harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
j_{\mu}^{m a t t e r}= \bar{\varphi}_{2}\partial_{\mu}\bar{\varphi}_{1}- ( \bar{\varphi}_{1}+ \mathrm{\frac m e}) \partial_{\mu}\bar{\varphi}_{2}+ e A_{\mu}( \bar{\varphi}_{2}^{2}+ ( \bar{\varphi}_{1}+ \mathrm{\frac m e})^{2}) + O ( \hbar )
\begin{array}{lll}{p_\delta}^*= b_{\delta}(p) \cdot z_{\delta}& 0_\delta \le z_{\delta}\le u_{\delta}, & 1_\delta \cdot z_{\delta}=1 , & \tilde{A}_\delta z_{\delta}\le \tilde{c}_\delta. & b_{\delta}(g_i) \cdot z_{\delta}\le 0, \; \forall i=1,\dots,m.\end{array}
\Gamma_c:=\begin{bmatrix}\Big(\sum_l\Gamma_{ij}^l\Gamma_{mb}^l\Big) \Big(\sum_l\Gamma_{ij}^l\Gamma_{mm}^l\Big) \Big(\sum_l\Gamma_{mb}^l\Gamma_{ii}^l\Big) \Big(\sum_l\Gamma_{ii}^l\Gamma_{mm}^l\Big) \end{bmatrix}, i\neq j, \, m\neq b.
D(W_\theta^{\times n}\| W_{\theta'}^{\times n}|\overline{P}^1_\theta) :=D(W_\theta^{\times n}\times \overline{P}^1_\theta\| W_{\theta'}^{\times n}\times \overline{P}^1_\theta) = n((\theta -\theta') \frac{d \phi}{d\theta}(\theta) - \phi(\theta) + \phi(\theta'))=n D(W_\theta \| W_{\theta'}).
\sum_{k=0}^{\infty}\bigg(\frac{-1}{4}\bigg)^k\frac{(\frac{1}{2})_k(d)_k^2(1-d)_k^2}{(1)_{k}^3(\frac{1}{2}+d)_{k}(\frac{3}{2}-d)_{k}}(d-d^2+2k+5k^2)=\frac{1-2d}{\pi}\tan(d\pi).
d s^{2}= - 2 d x^{+}d x^{--}\beta^{2}\sum_{k = 1}^{n}a_{k}^{2}\left[ ( \tilde{x}^{2 k - 1})^{2}+ ( \tilde{x}^{2 k})^{2}\right] + \sum_{i = 1}^{2 n}( d \tilde{x}^{i})^{2}+ \sum_{i = 2 n + 1}^{9}( d x^{i})^{2},
\Omega=\frac{V}{A}=\frac{1}{S}=\frac{W}{A^{2}}=\frac{V^{2}}{W}=\frac{s}{F}=\frac{H}{s}=\frac{J\cdot s}{C^{2}}=\frac{kg\cdot m^{2}}{s\cdot C^{2}}=\frac{J}{s\cdot A^{2}}=\frac{kg\cdot m^{2}}{s^{3}\cdot A^{2}}
\int_{\mathfrak{M}_{i}}f_{i}(\alpha)S_{i}^{2}(\alpha)T_{i}^{2}(\alpha)e(-n\alpha)d\alpha=\frac{1}{3^{4}}\mathfrak{S}(n)\mathfrak{J}_{i}(n)+O(N_{i}^{\frac{11}{9}}L^{-1})
\int_0^1\left(\frac{1}{x}\frac{d}{dx}\right)^k\left(\frac{J_{\alpha}(cxy)}{(cxy)^{\alpha}}\right)y^{\alpha+1/2}\psi_{n,c}^{\alpha}(y)dy=\mu_n^{\alpha}(c)\left(\frac{1}{x}\frac{d}{dx}\right)^k\left(\frac{\psi_{n,c}^{\alpha}(x)}{(cx)^{\alpha+1/2}}\right).
\begin{array}{ll}\mbox{minimize}\displaystyle f(x):=g\left(x_1,\cdots, x_K\right)+\sum_{k=1}^{K}h_k(x_k) [10pt]\mbox{subject to}\displaystyle E_1x_1+E_2x_2+\cdots+E_Kx_K=q, & x_k\in X_k,k=1,2,...,K,\end{array}
W ( C )_{\mathrm{m o n}}^{\cal N}= \frac{1}{N}{}\mathrm{t r}{}\exp \left( \frac{i g}{2}\vec{H}\int d^{3}x \vec{F}_{\mu \nu}^{\cal N}\Sigma_{\mu \nu}\right) = \frac{1}{N}\sum_{\alpha = 1}^{N}W_{\alpha}^{\cal N}.
\begin{aligned}\varphi_x(x_1) = \frac{x_1^6+5x_1^4+16x_1^3-5x_1^2-1}{x_1(x_1+1)^2(x_1-1)^2}, \varphi_y(x_1) = \frac{(x_1^2+1)(x_1^2-4x_1-1)(x_1^4+4x_1^3+6x_1^2-4x_1+1)}{x_1^2(x_1+1)^3(x_1-1)^3}. \end{aligned}
\frac{\delta{\cal W}}{\delta\psi^\alpha}=\frac{\delta\Gamma}{\delta\psi^\alpha}\,,\;\;\; \frac{\delta{\cal W}}{\delta\phi^*_{Aa}}=\frac{\delta\Gamma}{\delta\phi^*_{Aa}}\,,\;\;\; \frac{\delta{\cal W}}{\delta\bar{\phi}_A}=\frac{\delta\Gamma}{\delta\bar{\phi}_A}\,,
\bigl(T_1(k,l),T_1(k',l')\bigr)=(r,s),\bigl(T_2(k,l),T_2(k',l')\bigr)=(r,s'), &\bigl(T_3(k,l),T_3(k',l')\bigr)=(r',s),\bigl(T_4(k,l),T_4(k',l')\bigr)=(r',s'),
V=\begin{pmatrix}\left(v^{1k'}_{1k}\right)_{1 \leq k \leq N}^{1 \leq k'\leq N}\left(v^{1k'}_{2k}\right)_{1 \leq k \leq N}^{1 \leq k'\leq N}\left(v^{2k'}_{1k}\right)_{1 \leq k \leq N}^{1 \leq k'\leq N}\left(v^{2k'}_{2k}\right)_{1 \leq k \leq N}^{1 \leq k'\leq N}\end{pmatrix}\in \mathcal{M}_{2N^{2}\times 2N^{2}}\left(\mathbb{C}\right).
