harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\prod_{m=1}^n\frac{\left(1-\frac{\delta_{kj}}{m}\right)^{-1}}{\left(1-\frac{\delta_{ij}}{m}\right)^{-1}}=\prod_{m=1}^n\frac{m-\delta_{ij}}{m-\delta_{kj}}\sim\frac{\Gamma(1-\delta_{ij})}{\Gamma(1-\delta_{kj})}n^{-(i-k)c_1}\rightarrow0,\mbox{as}n\rightarrow\infty,
\limsup_{V \setminus \{0\}\ni h \to 0}\sup_{\mathbf{u}=(u_1,u_2,\ldots,u_n) \in (V \setminus \{0\})^n}\bigg[ \frac{\| f^{(n)}(x+h) \mathbf{u}- f^{(n)}(x) \mathbf{u}- g(x)(u_1, u_2, \ldots, u_n,h) \|_W}{\|h\|_V \prod^n_{i=1}\|u_i\|_V}\bigg] =0.
D_\lambda(p_{W_{T}, \hat W_{T\times \{d\}}}\|p_{W_{T}}s_{\hat W_{T\times \{d\}}}) \le \sum_{k=1}^n D_\lambda( q_{Y_{T^c}|X_{\mathcal{I}}}\|p_{Y_{T^c,k}|X_{T^c,k}}^{(\lambda, T)}|p_{X_{\mathcal{I},k}}^{(\lambda, T)}).
\frac{(axt, byt; q)_\infty}{(xt, yt; q)_\infty}{_3\phi_2}\left({{q^{-m}, xt, yt}\atop{axt, byt}}; q, q \right)=\sum_{n=0}^\infty \lambda_n \Phi_n^{(a, b)}(x, y|q).
CD_{n}^{\left(r,s\right)}\left(x\right)=\begin{cases}C_{n}^{\left(r-s\right)}\left(x\right) \mbox{if}r>s D_{n}^{\left(s-r\right)}\left(x\right) \mbox{if}r<s \left(x\right)_{n}\mbox{if}r=s\end{cases}
L(2,\chi_{4D}) = \prod_p \left(1 - \frac{\chi_{4D}(p)}{p^2}\right)^{-1}= \prod_{p \neq 2, 3}\left(1 - \frac{1}{p^2}\right)^{-1}= \left(1 - \frac{1}{2^2}\right)\cdot \left(1 - \frac{1}{3^2}\right)\cdot \frac{\pi^2}{6}.
\eta(g_1 g_2)=\; S(\psi(h_{g_1}\phi_{R(h_{g_1})}(h_{g_2}))) =\; S(\psi(h_{g_1}) \varphi_{S(\psi(h_{g_1}))}(\psi(h_{g_2}))) =\; S(\psi(h_{g_1})) S(\psi(h_{g_2})) =\; \eta(g_1) \eta(g_2).
T^{\mathrm{\scriptsize{r e g}}\: ( 1 )}( s , l ) \; = \; - \frac{1}{3 2 \pi^{3}}\: \lim_{a \rightarrow 0}\int_{- \infty}^{\infty}\left[ \frac{1}{u}\; e^{- \frac{i a l}{u}}\: \Theta ( u ) \: - \: ( 1 + \log a ) \: \delta ( u ) \right] e^{- i u s}\: d u \; .
1={72n^2(c^{-1}+1)\beta\sqrt{1-{\tilde\rho_0^2\over c^2}}(1+{1\over{\tilde\rho_0\over 2^{3\over 2}}-r})\over \alpha r({\tilde\rho_0\over 2^{3\over 2}}-r)({1\over 2}+{|x_-|\over\sqrt{2}})^\alpha},\ \tilde \rho_0\in(2^{3\over 2}r,c).
