harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
dR^{N,W}_{\beta,M}(\lambda):=\frac{1}{Z^{N,W}_{\beta,M}}\prod_{i<j}|\lambda_i-\lambda_j|^\beta e^{-N\sum_{i=1}^N W(\lambda_i)}\prod_{i=1}^N 1_{|\lambda_i|\le M}d\lambda_i\,,\lambda=(\lambda_1,\ldots,\lambda_N).
\Phi_{2}( z_{1}, z_{2}) = \langle ( \tilde{m})^{\infty}| u_{k j}( z_{1}) u_{i l}( z_{2}) | ( m )^{0}\rangle = ( z_{1}z_{2})^{\Delta}( T_{2}\chi_{1}^{+}( z ) + T_{1}\chi_{2}^{+}( z ) )
\Omega_{\rho}(z_1,z_2)=\sum_{n_1=0}^{\infty}\sum_{n_2=0}^{\infty}A(n_1\!+\!\rho_1,n_2\!+\!\rho_2)z_1^{n_1+\rho_1}z_2^{n_2+\rho_2},
\frac{\partial \Phi_{n}^{*}(z^{-1})}{\partial \bar{t}_{1}}=-\frac{S_{n}^{*}}{S_{n-1}^{*}}\frac{h_{n}}{h_{n-1}}(\Phi_{n}^{*}(z^{-1})-z^{-1}\Phi_{n-1}^{*}).
\begin{aligned}\int_X|\nabla^{g^+}u|_{g^+}^pdV_{g^+}=\int_{X\varepsilon}x^{p-n-1}|\nabla^gu|_g^pdV_g &=\int_{X\varepsilon}x^{p-n-1}|f'(x)|^p\cdot|\nabla^gx|_g^pdV_g &=\int_0^\varepsilon x^{p-n-1}|f'(x)|^p\cdot Vol(\Sigma_x)dx\end{aligned}
R=a^6 D^6 + R^{(5)}D^5 + R^{(4)}D^4 + R^{(3)}D^3 + R^{(2)}D^2 + R^{(1)}D + R^{(0)}&+\sigma^{(1)}D^{-1}\gamma^{(1)}+\sigma^{(2)}D^{-1}\gamma^{(2)},
\begin{array}{r c l}{\mathrm{H t r}[ T_{a}, \cdots ] = 0 ,}{{\qquad}}{\mathrm{H t r}[ \Sigma_{\alpha}, \cdots ] = 0 ,}{\mathrm{S t r}[ T_{a}, \cdots ] = 0 ,}{{\qquad}}{\mathrm{S t r}\{\Sigma_{\alpha}, \cdots \}= 0 ,}\end{array}
s = (\underbrace{\sigma_1,\ldots ,\sigma_1}_{k_1},\underbrace{\sigma_2,\ldots ,\sigma_2}_{k_2-1},0,\underbrace{\sigma_3,\ldots,\sigma_3}_{k_3-1},0,\ldots ,\underbrace{\sigma_{\nu},\ldots ,\sigma_{\nu}}_{k_{\nu}-1},0)
q(y_2|y_1)= &\begin{cases}a(y_1,\sigma)\dfrac{\psi_{-1,\sigma}(y_2)}{\Psi_{-1,\sigma}(-\infty,d_2(y_1,\sigma))},\,y_2<d_2(y_1,\sigma), begin{align*}10pt]b(y_1,\sigma)\dfrac{1}{\Delta(y_1,\sigma)},y_2\in[d_2(y_1,\sigma),d_1(y_1,\sigma)], begin{align*}8pt]c(y_1,\sigma) \dfrac{\psi_{+1,\sigma}(y_2)}{\Psi_{+1,\sigma}(d_1(y_1,\sigma),+\infty)},\,y_2>d_1(y_1,\sigma),\end{cases}
\left|\int_{2^N}^{2^N+n}f(r)P(r,k)dr\right|^2=\left|\sum_{j=1}^N\int_{2^N+n_{j-1}}^{2^N+n_j}f(r)P(r,k)dr\right|^2\le N\sum_{j=1}^N\left|\int_{2^N+n_{j-1}}^{2^N+n_j}f(r)P(r,k)dr\right|^2\leq N\sum_{j=1}^N\sum_{p=0}^{2^j-1}\left|\int_{2^N+p2^{N-j}}^{2^N+(p+1)2^{N-j}}f(r)P(r,k)dr\right|^2
\Phi(\alpha, \alpha) = (1,1), \Phi(\beta, \beta, \alpha) = (2,1), \Phi(\beta, \alpha, \alpha) = (3,1), \Phi(\alpha, \beta) = (1,2), \Phi(\beta, \beta, \beta) = (2,3), \Phi(\beta, \alpha, \beta) = (3,3)
\zeta ( s ) ={\frac{1}{1 6 \pi^{2}}}[{\frac{\sigma^{2 - s}}{( s - 1 ) ( s - 2 )}}-{\frac{1}{6}}\sigma_{i i}\sigma^{- s}+{\frac{1}{1 2}}s \sigma^{- s}{\frac{\sigma_{i}^{2}}{\sigma}}] + \cdots .
A=\begin{bmatrix}p_0 p_1 p_2 \hdots p_N 0 \frac{p_1}{q_1}\frac{p_2}{q_1}\hdots \frac{p_N}{q_1}0 0 \frac{p_2}{q_2}\hdots \frac{p_N}{q_2}\vdots \vdots \vdots \ddots \vdots 0 0 0 \hdots \frac{p_N}{q_N}\end{bmatrix}.
