harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\sum_{\gamma \in \Gamma}\frac{| \gamma^{\prime}( f ({\bf y}) ) |^{\Delta}}{| f ({\bf x}) - \gamma f ({\bf y}) |^{2 \Delta}}= | f^{\prime}({\bf x}) |^{- \Delta}| f^{\prime}({\bf y}) |^{- \Delta}\sum_{\gamma \in f^{- 1}\Gamma f}\frac{| \gamma^{\prime}({\bf y}) |^{\Delta}}{|{\bf x}- \gamma{\bf y}|^{2 \Delta}}.
(s_{\alpha}p_{r(\beta \alpha)}s_{\alpha}^*) (s_{\alpha'}p_{r(\beta' \alpha')}s_{\alpha'}^*) = \delta_{\alpha, \alpha'}s_{\alpha}p_{r(\beta \alpha) \cap r(\beta' \alpha')}s_{\alpha'}^* & =\delta_{\beta \alpha, \beta' \alpha'}s_{\alpha}p_{r(\beta \alpha)}s_{\alpha}^*.
\{s_m^a, s_n^b{}^\dagger \}= \delta^{a,b}\delta_{m,n}, ~~~\{\tilde{s}_m^a, \tilde{s}_n^b{}^\dagger \}= \delta^{a,b}\delta_{m,n},~~~ \{s_0^a, s_0^b{}^\dagger \}= \delta^{a,b}
|V_Y(t, x; \xi)|\leq \begin{cases}Ct^{\alpha -1+(\gamma-1)\frac\alpha2}p(t, x-\xi)\,, \d=1\,; Ct^{\nu_0 \alpha-1}|x-\xi|^{-d+\gamma-\nu_1+2-\nu_0}p(t, x-\xi)\,, \d\geq 2 \end{cases}
\frac{d^n\hat{\delta}}{dv^n}=\sum_{k=0}^n\binom{n}{k}\frac{d^k}{dv^k}\left(\frac{1}{v^3}\right)\frac{d^{n-k}\delta}{dv^{n-k}}&=\frac{1}{2}\sum_{k=0}^n\frac{n!(-1)^k(k+1)(k+2)}{(n-k)!v^{k+3}}\frac{d^{n-k}\delta}{dv^{n-k}}&=\frac{1}{v^{n+3}}L_n\delta,
\overline{{\psi}}^{( 1 )}( x ) P_{+}\psi^{( 1 )}( x ) \longleftrightarrow \frac{1}{2 \pi}c^{( 1 )}: e^{- i a \Phi^{( 1 )}( x )}:_{M^{( 1 )}}: e^{- i b \Phi^{( 2 )}( x )}:_{M^{( 2 )}}e^{i \frac{\theta}{2}},
D_{1.2}\le C\int_{t/2}^{t}\|G(\cdot, t-\tau)\|_{L^{1}}\left\|\psi^{2}(\cdot, \tau)\right\|_{L^{\infty}}d\tau \le C\int_{t/2}^{t}\|\psi(\cdot, \tau)\|_{L^{\infty}}^{2}d\tau &\le C\int_{t/2}^{t}(1+\tau)^{-\alpha}d\tau \le C(1+t)^{-\alpha+1}, \ \t\ge0.
\left(B^{F}(T_{N,n_N-1})\right)^{n_N-1}\times A^{(N-n_N+1)^{F}}_{\varepsilon_{N,n_N-1}}\subset A^{N^{F}}_{\varepsilon_N}, \lim_{N\to \infty}T_{N,n_N-1}=\infty, \frac{1}{\theta_N^{2}}<1+\frac{(1-\varepsilon_{N,n_N-1})^2((N-n_N+1)^F-1)}{(n_N-1)T_{N,n_N-1}^2}.
\lim_{k\rightarrow\infty}k(E(x,t,k)-I)=-\frac{1}{2\pi i}\int_{\Gamma_E}\mu(x,t,s)w(x,t,s)ds =-\frac{1}{2\pi i}\left(\oint_{|s+k_0|=\epsilon}+\oint_{|s-k_0|=\epsilon}\right)\mu(x,t,s)w(x,t,s)ds-\frac{1}{2\pi i}\int_{\Gamma}\mu(x,t,s)w(x,t,s)ds
g' \kappa g'^{-1}= g'(I+Ap)g'^{-1}&= (g+Bp)(I+Ap) (g^{-1}-g^{-1}Bg^{-1}p) &= I + (Bg^{-1}+ gAg^{-1}- Bg^{-1}) p &= I + gAg^{-1}&= g(I+Ap)g^{-1}&= g \kappa g^{-1}.
\boldsymbol{w}_{k+1}= \boldsymbol{w}_k + \mu e_k \boldsymbol{x}_k- \rho_{\rm p}\frac{\left( \| \boldsymbol{w}_k \|_{p}\right)^{1-p}( \boldsymbol{w}_k)}{| \boldsymbol{w}_k|^{(1-p)}}
{{\mathcal B}_i^{AF}}\left(\pmb \rho \right) = \frac{{\sqrt{{\left({{X_i}+{Y_i}+ 1}\right)\left({{W_i}+ 1}\right)}}}}{{\sqrt{\left({{X_i}+{Y_i}+ 1}\right)\left({{W_i}+ 1}\right)}+ \sqrt{Z_i +W_i+1}}},
P^{0}= E = \frac{1}{2 e^{2}{\phi_{0}}^{2}}\int d^{2}x \left[ \frac{1}{2}F_{i j}^{2}+ \vert D_{i}\Phi \vert^{2}+ \frac{e^{2}}{4}( \vert \Phi \vert^{2}-{\Phi_{0}}^{2})^{2}\right]
(\alpha_m,\beta_m)_{\rho,E_{\rho}^m, m}= \Phi_{E_{\rho}^m}(\beta_m)(\xi)-\xi = \Phi_{E_{\rho}^n}(\operatorname{N}_{m,n}(\beta_m))(\xi)-\xi =(\alpha,\beta)_{\rho,E_{\rho}^n, n},
\Omega_{J}:=\Omega\cap(B^{*}\times T_{0}\times T_{J}), Y_{J}:=(B^{*}\times T_{0}\times T_{J})\setminus \Omega_{J}&=\{(t,(x_{1},\dots,x_{n}),[\alpha_{1}:\dots:\alpha_{k}])\in B^{*}\times T_{0}\times T_{J}\ | \ \sum_{j\in J}\alpha_{j}f_{j}(t,x)=0\}.
