harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\P \big[ (\omega_1^j, \omega_2^j) = (\ell_1, \ell_2) \big] = C_N \min\left(C, \tfrac{1}{\ell_1^2 + \ell_2^2}\right),\ -\tfrac{N}{2}+1 \leq \ell_1, \ell_2 \leq \tfrac{N}{2},
I_g(u) = \int_{M \setminus \{p \}}(\eta r^{2-n}+u) L_g( \eta r^{2-n}+u)\,dv^g , J_g(u) = \int_{M\setminus\{p\}}\eta r^{2-n}F_{\eta}\,dv^g+2\int_{M}uF_{\eta}\,dv^g+\int_{M}uL_g u\,dv^g.
\begin{aligned}d\hat F_{w_0}(w_1) =d\tilde F_{(1+\gamma)(w_0)}d(1+\gamma)_{w_0}(w_1) &=dF_{\Gamma_{(1+\gamma)(w_0)}}d\Psi_{(1+\gamma)(w_0)}d(1+\gamma)_{w_0}(w_1). \end{aligned}
A^{31}_n:= \sum_{k: n\leq |\widehat{a}_k|\leq 2n+1n\leq |\widehat{b}_k|\leq 2n+1}\frac{\left(\widehat{f}(\widehat{a}_k)-\widehat{f}(\widehat{b}_k)\right)^2}{|\widehat{b}_k-\widehat{a}_k|}\rightarrow 0,
\lim_{y\to \infty}\frac{\#\{\gamma \mid \ell(C_\gamma) \leq y,\a \leq \dfrac{\mathrm{lk}(C_\gamma, K_{2,3})}{\ell(C_\gamma)}\leq b\}}{\#\{\gamma \mid \ell(C_\gamma) \leq y\}}= \frac{\arctan \dfrac{\pi b}{3}- \arctan \dfrac{\pi a}{3}}{\pi}.
\mathcal{I}\left(x,\pi/2\right) =\frac{x\sqrt{2\Delta_1\bar{\gamma}}\exp\left(-\Delta_2\right)}{\pi (x^2\bar{\gamma}+2\Delta_1)}\Phi_1^{(2)}\left(1;1;1.5; \frac{2\Delta_1}{x^2\bar{\gamma}+2\Delta_1},\frac{2\Delta_1\Delta_2}{x^2\bar{\gamma}+2\Delta_1}\right),
V_{l}{}^{i j}= e^{2 U}\delta_{l \, [ i}\partial_{j}\left( \ln \chi^{( 1 )}\chi^{( 2 )}\right) \qquad V_{l}{}^{i 4}={\frac{1}{2}}e^{2 U}\epsilon_{l i m}\partial_{m}\left( \ln{\frac{\chi^{( 1 )}}{\chi^{( 2 )}}}\right)
\Omega = \sum_{n_{2}\ll N_0/n^2_1}^{\infty}\,\left|\displaystyle\sum_{q_2\sim C/q_1}\,\,\sum_{m \sim M_1}\frac{\lambda_f(m)}{m^{1/4}}\,\mathcal{C}^{+}(n^2_1n_2,m;q)\, \mathcal{J}^{+}_1(n^2_1n_2,m,q)\right|^2.
\hat{B}_{n,k}(x_{1},x_{2},...,x_{n-k+1})=\sum\frac{k!}{j_{1}!j_{2}!\cdot\cdot\cdot j_{n-k+1}!}x_{1}^{j_{1}}x_{2}^{j_{2}}\cdot\cdot\cdot x_{n-k+1}^{j_{n-k+1}},
(L_{\frac{1}{2}}L_{\frac{1}{2}}^{\dagger}+\frac{a\mu sin\theta}{\lambda+a\mu cos\theta}L_{\frac{1}{2}}^{\dagger}+\lambda^{2}-a^{2}\mu^{2}cos^{2}\theta)S_{-\frac{1}{2}}=0.
\frac{\phi(t,r,v)}{t}=\frac{\phi(t,r,v)-\phi(t, r(t),v(t))}{t}&=(r(t)^2-r^2) \cot t +v(t)-v &= v(t)-v -\big(r-r(t)\big) 2 r(t) \cot t-(r-r(t))^2 \cot t
E_0(\rho_{j, j}, P_{Y_j|U_j}) \ge 0 ; \,\rho_{j, j}\ge 0\,. -\kappa\le\frac{\partial^2 E_{0}(\rho_{j, j}, P_{Y_j|U_j})}{\partial \rho_{j, j}^2}\leq 0; \,\rho_{j, j}\in [0, 1]\,. \left.\frac{\partial E_{0}(\rho_{j, j}, P_{Y_j|U_j})}{\partial \rho_{j, j}}\right\vert_{\rho_{j, j}= 0}= I(U_j;Y_j|U_{j+1})\,.
\|\sum_{j=1}^{\ell-1}\binom{2\ell-1}{2j}(-\partial(m_{s_1},...,m_{s_{2^{\ell-1}}}))^{2j}(\partial(m_{t_1},...,m_{t_{2^{\ell-1}}}))^{2(\ell-j)-1}\alpha-\beta_1\|<\frac{\epsilon}{10},
a = \left[\begin{array}{c}4 1 8 4 6 7 \end{array}\right],B = \left[\begin{array}{cccccc}4 0 0 3 0 3 0 0 0 6 6 0 0 0 5 0 3 6 3 6 0 4 4 3 0 6 3 4 4 5 3 0 6 3 5 2\end{array}\right],d = \left[\begin{array}{c}8 7 7 8 6 2 \end{array}\right].
