harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\mathbb E X^s = \frac1{E_{\alpha, \beta}^\gamma(t^\alpha)}\sum_{j=0}^s (\gamma)_j\,{s \brace j}\,t^{\alpha j}\, E_{\alpha, \alpha j+\beta}^{\gamma+j}(t^\alpha), s \in \mathbb N_0,\, t \geq 0.
R\Gamma(W_{(\bar w)}\times_{Y_{(f(\bar w))}}\bar t,h^\ast\mathcal F|_{W_{(\bar w)}\times_{Y_{(f(\bar y))}}\bar t})=\varprojlim_n R\Gamma(W_{(\bar w)}\times_{Y_{(f(\bar w))}}\bar t,h^\ast\mathcal F_n|_{W_{(\bar w)}\times_{Y_{(f(\bar y))}}\bar t}).
R = |q|^2 \sum_{1\leq \beta<\gamma \leq s}(|b_\beta| - |b_\gamma|)^2 (|u_{2\beta-1}|^2 + |u_{2\beta}|^2 )^2 (|u_{2\gamma-1}|^2 + |u_{2\gamma}|^2 )^2.
I_{\mathbb{C}^{o(G)}}=\frac{o(\Lambda)}{o(G)}\sum_{\mu\in \Lambda^0}f(\pi(\mu)^{-1}\tau) \pi(\mu)=\frac{o(\Lambda)}{o(G)}f(\tau)I_{\mathbb{C}^{o(G)}}+\frac{o(\Lambda)}{o(G)}\sum_{\mu\in \Lambda^0\setminus \{(0,0)\}}f(\pi(\mu)^{-1}\tau) \pi(\mu).
0 \le \int_\Omega - 2\cos R^{u_0}\cos R^{u_1}\nabla \left(\frac{d}{\sin d}\eta F\right) \cdot \nabla F\, d\mu_g =2 \int_\Omega \left(\frac{d}{\sin d}\eta F\right) \textup{div}\left(\cos R^{u_0}\cos R^{u_1}\nabla F\right)\, d\mu_g.
\frac{\min(j,\ell)}{\min(j,\ell)+|j^2-\ell^2|}=\frac{j}{j+|j^2-\ell^2|}\geq \frac{j}{j+|k^2-\ell^2|}\geq \frac{k}{k+|k^2-\ell^2|}=\frac{\min(k,\ell)}{\min(k,\ell)+|k^2-\ell^2|}
\times \exp \left( \frac{1}{2}\sum_{b = 1}^{N}\sum_{j = 1}^{m_{b}}\sum_{j^{\prime}= 1}^{m_{b}}\Big ( 1 - \delta_{j j^{\prime}}\Big ) \Big [ \ln ( y_{j}^{( b )}- y_{j^{\prime}}^{( b )})^{2}+ 2 \gamma - \ln ( 4 ) \Big ] \right) ,
\int_{0}^{t}\left(\sum_{j=1}^{m}\frac{1}{{\lambda_{j}^{\mu}(\tau)}}\right)d\tau\lesssim\begin{cases}C_{\mu}, \mu<2, \sum_{j=1}^{m}\frac{|\ln \lambda_{j}(t)|^{101}}{\lambda_{j}(t)^{\mu-2}}, \mu \geq 2.\end{cases}
\Vert{\mathbf D}^2 (B^0-\zeta \overline{Q_0})^{-1}\Vert_{L_2\rightarrow L_2}\leqslant c_*^{-1}\|f\|_{L_\infty}\|f^{-1}\|_{L_\infty}\sup \limits_{\nu \geqslant c_\flat}\frac{\nu}{\vert \nu -\zeta \vert}&\leqslant c_*^{-1}\|f\|_{L_\infty}\|f^{-1}\|_{L_\infty}( c_\flat +2) \rho(\zeta)^{1/2}.
\prescript{RL}{x}{\mathcal{D}}_{1}^{s}u_N(x_k) =\prescript{RL}{x}{\mathcal{D}}_{1}^{s}u(x_k)=f(x_k),u_N(1)=u(1)=0, &k=1,2,\cdots,N;\quad-1\leq x_1<\cdots<x_N\leq1,
\lVert G(v,\xi) \rVert_{L^2(D)}= \lVert \lambda_2v - p_v(\xi) \rVert_{L^2(D)}\leq \lambda_2 \lVert v \rVert_{L^2(D)}+ C_1(\xi)\lVert y_D - y_v(\xi)\rVert_{L^2(D)}\leq \lambda_2 \lVert v \rVert_{L^2(D)}+ C_1(\xi) \lVert y_D \rVert_{L^2(D)}+ (C_1(\xi))^2 \lVert v\rVert_{L^2(D)}=:C(\xi).
\sigma_i^{}(t) = \frac{1}{x_i \Delta x_i}\bigg[\sum_{j=1}^{i-1}x_j n_j \Delta x_j f'(x_j,x_{\alpha_{i-1,j}-1}) \bigg(\frac{\Delta x_{\alpha_{i,j}-1}^2}{8}- \frac{\Delta x_{\alpha_{i-1,j}-1}^2}{8}\bigg) \bigg] +{\cal O}(\Delta x^2).
=-2\pi b_0\int_{\Sigma}+\beta^{-1}\int_{\Sigma}\left[\gamma_1 + \gamma_2 R_{\Sigma}+(\gamma_3 + \gamma_4 \beta^{-2}) R_{\mu\nu}n^{\mu}_in^{\nu}_i+(\gamma_5 + \gamma_6 \beta^{-2})R_{\mu\nu\lambda\rho}n^{\mu}_in^{\lambda}_in^{\nu}_jn^{\rho}_j\right]~~~
s(\varphi,I_\Gamma(t,\pi))=\sup_{P\in \Gamma(t,\pi)}\left\langle \varphi,\int_X\imath_X(x)dP \right\rangle=\sup_{P\in \Gamma(t,\pi)}\int_X\langle \varphi,\imath_X(x)\rangle dP
\big\|u_n(t)\big\|^2 =\big\|P_nu_0\big\|^2-2\int_{0}^{t}\big\|u_n(s)\big\|_V^2ds+2\int_{0}^{t}(u_n(s)\log|u_n(s)|,u_n(s))ds &+2\int_{0}^{t}\big(\sigma(u_n(s)),u_n(s)\big)h(s)ds.
