+$ is the function defined in Lemma \ref{convergenzaVp})}.
\sqrt{V_j(r)}=\frac{1}{r}\left|j-\Phi(r) \right| \ge \frac{1}{r}\left( |j|-|\Phi(r)| \right) \ge \frac{1}{r}\big(|j|- \lambda_+(1+\varepsilon^{\sigma_+}|j|\big)\big) &= \frac{1}{r}\big( |j|(1-\lambda_+\varepsilon^{\sigma_+})-\lambda_+ \big)
R^e_n(s_{v_0})(w_{0}) =s_{w(-e+1,0)}s_{w(0,e-1)}(w_0) &=s_{w(-e+1,0)}\left(w_0+(w(0,e-1), w_0)w(0,e-1)\right) &=s_{w(-e+1,0)}(w_0-w(0,e-1))=-s_{w(-e+1,0)}(w_1+\cdots w_{e-1}) &=-\left(w_2+\cdots w_{e-1}+w_1+ (w_1,w(-e+1,0))w(-e+1,0)\right) &=-\sum_{i=-e+1}^{e-1}w_i.
\omega_{n\pm}:=\omega_{\pm}(1/n)=\left(\sqrt{\omega}\pm \frac{1}{n}\right)^2,\ \ \mathbf{c}_{n\pm}:=\mathbf{c}_{\pm}(1/n)=\frac{\mathbf{c}}{\sqrt{\omega}}\left(\sqrt{\omega}\pm \frac{1}{n}\right).
\Psi_{I}^{( n )}( \hat{\alpha}_{a_{0}}, \ldots , \hat{\alpha}_{a_{n}}) = ( - 1 )^{n + \sum_{i > 0}| \alpha_{a_{0}}| | \alpha_{a_{i}}|}\Psi_{I}^{( n )}( \hat{\alpha}_{a_{1}}, \ldots , \hat{\alpha}_{a_{n}}, \hat{\alpha}_{a_{0}}) .
\begin{aligned}\liminf_{m\to\infty}\sum_{n=1}^N \Big( a_k^{(n)}\cos2m\phi_n+b_k^{(n)}\sin2m\phi_n &-\frac1r a_k^{(n)}\cos(2m-1)\phi_n-\frac1r b_k^{(n)}\sin(2m-1)\phi_)\Big)\ge0.\end{aligned}
\partial_{p}^{\omega}\, G_{l , s}^{\, \alpha}( \vec{\alpha}, \vec{\lambda}, p_{0 , 1}+ i k_{0 , 1}, \underline{{{p}}}_{1}, \ldots , p_{0 , n - 1}+ i k_{0 , n - 1}, \underline{{{p}}}_{n - 1})
0 > \sum_{j=p^i+1}^{\alpha p^i}\nu_p\left( \frac{n+1-j}{j}\right) = \left( \sum_{k=n+1-\alpha p^i}^{n-p^i}\nu_p(k)\right) - \left( \sum_{j=p^i+1}^{\alpha p^i}\nu_p(j)\right), = \left( \sum_{k\in N}\nu_p(k)\right) - \left( \sum_{j\in N^{\prime}}\nu_p(j)\right),
\forall\:0\le k<2,\quad\mbox{Prod}_{\mathbf{P}_{k}}\left(\mathbf{x}^{\top^{2}},\mathbf{y}^{\top},\mathbf{z}\right)=\mathbf{x}^{T}\cdot\mathbf{M}_{k}\left(\mathbf{z}\right)\cdot\mathbf{y}
p^{2}\xi_{1 1}( p , q ) - p . q \xi_{0 2}( p , q ) ={\frac{1}{2}}\left\{-{\frac{1}{2}}Z_{0}( ( p + q )^{2}; m^{2}) +{\frac{1}{2}}Z_{0}( q^{2}; m^{2}) + p^{2}\xi_{0 1}( p , q ) \right\}\, .
\left(\sum_{i=0}^{p-1}YX^i\zeta_p^{-i}\right)^p =\left(Y\sum_{i=0}^{p-1}X^i\zeta_p^{-i}\right)^p &= Y^p\prod_{j=0}^{p-1}\left(\sum_{i=0}^{p-1}X^i\zeta_p^{-(2j+1)i}\right) &= Y^p\prod_{j=0}^{p-1}\left(\frac{1-X^p\left(\zeta_p^{-(2j+1)}\right)^p}{1-X\zeta_p^{-(2j+1)}}\right) &= Y^p(1-X^p)^p\prod_{j=0}^{p-1}\left(1-X\zeta_p^{-(2j+1)}\right)^{-1}.
a(n) \bigl\lbrack \hat{a}(n-1) \hat{y}_{n-1}+ \hat{b}(n-1) \hat{y}_{n}\bigr\rbrack + b(n) \bigl\lbrack \hat{a}(n) \hat{y}_{n}+ \hat{b}(n) \hat{y}_{n+1}\bigr\rbrack =d(x){\hat{d}}(x) \hat{y}_{n}.