p_{c}=\frac{(\sqrt[3]{2}-1)^{7/3}}{3^{1/3}}R^{1/3}\frac{a^{2/3}}{b^{5/3}},T_{c}=3^{2/3}(\sqrt[3]{2}-1)^{4/3}(\frac{a}{bR})^{2/3},V_{m,c}=\frac{b}{\sqrt[3]{2}-1},Z_{c}=\frac{1}{3}
{L}_{n}( \xi ) = \left( \begin{array}{c}{{1 +{\frac{N ( n )}{\Delta}}\xi}\qquad{- i \kappa^{\frac{1}{2}}\phi ( n ) \xi}}\\{{i \kappa^{\frac{1}{2}}\psi ( n ) \xi}\qquad{{\frac{1}{\Delta}}\xi}}\\ \end{array}\right) . \ll{n l 3 l}
\begin{array}{l}\overline{Ric}_{00}=4(\beta-1), \overline{Ric}_{11}=\overline{Ric}_{22}=\overline{Ric}_{33}=\overline{Ric}_{44}=\beta^2-\beta-3\alpha^2, \overline{Ric}_{13}=\overline{Ric}_{24}=3\alpha-2\alpha\beta. \end{array}
\pi(1,X_i)= \int \frac{e^{X_i^T\beta+\sqrt{\tau}z}}{1+e^{X_i^T\beta +\sqrt{\tau}z}}\phi(z)dz =\int \frac{e^{X_i^T\beta_0 +\sqrt{\tau_0}z}}{1+e^{X_i^T\beta_0+\sqrt{\tau_0}z}}\phi(z)dz = \pi^0(1,X_i).
\begin{cases}d^+b- 2^{-\frac{3}{2}}r^{\frac{1}{2}}(q(\psi, \eta) + q(\eta, \psi)) D_A \eta + (2r)^{\frac{1}{2}}\mathfrak{cl}(b) \psi *d*b - 2^{-\frac{1}{2}}r^{\frac{1}{2}}(\eta^{\dagger}\psi - \psi^{\dagger}\eta), \end{cases}
\|h_\beta\|_{L^q_\omega}\lesssim \|g^{1/r}\|_{L^q_\omega}=\|g\|^{1/r}_{L^{\tilde q}_\omega}&\lesssim \|(h_\alpha)^r\|_{L^{\tilde p}_\omega}&=\|h_\alpha\|_{L^p_\omega},
\dot{x}_i = r_i^+-r_i^-=\frac{r_i^+-r_i^-}{\ln r_i^+ - \ln r_i^-}\left(\ln r_i^+ -\ln r_i^-\right) &= \mathsf{lgm}(r_i^+,r_i^-) \left( \sum_{j\in N_i^+}w_{ij}\ln x_j - \sum_{j\in N_i^-}w_{ji}\ln x_i\right) &= \mathsf{lgm}(r_i^+,r_i^-) \sum_{j\in N_i^+}w_{ij}\left(\ln x_j-\ln x_i \right) .
\sum_{n\ge0}\left(\frac{\left(-1\right)^{n+p}}{n!}\int_{0}^{1}\dots\int_{0}^{1}\left(px-p+u_{1}+\dots+u_{p}\right)^{n}du_{1}\dots du_{p}\right)z^{n}=\left(-1\right)^{p}e^{-z\left(px-p\right)}\left(\frac{1-e^{-z}}{z}\right)^{p}=\left(-1\right)^{p}e^{-zpx}\left(\frac{e^{z}-1}{z}\right)^{p};
f_{Cayley}'(x_d^{\ast}) \bigg|_{\genfrac{}{}{0pt}{}{s_k = t_k = x_k^{\ast}}{1\leq k\leq d-1}}= \sum_{\sigma \in S_d}(-1)^\sigma \prod_{i=1}^d \frac{\partial p_{\sigma_i}}{\partial x_i}(\underline{x}^{\ast}).