\frac{\partial L_{d}D2(p\|q)}{\partial q_{j}}= \left\{\left[\frac{a-1}{a-b}A^{a-2}-\frac{b-1}{a-b}A^{b-2}\right]\frac{\partial A}{\partial q_{j}}\right\}&-\left\{\left[\frac{a-1}{a-b}(X.Y)^{a-2}-\frac{b-1}{a-b}(X.Y)^{b-2}\right]\frac{\partial (X.Y)}{\partial q_{j}}\right\}
{d''_{0}}_{\bar{\lambda}\bar{\mu}}(M)=\frac{1}{\left| B_{\frac{1}{8}\cdot 2^{n-1}}(M)\right|}\int\limits_{B_{\frac{1}{8}\cdot 2^{n-1}}(M)}\left(\bar{\mu}, \,\bar{\nu}(M) \right){d'_{2}}_{\bar{\lambda}}(\widetilde{M})\, dS(\widetilde{M}),
dJ_t = \sum_{j=1}^d \nabla V_j (y_t) J_t dw_t^j + \nabla V_0 (y_t) J_t dt, J_0 =\mathrm{Id}_e, dK_t = - \sum_{j=1}^d K_t \nabla V_j (y_t) dw_t^j -K_t \nabla V_0 (y_t)dt, K_0 =\mathrm{Id}_e.
g(z)=\sum_{ij}\frac{\sigma(z+s_{ij})}{\sigma(z)\sigma(s_{ij})}e^{-\frac{\zeta(\pi)}{\pi}zs_{ij}}e^{is_{ij}\frac{z-\bar z}{\Delta -\overline\Delta}}Y_{ij}E_{ij}=\sum_{ij}W(z,s_{ij}) Y_{ij}E_{ij}.
D^2_0f_\lambda(w_1,w_2) =\int_\Omega{\langle\nabla u_1(x),\nabla u_2(x)\rangle\,dx}-\int_\Omega{\langle \nabla v_1(x),\nabla v_2(x)\rangle\,dx}&+\int_\Omega{\langle S_\lambda(x)w_1,w_2\rangle \, dx},w_1=(u_1,v_1), w_2=(u_2,v_2),
P_{l}T^{\,j}\,e_{k}=\pm\, \Bigl(\prod_{s=k+1}^{b_{l+1}-1}w_{s}\Bigr)\ \Bigl( \prod_{s=b_l+1}^{b_{l}+n}w_s\Bigr)\ \Bigl( \prod_{s=b_l+1}^{b_{l+1}-1}w_s\Bigr)^{-1}\e_{b_{l}+n}&= \pm\,\Bigl( \prod_{s=k+1}^{b_{l+1}-1}w_{s}\Bigr)\ \Bigl( \prod_{s=b_{l}+n+1}^{b_{l+1}-1}w_{s}\Bigr)^{-1}e_{b_{l}+n}
(\Delta^{(i+1)}{\bf a}(r+1,T+1))_{k}=\sum^{T-1}_{t=0}\binom{T}{t}(\Delta^{(i)}{\bf a}(r+1,t+1))_{k}, (\Delta^{(i+1)}{\bf b}(r+T-s+1,r+1, T+1, s+1))_{k}= \sum^{T-1}_{t=s}\binom{T}{t}(\Delta^{(i)}{\bf b}(r+t-s+1,r+1, t+1, s+1))_{k}+\sum^{s-1}_{t=s-r}\binom{T}{t}(\Delta^{(i)}{\bf b}(r+t-s+1,r+1, t+1, s+1))_{k}.
\eta = l o g 2 + \frac{1}{2}l o g \rho + \frac{1}{l o g \rho}+ \frac{1}{2 l o g^{2}\rho}( 9 l o g l o g \rho - 2 ) + O ( \frac{l o g^{2}l o g \rho}{l o g^{3}\rho})
V^{CDLP}\equiv \min \,\,\,\,\,\,\, \sum_j C_j\pi(j)+\sum_k \sigma(k) \Lambda_k \sum_{n \in S, n>0}[r_n - \pi(l_n)] P^k(n,S) - \sigma(k) \leq 0 , \,\,\,\forall k = 1,2,...,K, \ \forall S \in \cal{S}, \pi(j) \geq 0, \,\,\, \forall j=1,2,...,L \sigma(k) \geq 0, \,\,\, \forall k =1,2,...,K.
d s_{\mathrm{h y p e r}}^{2}= d D^{2}+ D^{2}\sigma_{1}^{2}+ D^{2}\sigma_{2}^{2}+ D \tanh D \, \check{\sigma}_{1}^{2}+ D \coth D \, \check{\sigma}_{2}^{2}+ \check{\sigma}_{3}^{2}.