\nabla_i F(x_k) = \sum_{j=1}^J c^f_j (A^f)^\top_{j,i}\nabla f_j(A^f_j x_k - b^f_j) + Q x_k= \sum_{j \in \mathcal J^f(i)}c^f_j (A^f)^\top_{j,i}\nabla f_j((r^{f,x}_k)_j) + \sum_{j \in \mathcal J^Q(i)}U_i r^{Q,x}_k
\lim_{q\to\zeta}\frac{(-q^2;q^4)_{\ell d+k}^2}{(1-q)(-q^4;q^4)_{\ell d+k}^2}=\frac{(-\zeta^2;\zeta^4)_{\ell d+k}^2}{(1-\zeta)(-\zeta^4;\zeta^4)_{\ell d+k}^2}=\frac{(-\zeta^2;\zeta^4)_k^2}{(1-\zeta)(-\zeta^4;\zeta^4)_k^2}
Y_{0}=\mathbb{E}\left[ \Psi_{T}\left\langle GX_{T},X_{T}\right\rangle+\int_{0}^{T}\Psi_{s}(\left\langle Q_{s}X_{s},X_{s}\right\rangle+\left\langle R_{s}u_{s},u_{s}\right\rangle )\mathrm{d}s\right] .
\begin{array}{r c l}{{8 \partial \left[ C^{( 7 )}+ 3 \cdot 5 \cdot 7 C^{( 3 )}B^{2}- 3 \cdot 7 C^{( 5 )}B \right] = \hspace{- 4 c m}}}&{{}}&{{}}\\{{}}&{{}}&{{}}\\{{}}&{{}}&{{={}^{\star}G_{( 0 )}^{( 2 )}- 2 8 \left[{}^{\star}G_{( 0 )}^{( 4 )}B - 3 0 \partial C^{( 3 )}B^{2}\right] \, ,}}\\ \end{array}
\pi_1(\alpha_1)(\gamma_1) \cdot \pi_1(\alpha_1)(\gamma_2) = \pi_1(q) (\gamma_1 \otimes \gamma_2) = \pi_1(q \circ \tau) (\gamma_1 \otimes \gamma_2) = \pi_1(q) \pi_1(\tau)(\gamma_1 \otimes \gamma_2) = \pi_1(q) (\gamma_2 \otimes \gamma_1) = \pi_1(\alpha_1)(\gamma_2) \cdot \pi_1(\alpha_1)(\gamma_1).
I(t)\leq (t+1)^{2-3(\gamma-1)}e^{-\frac{a_1}{t+1}}\left(e^{a_1}I(0)-\frac{a_2e^{a_1}}{3(\gamma-1)-1}\right) +\frac{a_2(t+1)}{3(\gamma-1)-1}e^{-\frac{a_1}{t+1}}e^{a_1}\displaystyle\leq C\left(t^{2-3(\gamma-1)}+ t+1\right), \t\in [0,T_m).
D_1 = \frac{\tfrac{1}{2}\gamma^2 \mathcal{L}'}{\frac{\Phi(\gamma)}{\phi(\gamma)}+ \mathcal{L}}- \frac{\mathcal{L}( \tfrac{1}{3}\bigl( 2 + \gamma^2 + \gamma^3 \frac{\Phi(\gamma)}{\phi(\gamma)}\bigr) + \tfrac{1}{2}\gamma^2 \mathcal{L}' - \tfrac{1}{2})}{( \frac{\Phi(\gamma)}{\phi(\gamma)}+ \mathcal{L})^2}.
<J_{\alpha}(x)~{\cal J}_{\mu \nu \rho}^{(3)}(0)>=(\pi_{\mu \nu}\pi_{\rho \alpha}+\pi_{\mu \rho}\pi_{\nu \alpha}+\pi_{\nu \rho}\pi_{\mu\alpha})\left( \frac{f_{31}(t)}{|x|^{4}}\right)
\sum_{S\in \mathcal{R}(G,r)}|E(G \setminus S)| \leq q \big( h_{k-1}(n_1-1, n_2-1,\ldots, n_2 - 1) \big) + (n_1n_2^{r-1}- q) \big( h_k(n_1-1,n_2-1,\ldots,n_2 - 1) \big) &= q(1-n_2) + n_1n_2^{r-1}(h_k(n_1-1,n_2-1,\ldots,n_2-1))
\sum_{j = 0}^k{\sum_{s = 0}^{k - j}{( - 1)^{ns + j}\binom kj\binom{k-j}s\frac{{G_0^j G_n^s}}{{G_r^{j + s}}}H_{m - nk + rj + (n + r)s}}}= ( - 1)^{nk}\left({\frac{{G_{n + r}}}{{G_r}}}\right)^k H_m,G_r\ne 0\,,
{\mathbf{A}}_{Q}=\prod_{1\le k\le n-1}^{\gets}\left ( \prod_{k+1\le m\le n}^{\gets}\left({\mathbf{I}}-({\mathbf{T}}_{k})^{2}\,{\mathbf{T}}_{m,k+1}\right)\cdot \prod_{k+1\le m\le n}^{\to}\left({\mathbf{I}}-{\mathbf{T}}_{m,k}\right)^{-1}\right).
d_j = \frac{\sigma^2_j}{2(1-w_{jj})}, \sigma_j^2= \sum_{i< j}\alpha_i\bigl ( w_{ij}^2c_{e,i}^2 + w_{ij}(1-w_{ij}) \bigr ) + \alpha_jc_{e,j}^2 + \sum_{i>j}\lambda_i \bigl ( w_{ij}^2c_{s,i}^2 + w_{ij}(1-w_{ij})) + \lambda_j\bigl ( c_{s,j}^2(1-w_{jj})^2 + w_{jj}(1-w_{jj}) \bigr ).
2^{r_n}\sup_{s \in [0,T]}|G_p(s)-G_p^{(l),n}(s)|=o_{\mathbb{P}}(1), &2^{r_n}\sup_{s,t \in [0,T],|t-s|\leq |\pi_n|_T}|G_p^{(l),n}(t)-G_p^{(l),n}(s)|=o_{\mathbb{P}}(1).