R_\lambda (h,k) = \phi_\lambda(h) (1-\langle h\cdot z_0, k\cdot z_0\rangle)^{-\lambda}\overline{\phi_\lambda(k)}= e^{-2\pi i \lambda (x-y)}\left(1-\frac{1}{2n}\sum_{i=1}^n t_i \overline{s_i}\right)^{-\lambda}= e^{-2\pi i \lambda (x-y)}\left(1-\frac{1}{2n}\sum_{i=1}^n t_i (s_i)^{-1}\right)^{-\lambda}
\|u\|_{X^{s,b}_{1,\delta}}= \inf_{u = \tilde{u}, \; |t| \leq \delta}\| \tilde{u}\|_{X_1^{s,b}}\|v\|_{X^{s,b}_{\alpha,\delta}}= \inf_{v = \tilde{v}, \; |t| \leq\delta}\| \tilde{v}\|_{X^{s,b}_\alpha}.
\begin{aligned}\sum_{i=1}^{n}\sum_{j=0}^{k_i-2}Z^{\star}_{I}(j+1,k_{i+1},\ldots,k_n,k_1,\ldots,k_{i-1},k_i-j;(\alpha,\beta)) = (k-n)Z(n|k-n+1;(\alpha,\beta))+nZ(n+1|k-n;(\alpha,\beta))\end{aligned}
\binom{f\left((uk+vl)/\mathcal{N}(q_2)\right)}{f\left((-vk+ul)/\mathcal{N}(q_2)\right)}=\binom{f\left((\tilde{u}k+\tilde{v}l)/\mathcal{N}(q_2)\right)}{f\left((-\tilde{v}k+\tilde{u}l)/\mathcal{N}(q_2)\right)}
T_{\alpha,\beta, m}[y](x) :=(1-x^2)y''(x)+\left(\beta-\alpha-(\beta+\alpha+2)x-2(1-x^2)\left(\log(P_m^{(-\alpha-1,\beta-1)}(x))\right)'\right)y'(x) &+\left((\alpha-\beta-m+1)m-2\beta(1-x)\left(\log(P_m^{(-\alpha-1,\beta-1)}(x))\right)'\right)y(x)
12!!x^{\otimes(6)}=M_{[i]}^{(6)}+10r^{2}\langle\langle M_{[i]}^{(4)}\delta_{[..]}\rangle\rangle+72r^{4}\langle\langle M_{[i]}^{(2)}\delta_{[..]}^{\otimes2}\rangle\rangle+240r^{6}\langle\langle\delta_{[..]}^{\otimes3}\rangle\rangle
\Delta_q(u\cdot\nabla v)= \sum_{|q-p|\leq 2}\Delta_q(u_{\leq{p-2}}\cdot\nabla v_p)+\sum_{|q-p|\leq 2}\Delta_q(u_{p}\cdot\nabla v_{\leq{p-2}}) &+\sum_{p\geq q-2}\Delta_q(\tilde u_p \cdot\nabla v_p).
|P^{\vec{h}}_{\mathbb{Z}^d,\beta,H}(\sigma_x=s_1,\sigma_y=s_2)-P^{\vec{h}}_{\mathbb{Z}^d,\beta,H}(\sigma_x=s_1)P^{\vec{h}}_{\mathbb{Z}^d,\beta,H}(\sigma_x=s_2)|
P_{w}=\frac{C_{d}}{2C_{SO}}\rho u^{3}\pi R_{0}^{2}(\frac{P_{RMF}}{u^{2}R_{0}\rho\mu_{0}})^{1/f_{o}}(\frac{\mu_{0}^{2}}{4\pi\eta_{p}}\frac{2f_{o}-1}{f_{o}^{2}})^{1/f_{o}}
x(t)=x_{0}+\frac{c^{2}}{g}[\frac{1}{\sqrt{1-\frac{v_{0}^{2}}{c^{2}}}}-\sqrt{\frac{1}{1-\frac{v_{0}^{2}}{c^{2}}}+\frac{g^{2}t^{2}}{c^{2}}-\frac{2v_{0}gt}{c^{2}\sqrt{1-\frac{v_{0}^{2}}{c^{2}}}}}].
\begin{aligned}\int_{\Omega}\ast dw\wedge [\eta,u]_{1}=\int_{\Omega}dw\wedge\ast\delta(\eta\wedge u)=\int_{\partial\Omega}(\langle dw,\eta\rangle_{\Lambda^{1}}i_{\mathcal{N}}u)v_{\partial\Omega}. \end{aligned}
w_n:=\varepsilon \gamma^{-2}\lVert \mathcal{F}(U_n) \rVert_{s_0}, B_n:=\varepsilon\gamma^{-1}\lVert \mathfrak{I}_n \rVert_{s_0+\beta_1}+\varepsilon\gamma^{-2}\lVert \mathcal{F}(U_n) \rVert_{s_0+\beta_1}.
\delta_{n}(x)~=~ \sqrt{2x^{2}~+~x\bigg(\frac{p^{n}_n+x^2[n](p_n-q_n)}{2(p^{n-1}_nq_n-p^{n}_n+q^{2}_n[n-1])}~-~\frac{\sqrt{{p^{2n}_n+x^{4}[n]^{2}(p_n-q_n)^{2}+2x^{2}[n](2p_nq_n([n]-1)-p^{n}_n(p_n-q_n))}}}{2(p^{n-1}_nq_n-p^{n}_n+q^{2}_n[n-1])}\bigg)}\tiny
\mathbf{d}^\bullet(r) =\sup_{x\in X}\inf_{y\leq^\mathbf{d}x^\bullet_r}\mathbf{d}(x,y). \mathbf{d}_\bullet(r) =\sup_{x\in X}\inf_{x_\bullet^r\leq^\mathbf{d}y}\mathbf{d}(y,x).