S = \int d t \, d x{\cal L}= \int d t \, d x \left[ \frac{i \hbar}{2}( \hat{\phi}^{*}\partial_{t}{\phi}-{\phi}\partial_{t}{\phi}^{*}) - \frac{\hbar^{2}}{2 m}\left| \left( \partial_{x}+ i{\frac{\lambda}{2}}\rho^{2}\right) \phi \right|^{2}- V ( \rho^{2}) \right] ;
\begin{aligned}\omega(B(x_0,2^{j_k}2^{i_0}\sigma)) \geqslant 2^k \omega(B(x_0,2^{i_0}\sigma)) \geqslant{C_\omega^{-1}}2^{-\bf N}2^k \omega(B(x_0,2^{i_0+1}\sigma)) \geqslant{C_\omega^{-1}}2^{-\bf N}2^k \sigma^{\bf N}. \end{aligned}
\frac{1-E_{4}}{E_{1}^{2}+E_{2}^{2}+E_{3}^{2}}[\begin{matrix}E_{1}E_{1}&E_{1}E_{2}&E_{1}E_{3}\\E_{2}E_{1}&E_{2}E_{2}&E_{2}E_{3}\\E_{3}E_{1}&E_{3}E_{2}&E_{3}E_{3}\end{matrix}]+[\begin{matrix}E_{4}&-E_{3}&E_{2}\\E_{3}&E_{4}&-E_{1}\\ -E_{2}&E_{1}&E_{4}\end{matrix}],
\hat\beta_i =[0,\cdots,1](X_{\mathcal S\cup \{i\}}^\top X_{\mathcal S\cup \{i\}})^{-1}X_{\mathcal S\cup \{i\}}^\top Y &=[-(X_i^\top M_{\mathcal S}X_i)^{-1}X_i^\top X_{\mathcal S}(X_{\mathcal S}^\top X_{\mathcal S})^{-1}X_{\mathcal S}^\top +(X_i^\top M_{\mathcal S}X_i)^{-1}X_i^\top ]Y &=(X_i^\top M_{\mathcal S}X_i)^{-1}X_i^\top M_{\mathcal S}Y.
T_{0}^{t}h_{3 k , 3 k + 1}( q , J ) T_{0}= \frac{J}{4}\xi^{2}[ \sigma_{k}^{x}\sigma_{k + 1}^{x}+ \sigma_{k}^{y}\sigma_{k + 1}^{y}+ \frac{q + q^{- 1}}{2}\sigma_{k}^{z}\sigma_{k + 1}^{z}
\theta_hH_{h{\sf E}}g=\{\langle h_j, H_{h{\sf E}}g\rangle\}_{j\in\mathbb{J}}=\{\langle H_{h{\sf E}}h_j, g\rangle\}_{j\in\mathbb{J}}=\{E_j\langle h_j, g\rangle\}_{j\in\mathbb{J}}=\mathcal{E}\theta_h{g},
\widetilde{e}(T)=\frac{\int_{\mathbb{R}^3}\int_0^\infty \sqrt{1+|p|^2}e^{-\left(mc^2+ \mathcal{I}\right)\frac{1}{k_B T}\sqrt{1+|p|^2}}\left(mc^2+ \mathcal{I}\right)\phi(\mathcal{I})\, d\mathcal{I}dp}{\int_{\mathbb{R}^3}\int_0^\infty e^{-\left(mc^2+ \mathcal{I}\right)\frac{1}{k_B T}\sqrt{1+|p|^2}}\phi(\mathcal{I}) \,d\mathcal{I}dp}=\frac{e}{n}.
||\frac{|\partial_{r}v_{5}(t,r)| v_{5}(t,r)^{2}}{r^{2}}\vert_{r = R\lambda(t)}||_{L^{2}(R dR)}\leq ||(\partial_{2}v_{5})(t,\cdot \lambda(t))||_{L^{2}(R dR)}\cdot ||\frac{v_{5}^{2}}{r^{2}}||_{L^{\infty}}&\leq \frac{C \log^{11+b}(t)}{t^{31/4}}
\chi_f(s)= (-1)^{\frac{k}{2}}(2\pi)^{2s-1}\frac{\Gamma(1-s+\frac{k-1}{2})}{\Gamma(s+\frac{k-1}{2})}= (-1)^{\frac{k}{2}}(2\pi)^{2\sigma-1}|t|^{1-2\sigma}e^{i\left(\frac{\pi}{2}(1-k){\rm sgn}(t)-2t\log\frac{|t|}{2\pi e}\right)}(1+O(|t|^{-1}))
M_{n}=\frac{M_n^{(-\sqrt{2},2)}}{(-\sqrt{2})^{n}}=\frac{M_n^{(3-\sqrt{2}-3,2)}}{(-\sqrt{2})^{n}}=\frac1{(\sqrt{2})^{n+2}}\sum_{k=0}^n(-1)^k\binom{n}{k}(\sqrt{2}+3)^{n-k}S_{k+1}.
\begin{array}{r c l}{{\hat{\lambda}_{( p = 1 )}[ \gamma ( 1 ) ]}}&{{=}}&{{\hat{\lambda}_{( p = 1 )}[ \gamma ( 0 ) ] + 1 = \left( M_{( p = 1 )}\hat{\lambda}_{( p = 1 )}\right) [ \gamma ( 0 ) ] \, ,}}\\{{}}&{{}}&{{}}\\{{M_{( p = 1 )}}}&{{=}}&{{\left( \begin{array}{c c}{{1}}&{{1}}\\{{0}}&{{1}}\\ \end{array}\right) = T \, ,}}\\ \end{array}
\bar{X}_{L,t_{k}}=\frac{1}{N}\sum^{N}_{j=1}\tilde{X}_{L,t_{k}}^{j}=\frac{1}{N}\sum^{N}_{j=1}\sum^{N}_{i=1}T_{i,j}NX_{L,t_{k}}^{i}=\sum^{N}_{j=1}\sum^{N}_{i=1}T_{i,j}X_{L,t_{k}}^{i}=\sum^{N}_{i=1}\tilde{w}_{L}(X_{L,t_{K}}^{i})X_{L,t_{k}}^{i}
\Lambda_{\chi}(P_{\chi^{-1}}(f))=\int_{B_{n,*}H}\delta^{1/2}_{B_n}(b_g)\chi(\mathfrak{s}(b_g))f(\mathfrak{s}(b_g)\mathfrak{h}(h_g))\gamma_{\psi'}^{-1}(\det c_{h_g})\psi^{-1}(u_{h_g})\, dg.