\frac{1}{\tau}\int_0^\tau x h(x)\, dx = \frac{1}{\tau}\int_0^\tau x G'(x)U(G(x))\, dx = - \frac{1}{\tau}\int_0^\tau \Bigl[G(x)U(G(x))+xG(x)G'(x)U'(G(x))\Bigr]\, dx + G(\tau)U(G(\tau))\, ,
\Gamma_{I}^{\phi_S^0}(-i\Omega_m, \stackrel{\rightharpoonup}{p})=-G/\sum_{Q}g_{QQ}(-\Omega_m^2-{\stackrel{\rightharpoonup}{p}}^2-4m_Q^2) K^T_Q(-i\Omega_m, \stackrel{\rightharpoonup}{p}).
d (\sum_{\sigma \in S_n}sgn(\sigma)^{m - 1}a[x_{\sigma(1)}| x_{\sigma(2)}| \dots | x_{\sigma(n)}] ) & = \sum_{\sigma \in S_n}sgn(\sigma)^{m- 1}(1 + (-1)^{\epsilon_n'}) a x_{\sigma(1)}[x_{\sigma(2)}| x_{\sigma(1)}| \dots | x_{\sigma(n)}]
E(y^{m})=\frac{B(\frac{1}{2}p+m,\frac{1}{2}\nu-m)}{B(\frac{1}{2}p,\frac{1}{2}\nu)}=\frac{\Gamma(\frac{1}{2}p+m)\Gamma(\frac{1}{2}\nu-m)}{\Gamma(\frac{1}{2}p)\Gamma(\frac{1}{2}\nu)}
\|[(L_{b}^*)^{-1}-(L_{b_0}^*)^{-1}][f_h]\|_{L^2}\lesssim \|b-b_0\|_\infty (\|\hat b_T - b_0\|_{\infty}+ \|b-b_0\|_{\infty})=o_{P_{b_0}}(1/\sqrt T).
ln(\gamma_{i})=-\frac{z_{i}^{2}q^{2}\kappa}{8\pi\epsilon_{r}\epsilon_{0}k_{B}T}=-\frac{z_{i}^{2}q^{3}N_{A}^{1/2}}{4\pi(\epsilon_{r}\epsilon_{0}k_{B}T)^{3/2}}\sqrt{10^{3}\frac{I}{2}}=-Az_{i}^{2}\sqrt{I},
\mathbb{I}_{\mathrm{BrAlg}}^{s}\epsilon_{\mathrm{Br}}^{s}=\epsilon_{\mathrm{Br}}\mathbb{I}_{\mathrm{BrAlg}}^{s}\mathbb{I}_{\mathrm{Br}}^{s}\eta_{\mathrm{Br}}^{s}=\eta_{\mathrm{Br}}\mathbb{I}_{\mathrm{Br}}^{s}.
\sum_{\begin{subarray}{c}(e_k, \dots , e_m) \sum_k^m e_j=\mu\end{subarray}}\Bigl(\sum_{\begin{subarray}{c}(e_0, \dots , e_{k-1}) \sum_0^{k-1}e_i =d-\mu\end{subarray}}\alpha_{(e_0, \dots , e_{k-1}, e_k, \dots , e_m)}a_0^{e_0}\cdots a_{k-1}^{e_{k-1}}\Bigr) b_k^{e_k}\cdots b_m^{e_m}= 0
D_{1+s}(P_{X^{n}Y^{n}|U_{n}}\|\pi_{X^{n}Y^{n}}|P_{U_{n}}) =\frac{1}{s}\log\mathbb{E}_{U_{n}}\bigg[\sum_{x^{n},y^{n}}P_{X^{n}Y^{n}|U_{n}}(x^{n},y^{n}|U_{n}) \qquad\times\left(\frac{P_{X^{n}Y^{n}|U_{n}}(x^{n},y^{n}|U_{n})}{\pi_{X^{n}Y^{n}}(x^{n},y^{n})}\right)^{s}\bigg],
\sum\limits_{j=0}^{l-1}\|(Sf-Sg) (t_j)-(Sf-Sg)(t_{j+1})\| &\leq \alpha_{\max}\sum\limits_{j=0}^{l-1}\|(f-g)(P_k^{-1}(t_j))- (f-g)(P_k^{-1}(t_{j+1}))\| & \leq \alpha_{\max}\|f-g\|_{\mathcal{BV}}.