S_{E}^{vol}= \sum_{a}\frac{V_{a}}{4 G_{d}}+ \frac{\beta}{16 \pi G_{d}}\sum_{a}\int_{M_{a}^{d-2}}F \wedge \Psi - \frac{\beta}{8 \pi G_{d}}\int_{\partial\Sigma_{\infty}}J^{i}_{D}d\sigma_{i}.
(\alpha_{jk})_{jk}\mapsto \frac{2}{3}\begin{pmatrix}\alpha_{11}+\alpha_{22}0 0 0 \alpha_{22}+\alpha_{33}0 0 0 \alpha_{33}+\alpha_{11}\end{pmatrix}-\frac{1}{3}(\alpha_{jk})_{jk}\ \ \ \ \ \ \ \alpha_{jk}\in \mathbb{C}
\mathbb N_\mu(\|W_t\|\neq 0) = -\log \mathbf P_{\mu}(\|X_t\|=0) = \lim_{\lambda \to \infty}(- \log \mathbf P_\mu [e^{-\lambda X_t(\mathbf 1_E)}]) &= \lim_{\lambda \to \infty}\langle \mu, V_t(\lambda \mathbf 1_E)\rangle = \mu(v_t).
Q_k^{\Pi_2}(t) =\sum_{i=1}^k Q_i^{\Pi_2}(t)-\sum_{i=1}^{k-1}Q_i^{\Pi_2}(t) &\leq \sum_{i=1}^{k}Q_i^{\Pi_1}(t)-\sum_{i=1}^{k-1}Q_i^{\Pi_1}(t)+kn(N) &=Q_k^{\Pi_1}(t)+kn(N).
\begin{aligned}e^{H(x^{[f/g]})}\psi(z)e^{-H^\ast(-\beta)}=\left(\frac{1}{1+\beta z^{-1}}\prod_{j=g}^f\frac{1+\beta x_i}{1-x_iz}\right)e^{-H^\ast(-\beta)}\psi(z)e^{H(x^{[f/g]})},\end{aligned}
S\! =\!\! \int\!\!\! \sqrt{-g}d^{D}\! x\! \left[ -\sum^{N}_{k=1}\bar{\psi}_{k}\gamma^{\alpha}\nabla_{\alpha}\psi_{k}+\frac{\lambda_0}{2N}\left(\sum^{N}_{k=1}\bar{\psi}_{k}\psi_{k}\right)^{2}\right] ,
= \; \frac{1}{( 2 \pi )^{l}}\left( \frac{- 1}{2 L}\right)^{l}\exp \left( \frac{1}{2}l^{2}e^{2}\frac{1}{\pi^{2}}\int d^{2}p \; \frac{1}{p^{2}+ m_{d}^{2}}\; \frac{1}{p^{2}}\; \sin^{2}( p_{1}L ) \right)
\begin{array}{c c c}{{K_{a b}^{- 1}=}}&{{p \left( \begin{array}{c c}{{\sqrt{g}}}&{{{\frac{1}{\sqrt{g}}}}}\\{{{\frac{1}{\sqrt{g}}}}}&{{\sqrt{g}}}\\ \end{array}\right)}}&{{+ ( 1 - p ) \left( \begin{array}{c c}{{{\frac{1}{\sqrt{g}}}}}&{{\sqrt{g}}}\\{{\sqrt{g}}}&{{{\frac{1}{\sqrt{g}}}}}\\ \end{array}\right)}}\\ \end{array}
|g'(t)|=C\Bigg|\int_{\partial B}d\mathcal H^{n-1}_x\!\int_{\partial B}d\mathcal H^{n-1}_y\!\int_{u(x)}^{u(y)}dr \int_{u(x)}^{u(y)}\frac{2t(r-\rho)^2 + \big[(1+t\rho)r + (1+tr)\rho\big]|x-y|^2}{\big(t^2(r-\rho)^2 + (1+tr)(1+t\rho)|x-y|^2\big)^{\frac{n+s}{2}+1}}\,d\varrho\Bigg|\,.
[y_i+y_j,(\mathrm{ad}x_i)^\alpha(t_{ij})]=\sum_{\alpha'+\alpha''=\alpha-1}(\mathrm{ad}x_i)^{\alpha'}([-\sum_{k\neq i,j}t_{ik},(\mathrm{ad}x_i)^{\alpha''-1}(t_{ij})]) =\sum_{k\neq i,j}\sum_{\alpha'+\alpha''=\alpha-1}-(\mathrm{ad}x_i)^{\alpha'}(-\mathrm{ad}x_j)^{\alpha''}([t_{ik},t_{ij}])=\sum_{k\neq i,j}\mathrm{Term}_k,
\leq \;\left\langle \prod_{b=1}^N \; 2 \cosh \left( -2 \sum_{b=1}^N \beta^{(b)}\int_{\Lambda}d^2 x \; t(x): \cos\Big[ 2 \sqrt{\pi}\varphi^{(b)}(x) \Big] :_{C^1}\right)\right\rangle_{C^1}\; .