T_u(x,b,0) :=\frac{1+t}{2}\int_0^{|u|}\frac{\xi''(s)}{{b}-{d}(s)}ds+ \frac{1-t}{2}\int_0^{|u|}\frac{\xi''(s)}{{b}-{\phi}_{|u|}(s)}ds &\quad+\int_{|u|}^1 \frac{\xi''(s)}{{b}-{d}(s)}ds+{b}-1- \log{b}- \int_0^1 \xi''(q){x}(q)dq.
S_{-}( \omega ) = \frac{( m + i k ) \left[ ( \omega + G \phi_{1}( 0 ) ) f^{\ast}( 0 ) +{f^{\prime}}^{\ast}( 0 ) \right]}{\omega \bigl [ ( \omega + G \phi_{1}( 0 ) ) f ( 0 ) +{f^{\prime}}( 0 ) \bigr ]}.
S|_V=\left(\begin{array}{cc}0 I_2 I_2 0 \end{array}\right), J|_V=\left(\begin{array}{cc}J_2 0 0 -J_2 \end{array}\right), \quad\hbox{and}L|_V=\left(\begin{array}{cc}J_2 0 0 J_2 \end{array}\right).
S_{2,T}=\frac{1}{a_{d,2}^{\chi_{r}^{2}}(\mathcal{D})\mathcal{L}^{2}(T)T^{d-2\alpha}}\int_{\mathcal{D}(T)}e_{2}^{(r/2)}(\chi_{r}^{2}(\mathbf{x}))~d\mathbf{x}
g_{n}(p^{n+1}) = \frac{1}{2(p-1)^2(p+1)}\bigg(2p^{2n-3}+ (n^2-n) p^{n+1}&-(n^2-n)p^n -(n^2-n+2)p^{n-1}+(n^2-n-2) p^{n-2}+2\bigg)
B_{R}^{2}[u] =:-\big<u,[\mu|u|^{q-2}u,\nabla\varphi_{R}\cdot\nabla+\nabla\cdot\nabla\varphi_{R}]u\big>=2\mu\int_{\mathbb{R}^{N}}|u|^{2}\nabla\varphi_{R}\nabla(|u|^{q-2}) &=-\mu\frac{2(q-2)}{q}\int_{\mathbb{R}^{N}}\Delta \varphi_{R}|u|^{q}.
{\mathcal I}_{12}= -K\int_{\mathbb T}\langle\nabla\varphi,\nabla\bigg[\sigma\cdot\int_{\mathbb T}\nabla\varphi\rho ~ d\theta_* \bigg] \rangle\rho ~ d\theta = -K\bigg|\int_{\mathbb T}\nabla\varphi\rho ~ d\theta\bigg|^{2}.
\begin{array}{rcl}{\tilde S}[X^{\mu},b_i] = -T_{M2}l\int d^2\xi \sqrt{|D_{i}X^{\mu}D_{j}X^{\nu}g_{\mu\nu}|}& & -{\textstyle\frac{T_{M2}l}{2}}\int d^2\xi \epsilon^{ij}D_{i}X^{\mu}D_{j}X^{\nu}B_{\mu\nu}\, . \end{array}
\partial_\tau \bar{\mathcal{V}}_2 = \Delta \bar{\mathcal{V}}_2 + p \hat U_{K_0}^{p-1}(\tau) \bar{\mathcal{V}}_2 - 2 (\mathcal{V}_2 \nabla \psi_*) + \mathcal{V}_2 \Delta \psi_* + p (U_1^{p-1}- \hat U_{K_0}^{p-1}) \psi_* V_2 + \psi_* \mathcal{G}_2,
r (\phi\circ Q)([\mu]) + s (\phi\circ Q)([\lambda]) = (\phi\circ Q)([\lambda_c]) = \left ((n_1, \ldots, n_{c-1}, 0) \circ \alpha_L\right ) ([\lambda_c]) = \mathrm{lk}(L_1,L_c) n_1 + \ldots + \mathrm{lk}(L_{c-1},L_c) n_{c-1}.
w^h(z) := \left( \bar{w}_1(z') - \frac{z_3}{h}\partial_1 \bar{w}_3(z'),\, \bar{w}_2(z') - \frac{z_3}{h}\partial_2 \bar{w}_3(z'),\, \frac{1}{h}\bar{w}_3(z') \right) \,z=(z',z_3) \in \Omega^h,
{\frac{\partial \rho}{\partial y}}= -{\frac{2}{\sqrt{6}}}\left( \rho^{5}-{\frac{1}{\rho}}\right) , \hspace{1 c m}{\frac{\partial A}{\partial y}}={\frac{1}{3}}\left( \rho^{4}+{\frac{1}{2 \rho^{2}}}\right)
\mu_{X,b}(f):=\sup\{p\in\mathbb{R}:|f(x)|\leq const|x-b|^{p}\,,x\in X\}\nu_{X,b}(f):=\max\{l\in\mathbb{N}:f\in\mathcal{F}_{b}(X)+\widehat{m}_{b}^{l}\},
\underline{\Psi}_{0}(\xi, \tau) =\left({\psi_{+}(\xi) \atop \psi_{-}(\xi)}\right)_{0}, \underline{\Psi}_{1}(\xi, \tau) = \left({[\psi_{c}^{+}(\xi) + \psi_{s}^{-}(\xi)] e^{i\tau (1 - \mu_{0}\xi)}\atop [\psi_{c}^{-}(\xi) + \psi_{s}^{+}(\xi)] e^{-i\tau (1 + \mu_{0}\xi)}}\right)_{0},
\left|\int_{\mathbb T^2}u_{>Q}\theta_{>Q}\cdot\nabla\theta_{\leq Q}\,dx\right|\lesssim \sum_{p>Q}\|\theta_p\|_2\left(\sum_{p'\leq Q}\lambda_{p'}^2\|\theta_{p'}\|_2^2\right)^{\frac 12}\lesssim \sum_{p> Q}\lambda_{Q-p}^{\frac 12}\lambda_p\|\theta_p\|_2^2+\sum_{p'\leq Q}\lambda_{p'-Q}\lambda_{p'}\|\theta_{p'}\|_2^2.