\pi_1(T_0) = \langle v_1, v_2 \rangle \pi_1(T_\infty) = \langle v_3, v_4 \rangle \pi_1(T_1) = \langle v_1 + v_3, v_2 + v_4\rangle \pi_1(T_\zeta) = \langle v_1 + v_3 + v_4, v_2 - v_3 \rangle
\mathbb{E}_{f}\!\left[T^{2}\right]=\frac{1}{(q-1)^{2}}\left(1-\frac{2\Gamma(mn)}{\Gamma(mn+q)}\mathbb{E}_{g}\!\left[L\right]+\frac{\Gamma(mn)}{\Gamma(mn+2q)}\mathbb{E}_{g}\!\left[L^{2}\right]\right).
b(s)=2e^{si\pi/2}\int_{0}^{\infty}\frac{st^{s}}{1-e^{2\pi t}}\frac{dt}{t}=\frac{s!}{2^{s-1}}\frac{\zeta(s)}{\pi^{s}}(-i)^{s}=\frac{2s!\zeta(s)}{(2\pi i)^{s}}
\langle(\Delta\alpha)^2 \rangle^{1/2}\simeq 1.6^{0}\left(\frac{B(t_{dec})}{B_{c}}\right)\left(\frac{\omega_{M}}{\omega}\right)^2,~~~B_{c}=10^{-3}~Gauss,~~~\omega_{M}\simeq 3\times10^{10}~Hz
E(S_{\lfloor nt \rfloor}(r)\overline{S_{\lfloor nu \rfloor}(s)})=E(S_{\lfloor nt \rfloor}(r)\overline{S_{\lfloor nt \rfloor}(s)}) + E(S_{\lfloor nt \rfloor}(r)(\overline{S_{\lfloor nu \rfloor}(s)-S_{\lfloor nt \rfloor}(s)})
\mathcal{D}(K-1)\log \widetilde{b}<4M\Omega L \log \widetilde{b}&=32ML(\rho+3)^2\log \widetilde{b}\log \alpha_1 \log \alpha_2\log c &+32ML(\rho+3)^2\log \widetilde{b}\cdot 0.08675(\rho-1) \log \alpha_1 \log \alpha_2.
( E_{i}, M_{i}) \in \{( D p_{i}, D ( 6 - p_{i}) ) , ( \overline{{{D p_{i}}}}, D ( 6 - p_{i}) ) , ( D p_{i}, \overline{{{D ( 6 - p_{i})}}}) , ( \overline{{{D p_{i}}}}, \overline{{{D ( 6 - p_{i})}}}\}
\left[ \sum_{i = 1}^{N}\frac{\partial^{2}}{\partial{x_{i}}^{2}}+ 2 ( a + \frac{1}{2}) \sum_{i < j}^{N}\frac{1}{( x_{i}- x_{j})}( \frac{\partial}{\partial x_{i}}- \frac{\partial}{\partial x_{j}}) \right] P_{k}( x ) = 0 \, \, \, .
A(1) = \frac{1}{2}\left( 4 \cdot 1 + 1 \cdot 2\right) = 3, L(1) = -\frac{1}{2}\left((4 \cdot 1 \cdot (1-0-0) +1 \cdot 2 \cdot (2-1-1) \right) = -2, D(1) = -\left( 4 \cdot (0 + 0 \cdot (1-0)) + 1 \cdot (0 + 1 \cdot (1-1)) \right) = 0, C(1) = - \left( 4 \cdot (-1) \cdot (1-0-0) + 1 \cdot 0 \cdot (2-1-1) \right) = 4.
H = \left(\begin{matrix}\nu \sigma(\nu) \cdots \sigma^{n-\delta}(\nu) \sigma(\nu) \sigma^2(\nu) \cdots \sigma^{n-\delta+1}(\nu) \vdots \vdots \ddots \vdots \sigma^{n-1}(\nu) \nu \cdots \sigma^{n-\delta-1}(\nu)\end{matrix}\right).
J(t)=t^{-n/2}\int_{T^{\ast,r}M}\overline{\psi^{\prime}\circ\pi}e^{-E/t\hbar}\psi\circ\pi\circ\Phi_{i}\left( \sqrt{d\mathrm{vol}_{g}},\sqrt{\Phi_{i}^{\ast}d\mathrm{vol}_{g}}\right)_{}\frac{\omega^{n}}{(2\pi\hbar)^{n}n!}.
\left\{\widehat{A}(TX){\rm exp}(\frac{c}{2}){\rm ch}(\widetilde{T_CM}-\widetilde{L_C}-488+W_i)\right\}^{(10)}&=A_1\left\{e^{\frac{1}{24}A_1}\widehat{A}(TX){\rm exp}(\frac{c}{2})-\frac{e^{\frac{1}{24}A_1}-1}{A_1}\widehat{A}(TX){\rm exp}(\frac{c}{2}){\rm ch}(\widetilde{T_CM}-\widetilde{L_C}-488+W_i)\right\}^{(6)}.
\Pi_1^{n-\frac{1}{2}}-\sum\limits_{n=1}^{N}\Pi_2^{n-\frac{1}{2}}-\sum\limits_{n=1}^{N}\Pi_3^{n-\frac{1}{2}}-\sum\limits_{n=1}^{N}\Pi_4^{n-\frac{1}{2}}=\sum\limits_{n=1}^{N}\Pi_5^{n-\frac{1}{2}},
\nu\circ\lambda_1^X=\nu\circ\lambda_2^X\circ\Delta_X=\mu\circ\rho\circ\lambda_2^X\circ\Delta_X=\mu\circ\lambda_2^Y\circ(1\otimes\rho)\circ\Delta_X \phantom{\nu\circ\lambda_1^X}=\mu\circ\lambda_1^Y\circ\Delta_Y^{-1}\circ(1\otimes\rho)\circ\Delta_X.