z ( 1 - z ) \partial_{z}^{2}g ( z ) + ( 1 + 2 \kappa ) ( 1 - z ) \partial_{z}g ( z ) + \left[ \frac{1}{z}\left( \kappa^{2}+ \frac{\omega^{2}\ell^{4}}{4 r_{H}^{2}}\right) - \kappa^{2}\right] g ( z ) = 0 ,
\begin{array}{lll}p_1:= \sigma_2 \epsilon_{v_1}c_{v_1, -u_2}, p_2 := \sigma_2 \epsilon_{s_1}c_{s_1, -u_2}, p_3:= \sigma_2\epsilon_{t_1}c_{t_1,-u_2}, p_4:= \sigma_2 \epsilon_{v_2}c_{u_1, -v_2}, q_1 :=c_{u_1, u_2}, q_2 := c_{t_1,-u_2}c_{u_1,-v_2}c_{t_1,-v_2}. \end{array}
Q_{\mathrm{g r a v}}^{2}= R^{2}+{\frac{1}{1 2}}\left( F_{\mu \nu}F^{\mu \nu}\right)_{T^{2}}+{\frac{5}{3 6 N_{V}}}\left( F_{\mu \nu}F^{\mu \nu}\right)_{G_{( N_{V})}}.
\varphi(g) = \frac{u^{-3}v^{-1}(1-u^2)(1+u^2)}{u^{-3}v^{-2}(1-u^2)(u^2 + uv^2 - 1)}&= \frac{(1+u^2)v}{(u^2 + uv^2 - 1)}&= g.
{\frac{\langle : \dot{p}_{i}\dot{p}^{i}: \rangle}{r_{1}^{2}}}\sim{\frac{\kappa^{2}N^{2}}{L^{4}( Q_{1}Q_{5})^{2}Q_{5}}}; \ \{\frac{\langle : \dot{\gamma}^{2}: \rangle}{r_{1}^{2}r_{5}^{2}}}\sim{\frac{\kappa^{2}N^{2}}{L^{4}( Q_{1}Q_{5})^{2}}}
l^{ab}n_{ab}= l_{ab}\sigma^{ac}\sigma^{bd}n_{cd}= (l_{ab}+\frac{\sigma_{ab}}{r}) \sigma^{ac}\sigma^{bd}(n_{cd}- \frac{\sigma_{cd}}{2r}) + \frac{1}{2}r^{-1}\sigma^{ab}l_{ab}- r^{-1}\sigma^{ab}n_{ab}+ r^{-2}.
{\cal M}=\left(\begin{array}{cc}\frac{1}{\lambda}+{\cal G}_0(0,0){\cal G}_0(0,\ell){\cal G}_0(\ell,0) \frac{1}{\lambda}+{\cal G}_0(\ell,\ell)\end{array}\right)=\left(\begin{array}{cc}\frac{1}{\lambda}+\frac{1}{2\sigma}\frac{e^{-\sigma\ell}}{2\sigma}\frac{e^{-\sigma\ell}}{2\sigma}\frac{1}{\lambda}+\frac{1}{2\sigma}\end{array}\right).
F_{k_1,k_2,k_3,k_4}(x) = \sum_{(i,j,r,s) \in [n]^4}(\overline{c}_{i,j}d_{r,s}+ \max \{0, \Delta{c}_{i,j}d_{r,s}- \Delta c_{k_1,k_2}d_{k_3,k_4}\})x_{i,r}x_{j,s}
\mathcal{A}:=\left.\bigsqcup_{v\in \mathfrak{T}}\mathcal{A}_v\right/\left\{\mu_\gamma\right\}\mathcal{X}:=\left.\bigsqcup_{v\in \mathfrak{T}}\mathcal{X}_v\right/\left\{\mu_\gamma\right\}.
(- \Delta_{g_{\sigma_n}}- \tau_n )u_n = f_n, 1= \| u_{n}\|_{L^2( B_1)}> n \big( \Vert u_n \Vert_{L^2(B_2\setminus B_1)}+ \frac{1}{1+ |\tau_n|^{1/2}}\Vert f_n \Vert_{L^2(B_2)}\big)
\partial_t \begin{pmatrix}w \overline{w}\end{pmatrix}= it^{-1}|W_1|^2 P\begin{pmatrix}i\mu 0 0 - i\mu\end{pmatrix}P^{-1}\begin{pmatrix}w \overline{w}\end{pmatrix}+ \begin{pmatrix}S_2 + e^{3i\lambda_1 (\Phi_1-\theta_1)}R_2 \overline{S_2 + e^{3i\lambda_1 (\Phi_1-\theta_1)}R_2}\end{pmatrix}.
\sum_{j = 1}^k{( - 1)^{j - 1}jG_{2j - 1}}= - \frac{1}{5}G_0 + \frac{1}{5}G_1 + \frac{1}{5}( - 1)^k (k + 1)G_{2k}&- \frac{1}{5}( - 1)^k (2k + 1)G_{2k + 1}\,.