\|\varphi_n\big(\rho_{n,n-1}(1_{F_{n-1}})\psi_n(a_k)\big)-\varphi_n(\rho_{n,n-1}(1_{F_{n-1}}))\ \varphi_n(\psi_n(a_k))\| <\big(3\varepsilon_n^2/3\big)^{1/2}=\varepsilon_n.
\begin{array}{l}\Delta(\widehat{x}^{\mu})=\widehat{\lambda}{^{\mu}}_{\rho}\otimes\widehat{x}^{\rho}+\widehat{x}^{\mu}\otimes I \Delta(\widehat{\lambda}{^{\mu}}_{\nu})=\widehat{\lambda}{^{\mu}}_{\rho}\otimes\widehat{\lambda}{^{\rho}}_{\nu}\end{array}
\sum_{k_1 = 0}^{\tau}\sum_{k_2 = 0}^{\tau}\frac{1}{1+k_1+k_2}= \sum_{k = 0}^{2\tau}\min\{1+k, 2\tau+1-k\}\cdot \frac{1}{1+k}\le 2\tau+1 < 2T.
2 \kappa^{2}{\cal L}= \sqrt{- g}\left( R +{\frac{1}{2}}m^{2}( T^{2}- 2 T_{i j}T^{i j}) - T r ( P_{\mu}P^{\mu}) -{\frac{1}{2}}( V_{I}{}^{i}V_{J}{}^{j}F_{\mu \nu}^{I J})^{2}\right) .
|I_4| \leq\int_{\frac{1}{\beta}\leq|x|\leq\frac{2}{\beta}}\frac{1}{|x|^{n-\beta}}\,dx+\int_{\frac{1}{\beta}\leq|x-z|\leq\frac{2}{\beta}}\frac{1}{|x-z|^{n-\beta}}\,dx &\leq C\frac{2^\beta-1}{\beta}\beta^{-\beta}.
\bigg| \frac{f(z)}{\phi_t'(z)}\bigg| \le M \begin{cases}\exp{-\rho |\phi_t(z) |}\forall z \in \psi_t\big((-\infty, 0)\big) \exp{-\mu |\phi_t(z) |}\forall z \in \psi_t\big([0, -\infty)\big) \end{cases}.
u_k(t,\rho,s)= \Bigl(-\int_0^\rho m'(y)\widetilde{A}^k(t,y,s)dy\Bigr)m'(\rho)\int_0^\rho\frac{dy}{m'(y)^2}&+ \Bigl(\int_0^\rho m'(y) (\int_0^y \frac{dz}{m'(z)^2}) \widetilde{A}^k(t,y,s)dy \Bigr)m'(\rho),
\frac{1}{\mu (E(\lambda,s))}\int_{E(\lambda,s)}\log (1-|\xi|^2) d\mu(\xi) = \frac{1}{\mu (B(0,s))}\int_{B(0,s)}\log (1-|\phi_\lambda(\xi)|^2) d\mu(\xi) = \log(1-|\lambda|^2)+\frac{1}{\mu (B(0,s))}\int_{B(0,s)}\log (1-|\xi|^2) d\mu(\xi).
\psi_{\mu \vert A}\leftrightarrow \mathrm{\begin{array}{| c |}\hline{\stackrel{}{A}}\\ \hline{}\\ \end{array}}\equiv{\bf 8}\quad ; \quad \chi_{A B C}\leftrightarrow \mathrm{\begin{array}{| c |}\hline{\stackrel{}{A}}\\ \hline{\stackrel{}{B}}\\ \hline{\stackrel{}{C}}\\ \hline{}\\ \end{array}}\equiv{\bf 5 6}
S = - \frac{L}{4 \pi \lambda^{2}}\int_{0}^{\infty}\frac{d t}{e^{2 \pi t}- 1}\left( ( B \lambda^{2}- i t ) \frac{( \lambda^{2}m^{2}+ v^{2})}{v}\ln \frac{\lambda m - i v}{\lambda m + i v}+ c . c . \right)
\begin{aligned}\|(\check{\rho},\check{u})\|_{L^2(L^2)\times L^2(L^2)}=\|\Lambda_2^*(\rho_0,u_0)\|_{L^2(L^2)\times L^2(L^2)}&\leq \|\Lambda_2^*\| \ \|(\rho_0,u_0)\|_{L^2(0,1)\times L^2(0,1)}.\end{aligned}
\eta_{X, Y}(C(\mathsf{P})) = \sum_{i, j, k, \ell}\pi(K_{2 \rho}^{-1})^i_j (2 \varepsilon(\mathsf{P}^j_k) - \delta^j_k) \varepsilon(X \triangleright \mathsf{P}^k_\ell) \varepsilon(Y \triangleright \mathsf{P}^{\ell}_i).
{\bf J}\cdot \widehat{\bf p}\, \, \Theta_{[ j ]}\, \left[ \phi_{_{L , R}}^{h}({p}^{\mu}) \right]^{\ast}\, = \, - \, h \, \Theta_{[ j ]}\, \left[ \phi_{_{L , R}}^{h}({p}^{\mu}) \right]^{\ast}\quad .
\mathrm{E x p ( e r i m e n t a l ) R e s ( u l t s )}= F_{\mathrm{e x p .}}= 1 \times \: \mathrm{( g o t ~ 5 )}, \: \mathrm{a n d}\: 0 \times \: \mathrm{( g o t ~ a l l ~ o t h e r s )},
\operatorname{Re}\left(e^{-i\theta}[\frac{1}{|z_\alpha|},\partial^2_t]z_\alpha +e^{-i\theta}\partial_t^2\bigl(\frac{z_\alpha}{|z_\alpha|}\bigr)\right) =\bigl(\frac{\partial_\alpha}{|z_\alpha|}\bigr)^3\theta -a\frac{\partial_\alpha}{|z_\alpha|}\theta.
