harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\mathrm{E}_{\theta}\big(\sum_{i\in I}\xi_i^2 \big)^2=\frac{4}{t^2}\mathrm{E}_{\theta}\big(\tfrac{t}{2}\sum_{i\in I}\xi_i^2 \big)^2 \le \frac{4}{t^2}\mathrm{E}_{\theta}e^{t\sum_{i\in I}\xi_i^2}\le \frac{4}{t^2}\big[\mathrm{E}e^{\beta\sum_{i\in I}\xi_i^2}\big]^{t/\beta}\le \frac{4}{t^2}e^{Bt\beta^{-1}|I|}.
S_{\bar{\psi}_r^- \bar{\psi}_r^+}= S_{\psi_r^+ \psi_r^-}=S_{\bar{\Phi}_r^- \bar{\Phi}_r^+}=S_{\Phi_r^+ \Phi_r^-}= \frac{M_r C_D C \Gamma_{11}}{P^2 + M_r^2},
\left\lvert1-4\mu\left(\sum_{l=1}^{M}\lvert\hat{\phi}^{\left(l,j\right)}\rvert^2+\sigma_{e,i}^2\lVert\mathbf{p}_j\rVert^2+\sum_{q=1}^{M-1}\sum_{r=q+1}^{M}f_{q,r}^{\left(j\right)}\right)\right\rvert<1
\frac{s-t}{1-s-t}= \int_{\mathcal{C}_{r_4}}\frac{dw}{2\pi i}\frac{1}{w^2 -1}\frac{(w+1)s-w}{1 -(w+1) s}\frac{(\frac{1}{w}+1)s-\frac{1}{w}}{1 -(\frac{1}{w}+1) s}.
\eta_{+}^{A}= \epsilon_{\pi}( y ) e^{- \frac{1}{2}k | y |}\left( \begin{array}{c}{{\hat{\eta}_{R}}}\\{{- \hat{\eta}_{L}}}\\ \end{array}\right)^{A}\ , \qquad \eta_{-}^{A}= \epsilon_{0}( y ) e^{- \frac{1}{2}k | y |}\left( \begin{array}{c}{{\hat{\eta}_{L}}}\\{{\hat{\eta}_{R}}}\\ \end{array}\right)^{A}\ ,
M:=[\begin{matrix}0&1&0&0&0&\cdot\cdot\cdot\\ 1-p_{1}&0&p_{1}&0&0&\cdot\cdot\cdot\\ 0&1-p_{2}&0&p_{2}&0&\cdot\cdot\cdot\\ 0&0&1-p_{3}&0&p_{3}&\cdot\cdot\cdot\\ \cdot&\cdot&\cdot&\cdot&\cdot\end{matrix}].
P_{x_{12}^{n}x_{23}^{m}x_{34}^{k}}(\beta)=P_{x_{12}^{n}x_{23}^{m}}(\beta) \hphantom{P_{x_{12}^{n}x_{23}^{m}x_{34}^{k}}(\beta)=}{}+ \sum_{a \ge 1 \atop b,p \ge 0}{m \choose p}{k \choose a}{a-1 \choose b}{n+1 \choose p+a-b}{m+a-1-b \choose a}(\beta+1)^{p+a}.
E_{C a s}( v ) = \frac{1}{2 \pi}\int_{0}^{\infty}\mathrm{l n}\Bigl ( 1 - e^{- 2 \pi x}\left( \frac{x - v}{x + v}\right)^{2}\Bigr ) d x - \frac{v}{\pi}\mathrm{l n}( v T ) - \frac{v}{\pi}\gamma \, .
F\big(3^{-1/3}x,3^{-2/3}t;\beta=6\big) = (1-a^2)\kappa u^{\frac{1}{2}}\left[u^{-1}\left(\frac{1+q_2}{2}x -\alpha\right)Y_{12}^{(3)}(x,t) + Y^{(3)}_{22}(x,t)\right] \qquad{}-a\kappa u^{\frac{1}{2}}e^{\frac{x^3}{3}- xt}\left[u^{-1}\left(\frac{1+q_2}{2}x -\alpha\right)Y_{11}^{(3)}(x,t)+Y^{(3)}_{21}(x,t) \right].
\Big( (U_2^n XU_1^{-n}-U_2^{-n}XU_1^n)\,U_1^k e, U_2^l \bar e\Big)=\sum_{m=1}^{k+l}(U_2^nKU_1^{-n}\, U_1^{k-m}e, U_2^{l-m+1}\bar e).
|a r_i\theta_i+b r_j\theta_j|^2-r_k^2 = (a^2 r_i^2 + b^2 r_j^2 -r_k^2) + 2a b r_ir_j \theta_i\cdot\theta_j= 2a br_ir_j \big( s + \theta_i\cdot\theta_j\big)
D_{j_{1}, m_{1}, \overline{{{m}}}_{1}; j_{2}, m_{2}, \overline{{{m}}}_{2}}^{j_{3}, m_{3}, \overline{{{m}}}_{3}}= \sqrt{\frac{( 2 j_{1}+ 1 ) ( 2 j_{2}+ 1 )}{( 2 j_{3}+ 1 )}}\; \langle j_{1}j_{2}m_{1}m_{2}| j_{3}, m_{3}\rangle \; \langle j_{1}j_{2}\overline{{{m}}}_{1}\overline{{{m}}}_{2}| j_{3}, \overline{{{m}}}_{3}\rangle \; d_{j_{1}, j_{2}}^{j_{3}},
p(M)=(\prod_{i=1}^{m-1}(1+(v_i^{(0)})^2)\cdots(1+(v_i^{(n_i)})^2))(1+(v_{m}^{(0)})^2)\cdots(1+(v_{m}^{(n_m)})^2) =(\prod_{i=1}^{m-1}(1+(v_i^{(0)})^2)\cdots(1+(v_i^{(n_i)})^2))(1+u_m^2)\cdots(1+(u_m+y_{n_m})^2).