\mathcal{Z}^\alpha(\partial{_t^\varphi}+v\cdot\nabla^\varphi)v=(\partial{_t^\varphi}+v\cdot\nabla^\varphi)\mathcal{Z}^\alpha v-\partial{_z^\varphi}v(\partial{_t^\varphi}+v\cdot\nabla^\varphi)\mathcal{Z}^\alpha\eta+\mathcal{C}^\alpha(T),
(J_{\omega_{\psi}}- nI )(\varphi) = \sum_{i=0}^{n-1}\frac{n-i}{n+1}\int \sqrt{-1}\partial \varphi \wedge \bar{\partial}\varphi \wedge \omega_{\psi}^i \wedge \omega_{\varphi}^{n-1-i}\geq(J_{\omega_{\psi}}- nI )(0) = 0
d^{3}_{r}\gamma(x,y,z,t) =B(d^2f(x,y,z)),g(t)) + B(d^2_{c}f(x,y,t)),g(z))+B(d^2_{c}f(x,z,t)),g(y))+B(d^2_{c}f(y,z,t)),g(x)) + B((f\circ \alpha) \wedge d^{1}_{c}g )(x,y,z,a).
- \int_P \log \det (D^2 u) ~ dx dy = -\int_{-1}^1 \int_{-1}^1 2 \ln 3 + 2 \ln |x| + 2 \ln |y| ~ dx dy = - 8 \ln 3 - 16 \int_0^1 \ln x ~ dx = - 8 \ln 3 + 16.
H'(u, \vec{x})=\begin{cases}\bigwedge \Gamma(\vec{x}) \wedge s(\vec{x}) \leq p(\vec{x}) \wedge \forall z \leq p(\vec{x}) B(\vec{x}, z) u=0 H(u, y, \vec{x}) 0<u \leq t'(\vec{x})+1 \end{cases}
s'((\mathcal{S},\mathcal{T}),\gamma)=\limsup_{n\rightarrow\infty}\frac{1}{n}\log s'(n,\gamma),s'((\mathcal{S},\mathcal{T}))=\lim_{\gamma\rightarrow 0}s'((\mathcal{S},\mathcal{T}),\gamma).
\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},
\Lambda_{\{1,4\}}=\alpha \big[\Lambda_{\{1,2,3\}},\Lambda_{\{2,3,4\}}\big]_q + \beta \big( \Lambda_{\{2,3\}}\Lambda_{\{1,2,3,4\}}+\Lambda_{\{1\}}\Lambda_{\{4\}}\big), \Lambda_{\{1,3,4\}}=\alpha \big[\Lambda_{\{1,2\}},\Lambda_{\{2,3,4\}}\big]_q + \beta \big( \Lambda_{\{2\}}\Lambda_{\{1,2,3,4\}}+\Lambda_{\{1\}}\Lambda_{\{3,4\}}\big).
\Phi(x_1, x_2; z_1, z_2) = \bigl(\delta(z_1, z_2) + \sum\limits_{i_1 + i_2 < \mu}c_{i_1, i_2}(x_1, x_2) z_1^{i_1}z_2^{i_2}\bigr) \cdot \exp(x_1 z_1 + x_2 z_2),
\rho_{K}( x;\mathfrak{p,}N ) = ( 1-\mathfrak{p})^{N}\frac{( -N)_{x}}{x!}\left( \frac{\mathfrak{p}}{\mathfrak{p}-1}\right)^{x}=( 1-\mathfrak{p})^{N}\rho_{M}\left( x;-N,\frac{\mathfrak{p}}{\mathfrak{p}-1}\right) .
\alpha \leq \frac{-\lambda_{min}}{r-\lambda_{min}}\binom{n+m-1}{n}= \frac{\binom{m}{2}\binom{n+m-1}{n}}{n(m-1)+\binom{m}{2}}\leq \frac{\binom{m}{2}\binom{n+m-1}{n}}{\binom{m}{2}(m-1)+\binom{m}{2}}= \frac{\binom{n+m-1}{n}}{m}.
\Psi(E_j(x, a)) = 4\sqrt{a}\sum_{h = 1}^{\lfloor\frac{j}{2}\rfloor}\frac{\zeta^h + \zeta^{-h}}{2}= 2\sqrt{a}\sum_{h = 1}^{\lfloor\frac{j}{2}\rfloor}\left(\zeta^h + \zeta^{-h}\right) = 2\sqrt{a}\left( \left( \sum_{h = -\lfloor\frac{j}{2}\rfloor}^{\lfloor\frac{j}{2}\rfloor}\zeta^h \right) - 1 \right) .
\left[ \begin{array}{cccc}I I \dots I A_1 A_2 \dots A_n A_1^2 A_2^2 \dots A_n^2 \vdots \vdots \vdots \vdots A_1^{n-2}A_2^{n-2}\dots A_n^{n-2}\end{array}\right]\left[ \begin{array}{c}B_1 B_2 \vdots B_n \end{array}\right]=0,
T_{w}(x):=\sum_{n=0}^{\infty}w^{n}s(2^{n}x)=\frac{1}{4}\sum_{n=0}^{\infty}w^{n}-\frac{2}{\pi^{2}}\sum_{n=0}^{\infty}\sum_{k=0}^{\infty}\frac{w^{n}}{(2k+1)^{2}}cos(2\pi2^{n}(2k+1)x):
\partial_{(0)}^n(f)(g_0,\dots,g_n) =g_0f(g_1,\dots,g_n), \partial_{(i)}^n(f)(g_0,\dots,g_n) =f(g_0,\cdots,g_{i-1}g_i,\dots,g_n)~~1\leq i\leq n, \partial_{(n+1)}^n(f)(g_0,\dots,g_n) =f(g_0,\dots,g_{n-1}).