(\mathcal{H}(s, q)f)(x)=\sum_{i=1}^{M}\sum_{r=1}^{c_{i}}\frac{s^{r-1}}{[r]}\prod_{p=0}^{r-1}\frac{[c_{i}-p]}{1+s[c_{i}-1-p]}\left( f(x-\sum_{p=0}^{r-1}v_{c_{1}+\cdots +c_{i}-p})-f(x) \right),
P(Y_n=j, T_n \le t|Y_0=i_0,T_1=t_1,\ldots,T_{n-1}= t_{n-1},Y_{n-1}=i_{n-1}) =P(Y_n=j, S_n \le a|Y_0=i_0,S_1=s_1,\ldots,S_{n-1}= s_{n-1},Y_{n-1}=i_{n-1}) =P(Y_n=j, S_n \le t-t_{n-1}|Y_{n-1}=i_{n-1}).
C_{[m,j]}=[w^{m(p+1)+j+1}]\,h(w)^{j+1}& =\frac{j+1}{m(p+1)+j+1}\,[t^{m(p+1)}]\,(t^{p+1}+1)^{m(p+1)+j+1}=\frac{j+1}{m(p+1)+j+1}\binom{m(p+1)+j+1}{m}=\frac{j+1}{mp+j+1}\binom{m(p+1)+j}{m}
{\Gamma'}^{{(2)}}_{\mathrm{eff}}= \int \frac{d^4 \! p}{(2 \pi)^4}\, \phi(p) \phi(-p) \left(p^2 - \frac{\hbar g^2}{96 \pi^2}\left( \frac{1}{\tilde{p}^2}- \frac{1}{\tilde{p}^2 + \frac{1}{\Lambda^2}}\right) + O(g^4) \right)
A_{i,jl}^{(k)}=\sum^{ib_k}_{s=(i-1)b_k+1}(\upsilon_{s,jl}^{(k)}-\bar{\upsilon}_{s,jl}^{(k)}),\bar{\upsilon}_{s,jl}^{(k)}=\frac{1}{n_k}\sum^{n_k}_{s=1}\upsilon_{s,jl}^{(k)}.
\tilde{C}\cap \mathrm{H}_{A_{F_J}- V}\cap N_{M,\leq \delta}=(\tilde{C}\cap \mathrm{H}_{A_{F_J}- V,\ge 0}\cap N_{M,\leq \delta}) \cap \sigma_{A_{F_J}}= (C \cap N_{M,\leq \delta}) \cap \sigma_{A_{F_J}}.
\alpha'(x_1, x_2, x_3, \dots, x_k) + \alpha'(x_1, x_2, x_3, \dots, x_k) = \sum_{i \in [r_2]}\gamma_i(x_1, x_{J_{i, 1}}) \gamma'_i(x_2, x_{J_{i, 2}})\gamma''_{i, 1}(x_{J'_{i, 1}}) \cdots \gamma''_{i, d'_i}(x_{J'_{i, d'_i}}).
\prod_{\substack{(a,b,c,d)=1 a,b,c,d \geq 1}}\left( \frac{1}{1-w^a x^b y^c z^d}\right)^{\frac{1}{a^r b^s c^t d^u}}= \exp\left\{Li_r(w) Li_{s}(x) Li_{t}(y) Li_u(z)\right\}.
e^{-C\|h\|_\infty}\lesssim\mu_{N,\beta}^{V;\boldsymbol{A}}(\exp(\sum_{i=1}^N ( h(\lambda_{i})-\int h(\eta)d\mu_{{\rm eq}}(\eta)))) \lesssim e^{C(\|h\|_{\mathcal{L}}+\|h\|_\infty+1)^2)}\,.
\|Y^\alpha\varphi\|_p=\|Y^\gamma Y_j \varphi\|_p\leq c_N\sup_{\{\gamma:|\gamma|=M\}}\|X^\gamma X_j\varphi\|_p\leq c_N\sup_{\{\beta:|\beta|=M+1\}}\|X^\beta\varphi\|_p
{\rm pdeg}\, e_i=-{\rm pdeg}\, f_i=1 (1\le i\le n-1),{\rm pdeg}\, q^h=0 (h\in P), &{\rm pdeg}\, e_0^{(l)}=-{\rm pdeg}\, f_0^{(l)}=1,{\rm pdeg}\, Z_r=nr(r\neq0)\,.
\frac{\det D ( \varphi )}{\det D ( 0 )}\; = \; \lim_{\Lambda \to \infty}\left[ \frac{\langle + \mid \varphi + \rangle}{\mid \langle + \mid \varphi + \rangle \mid}\; \frac{\langle \varphi + \mid \varphi - \rangle}{\langle + \mid - \rangle}\; \frac{\langle \varphi - \mid - \rangle}{\mid \langle \varphi - \mid - \rangle \mid}\right]
\times\exp \bigg( \; -\frac{i}{2}l^2 e^2 \; 2 \mbox{Re}\Big( \varepsilon_{\mu \nu}\partial_\nu j^{(L)}_\mu \; , \; \tilde{Q}\;[ \delta(-L,0) - \delta(L,0) ] \; \Big) \; \bigg) \; .