V^{0}(Q)=\dfrac{3B_1}{2B_2}^2 (\dfrac{\sqrt{B_1}}{2}Q), W^{0}(Q)=-\dfrac{3B_1^{3/2}}{2B_2}\tanh{(\dfrac{\sqrt{B_1}}{2}Q)}^2 (\dfrac{\sqrt{B_1}}{2}Q).
h(x) &\le \Bigl(g_3(x)+\frac{2}{6^4}(3^4x-33)(7-2^4x)+\frac{2}{6^5}(2^5x-13)(101-3^5x)\Bigr)\Bigr|_{x=\frac{302}{3^6}}& + \frac1{10\cdot 6^5}- \eta=-\frac{13250661085528974219050820575468766883}{155139838509919829304649483410259188312000}<0,
||Pf||_{\mathcal{B}(\mathfrak{E})}=\int_{-\infty}^\infty ||E_+^{-1}(t)P(t)f(t)||^2 dt & = \int_{-\infty}^\infty ||(E_+^o)^{-1}(t)P(t)^{-1}P(t)f(t)||^2 dt & = \int_{-\infty}^\infty ||(E_+^o)^{-1}(t)f(t)||^2 dt & =||f||_{\mathcal{B}(\mathfrak{E}^0)}.
[x_{i,r+1}^\pm,x_{js}^\pm]-[x_{ir}^\pm,x_{j,s+1}^\pm]=\pm \hbar d_{ij}(x_{ir}^\pm x_{js}^\pm+x_{js}^\pm x_{ir}^\pm ) \sum_{\pi \in S_{m}}\left[x_{i,r_{\pi(1)}}^{\pm}, \left[x_{i,r_{\pi(2)}}^{\pm}, \cdots, \left[x_{i,r_{\pi(m)}}^{\pm},x_{js}^{\pm}\right] \cdots\right]\right] = 0,
b^{(k)}= p^{(k)}t(b), \textit{where}p^{(k)}\textit{is a small piece (possibly empty)}, &U^{(k)}= A^{(k)}b^{(k)}B, \textit{where}A^{(k)}\textit{is a prefix of}U^{(k)},\B \textit{is a suffix of}U^{(k)}, &k = 1, \ldots, k_1.
\mathbb{P}(\Psi(x,\gamma,Z)=y)= \mathbb{P}(x[(1-\gamma)( \mathbf{1}_{[0, \frac{1}{2}]}- \mathbf{1}_{(\frac{1}{2},1]})(Z) +\gamma( \mathbf{1}_{[0, \frac{3}{4}]}- \mathbf{1}_{(\frac{3}{4},1]})(Z)]=y)
\sqrt{\zeta_{n}^{j}(x)^{2}+m^{2}\epsilon_n^2}-\tau_{n}(|\cdot|^{-1}\star w_{n}^{(j)})(x)+\epsilon_{n}V(\epsilon_{n}x+x_{j})-\epsilon_{n}\mu_{n}\begin{cases}=0 w_{n}^{(j)}(x)>0, \geq0 w_{n}^{(j)}(x)=0.\end{cases}
h_C(m) =\max\{|x_2^{im}-x^{(i+1)m}_2|: i=0,1,\cdots,\lfloor n_C/m\rfloor-1\}=\frac{m\pi}{2n_C}\sin(\xi)\;, \ \ \xi \in \left(\frac{2mi-1}{2n_C}\pi, \frac{2m(i+1)-1}{2n_C}\pi\right) \approx \frac{m\pi}{2n_C}
\hat{\eta}(x,u)=1_{\{(r,x)\notin B(t_0,x_0; \varepsilon/2) \vee w (r,x)\leq \psi(r,x) -\rho\}}\eta (x,r)+1_{\{(r,x)\in B(t_0,x_0 ; \varepsilon/2)\wedge w (r,x)> \psi(r,x) -\rho\}}h(u).
\mathcal{\wp}=\prod\limits_{l=\frac{1}{2}+\nu}^{\frac{n}{2}+\nu -1}\left(r^{2}-l^{2}\right) \prod\limits_{s=\frac{1}{2}-\nu}^{\frac{n}{2}-\nu-1}\left( r^{2}-s^{2}\right)
|u'(t)|^{2}+M\left(|A^{1/2}u(t)|^{2}\right)+4\delta\int_{0}^{t}\left|A^{\sigma/2}u'(s)\right|^{2}\,ds=|u_{1}|^{2}+M\left(|A^{1/2}u_{0}|^{2}\right)
\ddot{\chi}_{m}+ \frac{3 \Delta + 1 4}{3 \Delta + 8}t^{- 1}\dot{\chi}_{m}+ \left[ p^{2}\left( H_{0}t \right)^{- \frac{4}{3 \Delta + 8}}+ \frac{m^{2}}{H_{0}^{2}}t^{- 2}\right] \chi_{m}= 0 .
A_{[1]}= N R_{11}\frac{r^2\left(1+\frac{q}{r^3}\right)\sin^2\alpha\sin^2\beta \sin^2\gamma}{w^2 + r^2\left(1+\frac{q}{r^3}\right)\sin^2\alpha\sin^2\beta \sin^2\gamma}d\psi \ .