\prod_{v \in S \cup \{v_0 \}}\gamma(s, R\circ (\rho_v \otimes \mathcal{X}_v), \Psi_v) = \frac{L^{S \cup \{v_0 \}}(s, R\circ (\rho_v \otimes \mathcal{X}))}{L^{S \cup \{v_0 \}}(1-s, R^{\vee}\circ( \rho_v \otimes \mathcal{X}))}.
\int_0^\infty F(\alpha+iy)y^{s-\frac 12}\frac{dy}{y}=\sum_{a\in\{0,1\}}i^{-a}\Delta_f\bigl(s,\alpha,\cos^{(a)}\bigr)\begin{cases}1 k=1(-1)^a=\epsilon, 0 \end{cases}
G_{0,0}[n,m] =n\wedge m, G_{0,1}[n,m] =-n\cdot m, G_{0,2}[n,m] =-\tfrac{1}{6}(n\wedge m) &\quad+\tfrac{1}{6}(n\wedge m)^3+\tfrac{1}{2}n\cdot m\cdot(n\vee m), G_{0,3}[n,m] =\tfrac{5}{24}n\cdot m-\tfrac{1}{6}n^3\cdot m-\tfrac{1}{6}n\cdot m^3.
\varepsilon(\chi_{\lambda},\sigma_{n!/2}) =(-1)^{\frac{(p-1)(n-d_{\lambda})}4}\varepsilon\left(\sqrt{\prod_{h\in{\mathfrak{d}(\lambda)}}h},f\right) &=(-1)^{\frac{(p-1)(n-d_{\lambda})}4}\cdot(-1)^{\frac{(p-1)(n-d_{\lambda})}4}\cdot(-1)^{\frac{(p-1)|A_{}|}2}&=(-1)^{\frac{(p-1)n}{4}},
f_{0}(I_{n}(x))= (c_{n}^{11}x + \alpha_{n}^{11}y_{f_{0}}^{1}(x)+ \beta_{n}^{11}z_{f_{0}}^{1}(x)+ d_{n}^{11}, \gamma_{n}^{11}z_{f_{0}}^{1}(x)+ e_{n}^{11}x+ f_{n}^{11}) \; \mathrm{for}\; n=1, 2, \ldots, K^{11}
\begin{aligned}D_{M_{\mu}/G_\mu}: =\{(X, \alpha ) \in \mathfrak{X}(M_{\mu}/G_\mu ) \times \Omega^1(M_{\mu}/G_\mu )\mid \exists \,(Y, \beta )\in D_{M}, & \;\; T \pi_\mu \circ Y = X \circ \pi_\mu , \; \pi_\mu^\ast \alpha = \beta \},\end{aligned}
r^C_A(b)\stackrel{(\ref{spl.com.1})}= l^B_A(r^C_A(b)) r^B_A(r^C_A(b)) = \; l^B_A(r^C_A(b))b \stackrel{(\ref{spl.com})}=l^B_A(r^C_A(b)) l^C_A(b) r^C_A(b).
\int_{W^+_{e}}\mathcal{D}_\lambda l_{0,\mathrm{S}_{\mathrm{B}},\,\Omega_{\mathrm{B}}}(\delta(w^0))=\mathrm{pf}\left((\mathit{\Omega}_{\rho\sigma})_e \right)=\mathrm{pf}(\mathit{\Omega}_{e})\;.
(\begin{matrix}&h&\\v&&v^{\prime}\\ &h^{\prime}&\end{matrix})\circ_{1}(\begin{matrix}&h^{\prime}&\\w&&w^{\prime}\\ &h^{\prime\prime}&\end{matrix})=(\begin{matrix}&h&\\vw&&v^{\prime}w^{\prime}\\ &h^{\prime\prime}&\end{matrix})
<\lambda|\tilde \Phi^{V^{(1)}_2 \lambda}_{\mu}(z_2)\tilde \Phi^{V^{(1)}_1 \mu}_{\lambda}(z_1)|\lambda>=\sum_{m_1,m_2=0,1 \atop m_1+m_2=1}v^{(1)}_{m_1}\otimes v^{(1)}_{m_2}<\lambda|\tilde\Phi^{V^{(1)}_2\lambda}_{\mu,m_2}(z_2)\tilde \Phi^{V^{(1)}_1 \mu}_{\lambda,m_1}(z_1)|\lambda> .
- J_{4}= x^{i j}y_{j k}x^{k l}y_{l i}-{\frac{x^{i j}y_{i j}x^{k l}y_{k l}}{4}}+{\frac{\epsilon_{i j k l m n o p}}{9 6}}( x^{i j}x^{k l}x^{m n}x^{o p}+ y^{i j}y^{k l}y^{m n}y^{o p})
\frac{x_{1}^{\left(t+1\right)}}{x_{2}^{\left(t+1\right)}}=\frac{\gamma_{1}x_{1}^{\left(t\right)}+\delta_{1}x_{2}^{\left(t\right)}}{\gamma_{2}x_{1}^{\left(t\right)}+\delta_{2}x_{2}^{\left(t\right)}},\quad\forall t\geq t_{0},
\begin{bmatrix}0 I_{\widehat{n}}I_{\widehat{m}}0 \end{bmatrix}\left[\begin{array}{c}\widehat{L}(\lambda) \hline \phantom{\Big{(}}\widetilde{L}(\lambda) \phantom{\Big{(}}\end{array}\right] =\left[\begin{array}{c}\widetilde{L}(\lambda) \hline \phantom{\Big{(}}\widehat{L}(\lambda) \phantom{\Big{(}}\end{array}\right]
0\leq E_{\Delta f-f_t}(x_0,y_0,t_0)-m(\psi_{ss}-\psi_t)(r_0/2,t_0) = E_{|\nabla f|^2-q}(x_0,y_0,t_0)-m(\psi_{ss}-\psi_t)(r_0/2,t_0) \leq \left(m^2\psi_s\psi-m(\psi_{ss}-\psi_t)-\varphi\right)(r_0/2,t_0).