H = \mathrm{T r}\left\{\frac{1}{2 \mu}\, \Pi_{i}^{2}- \frac{\mu}{2 R^{2}}\, ( X^{i})^{2}- \frac{\mu}{4 g^{2}R^{4}}\, [ X^{i}, X^{j}]^{2}+ \cdots \right\}
\tilde{Z}_{p}= \frac{1}{( \sqrt{4 \pi})^{2 6}}\int_{0}^{\infty}\frac{d t}{2 \pi}\left( \frac{2 \pi}{t}\right)^{( 2 5 - p ) / 2}e^{2 t}f^{- 2 4}( t ) .
\begin{array}{l}0\neq t_1\,\overline{a\beta}= (t_1\,\overline a+ t_2\,\overline{\alpha}) \overline \beta\in E' 0\neq t_3\,\overline{a\beta}= (t_3\,\overline a+ t_4\,\overline \alpha)\overline \beta\in E''\,. \end{array}
\|f_{n_1}-f_{n_2}\|_{C^0([0,T],L^1(m))}\ = \ \sup\limits_{t\in(0,T)}\|f_{n_1}-f_{n_2}\|_{L^1(m)}& \leq \e^{(C_0+C)T}\|f_{n_1}^\mathrm{in}-f_{n_2}^\mathrm{in}\|_{L^1(m)}\underset{n\to\infty}{\longrightarrow}0,
\begin{array}{l}D_1^s=P^{-2}s_1\cap s_1^-, begin{align*}0.1cm] D_2^s=P^{-1}s_1\cap P s_1^-, begin{align*}0.1cm] D_1^r=(P^3r_1\cap P^3 r_1^-)\cup(P^{-2}r_1\cap P^2r_1^-)\cup(P^{-3}r_1\cap P^{-3}r_1^-)\cup(P^{-3}r_1\cap P^3r_1^-), begin{align*}0.1cm] D_2^r= (Pr_1\cap P^{-1}r_1^-)\cup(Pr_1\cap P^{-2}r_1^-)\cup(r_1\cap r_1^-)\cup(P^2r_1\cap P^{-1}r_1^-).\end{array}
\sum_{n=1}x^n = \frac{x}{1-x}, \; \; \; \sum_{n=1}^{\infty}nx^{n-1}= \frac{1}{(1-x)^2}, \; \; \; \sum_{n=1}^{\infty}n^2 x^{n}= \frac{x(x+1)}{(1-x)^3}
ds^{2}= (1 - \frac{r_+}{r})(1 - \frac{r_-}{r})^\sigma dt^2 - (1 - \frac{r_+}{r})^{-1}(1 - \frac{r_-}{r})^{-\sigma}dr^2 - (1 - \frac{r_-}{r})^{1-\sigma}r^2 (d \theta^2 + sin^2 \theta d \phi^2)
^{(1)}= \sum_{i=0}^{2d-2}\left(d-i+\left\lfloor \frac{i}{2}\right\rfloor\right)\beta=d^2\beta,^{(2)}=\sum_{i=0}^{q}\left(d-i+\left\lfloor\frac{i}{2}\right\rfloor\right) \beta,^{(3)}=\sum_{i=0}^{\min(k,2d-1)-1}\left(d-i+\left\lfloor\frac{i}{2}\right\rfloor\right)\beta,
Z = \int \prod_{x , i , a}d \bar{E}_{x , i}^{a}\, d \mu ( U_{x , i}) \, \exp \left[ -{\frac{\bar{H}_{\mathrm{e l}}+ \bar{H}_{\mathrm{m a g}}}{T_{\mathrm{G}}}}\right] \, \delta ( D_{x , i}^{a b}\bar{E}_{x , i}^{b}) \, \delta ( F ( U_{x , i}) ) \, ,
\sum\limits_{x\in \L}(-\alpha+2U(x, \xi))1_{\{\xi_x>0\}}\leq \sum\limits_{x\in \L}(-\alpha+2U(x, \xi))=-\alpha|\L| +2\sum\limits_{x\in \L}(\alpha+\beta\nu(x))\xi_x &\leq -\alpha|\L| +2(\alpha+\beta\max_{x\in \L}\nu(x))S(\xi) &\leq -\alpha|\L|+2C(\alpha+\beta\max_{x\in \L}\nu(x)),
B_{G_{13}}\cong\bigg\langle \rho_1, \rho_2, \rho_3 \ \bigg\vert\ \begin{matrix}\rho_2 \rho_3 \rho_1 = \rho_3^2 \rho_1 \rho_3 \rho_2=\rho_3^2\end{matrix}\ \bigg\rangle.
\int^\infty_0 \frac{r^n}{t^{n/2}}\cdot \exp\left(-\frac{r^2}{C_1t}\right)\frac{C_1}{rt}k(x_0,r) dr =\left(\int^\infty_{r_0}+ \int^{r_0}_0 \right) \frac{r^n}{t^{n/2}}\cdot \exp\left(-\frac{r^2}{C_1t}\right)\frac{C_1}{rt}k(x_0,r) dr =\mathbf{I}+\mathbf{II}.