\Delta_k(x)= \frac{1}{2\pi i}\int_{\frac12-\delta-iY}^{\frac12-\delta+iY}\zeta^k(s) x^s \frac{ds}{s}+I_k(x;\theta, \delta) &+O\Big(X^{\varepsilon}\Big(1+X^{\frac12-\delta}X^{k(\lambda+\delta-2\lambda \delta)-1}\Big)\Big).
\left[F_{\Omega}(x)\right]^{N-1}= &\left[\frac{(-1)^{M}(M-1)!}{\mathcal{B}(M,K)\prod_{m=1}^{M}(m-M-K)}\right]^{N-1}\sum_{k=N-1}^{(N-1)(M+K-1)}q_{k-(N-1)}\frac{x^k}{(1+x)^{(N-1)(M+K-1)}},
\begin{array}{lcl}\{r,r'\}= \{r,r'\}\{r,-r\}=\{r,-rr'\}&= \{rr'(r')^{-1},-rr'\}=\{rr',-rr'\}\{(r')^{-1},-rr'\}&= \{r',-rr'\}^{-1}=\{r',r\}^{-1}\{r',-r'\}^{-1}&= \{r',r\}^{-1}. \end{array}
{\sl L}= -\frac{1}{4 g^2}(F_{\mu \nu}- B_{\mu \nu})^2 +{{\theta}\over 2}P_\mu \epsilon^{\mu \nu \lambda}(F_{\nu \lambda}- B_{\nu \lambda})- \frac{1}{4}\theta P_\mu \epsilon^{\mu \nu \lambda}\partial_\nu P_\lambda + \Phi \epsilon^{\mu \nu \lambda}\partial_\mu B_{\nu \lambda}
\int_{\mathbb{R}^d}D_iv(\tau_x \omega)\partial_j\eta(x) dx = -\int_{\mathbb{R}^d}v(\tau_x \omega)\partial_i\partial_j \eta (x)dx &= -\int_{\mathbb{R}^d}v(\tau_x \omega)\partial_j\partial_i \eta (x)dx &= \int_{\mathbb{R}^d}D_jv(\tau_x \omega)\partial_i \eta (x)dx.
X^\varepsilon (t,\;\cdot\;) = \int^t_0 \Delta \left[ \left\{\psi (X^1(s,\;\cdot\;))-\psi (X^2(s,\;\cdot\;))\right\}\star \phi_\varepsilon \right] \d s + \int^t_0 \phi_\varepsilon \star (X \mu(\d s,\;\cdot\;)),
H ={\frac{4 \pi r}{\lambda}}\left[ - \left({\frac{2 \pi r}{\lambda}}+{\frac{\overline{{{l}}}}{r}}\right) \pm \sqrt{\vec{p}^{\ 2}+ \left({\frac{2 \pi r}{\lambda}}+{\frac{\overline{{{l}}}}{r}}\right)^{2}}\ \right] .
u^{\mu}( p_{3}, 0 ) \mid_{m \rightarrow 0}= \left( \begin{matrix}{{p_{3}}}\\{{0}}\\{{0}}\\{{{\frac{p_{3}^{2}}{E_{p}}}}}\\ \end{matrix}\right) \equiv \left( \begin{matrix}{{E_{p}}}\\{{0}}\\{{0}}\\{{E_{p}}}\\ \end{matrix}\right) \quad ,
\lambda(\epsilon,s+1)\chi^+_p + \delta(\epsilon, s+1)G_1 + \cdots{}= \big( q^{-(s+1)}\big(q^s+q^{-s}\big)\lambda(\epsilon,s) - q^{-2s}\lambda(\epsilon,s-1) + (-\epsilon)^{p-1}(-1)^{s-1}\hat qq^{-(s+1)}\delta(\epsilon,s) \big) \chi^+_p \quad{}+ \big({-}q^{-(s+2)}\big(q^s + q^{-s}\big)\delta(\epsilon,s) - q^{-2s}\delta(\epsilon,s-1) \big) G_1 +\cdots.