\frac{df}{dv}(v)-\frac{\lambda}{3\kappa^2}v=\frac{\lambda}{18\kappa^2}\left\{2v(y(v)+1)+\frac{1}{v^2}\int_0^vv'^3\frac{dy}{dv}(v')dv'\right\}+\mathcal{O}(v^2).
F(t_1,t_2) = P(T_1\leq t_1, T_2 \leq t_2) &= \Phi_2\bigg[\frac{1}{\alpha_1}((\frac{t_1}{\beta_1})^{\nu_1}-(\frac{\beta_1}{t_1})^{\nu_1}),\frac{1}{\alpha_2}((\frac{t_2}{\beta_2})^{\nu_2}-(\frac{\beta_2}{t_2})^{\nu_2});\rho \bigg] ; t_1 > 0, t_2 > 0.
\O(J_{_U}) \cap \bigcap \mathcal{U}= \O(J_{_U}) \cap \bigg[ \Big( \bigcup_{U \neq V \in \mathcal{U}}V \Big) \cup \Big( \bigcup_{U' \neq V \in \mathcal{U}}V \Big) \bigg] & = \bigg[ \O(J_{_U}) \cap \Big( \bigcup_{U \neq V \in \mathcal{U}}V \Big) \bigg] \cup \bigg[ \O(J_{_U}) \cap \Big( \bigcup_{U' \neq V \in \mathcal{U}}V \Big) \bigg] = \emptyset.
\left(1 + \frac{R_1\big(f_1^{-1}\big(\frac{y_2}{n^2}\big)\big)}{n}\right)-\left(1 + \frac{R_1\big(f_1^{-1}\big(\frac{y_2}{(n+1)^2}\big)\big)}{n+1}\right) = O\left(\frac{1}{n^2}\right),
G_{L_{n-1}}(z) = \textstyle \prod_{i=0}^{n-2}f_{n-1-i}^{K-1}(z) f_k(z) = \begin{cases}q_k + p_k f_{k+1}^{K}(z), ~1 \le k \le n-1 z^{\frac{1}{K}}, ~k=n,\end{cases}
Mf(0,h) >\sum_{n=m}^{+\infty}\frac{1}{h}\int_{\pi n}^{\pi(n+1)}\frac{\vert \sin(s)\vert}{\pi^2 (n+1)^2}ds=\sum_{n=m}^{+\infty}\frac{2}{h\pi^2(n+1)^2}\geq\sum_{n=m}^{+\infty}\frac{2(m-1)}{\pi (n+1)^2}&=\frac{2(m-1)}{\pi}\sum_{k=0}^{+\infty}\frac{1}{(k+m+1)^2}\geq \frac{2m}{\pi}\int_{m+2}^{+\infty}\frac{dx}{x^2}&=\frac{2}{\pi}\frac{m-1}{m+2}.
P_{g_0}^n\psi_0=\frac{(n-1)!\Lambda}{\Lambda_1}-\dfrac{\Lambda \tilde{K}e^{n(\psi_0-w_0\circ \pi)}}{\int_{\mathbb{S}^n}\tilde{K}e^{n(\psi_0-w_0\circ \pi)}dg_0}.
[ \delta ( \epsilon_{1}) , \delta ( \epsilon_{2}) ] \zeta^{A}= \frac 1 2 \partial_{a}\zeta^{A}\bar{\epsilon}_{2}\gamma^{a}\epsilon_{1}+ \frac 1 4 \Gamma^{A}\bar{\epsilon}_{2}\epsilon_{1}- \frac 1 4 \gamma_{a}\Gamma^{A}\bar{\epsilon}_{2}\gamma^{a}\epsilon_{1}.
P ( \theta^{2}, \bar{\theta}^{2}) = 1 -{\frac{i}{2 R}}( \theta^{2}- \bar{\theta}^{2}) +{\frac{1}{3 ! R^{2}}}\theta^{2}\bar{\theta}^{2}, \qquad \theta^{2}\equiv \theta^{A}\theta_{A}, \quad \bar{\theta}^{2}\equiv \bar{\theta}_{\dot{A}}\bar{\theta}^{\dot{A}},
\frac{\sqrt{\epsilon}e^{-\epsilon/kT}}{\int_{0}^{\infty}\sqrt{\epsilon}e^{-\epsilon/kT}}=\frac{\sqrt{\epsilon}e^{-\epsilon/kT}}{\frac{\sqrt{\pi}}{2}(kT)^{3/2}}=\frac{2\sqrt{\epsilon}e^{-\epsilon/kT}}{\sqrt{\pi(kT)^{3}}}
\begin{aligned}\|X_{t}-Z_{t}\|_{H}^{2}= \|x-Z_{0}\|_{H}^{2}+2\int_{0}^{t}(-\eta_{s}-G_{s},X_{s}-Z_{s})_{H}\,ds +\int_{0}^{t}(L^{b}X_{s}-P^{\ast}L^{b}PZ_{s},X_{s}-Z_{s})_{H}\,ds +2\int_{0}^{t}(X_{s}-Z_{s},B(X_{s}-PZ_{s})\,dW_{s})_{H}+\int_{0}^{t}\|B(X_{s}-PZ_{s})\|_{L_{2}(U,H)}^{2}.\end{aligned}
Z ( \beta ) = \pi^{\zeta / 2}\beta^{- \zeta / 2}d e t ( \widetilde{T})^{- 1 / 2}\prod_{k = 1}^{N}d e t ( \widetilde{T}_{k}^{*}\widetilde{T}_{k})^{\frac{1}{2}( - 1 )^{k - 1}}
F(t_{\mathtt{D}_{i}},\beta_{\mathtt{D}_{i}},\psi_{\mathtt{D}_{i}}^{(n)})|_{\psi_{\mathtt{D}_{i}}^{(n)}=t_{\mathtt{D}_{i}}/\beta_{\mathtt{D}_{i}}}=t_{\mathtt{D}_{i}}\beta_{\mathtt{D}_{i}}=f(t_{\mathtt{D}_{i}},\beta_{\mathtt{D}_{i}}).