r\geq 0 \quad\textrm{and odd}i_1\leq \ldots\leq i_r\leq -1, &s\geq 0\quad\textrm{and}1\leq p_1\leq \ldots\leq p_s, & j_s\leq -p_s, &p_l<p_{l+1}\quad\textrm{implies}j_l\leq -p_l -2p_{l}(s-l), &p_l=p_{l+1}\quad\textrm{implies}j_l\leq -2p_l + j_{l+1}
c_{m_1,m_2}=\begin{cases}q^{-2}\prod_{b\neq a}\omega_2(b/a)^{n_{i,b}}(i=1,\ldots,n-1), s_1^{-2}\prod_{b\neq a}\omega_1(b/a)^{n_{i,b}}(i=\bar1,\ldots,\overline{m-1}), -\prod_{b\neq a}\omega_0(b/a)^{n_{i,b}}(i=0), \end{cases}
I_{11}= 2r\int_{0}^{2\pi}\frac{\cos(x)\sin(y)(\cos(x)h(x)-\cos(y-x)h(y-x))}{((1+r^2)+(r^2-1)\cos(y))}dy I_{12}= 2r\int_{0}^{2\pi}\frac{\sin(x)(1-\cos(y))(\cos(x)h(x)-\cos(y-x)h(y-x))}{((1+r^2)+(r^2-1)\cos(y))}dy
[b,R_j](a)(g) =\chi_{r_oB}(g)[b,R_j](a)(g)+\chi_{(r_oB)^c}(g)(b(g)-b_B)R_j(a)(g) &-\chi_{(r_oB)^c}(g)\int_{\mathcal G}\left(K_j(g,g')-K_j(g,\tilde g)\right)\left(b(g')-b_B\right)a(g')dg' &\quad-\chi_{(r_oB)^c}(g)\int_{\mathcal G}K_j(g,\tilde g)\left(b(g')-b_B\right)a(g')dg' &=:I_1(g)+I_2(g)+I_3(g,\tilde g)+I_4(g,\tilde g),
[D_t^2-iA\partial_{\alpha},\mathcal{H}]\partial_{\alpha}^k\tilde{\sigma}= 2[D_t\zeta, \mathcal{H}]\frac{\partial_{\alpha}D_t\partial_{\alpha}^k\tilde{\sigma}}{\zeta_{\alpha}}-\frac{1}{\pi i}\int \Big(\frac{D_t\zeta(\alpha,t)-D_t\zeta(\beta,t)}{\zeta(\alpha,t)-\zeta(\beta,t)}\Big)^2 \partial_{\beta}^{k+1}\tilde{\sigma}(\beta,t)d\beta := I_1+I_2.
U_{mn}^{A}=H_{mn}+\sum_{\alpha\ne m}^{B}\frac{H_{m\alpha}H_{\alpha n}}{E-H_{\alpha\alpha}}+\sum_{\alpha,\beta\ne m,n;\alpha\ne\beta}\frac{H_{m\alpha}H_{\alpha\beta}H_{\beta n}}{(E-H_{\alpha\alpha})(E-H_{\beta\beta})}+...
[\phi_k \, , \, \phi_h]=-{\frac{i\pi \beta^2}{2}}s\left (\frac{k-h}{2N}\right) \quad , \quad[\bar \phi_k \, , \, \bar \phi_h]={\frac{i\pi \beta^2}{2}}s \left (\frac{k-h}{2N}\right) \quad , \quad [\phi_k \, , \, \bar \phi_h] =0 \quad .
\varphi_t(u) =E[e^{iuX(kT+s)}]=e^{\Lambda_{kT+s}(\varphi_J(u)-1)}=e^{(k\Lambda_T+\Lambda_s)(\varphi_J(u)-1)}&=\big(e^{\Lambda_T(\varphi_J(u)-1)}\big)^ke^{\Lambda_s(\varphi_J(u)-1)}=\big(\varphi_T(u)\big)^k\varphi_s(u),
(\widehat{X}_{t_{1}}\cdots \widehat{X}_{t_{r}}G(f))(x) =(\widehat{T}_{w_{y}}f)(w_{y}y)=(\widehat{T}_{u_{1}}\widehat{T}_{w_{x}}f)(w_{x}x-\!\!\sum_{j=l-r+1}^{l}\!\!v_{j}) =(\widehat{X}_{l-r+1}\cdots \widehat{X}_{l}\widehat{T}_{u_{1}}\widehat{T}_{w_{x}}(f))(w_{x}x).
I_{g h}^{\prime}= \int d^{3}x \mathrm{T r}( Z_{g h}^{- 1}\partial^{\mu}\overline{{{c}}}^{( r )}\, p a r t i a l_{\mu}c^{( r )}+ \tilde{Z}_{g}^{- 1}\partial^{\mu}\overline{{{c}}}^{( r )}[ A_{\mu}^{( r )}, c^{( r )}] ) .
J_{0}\le \|\theta_{\epsilon}\|_{L^{\infty}(|x|<R)}\|\nabla \cdot \varphi_3\|_{L^2}\|\varphi_1\|_{L^4}\|\varphi_2\|_{L^4}\le C\theta_{\epsilon}(R)\prod_{j=1}^3\|\varphi_j\|_{H^1}.
\Lambda_{0}^{1}= \frac{1}{\tilde{\Delta}_{( m , n )}^{1 / 2}}\left( \begin{array}{c c}{m \qquad}&{- n e^{- \phi_{0}}+ \chi_{0}( m - \chi_{0}n ) e^{\phi_{0}}}\\{n \qquad}&{( m - \chi_{0}n ) e^{\phi_{0}}}\\ \end{array}\right) ,
\begin{aligned}u(x+h)+u(x-h)-2u(x) \leq C |h|^2 \int_0^Te^{-s}\left( 1-\frac{s}{T}\right)^2 ds + C |h|^2 \int_0^T \frac{e^{-s}}{T}ds+C|h|^2 \int_0^T \frac{e^{-s}}{T^2}ds \leq C|h|^2 \int_0^T e^{-s}ds+ C\frac{|h|^2}{T}+C\frac{|h|^2}{T}\int_0^1e^{-sT}ds \leq C\left(1+\frac{1}{T}\right)|h|^2, \end{aligned}
L(x) = \frac{6}{\pi^2}\left( Li_2(x) + \frac{1}{2}\log(x) \log(1-x) \right) = \frac{6}{\pi^2}\left( \sum_{k=1}^{\infty}\frac{x^k}{k^2}+ \frac{1}{2}\log(x) \log(1-x) \right).