Q_{0}^{\alpha \beta \mu \nu}= - \frac{1}{4 \pi}\int d^{2}{\bar{\cal S}}\int_{0}^{\infty}d{\bar{r}}( \log{\bar{r}}) \frac{\partial}{\partial{\bar{r}}}\left[{\bar{r}}\frac{\partial}{\partial{\bar{r}}}\Bigl ({\bar{r}}^{2}{\bar{\nabla}}_{\bar{\mu}}{\bar{r}}\;{\bar{Z}}^{\bar{\mu}}\biggl |_{i^{-}}\Bigr ) \right] + O [ R_{. .}^{2}] \; .
W_{p}^{p}(\bar{\mu}_{u},\bar{\nu}_{v})=\int_{[u,u+2\pi)}(x-\theta_{u,v}(x))^{p}\bar{\mu}(dx)=\int_{[u',u'+2\pi)}(x-\theta_{u',v'}(x))^{p}\bar{\mu}(dx)=W_{p}^{p}(\bar{\mu}_{u'},\bar{\nu}_{v'}),
\tilde{\alpha}_{\infty,f}(x^{(j)}) = \int_{-\infty}^{\infty}f(t)\tilde{\alpha}_{\infty,t}(x^{(j)})~dt =x^{(j)}\int_{-\infty}^{\infty}f(t)e^{ipt}~dt =\widehat{f}(-p)x^{(j)}\, .
B=\begin{pmatrix}0 0 0 \cdots 0 -1 1 0 0 \cdots 0 -1 0 1 0 \cdots 0 -1 \vdots \vdots \vdots \ddots \vdots \vdots 0 0 0 \cdots 0 -1 0 0 0 \cdots 1 -1\end{pmatrix}
\widehat{\vec{u}}_{\lambda}= 2|\xi|^{-2}\left(I-\frac{2\lambda+1}{2(\lambda+1)}\Pi_\xi\right) \widehat{\vec{f}}, \widehat{\vec{u}}_{0}= 2|\xi|^{-2}\left(I-\frac{1}{2}\Pi_\xi\right) \widehat{\vec{f}}.
\Sigma_{P}^{(\epsilon,\eta)}(\lambda):=\begin{bmatrix}B_k(\lambda) B_{k-1}(\lambda) \cdots B_{\eta+1}(\lambda) 0 \cdots 0 \vdots \vdots \ddots \vdots B_2(\lambda) 0 \cdots 0 B_1(\lambda)\end{bmatrix},
\beta_{\lambda}= \frac{12 N}{(tr m^2)^2}(trm_{\perp}^{4}(e \beta_e - 2e^2 \sum_{i}\frac{m_i \beta_{m_i}}{tr m^2}) + 2 e^2\sum_i m_i\beta_{m_i}(m_{i}^2 - \frac{tr m^2}{N}))
k_1(n,m) = B(n-1) + \underset{j=1}{\overset{m-1}{\sum}}(-1)^j\binom{m-1}{j}B(n-(j+1)) &= B(n-1) + \underset{j=2}{\overset{m}{\sum}}(-1)^{j+1}\binom{m-1}{j-1}B(n-j) &= \underset{j=1}{\overset{m}{\sum}}(-1)^{j+1}\binom{m-1}{j-1}B(n-j)
F_{n + r}\sum_{j = 0}^k{( - 1)^{(r + 1)j}\left({\frac{{F_r}}{{F_n}}}\right)^j G_{m + r + (n + r)j}}= ( - 1)^r F_n G_m + ( - 1)^{(r + 1)k}F_n \left({\frac{{F_r}}{{F_n}}}\right)^{k + 1}G_{m + (k + 1)(n + r)},n\ne 0
\frac d{ds}\bigg|_{s=0}\widetilde{Vol}(\Sigma_s)= -\int_{\Sigma}\tilde{g}(X,\tilde{\textbf{H}})d\tilde{v}=-\int_{\Sigma}e^{-\frac 2{n}f}g(X,\tilde{\textbf{H}})e^{-f}dv,
\left[g_n\right]_{\alpha,\varphi_{n}(E_M)}\leq\left[\phi^{0}_{n}\right]_{\mathrm{Lip},\psi_n\circ f(E_M)}\left[f_n\right]_{\alpha,\varphi_{n}(E_M)}\left[\phi_{n}\right]_{\mathrm{Lip},\varphi_{n}(E_M)}^\alpha \ .
0 =\lim_{\rho\rightarrow0^{+}}\lim_{j\rightarrow+\infty}\frac{1}{\rho^{n}}\int_{Q_{\rho}^{\nu}}|u_{j}(x+x_{0})-w^{\nu}(x+x_{0})|~dx & =\lim_{\rho\rightarrow0^{+}}\lim_{j\rightarrow+\infty}\int_{Q_{1}^{\nu}}|v_{j,\rho}(x)-w^{\nu}(x)|~dx~.