\begin{cases}\displaystyle\exp\left(\frac{1}{2}\left(\frac{x}{\tan x}-1\right)\right)< \displaystyle\frac{\sin x}{x}< \displaystyle\exp\left(\left(\log\frac{\pi}{2}\right)\left(\frac{x}{\tan x}-1\right)\right) x\in(0,\pi/2), \sqrt{AX}<P<A\left(\frac{X}{A}\right)^{\log(\pi/2)}.\end{cases}
U_n = \{\, x \in \P^1 \mid \lvert x - d_0 \rvert \leq \lvert b_2 \rvert \, \}\setminus \bigcup_{\nu=0}^{N_n}D_{\nu}, D_{0}= \{\, x \in \P^1 \mid \lvert x - d_0 \rvert < \lvert b_1 \rvert \, \}, D_{\nu}= \{\, x \in \P^1 \mid \lvert x - d_\nu \rvert < \lvert d_0 - d_\nu \rvert \, \}(1 \leq \nu \leq N_n).
\sigma \left(\frac{L^S(r,\pi \boxtimes \chi, \varrho_{5})}{(2\pi i)^{2k+3r}G(\chi)^3 \langle F, F\rangle}\right) = \frac{L^S(r,{}^\sigma\pi \boxtimes{}^\sigma\!\chi, \varrho_{5})}{(2\pi i)^{2k+3r}G({}^\sigma\!\chi)^3 \langle{}^\sigma\!F,{}^\sigma\!F\rangle}.
[S(t)\mu](x) \le\sum_{k=1}^m\sum_{i=1}^\infty \int_{B_{k,i}}G(x-y,t)\,d\mu(y) \le Ct^{-\frac{N}{\theta}}\sup_{z\in{\bf R}^N}\mu(B(z,t^{\frac{1}{\theta}}))\sum_{k=1}^m \sum_{i=1}^\infty \sup_{y\in B_{k,i}}\left(1+t^{-\frac{1}{\theta}}|x-y|\right)^{-N-\theta}.
\sum_{n=1}^{N}\left[\begin{array}{c}N n\end{array}\right] \frac{\left(\frac{c}{d}\right)_n(d)^n(-1)^{n-1}q^\frac{n(n+1)}{2}}{(cq)_n}=-\sum_{n=1}^{N}\frac{\left(\frac{d}{c}\right)_{n}(q^{-N})_nq^n}{(q)_n\left(\frac{q^{-N}}{c}\right)_n},
\varphi ( x_{4}, r ) = \pi + \pi{\frac{( r^{2}- x_{4}^{2})^{3}+ 3 ( r^{4}- x_{4}^{4}- x_{4}^{2}) + r^{2}- 1}{[ ( r^{2}- x_{4}^{2})^{2}+ 2 ( r^{2}+ x_{4}^{2}) + 1 ]^{\frac{3}{2}}}}.
Z_n:=V^{-1/2}\left(\begin{array}{c}\sqrt{n}(T_n-g({{\beta_{n,0}}})) \Lambda_n + \frac{1}{2}h^TI({{\beta_{n,0}}})h\end{array}\right)\stackrel{{{\beta_{n,0}}}}{\rightsquigarrow}\mathcal N_2(0,I)\sim Z.
\varphi\left(t,x\right):=\theta^{\varepsilon}\left(t,x\right)-\left(\theta^{\varepsilon}_{0}\left(t,x\right)+\varepsilon m^{\varepsilon}\left(x\right)\bar{\theta}\left(x,\frac{x}{\varepsilon}\right)\cdot\nabla_{x}\theta^{0}\left(t,x\right)\right),
W(x,t)-\varphi(z,\tau) \le W_{\kappa,z,\tau}(x,t)-\varphi(z,\tau) &\le \kappa+M_\kappa W_{z,\tau}(x,t) &\le \kappa+M_\kappa (W_{z,\tau}(x,t)-W_{z,\tau}(z,\tau)) &\le \kappa+M_\kappa\omega(|z-x|\vee|\tau-t|),
B =C A^2E^2(B^2-C^2)=\lambda_1^2\lambda_5^2( \lambda_2^2-\lambda_3^2)=\lambda_1^2\lambda_5^2(0)=0 A^2BCD^2=\lambda_1^2\lambda_2 \lambda_3\lambda_4^2=\lambda_1^2\lambda_2^2\lambda_4^2.
\int_x^\infty f(t) \,d_{q}t = \frac{1}{1+q}\int_{x^2}^\infty \frac{f(\sqrt{t})}{\sqrt{t}}\,d_{q^2}t,\;\; D_{q,\rho}(f(\rho^2))= \rho (1+q) \left(D_{q^2}f\right)(\rho^2),
(-g)^{-1/2}\partial_{\mu}(\sqrt{-g}g^{\mu\nu}\partial_{\nu}\varphi)+\sigma,_{\mu}\varphi^{,\mu}+V^{\prime}(\varphi)+\tilde{e}^{2}\varphi g^{\alpha\beta}\tilde{A}_{\alpha}\tilde{A}_{\beta}=0,
\int_{\mathbb{R}}\psi(x)d\mathcal{N}(x)=\mathbb{E}\big(\langle \delta_n, \psi(H^{\omega})\delta_n\big)\leq \|\psi\|_{\infty}~Q_\mu(|s_\psi|), ~s_\psi=\mathrm{supp}~\psi.