\begin{cases}\,(\phi(\tilde{u}'))'=\lambda \biggl{(}F(t,\bar{u}+\tilde{u},\tilde{u}')- \dfrac{1}{T}\displaystyle \int_{0}^{T}F(\xi,\bar{u}+\tilde{u}(\xi),\tilde{u}'(\xi)) \,d\xi\biggr{)}, \,\tilde{u}\in\tilde{\mathfrak{D}}, \lambda\in\mathopen{[}0,1\mathclose{]},\end{cases}
\left|\left\langle y^*,f(x)\right\rangle\right| =\left|\left\langle y^*,T_{f}(\delta_{U}(x)\right\rangle\right| =\left|\left\langle (T_{f})^*(y^*),\delta_{U}(x)\right\rangle\right| \leq \left\|(T_{f})^*(y^*)\right\|
N_d = \sum_{d_1 + d_2 = d}\Big(d_1^2d_2^2 N_{d_1}N_{d_2}{3d - 4 \choose 3d_1 - 2}- d_1^3d_2N_{d_1}N_{d_2}{3d - 4 \choose 3d_1 - 1}\Big)
\Pi_{\mu \nu}^{M}( k ) = \frac{i e^{2}}{( 2 \pi )^{3}}\sum_{i = 0}^{n_{f}}c_{i}\int d^{3}p \frac{P ( M_{i})}{({M_{i}}^{2}-{p_{1}}^{2}) ({M_{i}}^{2}-{p_{2}}^{2})},
\langle P [ \{\Phi^{( I )}\}] \rangle \; = \; \frac{1}{Z}\sum_{n = 0}^{\infty}\frac{( - 1 )^{n}}{n !}\; \langle \; P [ \{\Phi^{( I )}\}] \; S_{i n t}[ \{\Phi^{( I )}\}]^{n}\; \rangle_{Q}\; .
||\left(\partial_{r}+\frac{1}{r}\right)N_{2}(f_{v_{5}})(t,r)||_{L^{2}(r dr)}\leq ||\partial_{r}N_{2}(f_{v_{5}})(t,r)||_{L^{2}(r dr)}+ ||\frac{N_{2}(f_{v_{5}})(t,r)}{r}||_{L^{2}(r dr)}&\leq \frac{C \log(t)}{t^{23/8}}
\lim_{\lambda\rightarrow\infty}\frac{P_{\lambda p(p+1)+\rho+1,0}(z^{p+1})}{z^{p}P_{\lambda p(p+1)+\rho,0}(z^{p+1})}=\lim_{\lambda\rightarrow 0}\frac{Q_{\lambda p(p+1)+\rho+1}(z)}{Q_{\lambda p(p+1)+\rho}(z)}=\frac{\widetilde{F}_{0}^{(\rho)}(z^{p+1})}{z^{p}},z\in\mathbb{C}\setminus(\Gamma_{0}\cup\{0\}).
\chi(a_3,a_2,i_3,\ldots,i_k) \chi(a_1,a_4,i_3,\ldots,i_k) = \varepsilon, \chi(a_2,a_4,i_3,\ldots,i_k) \chi(a_1,a_3,i_3,\ldots,i_k) = \varepsilon.
{{\bf Q}^{3 \nu}}_{Z}=\left[ -m_{Z}^{2}-\frac{g^{2}\sigma^{2}}{4\cos \theta_{W}}--\frac{g\sigma m_{Z}}{\cos \theta_{W}}+\frac{ig}{2\cos \theta_{W}}\left(\partial_{\mu}\sigma \right) I^{\mu}\right] Z^{\nu}+ \beta X_{0}(\partial_{\mu}K^{3\mu})A^{\nu}.
\mathcal{F}_{b}^{\prime}(0)=\frac{b^{2}\mu_{B}}{2}\sum_{j=0}^{r}e_{j}G_{B}(be_{j});\ \mathcal{F}_{b}^{\prime \prime}(0)=b^{2}\mu \sum_{j=0}^{r}e_{j}G_{B}(2be_{j}).
g_{v , 1}( x ) = \frac{k}{2}C_{1}^{2}{\hat{h_{v}}}^{- 1}\bigg [ \{( 5 + 4 k^{2}) - 9 k^{2}s n^{2}( x ) \}s n ( x ) c n^{2}( x ) d n ( x ) \bigg ]
e_{ij}e_{jk}e_{ij}= e_{ij}+\big(q x_i-q^{-1}x_j\big)\big(q x_i - q^{-1}x_k\big)\big(q x_j -q^{-1}x_k\big) u_{ij}u_{jk}u_{ij}, e_{ij}^2= \big(q+q^{-1}\big)e_{ij}.
c_{0}=a_{1}-\frac{a_{0}}{b_{0}}b_{1}=\frac{b_{0}a_{1}-a_{0}b_{1}}{b_{0}};c_{1}=a_{2}-\frac{a_{0}}{b_{0}}b_{2}=\frac{b_{0}a_{2}-a_{0}b_{2}}{b_{0}};...(32)
n_{i}=3n_{i-1}-3n_{i-2}+n_{i-3}+\kappa \left(n+a\right)_{i}=3\left(n+a\right)_{i-1}-3\left(n+a\right)_{i-2}+\left(n+a\right)_{i-3}+\lambda \left(n+b\right)_{i}=3\left(n+b\right)_{i-1}-3\left(n+b\right)_{i-2}+\left(n+b\right)_{i-3}-\lambda
\sigma_{\mathbf{0}}=\sigma_{\mathbf{1}}=(\mathbf{0}\mathbf{2}\mathbf{4}\mathbf{1}\mathbf{3}\mathbf{5}); \sigma_{\mathbf{2}}=\sigma_{\mathbf{3}}=(\mathbf{0}\mathbf{3}\mathbf{5}\mathbf{1}\mathbf{2}\mathbf{4}); \sigma_{\mathbf{4}}=\sigma_{\mathbf{5}}=(\mathbf{0}\mathbf{2}\mathbf{5}\mathbf{1}\mathbf{3}\mathbf{4}).
\int_{\Omega}\rho^{-1}\big|Lu+\nabla P+\nabla |H|^2-H \cdot \nabla H\big|^2x \geq C^{-1}|Lu|^2_2-C(|\nabla P|^2_2+|\nabla u|^2_2+|H|^2_{\infty}|\nabla H|^2_2) \geq C^{-1}|u|^2_{D^2}-C(|\nabla \rho|^2_2+|\nabla u|^2_2+|\nabla H|^2_2),
\sum_{k=1}^\infty \Big(\frac{1}{2k-1}\Big)^{2j}+\Big(\frac{1}{2k+1}\Big)^{2j}-2\Big(\frac{1}{2k}\Big)^{2j}&=1+2\sum_{k=1}^\infty\left(\Big(\frac{1}{2k+1}\Big)^{2j}-\Big(\frac{1}{2k}\Big)^{2j}\right)<1.