\frac{K_q(dx)}{dx}=\frac{q}{p}\Phi(p+q)\left(\frac{\Phi(p+q)}{\Phi(q)}-1\right)e^{-\Phi(p+q)x}-\frac{q\Phi(p+q)}{\Phi(q)}k_q(x),\ \x \in \mathbb R,
{(\begin{matrix}\infty\\r\end{matrix})}_{q}=lim_{m\rightarrow\infty}{(\begin{matrix}m\\r\end{matrix})}_{q}=\frac{1}{(1-q)(1-q^{2})\cdot\cdot\cdot(1-q^{r})}=\frac{1}{[r]_{q}!(1-q)^{r}}
\Theta(z)=e^{\pm iz(\sqrt{\frac{m_1}{2b}}\pm\sqrt{\frac{m_2}{2b}})}w_2\left(\frac{z}{\mathcal{Z}}\right)\left(\frac{z}{\mathcal{Z}}\right)^{s_1-2s_2-2s_3-2s_4-2s_5}.
\partial_{t_{i}}\partial_{t_{j}}\lambda_{S W}= \sum_{k = 1}^{r}{C_{i j}}^{k}( t ) \partial_{t_{k}}\partial_{t_{r}}\lambda_{S W}+{\frac{1}{2 \pi i}}\mu \partial_{\mu}\left({\frac{x^{- 2}Q_{i j}}{( \widetilde{W}_{B}^{2}- 4 \mu^{2})^{1 / 2}}}\right) d x + \partial_{x}( * ) \, d x ,
\Phi_{\alpha, \bar \alpha s}- \Phi_{\bar \alpha, \alpha s}+ \Phi_{\beta, \bar \beta s}- \Phi_{\bar \beta, \beta s}+ [(q^{-x}\cdot \Phi_{\bar \alpha,s}\Phi_{\alpha, rs}+ q^x \cdot \Phi_{\alpha, r}\Phi_{\bar \alpha, ss}) - (q^{-x}\cdot \Phi_{\bar \alpha, r}\Phi_{\alpha, ss}
V_f^{\pm}(y)=\pi^{-\frac{k}2}\left(W_{\frac{k}2,\nu}(4\pi y)\pm\epsilon\nu^kW_{-\frac{k}2,\nu}(4\pi y)\right)=\begin{cases}4\sqrt{y}K_{\nu}(2\pi y) k=0\epsilon=\pm1, 0 k=0\epsilon=\mp1, 4yK_{\nu\pm\frac{\epsilon}2}(2\pi y) k=1.\end{cases}
B_{1}= \{4,6,7,9,10,11\}, B_{5}= \{2,3,4,8,10,11\}, B_{9}= \{1,3,4,6,7,8\}, B_{2}= \{1,5,7,8,10,11\}, B_{6}= \{1,3,4,5,9,11\}, B_{10}= \{2,4,5,7,8,9\}, B_{3}= \{1,2,6,8,9,11\}, B_{7}= \{1,2,4,5,6,10\}, B_{11}= \{3,5,6,8,9,10\}. B_{4}= \{1,2,3,7,9,10\}, B_{8}= \{2,3,5,6,7,11\},
S [ A_{0}+ a_{0}, a_{j},{\bar{\psi}}, \psi ] = \int d^{3}x \left[ \frac{1}{4}F_{\mu \nu}( A + a ) F_{\mu \nu}( A + a ) + \sum_{b}{\bar{\psi}}_{b}( \not \! \partial + i e_{N}( \not \! \! A + \not \! a ) ) \psi_{b}\right] \; .
\mu_{\mathbf{v}}(t) = f^t_{\mathcal{X},\mathcal{Y}}(\gamma_{\mathbf{v}}(t))=\gamma_{\mathbf{v}}(t)+\mathcal{O}(t^{N+1}) = x_0+\tfrac{C}{2}t^N\mathbf{v}+\mathcal{O}(t^{N+1}).
\langle h_{2}| \phi^{( h )}( z ) | h_{1}\rangle = \lim_{u \rightarrow \infty \atop w \rightarrow 0}u^{2 h_{2}}\langle 0 | \phi^{( h_{2})}( u ) \phi^{( h )}( z ) \phi^{( h_{1})}( w ) | 0 \rangle = \frac{\sqrt{C_{h_{2}, h , h_{1}}}}{z^{h + h_{1}- h_{2}}}
U_{\varepsilon}(t,\xi):=\left(\begin{array}{c}i\langle\xi\rangle \widehat{u}_{\varepsilon}(t,\xi) \partial_{t}\widehat{u}_{\varepsilon}(t,\xi)\end{array}\right), U_{0}(\xi):=\left(\begin{array}{c}i \langle\xi\rangle\widehat{u}_{0}(\xi) \widehat{u}_{1}(\xi)\end{array}\right),
q_2(x) y''(x) + q_1(x) y'(x) + q_0(x) y(x) = 0, q_2(x) = 5x(302400x - 31)(373248000x^2 + 216000x + 1), q_1(x) = 1354442342400000x^3 + 64571904000x^2 - 61473600x - 31, q_0(x) = 300(902961561600x^2 - 240974784x - 4991).
u(x,\overline{t})= \int_{\mathbb{R}^{n}}g(x,y,\overline{t})\phi(y)\,\mathrm{d}y +\int_{0}^{\overline{t}}\int_{\mathbb{R}^{n}}g(x,y,\overline{t}-s) u^{p}(y,s)\,\mathrm{d}y\mathrm{d}s.