L_{1}=\frac{\Gamma\left((m+1)/2\right)\sqrt{m}\left[\xi_{p}\left(1+\frac{\xi_{p}^{2}}{m}\right)^{-(m-1)/2}-\xi_{q}\left(1+\frac{\xi_{q}^{2}}{m}\right)^{-(m-1)/2}\right]}{\Gamma(m/2)(m-1)\sqrt{\pi}F_{Y}(\xi_{p},\xi_{q})},
\int_{0}^{\infty}\frac{p_1}{a_1^{d_1}}\, \frac{x^{d_1-1}}{\Gamma \left(d_1/p_1\right)}\, e^{- \left( x / a_1 \right)^{p_1}}\, (d_1 - d_2) \, \ln x \, dx begin{align*}1ex]= \underbrace{\frac{p_1}{a_1^{d_1}}\, \frac{1}{\Gamma \left(\frac{d_1}{p_1}\right)}\, (d_1 - d_2)}_{B}\int_{0}^{\infty}x^{d_1-1}\, e^{- \left( x / a_1 \right)^{p_1}}\, \ln x \, dx
\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}J(s;z,r,h,k,w)ds = \frac{1}{2\pi i}\int_{\sigma_0-i\infty}^{\sigma_0+i\infty}J(s;z,r,h,k,w) ds + Res (J,s_0) + Res (J,0) Res(s_0;J) = \Phi(z^k,s_0+1, \frac{r}{k})\frac{\Gamma(s_0) A_{rh,k}}{( k w)^{s_0}}Res(0;J) =\Phi(z^k,1, \frac{r}{k})D_{rh,k}(0).
\mathcal{S}_{q}(z)=\sum_{n=0}^{\infty}\frac{(-1)^{n}q^{n(n+1)}}{(q;q)_{2n+1}}z^{2n+1}\quad\mbox{and}\mathcal{C}_{q}(z)=\sum_{n=0}^{\infty}\frac{(-1)^{n}q^{n^{2}}}{(q;q)_{2n}}z^{2n},
T_{\ell}(f_1) = \lambda(\ell)f_1, \mbox{\ \and}T_{\ell}(f_0) + p^{j-1-\nu_p(n)}\left(T_{\ell}(f_1E_{p-1}^n) - T_{\ell}(f_1)E_{p-1}^n \right) = \lambda(\ell)f_0.
\mathcal{D}_{X}X^{n}\equiv i [ P , X^{n}] = \frac{1}{q^{1 / 2}+ q^{- 1 / 2}}\frac{q^{n}- 1}{q - 1}\left( U + q^{- ( n - 1 )}U^{\dagger}\right) X^{n - 1},
\mathfrak i_q(2n-1,n-1) =\sum_{\pi\in I(2n-1,n-1)}q^{\ell_{PF_{2n-1}}(\pi)}= &=\left(\sum_{i=1}^{2n-1}q^{2n-1-i}\right)\cdot [2(n-1)-1]_q!!= &=\left(q^{2n-1-1}+q^{2n-1-2}+\cdots+q+1\right)\cdot [2n-3]_q!!= &=[2n-1]_q\cdot [2n-3]_q!!=[2n-1]_q!!\,\,.
\xi_d(x)(1) = \oplus_{k=0}^n \delta_{\gamma_{xv_0}(k)}\xi_i(x)\left (\oplus_{k=0}^n\delta_{\gamma_{yv_0}(k)}\right ) = \oplus_{k=0}^n\delta_{\gamma_{xyv_0}(k)}\xi_1(x)\left (\oplus_{k=0}^n\delta_{\gamma_{yv_0}(k)}\right ) = \langle\oplus_{k=0}^n\delta_{\gamma_{xyv_0}(k)}, \oplus_{k=0}^n \delta_{\gamma_{v_0}(n-k)}\rangle
{\rm det_{\it u}}[{\cal O}]^{{1\over 2}\sum_i c_{u_i}}\equiv\lim_{\{M_{u_i}^2\}\rightarrow\infty}\prod_{i=1}^{n_u}\hskip .1truecm{\rm det}\left[{\cal O}+M_{u_i}^2\, 1_u\right]^{{1\over 2}c_{u_i}},
g_{{\bf{k}},{\bf{q}}}({\bf{x}}) = -e^{-i\mbox{}{\bf{q.x}}}(\frac{1}{2\mbox{}N\mbox{}\epsilon_{{\bf{q}}}})\Lambda_{{\bf{k}}}(-{\bf{q}})\omega_{{\bf{k}}}({\bf{q}}) + i\mbox{}U_{{\bf{q}}}({\bf{x}})\Lambda_{{\bf{k}}}(-{\bf{q}})= -f^{*}_{{\bf{k}},{\bf{q}}}({\bf{x}})
\sigma^{\mu}\frac{\partial}{\partial x^{\mu}}\psi_{R}(x)\mapsto\sigma^{\mu}\frac{\partial}{\partial x^{\prime\mu}}\psi_{R}^{\prime}(x^{\prime})=(S^{-1})^{\dagger}\sigma^{\mu}\frac{\partial}{\partial x^{\mu}}\psi_{R}(x)
F^\times F_\infty^\times \cap (\hat{A}^\times\cap(1+I\hat{A})) \cong F^\times F_\infty^\times \cap \hat{A}^\times &\cong F^\times \cap (F_\infty^\times\cdot\hat{A}^\times) &= \{x \in F^\times \mid v\neq\infty, x\in\mathcal{O}_{v}\}&= \Gamma(C\backslash{\infty}, \mathcal{O}_C^\times) = A^\times,
\delta(C_4\wedge H_3\wedge G_3) =C_4\wedge H_3\wedge d\delta C_2 &=-d(C_4\wedge H_3\wedge \delta C_2)+G_5\wedge H_3\wedge \delta C_2 &=d(\ldots)+\delta C_2\wedge G_5\wedge H_3.