S=\frac{1}{16\pi}\int_{B}d^{D}x\,\sqrt{|g|}\left(R+\alpha^{2}\left(d(d-1)\right)\right) -\frac{1}{8\pi}\int_{\partial{B}}d^{d}x\sqrt{|\gamma|}K+\frac{1}{8\pi}S_{(ct)}(\gamma).
\gamma B^A_{\mu \nu}= \partial_\mu C^A_\nu - \partial_\nu C^A_\mu, \;\gamma C^A_\mu = \partial_\mu \rho^A, \; \gamma \rho^A = 0, \; \gamma B^{* \mu \nu}_A = 0, \; \gamma C^{* \mu}_A = 0, \;\gamma \rho^*_A = 0.
\frac{S_2(w+1-k\,|\,1,\tau)}{S_2(w+1\,|\,1,\tau)}= \prod\limits_{j=0}^{k-1}S_1\bigl(w-j\,|\,1,\tau\bigr), = \prod\limits_{j=0}^{k-1}2 \sin\pi\bigl(\frac{w}{\tau}-\frac{j}{\tau}\bigr).
\left({\frac{\Phi_{\alpha_{1}}\cdots \Phi_{\alpha_{N - 2}}{\cal B}_{\beta_{1}\dots \beta_{N - 4}}}{\Lambda^{3}}}\right)^{\frac{3}{2 N - 9}}= \left({\frac{\left[ \Phi_{\alpha_{1}}\cdots \Phi_{\alpha_{N - 2}}{\cal B}_{\beta_{1}\dots \beta_{N - 4}}\right]^{3}}{\Lambda^{9}}}\right)^{\frac{1}{2 N - 9}}.
\begin{array}{c}{{\Gamma^{M}\Gamma^{N}+ \Gamma^{N}\Gamma^{M}= 2 \eta^{M N}, \qquad \eta^{M N}= ( + , - , - , \dots ) ,}}\\{{( \Gamma^{M})^{*}= - \Gamma^{M}, \qquad ( \Gamma^{0})^{\mathrm{T}}= - \Gamma^{0}, \qquad ( \Gamma^{i})^{\mathrm{T}}= \Gamma^{i}, \quad i = 1 , \dots , 9 .}}\\ \end{array}
y_1^{n_1}y_2^{n_2}= y_1^{n_1-2}(y_1^2y_2^{n_2-1})y_2= y_1^{n_1-2}y_2^{n_2+1}y_2= \cdots= y_1y_2^{n_1+n_2-1}= y_1y_2^{n-1}.
\int_{r=r_0}|(d\mathcal{M}^{j+1})(\mathcal{M}^{j+1})^{-1}-(d\mathcal{M}^{j})(\mathcal{M}^{j})^{-1}|^2i_{\vec{n}}dVol(M)\leq C\gamma_{_T}^2\kappa_3^2s^2
R(e_{ij}) = \begin{cases}e_{i,j+1}+e_{i+1,j+2}+\ldots+e_{n-j+i-1,n}, i\leq j\leq n-1, -(e_{i-1,j}+e_{i-2,j-1}+\ldots+e_{i-j+1,1}), i>j, 0, j = n. \end{cases}
\#\{(x_{n_1+1},...,x_{n_3})\in (\mathbb{F}_q)^{n_3-n_1}: x_{n_1+1}...x_{n_3}=0\}&=\sum_{i=1}^{n_3-n_1}{n_3-n_1 \choose i}(q-1)^{n_3-n_1-i}=q^{n_3-n_1}-(q-1)^{n_3-n_1}.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (3.10)
W_{\frac{\overrightarrow{\boldsymbol{N}}}{N}}(x):=\mu_{N, \frac{\overrightarrow{\boldsymbol{N}}}{N}, \beta}^{V;\boldsymbol{A}}(\sum \frac{1}{x-\lambda_{i}}), W_{\boldsymbol{f}}^{\{-1\}}(x):=\mu_{\mathrm{eq}, \boldsymbol{f}}(\frac{1}{x-\lambda}).
f(A_1,B_1) -f(A_2,B_2) begin{align*}.2cm] =\iiint\frac{f(x_1,y)-f(x_2,y)}{x_1-x_2}\,dE_{A_1}(x_1)(A_1-A_2)\,dE_{A_2}(x_2)\,dE_{B_1}(y) begin{align*}.2cm] +\iiint\frac{f(x,y_1)-f(x,y_2)}{y_1-y_2}\,dE_{A_2}(x)\,dE_{B_1}(y_1)(B_1-B_2)\,dE_{B_2}(y_2).
\frac{\partial}{\partial\delta}\biggl(\frac{P(\delta,c,1)}{\delta}\biggr)=-e^{\mu r}\biggl(\frac{\mu}{\sigma}+\frac{\sigma}{\mu}\biggr)\frac{\partial}{\partial\delta}(\mu r)=\mu|c|\biggl(\frac{\mu^2+\sigma^2}{-\sigma\mu}\biggr)e^{\mu r}>0.
\Big( 1- \tilde{t}^2 \frac{\vert \hat{w}\vert^2 p_1^2}{\vert w_1 \vert^2}\Big)\vert w_1 \vert^{2m}{\tilde{\alpha}}^{2m}= - \frac{\vert \hat{w}\vert^2 \tilde{t}^2{\tilde{\alpha}}^2}{m},
\|f(t)\|_2^2 = e^{-2\Re \lambda_1 t}\|v_1 +\alpha e^{-(\lambda_2 -\lambda_1) t}v_2 \|_2^2 = \tfrac1{1-\alpha^2}e^{-2\Re \lambda_1 t}\Big( 1 -2\alpha^2 e^{-\Re(\lambda_2-\lambda_1)t}\cos\big(\Im(\lambda_2-\lambda_1)t\big) +\alpha^2 e^{-2\Re(\lambda_2-\lambda_1)t}\Big) \,.