\sum_{k=0}^n \binom{a}{k}\binom{b}{n-k}\binom{k}{c}\binom{n-k}{d}H_{a-k}H_{b-n+k}={}\binom{a+b-c-d}{n-c-d}\binom{a}{c}\binom{b}{d}[H^{(2)}_{a+b-n}-H^{(2)}_{a+b-c-d}+ (H_{a+b-n}-H_{a+b-c-d}+H_{a-c})(H_{a+b-n}-H_{a+b-c-d}+H_{b-d})]
\mathcal A_\mu \;(\delta \mu,\delta\b u):= \mathcal A_{p\mu\rho}(0,\delta\mu,0,\delta \b u)= \begin{pmatrix}\mathcal L_\mu \;(\delta \mu,\delta\b u) \mathcal B \;(\delta\b u)|_{\partial\Omega}\end{pmatrix}=\begin{pmatrix}\mathcal S 0\end{pmatrix}
\left< \eta_{t}^{n},\varphi\right> -\left< \eta_{0}^{n},\varphi\right> =\int_{0}^{t}\big( \left< \eta_{z}^{n},L_{z}\varphi\right> + A^{n}_{z}(\varphi) \big) dz+M_{t}^{n}(\varphi),
\chi_{k,n}^{(t_{1},...,t_{n+k})}(x_{1},...,x_{n+k}|\rho)=\allowbreak\frac{l_{k,n}^{(t_{1},...,t_{n+k})}(x_{1},...,x_{n+k}|\rho)}{w_{n+k}(x_{1},...,x_{n+k}|\rho)},
\frac{\partial^2 V}{\partial\lambda_i^2}= \frac{n \beta}{2}+ \beta \sum_{j \ne i}\frac{1}{(\lambda_i - \lambda_j)^2}, \frac{\partial^2 V}{\partial\lambda_i \partial \lambda_j}= - \beta \frac{1}{(\lambda_i - \lambda_j)^2}.
\displaystyle \prod_{k}d \tau_{k}\prod_{i}d s_{i}\left[ \det ( J_{i j}^{\sigma}) \det \left( k^{i}, ( 1 - P \frac{1}{P^{\dag}P}P^{\dag}) k^{j}\right){\cal D}\mathrm{e t}( P^{\dag}P ) \right]^{\frac{1}{2}},
e(A,B)=\frac{1}{2}\sum_{x\in A\cup B}\deg_{G[A,B]}(x) &\geq \frac{1}{2}\left(\left(\frac{1}{2}-\frac{3}{10h}\right)n-h-1\right)(|A|+|B|) &\geq \frac{1}{2}\left(\left(\frac{1}{2}-\frac{3}{10h}\right)10h-h-1\right)(|A|+|B|) &> \frac{h}{2}(|A|+|B|) &> \frac{(l-2)-1}{2}(|A|+|B|).
\Phi^{\prime\prime}(w)(1)=\|w\|^2 -(p+q-1)\lambda\int_{-1}^1f(x)|w_1(x, 0)|^p|w_2(x,0)|^q dx &-\int_{-1}^1 \left\{(\alpha+\beta+2w_1^2+2w_2^2)(\alpha+\beta-1+2w_1^2+2w_2^2)+4(w_1^2+w_2^2)\right\}|w_1|^\alpha|w_2|^\beta e^{w_1^2+w_2^2}dx.
\displaystyle\frac{1+{p}^{\prime}/c^{2}}{1-v^{2}/c^{2}}(\rho_{t}+v\rho_{r})v+\frac{q(1+v^{2}/c^{2})}{(1-v^{2}/c^{2})^{2}}v_{t}+\frac{2q}{(1-v^{2}/c^{2})^{2}}vv_{r}+{p}^{\prime}\rho_{r}+\frac{2q}{r(1-v^{2}/c^{2})}v^{2}=\frac{4\pi n}{\sqrt{1-v^{2}/c^{2}}}\phi_{r}
\xi_\theta=\left(\mathrm{sin}(\theta)\frac{\partial}{\partial r^0}+\mathrm{cos}(\theta)\wedge \frac{\partial}{\partial r^1}\right)\wedge\frac{\partial}{\partial x^1}\wedge\frac{\partial}{\partial x^2}\wedge\frac{\partial}{\partial x^3},\forall \theta\in\left[-\frac{\pi}{2},\frac{\pi}{2}\right].
c_{3}=c_{1}c_{2}\overline{c_{0}}-3^{-1}c_{1}^{3}\overline{c_{0}}^{2}+3^{-1}c_{0}(\overline{\alpha}^{3}+3b_{1}a_{3}+6b_{2}a_{1}a_{2}+3b_{3}a_{1}^{3}).
|<x^{*},f_{n}>| =|\sum_{k=p_{n}}^{q_{n}}<P_{k}x^{*},x^{(n)}_{k}>| &\leq (\sum_{k=p_{n}}^{q_{n}}\|x^{(n)}_{k}\|^{p})^{\frac{1}{p}}\cdot (\sum_{k=p_{n}}^{q_{n}}\|P_{k}x^{*}\|^{p^{*}})^{\frac{1}{p^{*}}}&=(\sum_{k=p_{n}}^{q_{n}}\|P_{k}x^{*}\|^{p^{*}})^{\frac{1}{p^{*}}},
P(x^k,x^*) \le \tfrac{\theta_1 (\eta_1 + L_f \tau_1)}{\theta_k(\mu+ \eta_k)}P(x^0, x^*) =\tfrac{(\sqrt{2 L_f \mu}+ L_f \sqrt{2 L_f/\mu})}{\alpha (\mu + \sqrt{2 L_f\mu})}\alpha^k P(x^0, x^*)= \tfrac{\mu + L_f}{\mu}\alpha^k P(x^0, x^*).