|\begin{matrix}\frac{\partial x}{\partial r}&\frac{\partial x}{\partial\theta}\\ \frac{\partial y}{\partial r}&\frac{\partial y}{\partial\theta}\end{matrix}|=|\begin{matrix}cos\theta&-rsin\theta\\sin\theta&rcos\theta\end{matrix}|=rcos^{2}\theta+rsin^{2}\theta=r
\widetilde\rho\left(\gamma_{\frak{h}}(v,g)\right) =\rho\left(\gamma_{\frak g}(v^{\prime},g)\right) =\overline\tau\left(\langle v^{\prime},v_g\rangle_{\beta}\right) =\overline\tau\left(\langle v,v_g\rangle_{\beta}\right)
X_{1}Z_{2}X_{3}=[\begin{matrix}c_{2}&-c_{3}s_{2}&s_{2}s_{3}\\c_{1}s_{2}&c_{1}c_{2}c_{3}-s_{1}s_{3}&-c_{3}s_{1}-c_{1}c_{2}s_{3}\\s_{1}s_{2}&c_{1}s_{3}+c_{2}c_{3}s_{1}&c_{1}c_{3}-c_{2}s_{1}s_{3}\end{matrix}]
[\begin{matrix}t\\x\\y\\z\end{matrix}]=[\begin{matrix}1&0&0&0\\ 0&cos(\lambda)&-sin(\lambda)&0\\ 0&sin(\lambda)&cos(\lambda)&0\\ 0&0&0&1\end{matrix}][\begin{matrix}t_{0}\\x_{0}\\y_{0}\\z_{0}\end{matrix}]
\begin{array}{ll}y_{{T_i}}^{[3]}= \sqrt{{P_R}}{h_i}{S_R}+ \sum_{k = 1}^{{L_{{T_i}}}}{\sqrt{P_{I,{T_i}}}{c_{{T_i},k}}}{\cal I}_{{T_i},k}^{[3]}+ n_{{T_i}}^{[3]}\end{array}
I_2 =d^{f(p)}\big(H_0(x)\cdot(\delta_tH_0(x))^{-1}\cdot\delta_t H_1(x),H_0(x)\big) &=d^{f(p)}\big((\delta_{t}H_0(x))^{-1}\cdot\delta_t H_1(x),0\big) &=d^{f(p)}\big(\delta_t H_1(x),\delta_t H_0(x)\big) &=td^{f(p)}\big(H_1(x),H_0(x)\big).
\mathcal J_1:=\int_0^t\!\!\!\int_{\mathbb R}\hat c\,\partial_s\alpha\,\beta\, dx_2 ds\quad\mbox{and}\mathcal J_2:=\int_0^t\!\!\!\int_{\mathbb R}\hat c\,\partial_s\alpha\,\varphi\, dx_2 ds\,.
|{\hat \xi}|^2 \leq \frac{4}{\hat s}\int_{W_1}|{\hat\xi}|^2\, d\|V_{\hat t}^{(\xi_1)}\| \leq \frac{32}{3\hat s}\int_{W_1}{\rm dist}(\cdot,J)^2+{\rm dist}(\cdot,J_{\hat\xi})^2\,d\|V_{\hat t}^{(\xi_1)}\| &\leq \frac{32}{3\hat s}\sum_{j=1}^2\int_{B_{2\hat s}(\xi_j)}{\rm dist}(\cdot, J_{\xi_j})^2\, d\|V_{\hat t}\|.
\,x_{f_1,\cdots,f_{2n-2}}=\left(\sum\limits_{k=1}^{n-1}\alpha(k)\,D_k\right)-\sum_{i=1}^{n}(-1)^{i+n}\left(\sum\limits_{q=1}^{n-1}\beta(i,q)\,D_q\right)\left(\sum\limits_{s=1}^{n-1}\gamma(i,s)\,D_s\right),
\left\{\intop_{\mu-iT}^{\mu+iT}+\intop_{\mu+iT}^{\frac{1}{2}-\mu+iT}+\intop_{\frac{1}{2}-\mu+iT}^{\frac{1}{2}-\mu-iT}+\intop_{\frac{1}{2}-\mu-iT}^{\mu-iT}\right\}\,\frac{\zeta_{p^{\prime}}(1-s)\,\zeta_{p}(2s)}{2\cos\left(\frac{\pi s}{2}\right)}\,\left(\frac{x}{2\pi}\right)^{-s}\,ds=2\pi i\,R_{p,p^{\prime}}(x),
|T_{1}(x)| \leq C \sum_{k=-\infty}^{-1}\frac{\rho(2^{k}R)}{(2^{k}R)^{Q}}\int_{|xy^{-1}|<2^{k+1}R}|f(y)|dy \leq C Mf(x) \sum_{k=-\infty}^{-1}\rho(2^{k}R) \leq C Mf(x) \sum_{k=-\infty}^{-1}\int_{2^{k}R}^{2^{k+1}R}\frac{\rho(t)}{t}dt = C Mf(x) \int_{0}^{R}\frac{\rho(t)}{t}dt.
y_{1L}'(y_3)=\frac{-1/2\alpha}{1+\lambda_s+y_{1L}(y_3)}\;,y_{2L}'(y_3)=\frac{1-\lambda_s-y_{1L}(y_3)}{1+\lambda_s+y_{1L}(y_3)}\;,y_{3L}'(y_3)=1\;.