\begin{aligned}i_{(\ast e_{v}^{1})^{\sharp}}dv\wedge e_{v}^{2}=\ast\big((\ast e_{v}^{1})\wedge(\ast dv)\big)\wedge e_{v}^{2}=-\ast\big((\ast e_{v}^{2})\wedge(\ast dv)\big)\wedge e_{v}^{1}=-i_{(\ast e_{v}^{2})^{\sharp}}dv\wedge e_{v}^{1}. \end{aligned}
\langle J^{ai}(x) \rangle_{\tilde{A},\phi}= - \frac{\delta S}{\delta \tilde{A}^{a}_i(r_b,x)}, \hspace{1em}\langle{\cal O}^I(x) \rangle_{\tilde{A},\phi}= - \frac{\delta S}{\delta \phi^I(r_b,x)}
\left[ \begin{array}{c}r i \end{array}\right]=(r-1)\left[ \begin{array}{c}r-1 i \end{array}\right]+\left[ \begin{array}{c}r-1 i-1 \end{array}\right], \left[ \begin{array}{c}r 0 \end{array}\right]=\begin{cases}1 r=0, 0 r\neq0,\end{cases}
\lVert A^{n+m}(x)P(x)\rVert =\lVert A^n(f^m(x))A^m(x)P(x)^2 \rVert &=\lVert A^n(f^m(x))P(f^m(x))A^m(x)P(x)\rVert &\le \lVert A^n(f^m(x))P(f^m(x) \rVert \cdot \lVert A^m(x)P(x)\rVert,
\vert \tau(y)-\tau_k(y)\vert <\varepsilon + \sum_{i=k}^{j-1}\vert \tau_{i+1}(y) - \tau_i(y) \vert \leq \varepsilon + \sum_{i=k}^{\infty}\vert \tau_{i+1}(y) - \tau_i(y) \vert.
C^{-1}_{\mu\nu}= -\frac{1}{m^2}\left (\begin{array}{cccc}0 \partial^x_1 \partial^x_2 -1 \partial^x_1 0 -m 0 \partial^x_2 m 0 0 1 0 0 0 \end{array}\right ) \delta^2 (x-y) .
\Im{(\sigma_{p}^{-1}z)}^{1-s}=1 - (s-1)\log\big(\Im{(\sigma_{p}^{-1}z)}\big)+ O\big((s-1)^{2}\big), \,\,\,\,\frac{1}{2s-1}= 1- 2(s-1) + O\big((s-1)^{2}\big).
\begin{cases}y(t) = e^{tA}\phi_0 + \displaystyle\int^t_0 e^{(t-s)A}\alpha A_1y(t-r)ds + \displaystyle\int^t_0\int^0_{-r}\beta(\theta)e^{(t-s)A}A_2y(t+\theta)d\theta ds \hskip 150pt + \displaystyle\int^t_0 f(s)ds,\,\,\,\,\,\,t\in [0, T], y(0) =\phi_0,\hskip 5pt y_0=\phi_1,\,\,\,\,\,\Phi=(\phi_0, \phi_1)\in{\cal X},\end{cases}
({\gamma}^{\mu}i{\partial}_{\mu}- m_{1}){\psi}_{1}( x , t ) - \frac{1}{2}e_{1}b ( t ){\gamma}^{1}{\psi}_{1}( x , t ) + e_{1}\int_{0}^{\mathrm{L}}d y D ( x , y | \mathrm{L}) J^{0}( y , t ){\gamma}^{0}{\psi}_{1}( x , t ) = 0 ,
W_{\phi,p}(a(y)wn(\zeta_p)) = \sum_{\mu_p\in{}_p\mathfrak{X}}\mu_p(y^{-1})B_{\mu_p^{-1}\pi_{p},\frac{1}{2}}(\varpi_{p}^{-a(\mu_p\tilde{\pi}_{p})}y^{-1})\epsilon(\frac{1}{2},\mu_p^{-1}\pi_{p})[\mathfrak{M}W_l^{\omega_{\pi,p}}](\mu_p).
\int_{Q_K}\Delta_2(x_k, y_k) \nu_q(dx, dy) = \frac{1}{n \beta}\Bigl( (\mathcal{H}_q^{-1})_{2k-1,2k-1}- (\mathcal{H}_q^{-1})_{2k, 2k}\Bigr) + O \left( \frac{1}{n k^2}\right).
\Psi(l_c)^{-1}(L^j\ltimes(L\cap\sigma_jN))=L^{j,c}\ltimes(L\cap\sigma_{j,c}N),(L^{j,c}\ltimes(L\cap\sigma_{j,c}N))\cap P_L=L_L^{j,c}\ltimes (N_L^{j,c}(N_L\cap\sigma_{j,c}N)).
\frac{\pi^2}{\alpha^2}- \lambda(\ell) = \begin{cases}\displaystyle \left( \frac{\pi^3}{4\alpha^3}\right)^2 \ell^4 + o(\ell^4), \alpha_+ \neq \alpha_-, begin{align*}12pt] \displaystyle \left( \frac{\pi^3}{2\alpha^3}\right)^2 \ell^4 + o(\ell^4) , \alpha_+ = \alpha_- ,\end{cases}\ell \to 0 .