\begin{aligned}A_N = A_1(w_{N-1})\;\! A_{N-1}= \prod_{i=0}^{N-1}A_1(w_{N-i-1}) D_N = D_1(w_{N-1})\;\! D_{N-1}= \prod_{i=0}^{N-1}D_1(w_{N-i-1}) begin{align*}0.3em]C_N = C_1(w_{N-1})\;\! A_{N-1}+ D_1(w_{N-1})\;\! C_{N-1}&= \sum_{i=0}^{N-1}D_i(w_{N-1-i})\, C_1(w_{N-1-i}) \, A_{N-1-i}. \end{aligned}
\Big ( \, g_{0}^{2}\, \big < 0 | \bar{\Psi}\Psi | 0 \big > \, \Big )^{[ 1 , 0 ]}= \ \frac{1 2}{\beta_{0}}\ \big ( w_{1}^{\, 3}- 2 w_{1}w_{3}- 2 w_{2}w_{3}\big ) \ \Lambda^{3}
\mathfrak{h}_0=[0,1), \mathfrak{h}_1=[0,1/2)-[1/2,1), &\mathfrak{h}_2=[0,1/4)-[1/4,1/2), \mathfrak{h}_3=[1/2,3/4)-[3/4,1), &\mathfrak{h}_4=[0,1/8)-[1/8,1/4), \mathfrak{h}_5=[1/4,3/8)-[3/8,1/2), &\mathfrak{h}_6=[1/2,5/8)-[5/8,3/4), \mathfrak{h}_7=[3/4,7/8)-[7/8,1), \,\,\ldots
\sigma_{\mathrm{a b s}}={\frac{\pi^{3}R^{8}}{6 4}}( \omega^{2}+ 1 6 \pi^{2}T_{L}^{2}) ( \omega^{2}+ 1 6 \pi^{2}T_{R}^{2}) \omega{\frac{e^{\frac{\omega}{T_{H}}}- 1}{( e^{\frac{\omega}{2 T_{L}}}- 1 ) ( e^{\frac{\omega}{2 T_{R}}}- 1 )}}\ .
\nabla_V \big(\nabla_Y \hat X - \omega(Y )\big) = \big( \nabla_Y \omega - R(Y, \hat X)\big) V- T_h (V, Y) \nabla_V \left(\nabla_Y \omega - R(Y, \hat X) \right) = - R\big(\nabla_Y \hat X - \omega (Y), V\big) &-R(\omega(Y), V) + \nabla_Y (T_h(V))+ R(V, Y) \omega &+ (\nabla_{\hat X}R) (V, Y)- R(Y, \omega(V)).
s(f,\Upsilon,\mathcal{P}^1_n) = \sum_{k}f(x_k^{n,1})\, |I^1_k|_\Upsilon=\sum_{k}f(x_k^{n,1})\, \int_{I_k^1}\lambda\,\psi &= \sum_{k}f(x_k^{n,1})\,\gamma \int_{I_k}\psi =\gamma\, S(f, \Psi, \mathcal P_n)\,,
\lim_{t\to\infty}\|w_{Z}^{(t,y_{u,t})}(t,\cdot)-u^+(t,\cdot)\|_{L^\infty((-\infty,z_{u,t}])}= 0 = \lim_{t\to\infty}\|\tilde w_{Z}^{(t,\tilde y_{u,t})}(t,\cdot)-u^+(t,\cdot)\|_{L^\infty([z_{u,t},\infty))}.
t^{00}=\sum\hspace{-1.4em}\int\limits_{\lambda}E_\lambda\psi^*_\lambda(\textbf{x})\,\psi_\lambda(\textbf{x})-(\xi-1/4)\triangle \sum\hspace{-1.4em}\int\limits_{\lambda}E^{-1}_\lambda\psi^*_\lambda(\textbf{x})\,\psi_\lambda(\textbf{x}),
l_{\langle\rangle}(\mathcal C_tP) =l_{\langle\rangle}(P), o_{\langle\rangle}(\mathcal C_tP) =\vartheta(t+\Omega^{o_{\langle\rangle}(P)}), d(\mathcal C_tP) =1, \langle h_0(\mathcal C_tP),h_1(\mathcal C_tP)\rangle =\langle t+\Omega^{o_{\langle\rangle}(P)},0\rangle
t_{11}=\alpha(v)\cdot \frac{v^2(1-\overline{v}^2)}{|v|^2}+2+\frac{v^2(1-|v|^2)(1-\overline{v}^2)}{2|v|^4}=\frac{(1+|v|^2)\cdot v\cdot (1+\overline{v}^2)}{|v|^2(\overline{v}-v)}, t_{12}=-\alpha(v)\cdot(v+\overline{v}){-}\frac{(1-|v|^2)({v-\overline{v}})}{2|v|^2}={2}\frac{(1+|v|^2)(1+\overline{v}^2)}{(1-\overline{v}^2)(v-\overline{v})}.
H t_{-}( \tau ) = \frac{1}{2}\log \mid \frac{\sin ( \frac{\pi ( \mu \tau - x )}{2 K})}{\sin ( \frac{\pi ( \mu \tau + x )}{2 K})}\mid + \frac{\mu \tau \pi}{2 K}\frac{\vartheta_{1}^{\prime}}{\vartheta_{1}}( \frac{\pi x}{2 K}) - 2 \sum_{n = 1}^{\infty}\frac{q^{2 n}}{n ( 1 - q^{2 n})}\sin ( \frac{n \pi \mu \tau}{K}) \sin ( \frac{n \pi x}{K}) ,
E_{x,s}^U(U_{\tau_s^S}) = x ~+~ \delta \cdot \mu_s ~+~ O(e^{-x^{1/4}}) ~~\mbox{and}~~E_{x,s}^V(V_{\tau_s^R}) = x ~+~ (1-\delta) \cdot \mu_s ~+~ O(e^{-x^{1/4}}).