\|x^{k+1}-x^*\|^2 = \|P_{X}(P_{H_k}(x^k))-P_{X}(x^*)\|^2\le \|P_{H_k}(x^k)-x^*\|^2 = \|P_{H_k}(x^k)-P_{H(\bar{x}_i^k,\bar{u}_i^k)}(x^k)+P_{H(\bar{x}_i^k,\bar{u}_i^k)}(x^k)-x^*\|^2 \le \|P_{H_k}(x^k)-x^k\|^2+\|P_{H(\bar{x}_i^k,\bar{u}_i^k)}(x^k)-x^*\|^2.
\mathfrak{h}_{s,p}(\Omega)>0\mbox{and}\inf_{\varphi\in C^\infty_0(\Omega)}\left\{[\varphi]^p_{W^{s,p}(\mathbb{R}^N)}\, :\, \int_\Omega |\varphi|^p\,dx=1\right\}=0.
\sum_{i=0}^{l_1(x)-1}\Psi_g(f)(\sigma_B^i(h(x)) ) = \sum_{i'=0}^{l_2^{l_1(x)}(h(x))-1}f(\sigma_C^{i'}(gh(x)))-\sum_{j'=0}^{k_2^{l_1(x)}(h(x))-1}f(\sigma_C^{j'}(g(\sigma_B^{l_1(x)}(h(x))))).
E_{pot}=\frac{e^2}{4\pi}\left[q^2\ln\frac{L^2}{4\zeta^2}+\left(\frac{1-q}{2}\right)^2\ln\frac{L^2}{4a_1^2}+\left(\frac{1+q}{2}\right)^2\ln\frac{L^2}{4a_2^2}\right.
A_\phi=\begin{bmatrix}-\kappa_{d-1}P_{d-1}+\kappa_d\beta_{d-1}P_d \alpha_{d-2}I_n -\kappa_{d-1}P_{d-2}+\kappa_{d}\gamma_{d-1}P_d \beta_{d-2}I_n \alpha_{d-3}I_n -\kappa_{d-1}P_{d-3}\gamma_{d-2}I_n \beta_{d-3}I_n \ddots \vdots \ddots \ddots \alpha_0I_n -\kappa_{d-1}P_0 \gamma_1 I_n \beta_0I_n\end{bmatrix}.
\Bigl(X^{aA,bB}(x,y)\Bigr)=\left(\begin{array}{cc}\frac{g}{2m^2}\,f^{abc}\bigl(D_i\pi^i\bigr)^c \delta^{ab}-\,\frac{1}{m^2}\,\delta^{ab}0\end{array}\right)\,\delta(x-y)\,.
\begin{aligned}z-y = 2^{pd}r_0^p, \phi_p(z, y) = r_1^p, x = 2^{d}r_0 r_1, z-x = p^{pe-1}s'^p, \phi_p(z, x) = p s_1^p, y = p^e s' s_1, x+y = t^p, \phi_p (x, -y) = t_1^p, z = t t_1,\end{aligned}
\begin{array}{r c l}{{G^{\pm \pm}}}&{{=}}&{{- 4 x^{\pm}x^{\pm}\left( \phi^{\prime}e^{( \phi - 2 \psi )}\right)^{\prime}e^{- \phi}\, ,}}\\{{G^{\pm y}}}&{{=}}&{{0 \, ,}}\\{{G^{\pm \mp}}}&{{=}}&{{4 \left( - \zeta \phi^{\prime}e^{\phi}\right)^{\prime}e^{- ( \phi + 2 \psi )}\, ,}}\\{{G^{y y}}}&{{=}}&{{4 \left( - \zeta \psi^{\prime}\right)^{\prime}e^{- 2 \psi}\, .}}\\ \end{array}
\mathrm{R e}\, \left[ \Pi_{0 0}( k_{0}, k ) - \Pi_{0 0}( 0 , k ) \right] ={\frac{1}{\pi}}\int_{- \infty}^{\infty}d \omega \, \left({\frac{1}{\omega - k_{0}}}-{\frac{1}{\omega}}\right) \, \mathrm{I m}\, \Pi_{0 0}( \omega , k )
v_n = \frac{\sqrt{k+1}+\sqrt{k-1}}{2\sqrt{k+1}}(k+\sqrt{k^2-1})^n + \frac{\sqrt{k+1}-\sqrt{k-1}}{2\sqrt{k+1}}(k-\sqrt{k^2-1})^n, w_m = \frac{1}{2\sqrt{16k^3-4k}}\big( (x_1\sqrt{16k^3-4k}+z_1 \sqrt{k-1})(4k^2-2k-1+\sqrt{(16k^3-4k)(k-1)})^m + & \, + (x_1\sqrt{16k^3-4k}-z_1 \sqrt{k-1})(4k^2-2k-1-\sqrt{(16k^3-4k)(k-1)})^m \big).
E(u_{0,n}) = \frac{\mu_n^2}{2}\|Q\|^2_{\dot{H}^1_c}-\frac{d\mu_n^{\frac{4}{d}+2}}{2d+4}\|Q\|^{\frac{4}{d}+2}_{L^{\frac{4}{d}+2}}&= \mu_n^2 E(Q) + \frac{d}{2d+4}\mu_n^2 \left(1- \mu_n^{\frac{4}{d}}\right) \|Q\|^{\frac{4}{d}+2}_{L^{\frac{4}{d}+2}}.