\frac{8\pi}{3}G_{N}\rho_{eff}= (\frac{\dot{\alpha}}{\alpha})^{2}=\frac{(C+E)^{2}g_{S}e^{2\Phi}-|g_{00}|(g_{S}g^{3}+\ell^{2}e^{2\Phi})}{4|g_{00}|g_{rr}g_{S}g^{3}}(\frac{g'}{g})^{2}
-\frac{\widetilde h_{\alpha,\theta}^\prime(s|\mu)}{\widetilde h_{\alpha,\theta}(s|\mu)}= \mu\dfrac{s^{\alpha-1}}{(1+s)^{\alpha-\theta}}\equiv \mu\,\rho_{\alpha,\theta}(s) \implies \widetilde h_{\alpha,\theta}(s|\mu) = \exp\left\{-\mu\int_0^s \rho_{\alpha,\theta}(t) dt \right\}
[{\cal J}^{\mu \nu},{\cal J}^{\rho \sigma}] = \eta^{\mu \rho}{\cal J}^{\nu \sigma}- \eta^{\nu \sigma}{\cal J}^{\mu \rho}+ \eta^{\nu \rho}{\cal J}^{\mu \sigma}- \eta^{\mu \rho}{\cal J}^{\nu \sigma},
\langle A \rangle_{0}= \lim_{R\rightarrow\infty}\frac{1}{Z_{R}[0]}\mbox{Tr}\, (e^{-\beta H_{R}}\, \delta(Q_R)\, A) =\lim_{R\rightarrow\infty}\frac{1}{Z_{R}[0]}\mbox{Tr}\, (e^{-\beta H_{R}}\,P_{s}\,A)\,,
| \delta{h_{\omega}}| = \sqrt{H_{0}/ M_{s}}z_{e q}^{- 1 / 4}z_{o u t}^{1 / 2}\exp{( \frac{1}{2}\phi_{e n d})}({\frac{\omega}{\omega_{m a x}}})^{\frac{1}{2}}\ln ({\frac{\omega}{\omega_{m a x}}})
\begin{array}{@{}l@{}}{\displaystyle \partial_{\tau}u^{k}= (u^{k}-u^{k-1})/\tau, \quad\partial_{\tau}^{2}u^{k}=(\partial_{\tau}u^{k}-\partial_{\tau}u^{k-1})/\tau,}begin{align*}2mm]{\displaystyle \overline{u}^{k}= (u^{k}+u^{k-1})/2, \widetilde{u}^{k}=(u^{k}+u^{k-2})/2,}begin{align*}2mm]\end{array}
\begin{aligned}b_1 =a_1^{k_{1,1}}, b_2 =(a_2a_1)^{k_{2,1}}a_2^{k_{2,2}}, b_3 =(a_3a_1)^{k_{3,1}}(a_3a_2)^{k_{3,2}}a_3^{k_{3,3}}, \vdotswithin{=}b_n =(a_na_1)^{k_{n,1}}(a_na_2)^{k_{n,2}}\dotsm(a_na_{n-1})^{k_{n,n-1}}a_n^{k_{n,n}}, \end{aligned}
\begin{aligned}[L^2, H^1_0]_\gamma=D(\L^\gamma), & [H^{-1}, L^2]_\gamma=\big([H^1, L^2]_\gamma\big)^*=\big([L^2, H^1]_{1-\gamma}\big)^*=D(\L^{1-\gamma})^*=D(\L^{\gamma-1}).\end{aligned}
2N-2\geq \sum_{\ell=1}^{r_1}(e_{1,\ell}-1) +\sum_{j=2}^r \sum_{\ell=1}^{r_j}(e_{j,\ell}-1) = N-p_1-u_1-v_1 + \sum_{j=2}^r(N-p_j-q_j).
\hat H^{(q)}_I(\hat x,\hat p)=\frac{f^2}{6}\sum_{k=1}^\infty \frac{V^{(k)}(0)}{k!}\hat x^k\biggl(k \hat x^2 \partial^2_{\hat x}+ k(k+2) \hat x \partial_{\hat x}+\frac{1}{6}k(k-1)(2k+5)\biggr).
w:=\rho\left(\frac{L_{1}y}{y^{\frac{\gamma}{\beta}}}+\frac{1}{\pi_{1}^{\frac{\gamma}{\beta}}}\right)=\rho\left(\frac{L_{1}y}{y^{\alpha}}+\frac{1}{\pi_{1}^{\alpha}}\right)\ \ \ \big[t_{1},\infty\big).
{\tilde{\gamma}_{5}}={\frac{\epsilon^{c d}}{4 \sqrt{g}}}{\hat{\epsilon}^{E F}}\partial_{c}X^{\mu}\partial_{d}X^{\nu}N_{E}^{\lambda}N_{F}^{\rho}\frac 1 4 [ \gamma_{\mu}, \gamma_{\nu}] \, [ \gamma_{\lambda}, \gamma_{\rho}] .
H_{1}= \exp \left[ i \theta_{1}\left( \begin{array}{c c c}{{1}}&{{0}}&{{0}}\\{{0}}&{{- 1}}&{{0}}\\{{0}}&{{0}}&{{0 \}}\\ \end{array}\right) \right] \quad ; \quad H_{2}= \exp \left[ i \theta_{2}\left( \begin{array}{c c c}{{1}}&{{0}}&{{0}}\\{{0}}&{{1}}&{{0}}\\{{0}}&{{0}}&{{- 2}}\\ \end{array}\right) \right]
S ={\frac{\pi \bar{r}_{+}^{2}}{G}}+{\frac{1}{1 2}}( 1 - ({\frac{\bar{r}_{-}}{\bar{r}_{+}}})^{2}) \ln{\frac{L}{( \bar{r}_{+}- \bar{r}_{-})}}+{\frac{1}{6}}\ln{\frac{\bar{r}_{+}}{z_{0}}}~ ~ .
\sum_{k=4}^\infty s_{b+1,k}\geq \sum_{k=4}^\infty \frac{y_{b+1}}{(b+1)^{\left\lfloor\frac{k}{2}\right\rfloor}}= 2y_{b+1}\sum_{k=2}^\infty \frac{1}{(b+1)^k}\geq \frac{2y_b}{b(b+1)}.
\ \v(n) = 1\; n_1 = 0 \; \; 1 \leq n_j \leq M_j \; \; j \ne 1, & \; k = 2, \dots, g & \ \v(n) = 0\; n_k = 0 \; \; 1 \leq n_j \leq M_j \; \; \j \ne k.
S_{si}= \oint_0{dz \over 2\pi i}{1 \over z^{s-1}}\oint_0{dw \over 2\pi i}{1 \over w^{i+2}}{(f'(z))^2 (f'(w))^{-1}\over f(z)-f(w)}\biggl({f(w) \over f(z)}\biggr)^3.