\lim_{\phi \rightarrow \infty}\Phi_{j}( x , \bar{x}) = e^{\frac{2 j}{\alpha_{+}}\phi}\delta^{( 2 )}( \gamma - x ) + \frac{2 j + 1}{\pi}\left| \gamma - x \right|^{- 4 j - 4}e^{\frac{- 2 j - 2}{\alpha_{+}}\phi}+{\cal O}( e^{\frac 1{\alpha_{+}}j \phi}) ,
R(x,y) = ((a_0 x^{q+1}+ b_0 x^qy + c_0 xy^q + d_0 y^{q+1}), (a_1 x^{q+1}+ b_1 x^qy + c_1 xy^q + d_1 y^{q+1})) = (f(x,y),g(x,y))
\hspace{1 . 7 5 i n}\times e^{\overline{{{\Lambda}}}_{1}}( ( - 1 )^{n - 1}q z )^{\partial_{\overline{{{\Lambda}}}_{1}}+ \frac{i}{n}}q^{i}( - 1 )^{i n + \frac{1}{2}i ( i + 1 )}
{}_{p}C_{p - 1}\hat{h}_{i - 1}^{( p - 1 )}\hat{e}_{i - 1}^{( 1 )}= p \hat{e}_{i - 1}^{( 0 )}\hat{h}_{i - 1}^{( p - 1 )}\hat{h}_{i - 1}^{( 1 )}.
\begin{multlined}[t][12.5cm]\widetilde{\Pi}^{P}_{\nu_{G},\nu_{G}+3b}\left((z_{0},z_{1}),(u_{0},u_{1})\right)=\frac{(2b+\nu_{G}+1)!}{\pi(b+\nu_{G})!b!}(z_{0}\overline{u}_{0})^{b+\nu_{G}}(z_{1}\overline{u}_{1})^{b}.\end{multlined}
I_{3k}^{(m)}(F(w_1))= \idotsint_{\mathcal{R}_{k-1}}\left( \int_0^{T_m\left(\frac{\log |w_1|}{\log R}\right)}F(x_1,\dots,x_k) \, dx_m\right)^2 dx_1 \dots dx_{m-1}dx_{m+1}\dots dx_k
\bar{\Psi}^{\prime \rho}{}_{r}( z^{\prime}) = \Omega ( z ; g )^{- \eta}D^{\rho}{}_{\sigma}( L^{- 1}( z ; g ) ) D_{r}{}^{s}( U^{- 1}( z ; g ) ) \bar{\Psi}^{\sigma}{}_{s}( z ) .
E^{\Pi^{D_T}}[e^{u\sqrt{T}G(b)}|X^{T}] =e^{\Lambda_T(u)}\frac{\int_{D_T}e^{S_T(b) + \ell_T(b_u)}d\Pi(b)}{\int_{D_T}e^{\ell_T(b)}d\Pi(b)},
\lim_{\lambda \to \infty}I^\lambda[u^\lambda] = \mu(B) \geq K(n,p) \sum_k \nu_k^\frac{p}{q}= K(n,p) \sum_j |Gx_j| \left(\frac{\tilde{\nu_j}}{|Gx_j|}\right)^\frac{p}{q}= K(n,p) \sum_j |Gx_j|^{1 - \frac{p}{q}}\tilde{\nu_j}^\frac{p}{q}.
\widehat{Var}_{n}(t_j(X_i,\theta))=Var_{P_n}(t_j(X_i,\theta))+o_{P_n}(1) &\widehat{Cov}_n(m_j(X_i,\theta),t_j(X_i,\theta))\le \hat \sigma_{n,j}(\theta_n)\widehat{Var}_{n}(t_j(X_i,\theta))^{1/2},
n(2\ln n)^{\frac{1}{2}}\int_{I_1}D_n(\sqrt{4-y^2}/S(I) \cdot G_n(x)/2)dy=n(2\ln n)^{-\frac{1}{2}}D_n(G_n(x)/2)\times 2S(I)\int_{0}^{(b_1/a_1-1)\ln n}\frac{D_n((1+z/\ln n) G_n(x)/2)}{D_n(G_n(x)/2)}\frac{(1+z/\ln n)a_1}{\sqrt{4-(1+z/\ln n)^2a_1^2}}dz.
R(w; k \lambda) = \prod_{j=0}^{k-1}R(k w + j; \lambda)= d_k \cdot \prod_{j=0}^{k-1}\dfrac{\left(w+ \frac{u_1 + j}{k}\right) \cdots \left(w+ \frac{u_m +j}{k}\right)}{\left(w+ \frac{v_1 + j}{k}\right) \cdots \left(w+ \frac{v_n +j}{k}\right)}.
U_v(\alpha, T(x^n)) = U_v\left(\alpha, \sum_{k=0}^{\deg T}\frac{T^{(k)}(1)}{k!}(x^n-1)^k\right) \leq \nu_v\left( \sum_{k=0}^{\deg T}\frac{T^{(k)}(1)}{k!}(x^n-1-f(x))^k\right).
\begin{aligned}w_{2s}= \frac{(-\Delta-j+1)_N(-j)_s(-\Delta-j+1)_s}{(-1)^{j}\binom{2j}{j}s!j!(\Delta)_{j}(1-\Delta)_s}, w_{2s+1}= \frac{(-\Delta-j+1)_N(-j+1)_s(\Delta-j)_s}{(-1)^{j+1}\binom{2j}{j}s!(j-1)!(-\Delta)_{j+1}(1+\Delta)_s}, \end{aligned}
A_{\phi}=\frac{\mu_{0}I}{\pi}\frac{R}{l}[\frac{\zeta}{\sqrt{(R+\rho)^{2}+\zeta^{2}}}(\frac{m+n-mn}{mn}K(m)-\frac{1}{m}E(m)+\frac{n-1}{n}\Pi(n,m))]_{\zeta_{-}}^{\zeta_{+}},
\lim_{r \to s}\int_{\gamma Q_k\cap\{r<|\bar{u}_k-a_k|+ \tfrac{|\tau_{L_k}({\bar v}_k)-{\bar v}_k|}{\lambda}<s\}}|\nabla\bar{u}_k| \,dy &=\lim_{r\to s}\int_{\gamma Q_k\cap\{r<|\bar{u}_k-a_k|+ \tfrac{|\tau_{L_k}({\bar v}_k)-{\bar v}_k|}{\lambda}<s\}}(|\tau_{L_k}({\bar v}_k)-{\bar v}_k|^p+ |\tau_{L_k}({\bar v}_k)|^p)\,dy =0.
y_0 y_i + y_i y_0 = \omega^{-\frac{i(i+1)}{2}}(x_0 x_i + x_i x_0) = \omega^{-\frac{i(i+1)}{2}}a_i x_{\frac{i}{2}}^2 = \omega^{-\frac{i(i+1)}{2}}a_i \omega^{\frac{i}{2}(\frac{i}{2}+1)}y_{\frac{i}{2}}^2 = \omega^{\frac{-i^2}{4}}a_i y_{\frac{i}{2}}^2.