\delta^{(2)}(\mbox{\rm\bf p}_1+...+\mbox{\rm\bf p}_n) \int d^2\mbox{\rm\bf q}_1\frac{q_n^{i_1}q_1^{i_2}\ldots q_j^{i_{j+1}}\ldots q_{n-1}^{i_n}}{(\mbox{\rm\bf q}_1')^2...(\mbox{\rm\bf q}_n')^2}
P_p(x_0)\ge h^{K_1}\prod_{i=1}^{K_1}\left(\varepsilon_0+\frac{\kappa (i-1) \varepsilon}l\right), P_e(x_0)\ge h^{K_2}\prod_{i=1}^{K_2}\left(\delta_0+\frac{\kappa (i-1) \delta}l\right),
\begin{cases}w_0^+=b_0+2\log\mu_\lambda-\log h_\gamma, \partial_n w_0^+ =-a_0\lambda\mu_\lambda,\end{cases}\ \hbox{and}\begin{cases}w_0^-=b_0+2\log\mu_\lambda-\log h_\gamma, \partial_n w_0^-=a_0\lambda\mu_\lambda.\end{cases}
\int_{B_{2\rho}}r^{4-n}|D^2 u| |\nabla u| \d x \leq C \left(\int_{B_{4\rho}}r^{2-n}|\nabla u|^2 \d x\right)^{1/2}\left( \int_{B_{4\rho}}r^{4-n}|\nabla u|^2 \d x\right)^{1/2}& + C \int_{B_{4\rho}}r^{3-n}|\nabla u|^2 \d x,
t(\mathbf{P}_{\!\sigma}) = t(\mathbf{p}_{\sigma_d};\ldots;\mathbf{p}_{\sigma_1}) = \sum_{n=1}^{d}C_{n}^{(d), \sigma}(\mathbf{P}_{\!\sigma}) \cdot\bigg(H(\mathbf{p}_{\sigma_{n}}) + \int\! \varphi_n^{\sigma}\,\mathrm{d}\mathbf{p}_{\sigma_n}\bigg),
u^\sigma(t) =\int_{a}^{\sigma(t)}w^\sigma(s)g(s)\big(G^\sigma(s)\big)^{\alpha-\gamma-1}\Delta_\alpha s \leq\frac{1}{\alpha-\gamma+\theta}\int_{a}^{\sigma(t)}T_{\alpha}^{\Delta}\Big(w(s)G^{\alpha-\gamma}(s)\Big)\Delta_{\alpha}s &=\frac{1}{\alpha-\gamma+\theta}w^\sigma(t)\big(G^\sigma(t)\big)^{\alpha-\gamma}.
\sum_{m=0}^{k+1}{c}_{k-m}(\underline{\Lambda})\hat{F}_{m-1}=\sum_{m=0}^{k+1}{c}_{k-m}(\underline{\Lambda})\sum_{l=0}^mF_{l-1}\hat{c}_{m-l-1}(\underline{\Lambda}) =\sum_{l=0}^{k+1}F_{l-1}\sum_{p=0}^{k+1-l}c_{k-l-p}(\underline{\Lambda})\hat{c}_{p-1}(\underline{\Lambda}) =F_k,
\mathbb{E}[V-\sigma^2_N] &= \mathbb{E}[X_A^2] \Delta \mathbb{E}[G] + 2 \mathbb{E}[X_A] \Delta \mathbb{E}[Z_1] + \mathbb{E}[Z_0] - \sigma^2_N &= P \Delta \mathbb{E}[G] \geq 0.
\lim_{n\to\infty}\frac{1}{n}\cdot\nu(g_1^{[\lambda s_1n]}\cdots g_k^{[\lambda s_kn]}) &=\lim_{n\to\infty}\frac{1}{pn}\cdot\nu(g_1^{[q s_1n]}\cdots g_k^{[q s_kn]}) &=\lim_{n\to\infty}\frac{q}{pn}\cdot\nu(g_1^{[s_1n]}\cdots g_k^{[s_kn]}) &=\lambda\lim_{n\to\infty}\frac{1}{n}\cdot\nu(g_1^{[s_1n]}\cdots g_k^{[s_kn]}).
p_u=\mathbf{P}((\overline{\omega}_i)_1=(u)_1)\mathbf{P}(\overline{\omega}_i=u|(\overline{\omega}_i)_1=(u)_1) = q_{(u)_1}\mathbf{P}(\overline{\omega}_i=u|(\overline{\omega}_i)_1=(u)_1)
\widehat{\widetilde{\nabla_x\times\Upsilon}}(s,{\scriptstyle X}, y)= \widehat{{}^\top \!\!\, \widetilde{J}}(s,{\scriptstyle X})\left(\nabla_{\scriptscriptstyle X}\times \widehat{\widetilde{\Upsilon}}(s,{\scriptstyle X}, y)\right).
\Psi( \{[\xi A]_{k,A}\}_{k\in{\mathbb N}}= \Psi ( \{A_k[\xi]_{k,A}\}_{k\in{\mathbb N}}& = (\Psi\circ \Gamma_A)(\{[\xi]_{k,A}\}_{k\in{\mathbb N}}& = (\Gamma_B\circ \Psi) (\{[\xi]_{k,A}\}_{k\in{\mathbb N}}& = \Gamma_B ( \{[\eta_k]_{k,B}\}_{k\in{\mathbb N}}& = \{B_k[\eta_k]_{k,B}\}_{k\in{\mathbb N}}& = \{[\eta_k B]_{k,B}\}_{k\in{\mathbb N}}.
\Delta\left(h(f_0,g_0);f,g\right) = \left\|{A}\xi- \hat{A}\xi \right\|_\alpha^\alpha =\int_{-\pi}^{\pi}\left| A(e^{i\theta}) -{h}^0(\theta) \right|^{\alpha}f(\theta)d \theta + \int_{-\pi}^{\pi}\left|{h}^0(\theta) \right|^{\alpha}g(\theta)d \theta.