H_{x,x'}^{(n)}=\frac{1}{(n+1)^2}\sum_{j,j'=0}^n\Phi\left(\frac{j}{n+1}, \frac{j'}{n+1}\right)\frac{\exp\left\{\frac{2 i\pi x}{n+1}\right\}-1}{\exp\left\{\frac{2 i\pi x}{n+1}\right\}-1}\exp\left\{\frac{2 i\pi jx}{n+1}\right\}\exp\left\{\frac{2i\pi j'x'}{n+1}\right\}
\left\|x\right\| \geqslant 0 , ~\left\|x\right\| = 0 \Leftrightarrow x=0 ; \left\|\alpha x\right\| \leqslant \left\|x\right\|, ~\forall~\alpha \in \mathbb{C}, |\alpha| \le 1 ; \lim_{\alpha \rightarrow 0}\left\|\alpha x\right\| = 0; \left\|x+y \right\| \le \left\|x\right\|+\left\|y\right\| .
l({x}) =l(\cap_{n=1}^{\infty}(x-\frac{1}{n},x]) &=\lim_{n\to\infty}l{(x-\frac{1}{n},x]}&=\lim_{n\to\infty}{F(x)-F(x-\frac{1}{n},x)}&=F(x)-F(x-) &=
W_{q}= - \frac{1}{2}\int d x \int_{0}^{\infty}\frac{d \tau}{\tau}\tau^{- \frac{2}{1 + \gamma}}\langle E [ \delta \left( \tilde{q}\left( 1 \right) \right) \exp \left( - \tau \int_{0}^{1}d s \alpha \phi_{c}^{2}\left( x + \tau^{\sigma}\left( \tilde{q}_{s}- x \right) \right) \right) ] \rangle
\Gamma_{M+1}\bigl(w\,|\,a,a_{M+1}\bigr) = \prod\limits_{k=1}^\infty \frac{\Gamma_M(w+ka_{M+1}\,|\,a)}{\Gamma_M(x+ka_{M+1}\,|\,a)}e^{\Psi_{M+1}(x,\,ka_{M+1}\,|\,a)-\Psi_{M+1}(w,\,ka_{M+1}\,|\,a)}\times &\times \exp{\bigl(\phi_{M+1}(w,x\,|\,a, a_{M+1})\bigr)}\, \Gamma_{M}(w\,|\,a).
\textstyle L_r = \textstyle - r \sum_{u \in \mathbb F_p^\times}u (1 + u)^{r-1}- r \delta_{s=p-1}&\textstyle = - r \left(\sum_{u \in \mathbb F_p^\times}u (1 + u)^{s-1}+ \delta_{s=p-1}\right) = \frac rs L_s,
\left\{(h, \bar{h}, g_s, g_t) \begin{tabular}{|l}$h\bar{h}=h_1$ $g_s\in c'(h, Y), g_t\in c'(\bar{h}, Y)$ $g_sX_{g_t, \bar{h}}\cap X_{g_s, h}\neq \emptyset$\end{tabular}\right\}
( \Gamma_{\mu_{1}\mu_{2}\cdots \mu_{p}})^{T}= ( - 1 )^{p}\, \Gamma_{\mu_{p}\cdots \mu_{2}\mu_{1}}= ( - 1 )^{\frac{p ( p + 1 )}{2}}\Gamma_{\mu_{1}\mu_{2}\cdots \mu_{p}}.
||s||^2_{h_\delta}\ast\rho_\nu\geq||s||^2_{h_\delta}\ast\rho_\mu \mathrm{and}||s||^2_{h_\delta}\ast\rho_\nu=||s||^2_h\ast\rho_\delta\ast\rho_\nu=||s||^2_{h_\nu}\ast\rho_\delta.
\sqrt[4]{\int_{0}^{\mathrm{T_0}}\int_{0}^{\mathrm{T_0}}|\partial_{t}\Phi^{\textbf{e}}(\xi,t;z,\tau)|^2d\tau dt}= \mathcal{O}\Bigg(\sqrt[4]{\mathcal{S}^{(\mathrm{T_0})}_{\mathrm{r}}}\frac{1}{|\xi-z|^{2-2\mathrm{r}}}\Bigg),
I = \int d t \left( \frac 1 2 q^{2}- \frac{1}{2 l}\varsigma^{1}{\varsigma^{1}}^{\prime}- \frac{1}{2 l}\varsigma^{2}{\varsigma^{2}}^{\prime}- \frac{g}{2 q^{2}}+ \frac{\sqrt g}{2 l q^{2}}[ \varsigma^{1}, \varsigma^{2}] + \frac{3}{c l^{2}}T_{B}q^{2}- \frac{3}{c l^{2}}T_{F}\varsigma^{1}q \right) \, .
\left. \mathrm{T r}V_{1 \nu}^{\mu}( \nabla ) \right|^{\mathrm{d i v}}= \frac{2 i}{4 - d}\frac{\kappa^{2}}{2 4 \pi^{2}}\int d^{4}x \sqrt{- g}\left[ - 5 R_{\mu \nu}{\cal E}^{\mu \nu}+ \frac{5}{2}R g_{\mu \nu}{\cal E}^{\mu \nu}\right] .