\left\Vert \left\{\xi_{j}-\varphi\left( z_{j}\right) \right\}_{j=1}^{J}\right\Vert_{\ell^{2}\left( \mu\right)}<\delta\left\Vert \left\{\xi_{j}\right\}_{j=1}^{J}\right\Vert_{\ell^{2}\left( \mu\right)}
t^{57}f(t) = \dfrac{c}{{(2\pi)}^{57}i}\prod_{k=0}^{56}\left(e^{i\frac{\pi t}{2k+1}}-e^{-i\frac{\pi t}{2k+1}}\right) = \dfrac{c}{{(2\pi)}^{57}i}\sum_{k}e_ke^{id_kt}
\tilde F_\lambda (x_1,\dots, x_n;t)= \frac{(1-t)^n}{\prod_{i=1}^{n-l}(1-t^i)}\sum_{\sigma \in S_n}\sigma\left((x_{1}|a)_{\lambda_1-1}\dots (x_{l}|a)_{\lambda_n-1}\prod_{i=1}^{n}{x_i}\prod_{i < j}\frac{x_{i}-tx_{j}}{x_{i}-x_{j}}\right),
\begin{dcases}\rho_t+\rho_x +u_{x}=0 (0,T)\times (0,1), u_{t}-u_{xx}+u_x +\rho_x =0 (0,T)\times (0,1), \rho(t,0)= 0 t\in (0,T) , u(t,0)=0, \u(t,1)= q(t) t\in (0,T) , \rho(0,x)=\rho_0(x),\u(0,x)=u_0(x) x\in(0,1).\end{dcases}
\theta_{k,m}^{\star}\geq 0,k=1,...,K 1-tB\sum_{k=1}^K\sum_{m=1}^M \theta_{k,m}^{\star}=0 \frac{t}{\omega}\cdot\frac{\sigma^2\ln 2}{\left(\mathbf{\Lambda}_k\right)_{m}}2^{\theta_{k,m}^{\star}}-\upsilon tB =0,k=1,...,K
CPT:\Psi(\vec{x},t)\rightarrow-iL^0e^{i(\alpha_P+\alpha_C+\alpha_T)}\gamma^0\gamma_5\bar{\Psi}^T(-\vec{x},-t)~,~~~L^0=\left(\begin{array}{cc}0 e^{i\eta}I_4 e^{-i\eta}I_4 0\end{array}\right)
F \approx - 6 4 \frac{\gamma \zeta ( 5 )}{\pi h^{2}}\sqrt{\frac{h}{2 M}}\left( \frac{M}{\beta}\right)^{5}- \frac{3}{8 \pi k}\gamma L^{3}\int_{0}^{\infty}d \omega \frac{\omega^{4}}{e^{\beta \omega}- 1}.
K_{\omega,c}(\lambda \Phi_{\omega}) =2\lambda^2L(\Phi_{\omega})+3\lambda^3N(\Phi_{\omega}) +2\lambda^2\omega Q(\Phi_{\omega}) +2\lambda^2cP(\Phi_{\omega}) =\lambda^2K_{\omega,0}(\Phi_{\omega}) +\lambda^2\{3(\lambda -1)N(\Phi_{\omega})+2cP(\Phi_{\omega})\}=2\lambda^2\{-3(\lambda -1)\mu_{\omega,0}+cP(\Phi_{\omega})\}.
R_{\mu \nu}- 2 \nabla_{\mu}\nabla_{\nu}\phi +{\textstyle \frac{1}{4}}H_{\mu}{}^{\rho \sigma}H_{\nu \rho \sigma}-{\textstyle \frac{1}{4}}e^{2 \phi}\sum_{n}{\textstyle \frac{( - 1 )^{n}}{( n - 1 ) !}}T^{( n )}{}_{\mu \nu},
S = \frac{T}{2 \pi}\int dx \sqrt{H^{-1}(U) + (\partial_x U)^2 +{U^2}/{9}(\partial_x \psi)^2 +{U^2}/{9}(\partial_x \psi) (\partial_x \phi_1) +{U^2}/{6}(\partial_x \phi_1)^2},
\phi(\sigma, \tau_1\tau_2) = \widetilde \sigma \widetilde{\tau_1}\widetilde{\tau_2}{\widetilde{\sigma}}^{-1}{\widetilde{\tau_2}}^{-1}{\widetilde{\tau_1}}^{-1}, \phi(\sigma, \tau_1) \phi(\sigma, \tau_2) = \widetilde \sigma \widetilde{\tau_1}{\widetilde \sigma}^{-1}{\widetilde{\tau_1}}^{-1}[\widetilde \sigma, \widetilde{\tau_2}]
M_{i j}^{- 1}= \left( \begin{array}{c c c}{{- ( 2 k^{2}+ \ell^{2})^{2}/ 4 k^{2}}}&{{- \ell^{2}/ 2 k^{2}}}&{{- ( 2 k^{2}\ell + \ell^{3}) / 2 k^{2}}}\\{{}}&{{}}&{{}}\\{{- \ell^{2}/ 2 k^{2}}}&{{- 1 / k^{2}}}&{{- \ell / k^{2}}}\\{{}}&{{}}&{{}}\\{{- ( 2 k^{2}\ell + \ell^{3}) / 2 k^{2}}}&{{- \ell / k^{2}}}&{{- ( k^{2}+ \ell^{2}) / k^{2}}}\\ \end{array}\right) \, .