\|e_t\varphi(a)e_t-\varphi(a)\| =\|v_tv_t^*\varphi(a)v_tv_t^*-\varphi(a)\| &\leq \|v_tv_t^*\varphi(a)v_tv_t^*-v_t\varphi(\alpha_{t^{-1}}(a))v_t^*\|+\|v_t\varphi(\alpha_{t^{-1}}(a))v_t^*-\varphi(a)\| &\leq\|v_t^*\varphi(a)v_t-\varphi(\alpha_{t^{-1}}(a))\|
\mathfrak{B}_{3}\left( u,v\right) =\left( \begin{array}{c}-\frac{1}{2}\left( u^{2}-v^{2}\right) -\frac{1}{4}\left( u^{4}-v^{4}\right) \frac{1}{2}\left( u^{2}+v^{2}\right) -\frac{1}{4}\left( u^{4}+v^{4}\right) \frac{2}{3}\left( u^{3}+v^{3}\right)\end{array}\right) ,
t_1 =s_1,\t_2=\frac{s_2-\frac{a}{s_2}}{s_2s_3-\frac{ab}{s_2s_3}},\t_3=\frac{s_3-\frac{b}{s_3}}{s_2s_3-\frac{ab}{s_2s_3}},\a=1+3s_1^2, b = \frac{-1-6s_1^2-9s_1^4+2s_2+10s_1^2s_2+4s_1^4s_2-s_2^2-3s_1^2s_2^2}{s_2}&=-A (1 + 3 s_1^2) + 4s_1^2 (1 + s_1^2), A =s_2+\frac{1+3s_1^2}{s_2}-2.
\chi_{\gamma_e}= \chi_{\gamma_e}^{sf}\chi_{\gamma_m}= \chi_{\gamma_m}^{sf}\cdot \exp \left[ \frac{i}{4\pi}\int_{l_+}\frac{d\zeta'}{\zeta'}\frac{\zeta'+\zeta}{\zeta'-\zeta}\ln(1-\textit{X}_{\gamma_e}(\zeta'))- \frac{i}{4\pi}\int_{l_-}\frac{d\zeta'}{\zeta'}\frac{\zeta'+\zeta}{\zeta'-\zeta}\ln(1-\textit{X}^{-1}_{\gamma_e}(\zeta'))\right]
\frac{\varphi_{n + 1}- 2 \varphi_{n}+ \varphi_{n - 1}}{T^{2}}+ \left( \mu^{2}- \nabla^{2}\right) \frac{\left( \varphi_{n + 1}+ 4 \varphi_{n}+ \varphi_{n - 1}\right)}6 ={}_{{}_{\! \! \! \! \! \! \! c}}\; \, j_{n}.
\epsilon(D_1 D_2) = \epsilon(D_1) \epsilon(D_2), \D_1,D_2, \epsilon(D_1 + D_2) = \epsilon(D_1) + \epsilon(D_2) \D_1,D_2 \mathrm{lhd}(D_1) = \mathrm{lhd}(D_2).
\begin{matrix}{{{\cal T}_{z z}= T_{z z}^{g r a v}+{\frac{1}{2}}T_{z z}^{g h},}}&{{{\cal J}_{z}= J_{z}^{g r a v}+{\frac{1}{2}}J_{z}^{g h},}}\\{{{\cal G}_{+ z}={G}_{+ z}^{g r a v}+{\frac{1}{2}}{G}_{+ z}^{g h},}}&{{{\cal G}_{- z}= G_{- z}^{g r a v}+{\frac{1}{2}}G_{- z}^{g h}.}}\\ \end{matrix}
C_V = \sum_{n,w}\int_{0}^{\infty}[dt] e^{-y}\left [ 4 t^2 ({{w^2 \beta^2}\over{4\pi^2 \alpha^{\prime}}}-{{\alpha^{\prime}\pi^2 n^2}\over{\beta^2}}))^2 - 2t ({{w^2 \beta^2}\over{4\pi^2 \alpha^{\prime}}}+ 3{{\alpha^{\prime}\pi^2 n^2}\over{\beta^2}})\right ] .
F^{2}\approx F_{\mathrm{N G}}^{2}( \beta ) \, = \, \beta^{2}M^{4}\left[ 1 \, - \, \frac{( d - 2 ) \pi}{3 M^{2}\beta^{2}}\right] \, \approx \, - \frac{( d - 2 ) \pi M^{2}}{3}.
C(Q^2)=\int_XQ^2e_1^{2n-4}=c_X+\frac{2(b_2-1)c_X}{2n-1}+\frac{3(b_2-1)c_X}{(2n-1)(2n-3)}+\frac{(b_2-1)(b_2-2)c_X}{(2n-1)(2n-3)}>0.
\lim_{y\rightarrow+\infty}\frac{\int_{2}^yf(x)x^{-1}\,dx}{\int_{2}^yf(x)\,dx}=0~~~~\lim_{y\rightarrow+\infty}\left|\frac{\int_{2}^yf(x)x\,dx}{y\int_{2}^yf(x)\,dx}\right|\le 1.
\left( \begin{array}{c c}{{k}}&{{j}}\\{{0}}&{{p}}\\ \end{array}\right) \ , \ \left( \begin{array}{c c}{{0}}&{{- p}}\\{{k}}&{{j}}\\ \end{array}\right) \ , \ \left( \begin{array}{c c}{{0}}&{{- p}}\\{{k}}&{{j + p}}\\ \end{array}\right) \ , \ \ 0 \leq j < k , \ \p \neq 0 \ .
P_{i}=\epsilon_{0}\sum_{j=1}^{3}\chi_{ij}^{(1)}E_{j}+\epsilon_{0}\sum_{j=1}^{3}\sum_{k=1}^{3}\chi_{ijk}^{(2)}E_{j}E_{k}+\epsilon_{0}\sum_{j=1}^{3}\sum_{k=1}^{3}\sum_{l=1}^{3}\chi_{ijkl}^{(3)}E_{j}E_{k}E_{l}+\cdot\cdot\cdot
\begin{pmatrix}1 1 -1 0 \ldots 0 0 1 0 1 -1 \ldots 0 0 1 0 0 1 \ldots 0 0 \vdots \vdots \vdots \vdots \ddots \vdots \vdots 1 0 0 0 \ldots 1 -1\end{pmatrix}