S_2 :=\{( Z^{(1)}, \xi^{(1)}, \cdots, Z^{(m)}, \xi^{(m)}) \in \wedge^p(T\Omega_{1}) \times \cdots \times \wedge^p(T \Omega_m): &\ \ \ \ \omega^p_{\Omega_1}(Z^{(1)})(\xi^{(1)}, \overline{\xi^{(1)}}) + \cdots +\omega^p_{\Omega_m}(Z^{(m)})(\xi^{(m)}, \overline{\xi^{(m)}}) =1\}.
p_{\hat{s}}= \frac{1}{M\sqrt{2\pi}\rho_n\sqrt{{\boldsymbol w}^\emph{H}_j[i]{\boldsymbol w}_j[i]}}\sum_{j=1}^{M}exp\big(-\frac{(\tilde{s}_j-\hat{s}_j)^2}{2\rho^2_n{\boldsymbol w}^\emph{H}_j[i]{\boldsymbol w}_j[i]}\big),
\begin{array}{l l}{{a^{\dagger}= \frac{x / m - i p_{x}}{\sqrt{2}},}}&{{a = \frac{x / m + i p_{x}}{\sqrt{2}},}}\\{{b^{\dagger}= \frac{y / n - i p_{y}}{\sqrt{2}},}}&{{b = \frac{y / n + i p_{y}}{\sqrt{2}}.}}\\ \end{array}
[\nabla f(x)]\cdot c=\frac{d}{d\alpha}f_{\varphi}(q^{1}+\alpha c^{1},q^{2}+\alpha c^{2},q^{3}+\alpha c^{3})|_{\alpha=0}=\frac{\partial f_{\varphi}}{\partial q^{i}}c^{i}=\frac{\partial f}{\partial q^{i}}c^{i}
\sum_{l\in L}\sum_{m \in M}x_{b,l,m}= 1 \forall b\in B, \underline{I}\le \sum_{b\in B}\sum_{l\in L}\sum_{m \in M}m \cdot |l| \cdot x_{b,l,m}\le \overline{I}, \sum_{l \in L}y_l \le \sum_{i=1}^n z_i, \sum_{m \in M}x_{b,l,m}\le y_l \forall b\in B, l\in L,
\big<J(u_k)-J(u^1),u_k-u^1\big> =\big<J(u_k)-J(u)+J(u)-J(u^1),u_k-u^1\big> &=\big<J(u_k)-J(u),u_k-u^1\big>+\big<J(u)-J(u^1),u_k-u^1\big>.
t_I = \prod_{i \in I}\Biggl(\sum_{j = 0}^n a_{ij}\lambda_{ij}\Biggr) \cdot \prod_{i' \notin I}\Biggl(\sum_{j = 0}^n\lambda_{i'j}\Biggr)= \sum_{e \in E}\Biggl(\prod_{i \in I}a_{ie(i)}\Biggr) y_e =: L_I(y) I \subseteq \{1, \ldots,d \}.
p_{i}^{\nu}=\frac{1}{2}\int f_{i}^{2}\left\{3a(u^{\nu}s_{\sigma}-s^{\nu}u_{\sigma})+\Theta \delta_{\sigma}^{\nu}+3(\sigma_{.\sigma}^{\nu}+\omega_{.\sigma}^{\nu})\right\}dx^{\sigma}dV.
\textrm{RRMSE}= \sqrt{\dfrac{1}{s}\sum_{i=1}^{s}\dfrac{\left|f(\tilde{\boldsymbol{x}}_i) -{\bar{\cal I}}(\tilde{\boldsymbol{x}}_i)\right|^2}{\left|f(\tilde{\boldsymbol{x}}_i)\right|^2}}.
I_q(\Gamma,b) =\sum_{j \in \frac12 \mathbb Z \setminus \mathbb Z}\chi(S_j) q^j - \frac{\chi(S)}{q^{\frac12}- q^{-\frac12}}- \frac12\sum_{d \in X}(q^{\mathrm{ind}_{\Gamma, b}(d) + \frac12}- q^{\mathrm{ind}_{\Gamma, b}(d) - \frac12})
\kappa^2_{\mathsf{A}}-\kappa^2_{\mathsf{J}_{\mathsf{f}-1}}=\sum\limits_{\mathsf{g}=1}^{\mathsf{f}-1}\big[\kappa_{\mathsf{J}_{\mathsf{g}-1}}^2-\kappa_{{\mathsf{J}_{\mathsf{g}}}}^2\big]\leq\frac{\mathsf{f}-1}{\mathsf{F}}\,\big[\kappa^{2}_{\mathsf{A}}-\kappa^{2}_{\mathsf{B}}\big]\,.
\frac{1}{x^{1/2}(1-x)^{1/2}}\left(\arcsin(x^{1/2}e^{(1-x)t}) - \arcsin(x^{1/2}) \right) = \sum_{n=0}^\infty 2^n F_n(x, 1/2) \frac{t^{n+1}}{(n+1)!}.