E_{x}=\frac{1}{j\omega\epsilon}\frac{dL}{dz}\frac{\partial T}{\partial x}^{TM}-L\frac{\partial T}{\partial y}^{TE}=\frac{-k_{z}}{\omega\epsilon}L\frac{\partial T}{\partial x}^{TM}-L\frac{\partial T}{\partial y}^{TE}(27)
\varepsilon=\frac{1}{R^4}\left\{\left[-\frac{\pi^2}{90}+\frac{(mR)^2}{24}-\frac{(mR)^3}{12\pi}+\ldots\right]+\lambda\left[\frac{1}{1152}-\frac{mR}{192\pi}+\ldots\right]\right\}+\ldots
\phi_1 := \phi_{12A}(\tau, z) = 2 \zeta^{\pm 1}- 4 + (-4\zeta^{\pm 2}+ 10 \zeta^{\pm 1}- 12) q + (2 \zeta^{\pm 3}- 12 \zeta^{\pm 2}+ 32 \zeta^{\pm 1}- 44) q^2 + O(q^3)
m_{\phi}^{2 \ ( - )}= \lambda_{o b s}^{- 2}\m_{\phi}^{2 \ ( + )}= \lambda_{o b s}^{- 2}\{\frac{d^{2}V_{\mathrm{\footnotesize \it ~ ~ e f f \,}}}{d \phi^{2}}}\approx \mp \lambda_{o b s}\left({\frac{5 \pi^{3}\zeta_{R}^{\prime}( - 4 )}{6 M^{3}l^{5}}}\right) .
\liminf_{\theta\to 0}\int_{U^{M}(\delta)}u_{\theta}^2\,dv^{g_{\theta}}\geq \liminf_{\theta\to 0}\sum_{m=1}^{N_0}\int_{A_{\alpha_m}}u_{\theta}^2\,dv^{g_{\theta}}&\geq \sum_{m=1}^{N_0}\liminf_{\theta\to 0}\int_{A_{\alpha_m}}u_{\theta}^2\,dv^{g_{\theta}}&\geq \frac{E N_0}{2}.
\mathbb{P}_\eta\left(\mathcal{T}_t(y)\geq\frac{1-p}{4}t,\mathcal{T}_t(0) \leq \frac{1-p}{4}\delta^d t\right)\qquad \leq \sum_{i=1}^d \mathbb{P}_\eta\left(\mathcal{T}_t(y^{(i-1)})\geq\delta^{i-1}\frac{1-p}{4}t, \mathcal{T}_t(y^{(i)}) \leq \frac{1-p}{4}\delta^i t\right).
|{\cal O}_{c o v}( \alpha^{+}, \alpha^{-}, \alpha^{i}) \rangle = \exp \Bigl ( - \frac{\alpha^{+}}{\partial^{+}}( \bar{\alpha}^{+}\partial^{-}+ \bar{\alpha}^{i}\partial^{i}) \Bigr ) |{\cal O}( \alpha^{-}, \alpha^{i}) \rangle^{\prime}
F_{p - 2}= - \frac{V_{p - 2}}{( 2 \pi )^{p - 2}\tilde{g}}\left( \frac{u}{R}\right)^{7 - p}\left[ \sqrt{\tilde{f}}- 1 + \frac{u_{0}^{7 - p}}{2 u^{7 - p}}\right] .
\tilde{S}^{\prime}\ [ X^{\mu}, b_{\xi}] = - | P_{y}| \int d \xi \e^{- \phi}\sqrt{| g_{\xi \xi}|}+ P_{y}\int d \xi \left( C^{( 1 )}{}_{\xi}-{\textstyle \frac{m}{2}}\ ( 2 \pi \alpha^{\prime}) \b_{\xi}\right) ,
\frac{d}{dx}\left( h_n(x) e^{-x^2/4}\right) = \left[ \frac{1}{2}h_{n+1}(x) + \frac{n}{2}h_{n-1}(x) - h_{n+1}(x) \right] e^{-x^2/4}= \left[\frac{n}{2}h_{n-1}(x)-\frac{1}{2}h_{n+1}(x) \right]e^{-x^2/4}.
\#\left(P_{g\oplus h}^{-1}\alpha\right)=\#\left(P_{g}^{-1}\gamma_{r}\right)+\#\left(P_{h}^{-1}\beta\right)-\left(1-\textbf{1}_{g\equiv 1}- \textbf{1}_{h\equiv1}+\textbf{1}_{g\oplus h\equiv 1}\right).
\langle \left[(\widehat x)^{2k}, (\widehat \xi)^{2k}\right] f , f \rangle_{L^2 (\mathbb{R})}= (-i )^{2k}\sqrt{\frac{2a}{\pi}}\int_{\mathbb{R}}e^{-a x^2}\left( x^{2k}\frac{d^{2k}}{dx^{2k}}- \frac{d^{2k}}{dx^{2k}}x^{2k}\right) e^{-a x^2}dx
\Gamma ( X , Y ) \Delta_{N}^{2}( X^{2}) \prod_{n = 1}^{N}x_{n}= \frac{1}{N !}\det [ \gamma ( x_{n}, y_{m}) ]_{n , m = 1 , \ldots , N}\frac{\Delta_{N}( X^{2})}{\Delta_{N}( Y^{2})}\prod_{n = 1}^{N}x_{n}
L(\hat{{\mathbf U}}^{\pm},\hat{\Psi}^{\pm})\dot{{\mathbf U}}^{\pm}+C(\hat{{\mathbf U}}^{\pm},\hat{\Psi}^{\pm})\dot{{\mathbf U}}^{\pm}-\frac{\Psi^{\pm}}{\partial_1\hat{\Phi}^{\pm}}\partial_1\{\mathbb{L}(\hat{{\mathbf U}}^{\pm},\hat{\Psi}^{\pm})\}=\mathbf{f^{\pm}}.
\sigma(\psi(S)\psi(iT)-\psi(iT)\psi(S)^{\ast}) =\sigma(S(iT)-(iT)S^{\ast})=i\sigma(ST-TS^{\ast}) &=i\sigma(\psi(S)\psi(T)-\psi(T)\psi(S)^{\ast}) &=\sigma(\psi(S)(i\psi(T))-(i\psi(T))\psi(S)^{\ast}).