T_{k}=\left[\begin{array}[c]{ccccc}\alpha \beta\sqrt{d_{2}-1}0 0 \beta\sqrt{d_{2}-1}\alpha d_{2}\ddots & \ddots \ddots \beta\sqrt{d_{k-1}-1}& \beta\sqrt{d_{k-1}-1}\alpha d_{k-1}\beta\sqrt{d_{k}}0 0 \beta\sqrt{d_{k}}\alpha d_{k}\end{array}\right] ,
\bar{W}( \phi ) = \pm 4 \frac{1}{\sqrt 2}( \frac{1}{4}\mathrm{s i n h}2 \phi - \frac{3}{2}\phi ) \hspace{0 . 5 c m}, \hspace{0 . 5 c m}\bar{W}( \phi ) = \pm \frac{\phi^{2}}{2}( \frac{\phi^{2}}{2}- 1 ) \, .
\frac{d}{ds}\Big|_{s=0}\frac1h \int \left( \tilde u^n_s-u^{n-1}\right) \cdot G_h\ast\left( \tilde u^n_s-u^{n-1}\right) dx = 2 \int G_h\ast\left( \frac{u^n-u^{n-1}}h\right) \cdot (Id-u^n\otimes u^n) \varphi\, dx.
{\cal{L}}={\frac{i}{2}}\ [ \{\bar{\psi}}\gamma^{\mu}( \overrightarrow{\partial}{}_{\mu}+ i e A_{\mu}) \psi -{\bar{\psi}}\gamma^{\mu}( \overleftarrow{\partial}{}_{\mu}- i e A_{\mu}) \psi \ ] - m{\bar{\psi}}\psi -{\frac{1}{4}}F^{\mu \nu}F_{\mu \nu}\ ,
\sqrt{2}M_{j}= \pm \sum_{i}\frac{\nu_{i}\phi(a_{i})}{a_{i}^{2}-m_{j}^{2}}\mbox{det}(a_{i}^{2}- m^{2})^{1/r_{i}}\prod_{j\ne i}(a_{j}^2-a_{i}^2)^{1-2r_{j}/r_{i}}\Lambda_{N=2}^{\frac{4(N_{c}-1)-2N_{f}}{r_{i}}}.
\lim_{n\rightarrow\infty}\frac{k_1}{n}= R_1\frac{1-\delta_2}{\delta_2(1-\delta_1\delta_2)}\lim_{n\rightarrow\infty}\frac{k_2}{n}= R_2\frac{1-\delta_1}{\delta_1(1-\delta_1\delta_2)},
\int_{\Omega}\left(1-\rho^{2}\right)\left|\nabla\varphi_{0}\right|^{2}\leq 2\pi \left(\sum_{i=1}^{m}d_{i}^2\right) I\left(\frac{R}{R_{0}}\right)+2\pi(1-a^{2}) \sum_{i\neq j}\left|d_{i}\right|\left|d_{j}\right|\log \dfrac{R}{\left|a_{i}-a_{j}\right|}+C.
\begin{array}{ll}[K_{A},K_{B}]=0~~~ ~~~[M_{AB},K_{C}]=i(\eta_{AC}K_{B}-\eta_{BC}K_{A}){}{}\displaystyle{[K_{A},S^{i}]=0}~~~ ~~~ \{S^{i},S^{j}\}=2{\cal E}^{ij}\tilde{\gamma}^{A}K_{A}{}{}\multicolumn{2}{c}{[M_{AB},S^{i}]=i\textstyle{\frac{1}{2}}\tilde{\gamma}_{[A}\gamma_{B]}S^{i}}\end{array}
\sum_{i=1}^{L}c_{i}H_{L+m-i}-\sum_{i=1}^{L}c_{i}G_{L+m-i}\geq 2\left( \sum_{i=1}^{L}c_{i}H_{L+m-1-i}-\sum_{i=1}^{L}c_{i}G_{L+m-1-i}\right).
\int_{\Omega}a(x)^{\frac{p+1}{p-1}}\Theta(s,x;t_0)^{\lambda - \frac{p+1}{p-1}}\,dx = \int_{\Omega_1(s)}a(x)^{\frac{p+1}{p-1}}\Theta(s,x;t_0)^{\lambda - \frac{p+1}{p-1}}\,dx + \int_{\Omega_2(s)}a(x)^{\frac{p+1}{p-1}}\Theta(s,x;t_0)^{\lambda - \frac{p+1}{p-1}}\,dx =: I(s) + I\!I(s).
P\int_0^{\infty{e}^{{i}\varphi}}\frac{t^{n-1}\,{e}^{at}}{z\,{e}^{t}-1}\,{d}t =\int_\Gamma \frac{t^{n-1}\,{e}^{at}}{z\,{e}^{t}-1}\,{d}t-{\rm sgn}(\varphi)\,{i}\,\pi\,(-\ln z)^{n-1}\,z^{-a}.
F\cdot\psi =\frac{1}{k!}\sum_{a_1,\ldots, a_k}F_{a_1\ldots a_k}\Gamma^{a_1\ldots a_k}\psi &=\sum_{a_1<\ldots<a_k}F_{a_1\ldots a_k}\Gamma^{a_1}\cdots\Gamma^{a_k}\psi,
\begin{aligned}Q_{\Psi(P_1,Q_l;F)}= R_{P_1}^*(Q_l) - R_{Q_l}^*(P_1) = D_t Q_l,l=1,2,3 & Q_{\Psi(P_2,Q_l;F)}= R_{P_2}^*(Q_l) - R_{Q_l}^*(P_2) = D_x Q_l,l=1,2,3\end{aligned}
g^* \o_{\tilde{F}}= \frac{1}{\mu(G)}\int_{G}g^*(h^* \o_1) d \mu(h) = \frac{1}{\mu(G)}\int_{G}(hg)^*(\o_1) d \mu(h) =\frac{1}{\mu(G)}\int_{G}k^*\o_1 d \mu(k)= \o_{\tilde{F}},
P_{x}= \frac{2}{\pi \alpha^{\prime}\sqrt{- K}( 1 - K L^{2})}\left\{- \frac{2}{3}+ \frac{1}{K L^{2}}- \frac{1 - K L^{2}}{K L^{3}\sqrt{- K}}[ \frac{\pi}{2}- \operatorname{a r c s i n}\frac{1}{\sqrt{1 - K L^{2}}}] \right\},
\frac{det(-\frac{d^{2}}{dx^{2}}+A)}{det(-\frac{d^{2}}{dx^{2}})}=\prod_{n=1}^{+\infty}(1+\frac{L^{2}A}{n^{2}\pi^{2}})=\frac{sinhL\sqrt{A}}{L\sqrt{A}}.