\int_{0}^{2 \pi}d \tau^{\prime \prime}\triangle^{- 1}( \tau - \tau^{\prime \prime}) \triangle ( \tau^{\prime \prime}- \tau^{\prime}) = \delta ( \tau - \tau^{\prime}) - \frac{1}{2 \pi}, \ \delta ( \tau - \tau^{\prime}) = \frac{1}{2 \pi}\sum_{n}e^{i n ( \tau - \tau^{\prime})}\ .
\begin{array}{l}{{W_{q}\simeq - \int d x \langle E [ \delta \left( \tilde{q}\left( 1 \right) \right) ] \int_{0}^{\infty}\frac{d \tau}{\tau}\tau^{- \frac{2}{1 + \gamma}}\Big ( \exp ( - \tau \alpha \phi_{c}^{2}( x ) ) - 1 + \tau \alpha \phi_{c}^{2}( x ) \Big ) \rangle}}\\{{\simeq \int d x \langle v ( x ) \vert \phi_{c}\vert^{\frac{4}{1 + \gamma}}\rangle}}\\ \end{array}
\int_\Omega \left(1+g(U_{\varepsilon,x_\varepsilon}) \right)\exp\left(U_{\varepsilon,x_\varepsilon}^2 \right) dy~ ~= \int_\Omega \left(1+g(u_\varepsilon) \right) \exp\left(u_\varepsilon^2 \right) dy+o\left(\gamma_\varepsilon^2 \check{\zeta}_\varepsilon\right)\,,
\left\| H_{n+1}- H \right\|_{\mathrm{F}}^2 \leq \left\|H_{n}- H \right\|_{\mathrm{F}}^2- \alpha_n \left(1-\alpha_n \right) \left\| H_n - T_n\left(H_n\right) \right\|_{\mathrm{F}}^2 &\leq \left\|H_{n}- H \right\|_{\mathrm{F}}^2.
\mathbf{f}_{n}:= \mathcal{A}\big(\mathbf{H}(\tilde{\mathbf{u}})\big) - \mathcal{A}\big(\mathbf{H}(\tilde{\mathbf{u}}_{n})\big) \tilde{\mathbf{u}}_{n}= \mathrm{div}\Big(\big(\mathbf{H}(\tilde{\mathbf{u}}) - \mathbf{H}(\tilde{\mathbf{u}}_{n})\big) \nabla \tilde{\mathbf{u}}_{n}\Big) n \in \mathbb{N}.
p(\hat{y}_2|y_2=0)p(\hat{y}_3|y_3=x) =p(\hat{y}_2,\hat{y}_3|x_1'=0,x_1''=x) p(\hat{y}_2|y_2=E)p(\hat{y}_3|y_3=1-x) =p(\hat{y}_2,\hat{y}_3|x_1'=0,x_1''=x) p(\hat{y}_2|y_2=1)p(\hat{y}_3|y_3=x) =p(\hat{y}_2,\hat{y}_3|x_1'=1,x_1''=x) p(\hat{y}_2|y_2=E)p(\hat{y}_3|y_3=1-x) =p(\hat{y}_2,\hat{y}_3|x_1'=1,x_1''=x).
[r] = \frac{r^\dagger r}{r r^\dagger}= \frac{1}{4}\left(\begin{matrix}1 1 \overline{\lambda}\end{matrix}\right) \left(1,1,\lambda\right) = \frac{1}{4}\left( \begin{matrix}1 1 \lambda 1 1 \lambda \overline{\lambda}\overline{\lambda}2 \end{matrix}\right).
Z_g\big(s, \chi, S(\Delta_{\gamma_2}\bigcap(\mathbb{N}^2\setminus \{0\}))\big)= \sum_{m=0}^{j_0-1}q^{-m-w(m)}\dfrac{\tilde{G}_2(q^{-s})}{1-q^{-1-s}}\sum_{b=1}^{\infty}q^{-b(i_0+j_0+i_0j_0s)}= \dfrac{G_2(q^{-s})}{(1-q^{-1-s})(1-q^{-i_0-j_0-i_0j_0s})},
\left(\begin{array}{lll}I_k 0 0 w_5 w_3 w_6\end{array}\right)^*\left(\begin{array}{lll}I_k 0 0 w_5 w_3 w_6\end{array}\right)=\left(\begin{array}{lll}2w_3 w_6 -\overline w_6^t \overline w_5^t F' w_5\end{array}\right).
\langle \Psi | \pi_{\rm R'}(x) |\Omega_{\rm R'}\rangle = \frac{1}{\sqrt{2}}\langle \Psi_+ | \pi_{E_{\rm R},+}(x) |\Omega_{E_{\rm R}}\rangle + \frac{1}{\sqrt{2}}\langle \Psi_- | \pi_{E_{\rm R},-}(x) |\Omega_{E_{\rm R}}\rangle = 0
I_j = \left( \frac{1}{\lambda_1 \lambda_2}\right) \left( \frac{\lambda_j^{-1}v_j}{(T)}\right) & = \left( \frac{1}{\lambda_1 \lambda_2}\right) \left( \frac{\lambda_j^{-1}}{(\lambda_1^{-1}\lambda_2^{-1}/ 2)}\right) & = \frac{2}{\lambda_j}.
( (0, \tau) + E_{2}) \cap B_{\bar{\rho}}((\bar{z},w)) &= \left((0,\tau) + ( F_{\ell}\cap \{z \geq \bar{z}\}) \right) \cap B_{\bar{\rho}}((\bar{z},w)) &\subset \left\{(z',w') \in \mathbb{R}\times \mathbb{R}^{n-1}: z' \geq \bar{z}, \ \ \frac{\epsilon}{2}< |w' - \tau | < r_{\ell}(z')\right\}\cap B_{\bar{\rho}}((\bar{z},w)).
\bar{V}_i = (\bar{a}_4^i - \bar{a}_5^i) \,\bar{Q}^{1/2}\Big[ \begin{array}{cc}0 0 \cr1 0 \end{array}\Big] \,\bar{Q}^{1/2}\Big[ \begin{array}{cc}0 1 \cr1 0 \end{array}\Big] ,
\xi = W^{\frac{1}{2}}\psi = e x p ( \frac{2 i}{F_{\pi}}\gamma_{5}\mathrm{\boldmath \tau}\cdot \mathrm{\boldmath \phi}+ \frac{g_{\omega}}{2 M_{\omega}^{2}}\sigma_{\alpha \beta}\omega^{\alpha \beta}+ \frac{g_{\rho}}{2 M_{\rho}^{2}}\sigma_{\alpha \beta}\mathrm{\boldmath \tau}\cdot \mathrm{\boldmath \rho}^{\alpha \beta}) \psi
\phi(x,t) = \Phi_1 (x - \alpha \t + \int_{-\infty}^{\sigma (x,t)}\frac{{\Phi'}_2^2 - p \Phi'_2}{a + q \Phi'_2}\d \tilde \sigma) \ + \Phi_2 (x - \beta \t - \int_{\rho (x,t)}^{\infty}\frac{{\Phi'}_1^2 + q \Phi'_1}{a - p \Phi'_1}\d \tilde \rho).