\int \delta w_h(x,y) \frac{(2-\sigma)\langle A^h(x)y,y\rangle\rho(x+he,y)}{|y|^{n+\sigma+2}}dy=\frac{f(x+he)-f(x)}{|h|^\beta}+k_h(x)|h|^{1-\beta}&-\int\delta u(x,y)\frac{\rho(x+he,y)-\rho(x,y)}{|h|^{\beta}}\frac{(2-\sigma)\langle A^h(x)y,y\rangle}{|y|^{n+\sigma+2}}dy &= \frac{f(x+he)-f(x)}{|h|^\beta}+k_h(x)|h|^{1-\beta}-I^1_h(x).
\Big{(}\mathcal{D}_tE_u^{\frac12},v\Big{)}-\gamma(\nabla E_\sigma^{\frac12},\nabla v)+(\nabla E_u^{\frac12},\nabla v)+(f(u(t_{\frac12}))-f^{\frac12}(u),v)= (R_3^{\frac12},v),~\forall v\in H_0^1,
\sum_{P}( - 1 )^{P}{\cal J}^{- 2}( \theta ){\frac{\exp \{N \beta \sum_{k}\cos ( \theta_{k}- \theta_{P ( k )}+{\frac{2 \pi}{L}}) \}}{\prod_{i , j}\cos \bigl ({\frac{\theta_{i}- \theta_{P ( i )}+ \theta_{j}- \theta_{P ( j )}}{2}}\bigr )}}~ ~ ,
\biggl ( - \frac{1}{2}\frac{d^{2}}{d r^{2}}+ \frac{( N^{2}- 1 ) ( N^{2}- 3 )}{8 r^{2}}+ \frac{r^{2}}{2}+ \sum_{K = 1}^{\infty}\frac{c_{K}g^{K}}{N^{4 K - 2}}( r^{2})^{2 K}\biggr ) \rho = \epsilon_{0}\rho .
N(m)=\frac{1}{2^{m+1}}\sum_{i=0}^{m}(-1)^{i}\sum_{a_{1},\cdots,a_{i+1}=1}^{\infty}\delta_{a_{1}+\cdots+a_{i+1},m+1}\prod_{j=1}^{i+1}\frac{(2a_{j})!}{a_{j}!},
h_\varphi(x,y) = h(\varphi^{-1}(x), \varphi^{-1}(y)) &= h(\varphi^{-1}(\ell_p^{-1})_*(x), \varphi^{-1}(\ell_p^{-1})_*(y)) &= (\ell_p)^*h_\varphi(x,y).
-\int_{\Omega_T}\nabla u_n G dxdt =-\int_{\Omega_T}u_n\varphi_tdxdt+\int_{\Omega_T}A^{*}(x,t)\nabla\varphi\nabla u_ndxdt & =-\int_{\Omega_T}u_n\varphi_tdxdt+\int_{\Omega_T}A(x,t)\nabla u_n\nabla\varphi dxdt &=- \int_{\Omega_T}\nabla\varphi F_n.
|h_3\rangle=({\bf P}^{\bar{z}})^k\sum_{n=0}^{k}\rho^n\frac{\Gamma(\frac{D}{2}+k-n-2)}{2^n n!\Gamma(\frac{D}{2}+k-2)}({\bf M}^{\bar{z}j}{\bf M}^{\bar{z}j})^nV_0|0\rangle_1|0\rangle_2|0\rangle_3\,,
{\cal Z}_n=\int \prod_{i,j,k}^{n}dx_i dy_j dz_k \int{\cal D}\phi{\cal D}\psi \left(\lambda \phi(x_i)\psi(y_j)\phi(z_k)\right)^ne^{-\left(S^E_{free}\left[\phi\right]+S^E_{free}\left[\psi\right]\right)}
\left.\frac{k^1}{(1+h^1_0)h^2}\right|_{S^0}=\left.\frac{f^1-g^1-z^1h^1_0}{(z^1+z^1h^1_0)(f^2-g^2)}\right|_{S^0}= = \left.\frac{z^1\circ g^{-1}\circ f-z^1-z^1h^1_0}{(z^1\circ g^{-1}\circ f)(z^2\circ g^{-1}\circ f-z^2)}\right|_{S^0},
g_B^i\left(\hat{s}\right) = \frac{1}{\mu_0^i\left(B\right)}\int\limits_{B}\prod\limits_{t=1}^{k}\prod\limits_{j=1}^{n}\ell^j(S_{t}^j|\theta)^{\left[A^{t-k}\right]_{ij}}d\mu_0^j\left(\theta\right)
{{\mathcal{E}}_{\mu\nu}}=-{{\left({{\tilde{\kappa}}\over{\kappa}}\right)}^4}\left[{\mathcal{U}}\left({u_\mu}{u_\nu}+{1\over{3}}{h_{\mu\nu}}\right)+{{\mathcal{P}}_{\mu\nu}}+{{\mathcal{Q}}_\mu}{u_\nu}+{{\mathcal{Q}}_\nu}{u_\mu}\right],
-\sum_{\tau\in \mathcal{F}_N}\nu_\tau^{(N)}(f)-\sum_{m=1}^\infty \biggl( \sum_{\tau\in \mathcal{F}_N}\nu_\tau^{(N)}(f)J_{N,m}(\tau)\biggr) q^m = f_{\theta}(z)-\mathcal{E}_2(z),
{\cal L}_{N J L}= \bar{q}[ i \gamma^{\mu}\partial_{\mu}- g ( \sigma + i \vec{\pi}\cdot \vec{\tau}\gamma_{5}) ] q -{\frac{\mu^{2}}{2}}( \sigma^{2}+ \vec{\pi}^{2}) +{\frac{M_{0}\mu^{2}}{g}}\sigma \, \, ,