\langle \sum_{1\leq i\leq j\leq 2m}b_{ij}x_ix_j+cx_n,\sum_{s=1}^{2m}x_s^3\rangle = &\sum_{\substack{1\leq i\leq j\leq 2m j\neq i'}}3b_{ij}(\sigma(i)x_jx_{i'}^2+\sigma(j)x_ix_{j'}^2)+\sum_{i=1}^m 3b_{ii'}(x_{i'}^3-x_i^3)+c\sum_{i=1}^{2m}x_i^3
p(v) = - \left[v^k f_j(v)\right]_+, q_{\alpha^\vee}(v) = \left[v^k d_j^{-1/2}d_{\alpha^\vee}\Big( \frac{\Delta_{v_j^\ast, f_{\alpha^\vee}f_j v_j}(v)}{\Delta_{v_j^\ast, v_j}(v)}- \frac{\Delta_{v_j^\ast, f_j v_j}(v) \Delta_{v_j^\ast, f_{\alpha^\vee}v_j}(v)}{\Delta_{v_j^\ast, v_j}(v)^2}\Big)\right]_+
\left\{\begin{array}[c]{ccc}l_{1}l_{2}l_{3}j j j\end{array}\right\}=l_{1}!l_{2}!l_{3}!\Delta(l_{1},l_{2},l_{3})\sqrt{\frac{(n-l_{1})!(n-l_{2})!(n-l_{3})!}{(n+l_{1}+1)!(n+l_{2}+1)!(n+l_{3}+1)!}}& \cdot\displaystyle{\sum_{k}}\frac{(-1)^{n+k}(n+1+k)!}{(n+k-l_{1}-l_{2}-l_{3})!R(l_{1},l_{2},l_{3};k)}\quad\quad,
\mathbb{P}_N\circ(\Phi_i, \widetilde \Phi_i)^{-1}(d\varphi,d\psi)=\mathbb{P}_N\circ(\Phi_i, \Phi_i)^{-1}(d\varphi,d\psi)=\delta_{\phi}(d\psi)\gamma(d\varphi).
H(x,t)= div_x(\frac{\nabla_x\Phi}{|\nabla_x\Phi|})(x,t) = -\frac{1}{|\nabla_x\Phi|}(\Sigma_{i,j=1}^{n-1}a_{ij}(S'(x),\Lambda(S'(x),t),\nabla_{s'}\Lambda(S'(x),t))\frac{\partial^2\Lambda}{\partial s_i\partial s_j}& +b(S'(x),\Lambda(S'(x),t),\nabla_{s'}\Lambda(S'(x),t)).
I(t_1u_1+t_2u_2+t_3u_3) =I(t_1u_1+t_2u_2+t_3u_3)-\sum_{i=1}^3\frac{t_i^4}{4}I'(u)[u_i] =\sum_{i=1}^3\left(I(t_iu_i)-\frac{t_i^4}{4}I'(u_i)[u_i]\right)-\frac{\lambda}{4}\sum_{i<j}(t_i^2-t_j^2)^2\int_{\mathbb R^3}\phi_iu_j^2dx.
\forall v \in \mathcal{C}_c^{\infty}(\mathbb{D},\mathbb{R}^{n+1}), \int_{\mathbb{D}^+}\left\langle\nabla{\tilde{u}},\nabla v \right\rangle + \int_{\mathbb{D}^-}\left(\frac{\left\langle\nabla{\tilde{u}},\nabla v \right\rangle}{\left|\tilde{u}\right|^2}- \sum_{i,j}\left\langle X_{i,j}, \nabla \tilde{u}_i \right\rangle v_j\right) = 0 \hskip.1cm.
\|Df^{m_{k+1}-m_k}|E^u_{f^{m_k}(z)}\| \leq 2e^{\lambda^u(\delta_Q)(m_{k+1}-m_k-r_k)}\min\{b^{\frac{r_k}{2}},3\delta\}& \leq e^{\lambda^u(\delta_Q)(m_{k+1}-m_k)}\min\{b^{\frac{r_k}{3}},3\delta\}.
\overline{{w}}_{1}\Gamma^{( 1 )}S_{F}^{( 1 )}\Gamma^{( 2 )}\ldots \Gamma^{( n )}w_{2}= ( - 1 ) \cdot \overline{{w_{2}^{c}}}\Gamma^{\prime ( n )}\ldots S_{F}^{\prime ( 1 )}\Gamma^{\prime ( 1 )}w_{1}^{c},
\left( P^{\mu_{1}\nu_{1}\, \mu_{2}\nu_{2}}\right)^{- 1}= d_{A}{\cal T}_{A \;{\mu_{1}\nu_{1}\, \mu_{2}\nu_{2}}}+ d_{B}{\cal T}_{A \;{\mu_{1}\nu_{1}\, \mu_{2}\nu_{2}}}+ d_{C}{\cal T}_{A \;{\mu_{1}\nu_{1}\, \mu_{2}\nu_{2}}}+ \sum_{i = 4}^{1 4}d_{i}T_{i \;{\mu_{1}\nu_{1}\, \mu_{2}\nu_{2}}}
\ell(\mu \lambda n_w) = \sum_{\alpha\in\Sigma^+}\lvert\langle\nu(\mu),\alpha\rangle\rvert + \sum_{\alpha\in\Sigma^+, w^{-1}(\alpha) > 0}\lvert\langle\nu(\lambda)\alpha\rangle\rvert + \sum_{\alpha\in\Sigma^+, w^{-1}(\alpha) < 0}\lvert\langle\nu(\lambda),\alpha\rangle - 1\rvert & = \ell(\mu) + \ell(\lambda n_w).