\frac{d}{dr}\left(e^{(\Gamma-\Lambda)/2}(\Gamma'-\phi') \,r^2 -\frac{\alpha'e^\phi}{g^2}e^{(\Gamma-\Lambda)/2}[(\phi'-\Gamma')\,(1-e^{-\Lambda})+ e^{-\Lambda}\phi' \Gamma' r]\right)=0
\rho(\alpha(x),\alpha(y))\circ\beta=\beta\circ\rho(x,y), &\rho(\alpha(x),\alpha(y))\rho(a,b)-\rho(\alpha(a),\alpha(b))\rho(x,y)=(\rho([x,y,a],\alpha(b))-\rho([x,y,b],\alpha(a)))\circ\beta, &\rho([x,y,a],\alpha(b))\circ\beta-\rho(\alpha(y),\alpha(a))\rho(x,b)=\rho(\alpha(a),\alpha(x))\rho(y,b)+\rho(\alpha(x),\alpha(y))\rho(a,b),
t\cdot \varphi(g,x) = \underset{=m(f+tg)}{\underbrace{\varphi(f+tg,x)}}- \underset{\geq m(f)}{\underbrace{\varphi(f,x)}}\leq m(f+tg) - m(f) & \stackrel{\eqref{m(a) bound}}{\leq}m(f)+t\cdot m_0(g) - m(f) = t\cdot m_0(g)
=\sum_{a^{n}\in T_{\delta}^{p^{n}}}Pr\{E_{a^{n}}\}Pr_{S}\{\exists E_{b^{n}}:b^{n}\in T_{\delta}^{p^{n}},b^{n}\ne a^{n},E_{a^{n}}^{\dagger}E_{b^{n}}\in N(S)\backslash S\}
\hbox{for}-1<N\leq0 \mbox{we have:}\, \int_{\vert x \vert \leq \frac{1}{\delta}}\vert x \vert^{2N}e^{u_k}= \int_{\vert x \vert \leq \delta}\frac{1}{\vert x \vert^{2(N+2)-\beta_k}}e^{\hat{u}_k}\to 0,
\varphi_{X}(\theta)(t):=\mathbb{E}[e^{i\theta X(t)}]=exp(t(ai\theta-\frac{1}{2}\sigma^{2}\theta^{2}+\int_{\mathbb{R}\backslash\{0\}}(e^{i\theta x}-1-i\theta x1_{|x|<1})\Pi(dx)))
(x_1^2 - q^{2i-1}) z_i - q^2 (x_1^2 - q^{2i-5}) z_{i-1}\qquad - q^d (q^2-1)(x_1^2 + q^{2i-3}) + q^{i-1}(q-1)(q^{d+1}+1)(x_1^2 + q^{d-1}) =0.
\left\| x_{0}^{\ast}- \left(\sum_{k=1}^{n}{\lambda_{0}\cdot \ldots \cdot \lambda_{k-1}y_{k}^{\ast}}\right) - \lambda_{0}\cdot \ldots \cdot \lambda_{n}x_{n}^{\ast}\right\| < \varepsilon \sum_{k=1}^{n}{\frac{1}{2^{k}}}.
\| (\psi, V) \|_{\mathcal{B}_0\times \mathcal{B}_1}= \|\psi \|_{\mathcal{B}_0}+ \| V \|_{\mathcal{B}_1}\| (h,w)\|_{\mathcal{B}_2\times \mathcal{B}_2}= \|h \|_{\mathcal{B}_2}+ \| w \|_{\mathcal{B}_2}\| (f,X)\|_{\mathcal{B}_4\times \mathcal{B}_3}= \|f \|_{\mathcal{B}_4}+ \| X \|_{\mathcal{B}_3}.
\int_{(0,1/2)\setminus\mathcal{A}}l(x)^Ldx= \sum_{j\leq j_0}\left(\int_{\mathcal{E}_1(K,j,n)}l(x)^Ldx+\int_{\mathcal{E}_2(K,j,n)}l(x)^Ldx\right) &+\int_{x_0}^{1/2}l(x)^Ldx\:.
H^1(\hat C,L_j|_{\hat C})=\left\langle \frac{1}{x_0^{db}x_1^{q_j+\langle q_j r \rangle+\sum_i\langle q_j(\vec l_0)_i \rangle-b}}\Bigg|\substack{0< b<q_j+\langle q_j r \rangle+\sum_i\langle q_j(\vec l_0)_i \rangle \langle b \rangle = \langle q_j+q_j r+\sum_i q_j(\vec l_0)_i \rangle}\right\rangle
V\left(x,u\right)=\frac{1}{2}\sum_{i=1}^{{n}}\sum_{\substack{j=1 j\neq{i}}}^{{n}}\frac{1}{\left|x_i-x_j\right|}+\sum_{i=1}^{{n}}\sum_{j=0}^{c}\frac{u_j}{\left|x_i-q_j\right|}\,.
|\sigma|^2 \le \| f_1\|^2_2 \cdot \sum_x \left| \sum_{g \in G}f_2 (g x) \right|^2=\| f_1\|^2_2 \cdot \sum_g r_{GG^{-1}}(g) \sum_x f_2 (x) \overline{f_2 (g x)}\,.
K_{S_2}(z,z',s)={1\over l^2_1}{\sqrt{2}e^{-{\bar{s}/ 4}}\over (4\pi \bar{s})^{3/2}}\sum_{n=-\infty}^\infty (-1)^n\int^\pi_\sigma{d\varphi (\varphi+2\pi n)e^{-{(\varphi+2\pi n)^2\over 4 \bar{s}}}\over \sqrt{\cos \sigma-\cos \varphi}}~~,
{\rm I}+{\rm II}:= \|u(r\,\cdot\,) - P_{X_\circ}(r\,\cdot\,)\|_{L^2(B_{1/2}(r^{-1}X_\circ),|y|^a)}+ \|u(r\,\cdot\,) - P_{Z_\circ}(r\,\cdot\,)\|_{L^2(B_{1/2}(r^{-1}X_\circ),|y|^a)}.
P_{B}-P_{C}=\frac{\rho}{2}(v_{C}^{2}-v_{B}^{2})=\frac{\rho}{2}v_{C}^{2}(1-\frac{A_{C}^{2}}{A_{B}^{2}})=\frac{\rho}{2}Q^{\prime2}(\frac{1}{A_{C}^{2}}-\frac{1}{A_{B}^{2}})
P(F(\nu^{j}),t)=\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1-te^{-x})^{-1-m_{j,1}+k+g_{Y}}\cdot exp\left(\frac{-t\theta}{e^{x}-t}\right)(1-te^{-2x})^{2+m_{j,2}+m'_{p-j,-2}-2g_{Y}}