E\left[ g(X(t))\right] = E\left[ g(\xi) \right] +E\left[ \int_0^t\int_A \Lambda_s^a g(X(s)) \rho_s(da) ds\right] &= E\left[ g(\xi) \right] +E\left[ E\left[\left. \int_0^t\int_A \Lambda_s^a g(X(s)) \rho_s(da)\right| X(s)\right]ds \right]
\gamma( h_{1}k_{1}h_{2}k_{2}) = \gamma( h_{1}h_{2}k_{1}^{h_{2}}k_{2}) &= \gamma'( h_{1}h_{2}) &= \gamma'( h_{2}) \gamma'(h_{1}) &= \gamma( h_{2}k_{2}) \gamma( h_{1}k_{1}),
\psi_{\theta}( 1 ) = e^{- i \Delta_{1}}e^{i p^{k}x_{1}^{k}}\hspace{2 c m}K_{\theta}( 2 , 1 ) = e^{- i \Delta_{2}^{\prime}}e^{i p^{k \, \prime}x_{2}^{k}}e^{i \Delta_{1}^{\prime}}e^{- i p^{k \, \prime}x_{1}^{k}}.
\sum_{j: j \neq k}S_{m j}(t) \sum_{\ell=1}^N S^{-1}_{j \ell}(t) dS_{\ell k}(t) =dS_{m k}(t) - S_{m k}(t) \sum_{\ell=1}^N S^{-1}_{k \ell}(t) dS_{\ell k}(t) &= dS_{m k}(t) - S_{m k}(t) dU_{kk}(t),1 \leq k, m \leq N, \, t \geq 0.
\prod_{i=1}^{n}\| A_{i}^{\frac{1}{2}}x\|^{2w_{i}}=\prod_{i=1}^{n}\langle A_{i}x,x\rangle^{w_{i}}\leq \sum_{i=1}^{n}w_{i}\langle A_{i}x,x\rangle =\langle \frak{A}(\omega; \mathbb{A})x,x\rangle.
\mathcal{C}_{1,\dots,r}^{n_{r}+1}\left(z_{1},\dots,z_{r}\right)=\frac{\left(\mathcal{C}_{1,\dots,r-1}^{n_{r-1}+1}\left(z_{1},\dots,z_{r-1}\right)+\mathcal{B}_{r}\left(z_{r}\right)\right)^{n_{r}+1}}{n_{r}+1}
3 D_{i \alpha}( W ( \theta ) \tilde{\gamma}^{A})^{[ \beta \gamma ]}= \delta_{\alpha}^{~ \beta}D_{i \delta}( W ( \theta ) \tilde{\gamma}^{A})^{[ \delta \gamma ]}- \delta_{\alpha}^{~ \gamma}D_{i \delta}( W ( \theta ) \tilde{\gamma}^{A})^{[ \delta \beta ]}
\mathbb{E}\left[H\left(\mu[t]\mid P_{Y^1,\cdots,Y^k}^{(k+1\mid k)}[t]\right)\right]=H\left(\mu^{\otimes k+1}[t]\mid P^{(k+1)}[t]\right)-H\left(\mu^{\otimes k}[t]\mid P^{(k)}[t]\right)=H_t^{k+1}-H_t^k.
\begin{cases}\eta_t + w_x+w_{xxx}+(\eta w)_x= 0, \,\, (0,L)\times (0,+\infty), w_t +\eta_x +\eta_{xxx}+ww_x=0, \,\, (0,L)\times (0,+\infty), \eta(x,0)= \eta_0(x), w(x,0)= w_0(x), \,\, (0,L),\end{cases}
\Pi_{g_{\mathbb{E}}}\circ\Phi^{V^R}_{(g_{\mathbb{E}}, 0)}(\gamma^R+h^R , \tau^R+ w^R) = \Pi_{g_{\mathbb{E}}}\circ\Phi^{V^R}_{(g_{\mathbb{E}}, 0)}(\gamma^R, \tau^R)+ \Pi_{g_{\mathbb{E}}}(2\psi^R, V^R)
( - \hbar \sp 2 \Delta -{\frac{e \sp 2}{\pi k \theta}}- E ) \psi + 2 e \epsilon_{i j}( y_{i}\chi ( y ) ) * \hat{P}_{j}\psi +{\frac{2 e}{\theta}}( y \sp 2 \chi ( y ) ) * \psi = 0 .
x^{(-n+s+t+u+...)}((1p_1p_2..p_s)(q_0q_1..q_t)(r_0r_1..r_u)....) \cong \left( \prod_{k=1}^s A_{1p_k}\right) \left( \prod_{k=1}^t A_{q_0q_k}\right)\left( \prod_{k=1}^u A_{r_0r_k}\right) .... E_0
\gamma(j_{1},j_{2},\boldsymbol{\theta}) = \prod_{i=1}^{2}\frac{2^{1-2d_{i}}\sigma^{2}_{\epsilon}}{\pi\sin^{2d_{i}}(\nu_{i})}\sin(d_{i}\pi) \Gamma(1-2d_{i}) \cos(j_{i}\nu_{i})\frac{\Gamma(j_{i}+2d_{i})}{\Gamma(j_{i}+1)}[1+\mathcal{O}(j_{i}^{-1})].