\eta(x, q, \beta) = -\sup_{\beta \in \mathbb{R}^n}\bigl[ - 0 \cdot \beta - \eta(x, q, \beta) \bigr] = \sup_{p \in \mathbb{R}^d}\bigl[ q \cdot p - \lambda(x, p, 0) \bigr] = \rho(x, q).
\sigma_{e_\beta}(z_k)=\!\!\!\sum_{\stackrel{j_1 \geq 0,}{\gamma_1(\beta)-j_1\alpha_1 \in \Delta_+}}c_{\alpha_1,\beta}^{\gamma_1,j_1}\, z_1^{j_1}\,\sigma_{e_{\beta_{\gamma_1(\beta)-j_1\alpha_1}}}^{(2)}(z_k)+\begin{cases}z_1\sigma_{h_{\alpha_1}}^{(2)}(z_k), \textrm{if}\beta=\alpha_1\textrm{and}\gamma_1=e, 0, \mbox{otherwise}.\end{cases}
\tilde{\phi}_{jk}^{n+1}=\sum_{p=-M/2}^{M/2 -1}\sum_{q=-M / 2}^{M/2-1}\frac{\widehat{(\mathcal{H}^n)}_{pq}}{1+\frac{\tau}{2}\eta^n+\frac{\tau^2}{2}\vartheta^n+\frac{\tau^2}{4}\left((\varrho^{x}_{p})^2+(\varrho^{y}_{q})^2\right)}\, e^{i \frac{2j p \pi}{M}}e^{i \frac{2k q \pi}{M}}.
\int_{\mathbb{R}^n}|v_c|^{p-2}v_c (x) \phi (x) dx &= \int_{\mathbb{R}^n}c (-\Delta + m^2 c^2)^{1/4}v_c (x) (-\Delta + m^2 c^2)^{1/4}\phi (x) - mc^2 v_c \phi (x) + \mu v_c (x) \phi (x) dx & = \int_{\mathbb{R}^n}c v_c (x) (-\Delta + m^2 c^2)^{1/2}\phi (x) - mc^2 v_c \phi (x) + \mu v_c (x) \phi (x) dx.
\langle w_{1, i_1+1}, w_{1,i_2+1}, w_{1, i_3+1}\rangle_d = \chi(\O_{ev_1^{-1}(g_1 \cdot L_{i_1})\cap ev_2^{-1}(g_2 \cdot L_{i_2})\cap ev_3^{-1}(g_3 \cdot L_{i_3})}).
\left[\widehat{|g|^2_r}(z)\right]^{1/2}= \left( \frac{1}{|D(z,r)|}\int_{D(z,r)}|f(w)-\widehat{f}_r(w)|^2 dV(w) \right)^{1/2}&\leq \left( \frac{1}{|D(z,r)|}\int_{D(z,r)}|f(w)-\widehat{f}_r(z)|^2 dV(w) \right)^{1/2}&+ \left( \frac{1}{|D(z,r)|}\int_{D(z,r)}|\widehat{f}_r(z)-\widehat{f}_r(w)|^2 dV(w) \right)^{1/2}&\leq MO_r(f)(z)+w_r(\widehat{f_r})(z).
\sum_{i}(m_i-1)\widetilde{R}_hG_i \ge \sum_{t=1}^{\eta_p}\sum_{k=1}^{\iota^t(F_p)}(m^{t,k}-1)\widetilde{R}_h\widehat{E}^{t,k}\ge (n-2)\sum_{t=1}^{\eta_p}\sum_{k=1}^{\iota^t(F_p)}(m^{t,k}-1) \ge (n-2)\sum_{t=1}^{\eta_p}(\iota^t(F_p)+2\kappa^t(F_p)) = \; (n-2)(\iota(F_p)+2\kappa(F_p)).
C_{\Pi_{1S_0}\times \Pi_{2S_0}}(s): =\left(\prod_{\textrm{infinite}w \in S_0}\prod_{i=1}^{n_1}\prod_{j=1}^{n_2}\left|\frac{1+\mu_{\Pi_{1iw}\times \Pi_{2jw}}+s}{2\pi}\right|_w\right)\prod_{\textrm{finite}w \in S_0}N_{\Pi_{1w}\times \Pi_{2w}}.
\omega^{(S)-1}(x) \{\tilde{A}_i^{(S)}(x) - S(x) \tilde{A}_i^{(N)}(x) S^{-1}(x) \}\omega^{(S)}(x) = 2 \sqrt{6}\{{\cal T}_i^{(N)}(x) -{\cal T}_i^{(S)}(x) \},
\lim_{n\to\infty}\,\frac{\ \Big\| f(e^{i Z_n}U_n)-f(U_n)-\frac{d}{dt}\bigl(f(e^{it Z_n}U_n)\bigr)_{\vert t=0}\Big\|_1\,}{\|Z_n\|_2^2}\,=\,\infty.