\partial_b (B(X)^{n})=\sum\limits_{\substack{1\leq q\le n j_1+\dots+j_q=b 0<j_1\le\dots\le j_q}}C_n(j_1,\dots,j_q)B(X)^{n-q}\prod\limits_{t=1}^q\left(\partial_{j_t}B(X)\right).
\begin{array}{l}\alpha_{i}\alpha_{j}^{*}(b_{i}b_{j}^{\dagger}- q_{ij}b_{j}^{\dagger}b_{i}) + \beta_{i}^{*}\alpha_{j}^{*}(b_{i}^{\dagger}b_{j}^{\dagger}- q_{ij}b_{j}^{\dagger}b_{i}^{\dagger}) + \alpha_{i}\beta_{j}(b_{i}b_{j}- q_{ij}b_{j}b_{i}) + \beta_{i}^{*}\beta_{j}(b_{i}^{\dagger}b_{j}- q_{ij}b_{j}b_{i}^{\dagger}) = \delta_{ij}. \end{array}
\bigl[ [s, b_1] b_2, b_3, x \bigr] = \bigl[ [s, y_1 \dots y_k] b_2, b_3, x \bigr] = \sum_{i=1}^k [ y_1 \dots y_{i-1}[s, y_i] y_{i+1}\dots y_k b_2, b_3, x ] .
\begin{vmatrix}R_{ab}\overline{R}_{ab}R_{cd}\overline{R}_{cd}\end{vmatrix}\begin{vmatrix}R_{pq}\overline{R}_{pq}R_{rs}\overline{R}_{rs}\end{vmatrix}= (R_{ab}\overline{R}_{cd}- R_{cd}\overline{R}_{ab})(R_{pq}\overline{R}_{rs}- R_{rs}\overline{R}_{pq}) = 2(R_{abpq}R_{cdrs}- R_{abrs}R_{cdpq}).
\mathbb{E}_x[(X_{\tau_{ab}})^{2N}] =x^{2N}-(x-a)^N(x-b)^N, \mathbb{E}_x[(X_{\tau_{ab}})^{2N+1}] =x^{2N+1}-[x+N(a+b)](x-a)^N(x-b)^N, \mathbb{E}_x[(X_{\tau_{ab}})^{2N+2}] =x^{2N+2}-\left[x^2+N(a+b)\,x+\left(\textstyle\frac12\,N(N+1)(a^2+b^2)+N^2ab\right)\!\right]\!(x-a)^N(x-b)^N.
U_n = \begin{pmatrix}C_1 E_1^T \end{pmatrix}^{-1}\begin{pmatrix}C_1U_n E_1^TU_n\end{pmatrix}\rightarrow u\begin{pmatrix}C_1 E_1^T \end{pmatrix}^{-1}\begin{pmatrix}0 1\end{pmatrix}=ue_1
A^{-1/2}Re_i\cdot n_i = A^{-1/2}Re_i\cdot a_{ii}^{-1/2}A^{1/2}e_i = a_{ii}^{-1/2}A^{1/2}A^{-1/2}Re_i\cdot e_i = a_{ii}^{-1/2}Re_i\cdot e_i = a_{ii}^{-1/2}r_{ii}= a_{ii}^{-1/2}.
\|u\|_{\mathcal{V}_2(\mathcal{O}_t)}+\|v\|_{0,2;\mathcal{O}_t}\leq C\left( \textrm{esssup}_{\omega \in \Omega}\|G(\omega,\cdot)\|_{L^2(\mathcal{O})}+\textrm{esssup}_{\omega \in \Omega}\|\hat{G}(\omega,\cdot)\|_{L^2(\mathcal{O})}\right. &\left.+B_2(f_0,g_0;\mathcal{O}_t)+B_2(\hat{f},\hat{g};\mathcal{O}_t)\right),
G_{\varepsilon}^{( N )}( \zeta_{1}, \cdots , - \zeta_{j}, \cdots , \zeta_{N})^{- \cdots - + \cdots +}= ( - 1 )^{N - j + H ( n \ge j )}G_{- \varepsilon}^{( N )}( \zeta_{1}, \cdots , \zeta_{j}, \cdots , \zeta_{N})^{- \cdots - + \cdots +},
\begin{aligned}T_a(\rho_{|a|},\theta_{|a|})\cap S_{\theta_N}\subset{\tilde T}_a(\rho_{|a|},\theta_{|a|})\cap S_{\theta_N}&\Subset{\tilde T}_a(\gamma^{-1}\rho_{|a|},\gamma\theta_{|a|})\cap S_{\theta_N}&\subset T_a({\gamma'_1}^{-1}\rho_{|a|},\gamma'_2\theta_{|a|})\cap S_{\theta_N},\end{aligned}
\begin{aligned}I_s ( (X_s - (T-t) V_s,V_s)) + E_s (Z_s) \geq & \geq \frac{1}{2}\left( \left| x_i - (T-t)v_i\right|^2 + |v_i|^2\right)+ E_{s-1}(Z_s^{(i)}) \end{aligned}
{\hat J}^0=e^{-2\L}(J^0-iD\L\Psi^0)+e^{-\L-{1\over 2}D\L\Psi^-}\partial e^{-\L}-e^{-2\L}\partial e^{-{1\over 2}D\L\Psi^-}+ e^{-2\L-D\L\Psi^-}g^+(Z).
\begin{aligned}\|u^{k}- q \|^2 = \|T_{\rho_k}(x^k) - T_{\rho_k}(q)\|^2 \leq \langle T_{\rho_k}(x^k) - T_{\rho_k}(q) , x^{k}- q \rangle = \langle u^k - q, x^k - q \rangle = \frac{1}{2}\big[\|u^k - q\|^2 + \|x^{k}- q \|^2 - \|u^k - x^k\|^2 \big].\end{aligned}