_{(V_\alpha \cap \mathbb{P}_\alpha)}\big( h \restriction \alpha \big) \Vdash_{(V_\alpha \cap \mathbb{P}_\alpha)}h(\alpha) \le_{\dot{Q}_\alpha}\pi^W_{\alpha+1}(f) (\alpha) = \dot{y}^W_{f(\alpha)}= \dot{\rho}\big( f(\alpha) \big).
g_{m,N}(\xi):=\prod_{i=1}^{m}\prod_{k=1}^{N}\xi_{i,k}^{\sum_{j=0}^{k-1}\binom{j+i-1}{i-1}\binom{N-k+m-i}{m-i}}=\prod_{i=1}^{m}\prod_{k=1}^{N}\xi_{i,k}^{\binom{k+i-1}{i}\binom{N-k+m-i}{m-i}},
\sum^r_{i=0}{r\choose i}\int^\infty_0 (\cosh t)^{-\beta-3}\tilde{c}_i(t)D(t)dt &\qquad\times\int_{[0,\pi/2]^2}J_i(\theta,\theta',t) (\sin\theta\sin\theta')^i (\cos\theta\cos\theta')^{r-i}\sin2\theta\sin2\theta'd\theta d\theta' \|\phi\|^2,
J = \int_{I_{0}}+ \sum_{k=1}^{\infty}\left( \int_{I_{k}}+\int_{I_{k}'}\right)v_0^{1-p'}(t)(1+t)^{-\frac{p'(n-1)}{2}}t^{n-1}\,dt & \le C\sum_{k=0}^{\infty}\int_{I_{k}}\le C\int_{|x|\in I}v^{1-p'}(x)(1+|x|)^{-\frac{p'(n-1)}{2}}\,dx<C.
b_s^k \ = \ \frac{f(s,X_s^{t,x},\check Y_s^{t,x},Z_s^{(k-1)})-f(s,X_s^{t,x},\check Y_s^{t,x},Z_s^{(k)})}{V_s^k}1_{\{V_s^k\neq0\}}, t \leq s \leq T,
\sum_{m \in \Delta(n)}\frac{1}{m^{d_k}}E_n^{C}(T_m) = \sum_{m\in \Delta(n)}\frac{2}{m^{d_k}(1 - m^2)}+ \sum_{m \in \Delta(n)}\frac{2}{m^{d_k}(4r^2 - 1)}.
\delta_h^{\alpha}= \prod_{i=1}^{d}\bigg( \frac{\Delta_{h_ie_{i}}}{h_i}\bigg)^{\alpha_i}= \bigg( \frac{\Delta_{h_1e_{1}}}{h_1}\bigg)^{\alpha_1}\dotsb\ \bigg( \frac{\Delta_{h_de_{d}}}{h_d}\bigg)^{\alpha_d}
\frac{1}{2\pi}\int_{-\infty}^\infty \frac{2c}{c^2+(t+b)^2}\int_0^\infty g(x)x^{-\frac12+it}dx\dt &=\int_0^\infty g(x)x^{-\frac12}\frac{1}{2\pi i}\int_{-\infty}^\infty \Big(\frac{c}{t+ic}-\frac{c}{t-ic}\Big)x^{i(t-b)}dt\dx,
\frac{\partial f_{j,m}}{\partial z_k}(x) = \frac{\partial f_{j,m}}{\partial \overline{z}_k}(x) = \frac{\partial \beta_{i_1, \dots , i_p}}{\partial z_k}(x) = \frac{\partial \beta_{i_1, \dots , i_p}}{\partial \overline{z}_k}(x) = 0
(\widetilde{E}\times E)[\prod_{i=0}^{n-1}F(S_i,\widehat{{S}}_i;S_{i+1},\widehat{S}_{i+1})] = \sum_{\overrightarrow{x}\in\Lambda_n}\sum_{\overrightarrow{y}\in\Lambda_n}\frac{1}{d^{2n}}E[\prod_{i=0}^{n-1}F(x_i,y_i;x_{i+1},y_{i+1})].
{\mathsf g}_{s}(d) -{\mathsf g}^{s+1}+ 1 = \tfrac{s^2-(2d+1)s+d^2-d+4}{2}< 0\ \Rightarrow\ \tfrac{2d+1-\sqrt{8d-15}}{2}< s < \tfrac{2d+1+\sqrt{8d-15}}{2},
a = \sqrt{\frac{m \omega}{2 \hbar}}\, \left[ \hat{q}+ i \frac{\hat{p}}{m \omega}\right] \ \ \ , \ \ \a^{\dagger}= \sqrt{\frac{m \omega}{2 \hbar}}\, \left[ \hat{q}- i \frac{\hat{p}}{m \omega}\right] \ \ \ , \ \ \
X_{0}^{n}(t) =Y^{n}(t)+\eta^{n}(t), X_{k}^{n}(t) =X_{k}^{n}(0)-\frac{1}{n}\int_{[0,t]\times \lbrack 0,1]}{{1}}_{[0,r_{k}(\boldsymbol{X}^{n}(s-)))}(y)\,N_{k}^{n}(ds\,dy),\:k\in \mathbb{N}.
\begin{cases}\partial_t m = \nabla \cdot \left(D(m) \nabla m\right) - \chi \nabla \cdot \left(f(m) \nabla c\right) + M(m), \tau \partial_t c = \alpha \Delta c + \lambda d - c + \beta m, \partial_t d = r h(m) \left(1-d\right), \end{cases}
I_3 \le \int_{A_3}\frac{|v(x) - v(y)|^p}{|x - y|^{N+sp}}\, dx dy + C \int_{A_3}\frac{|v(x) - v(y)|^{p-1}\, |v(x)|}{|x - y|^{N+sp}}\, dx dy &\qquad+ C \int_{A_3}\frac{|v(x)|^p}{|x - y|^{N+sp}}\, dx dy =: I_4 + C I_5 + C I_6.
\begin{array}{rl}Q_{ij}={\rm diag}(n_2 n_3 m_1, n_1 n_3 m_2, n_2 n_1 m_3 ) \,, P^{ij}={\rm diag}(m_2 m_3 n_1, m_1 m_3 n_2, m_2 m_1 n_3 ) \,, q_0 = n_1 n_2 n_3 \,, p^0 = - m_1 m_2 m_3 \,. \end{array}