B(F_p(x), \pi(\phi(\overrightarrow{x\xi})), f(\xi)) & = B(F_p(x), F_p(z), f(\xi)) + B(F_p(z), \pi(\phi(\overrightarrow{z\xi})), f(\xi)) +B(\pi(\phi(\overrightarrow{z\xi})), \pi(\phi(\overrightarrow{x\xi})), f(\xi)) & = B(F_p(x), F_p(z), f(\xi)) + B(F_p(z), \pi(\phi(\overrightarrow{z\xi})), f(\xi)) -B(x, z, \xi),
{n_{\max ,i}}= \max \left\{{(q - 1)!,\mathop{{\sup}}\limits_{0 \le k \le n + 1}\left({{{(q - 1)}_{n - k + 1}}\,{{\left|{x_{l,n,i}^{(\alpha )}- \zeta_{l,n,i}^{(\alpha )}}\right|}^{q - n + k - 2}}}\right)}\right\},
\{z\in \overline{\Omega}: |z|\leqslant m\}=\overline{\Omega}\cap \overline{D(0,m)}\subseteq \overline{\Omega \cap D(0,m+1)}= \overline{\Omega_{m+1}}&\subseteq \overline{\Omega}\cap \overline{D(0,m+1)}=\{z\in \overline{\Omega}: |z|\leqslant m+1\},
A_{\nu}( q ) = \frac{1}{2}\left( \begin{array}{c c c}{{0}}&{{\overline{{\psi}}( q ) \gamma_{\mu}\gamma_{\nu}}}&{{- ( \gamma_{\nu}\gamma_{\mu}\psi ( q ) )^{T}}}\\{{0}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{0}}\\ \end{array}\right)_{( \mu , \lambda )}\, ,
\sum_{i=1}^{M}\lambda_i2K\sigma^2+2\sigma^2\sqrt{KM\sum_{i=1}^{M}(\lambda_i-\bar{\lambda})^2}&=2FK\sigma^2+2\sigma^2\sqrt{KM\left[F\left(1-\frac{F}{M}\right)^2+(M-F)\left(-\frac{F}{M}\right)^2\right]}&=2FK\sigma^2+2\sigma^2\sqrt{K[F(M-F)]}.
CP_f(r)=\underset{i \in \psi}\prod\frac{1}{1+\max\{T, S_{th}\}r^\alpha d_i^{-\alpha}}e^{-\max\{T, S_{th}\}r^{\alpha}\frac{\sigma^2}{P}}+P[\hat{\eta}(r)>T]-P[\hat{\eta}(r)>\max\{\hat{S}_{th},T\}]
\omega^n(z)= n!\bigwedge_{j=1}^n \frac{1}{(1-|z_j|^2)^2}\frac{i}{2\pi}dz_j\wedge d\bar{z_j}=\frac{n!}{\pi^n}\frac{dm(z)}{[1-|z|^2]^2}=n!\; d\nu_n(z)\ ,
\iint_{Q_t}(u\mathbf{b})_x\cdot \mathbf{b}_{xx}\omega^2 dxds-\iint_{Q_t}\mathbf{w}_x\cdot \mathbf{b}_{xx}\omega^2 dxds \leq \frac{\nu}{4}\iint_{Q_t}|\mathbf{b}_{xx}|^2\omega^2 dxds+C\iint_{Q_t}|(u\mathbf{b})_x|^2\omega^2 dxds +C\iint_{Q_t}|\mathbf{w}_{x}|^2\omega^2 dxds.
S^{c}=\frac{e^{i\frac{\pi}{4}}}{2(4\pi)^{3/2}}\int_{0}^{\infty}de_{0}\left(\det\frac{\sinh\frac{ge_{0}F}{2}}{gF}\right)^{-\frac{1}{2}}e^{iI[x_{cl},e_{0}]}\Phi[x_{cl},e_{0}],
r_s \rightarrow r = \lim_{K \rightarrow \infty}\frac{\binom{K-2}{s-1}}{\binom{K-2}{s-1}+\sum_{i=0}^{K-1-s}\binom{K-1}{s+i}(N-1)^i N}&=\lim_{K \rightarrow \infty}\frac{\psi_2(N,K,s)}{\psi_2(N,K,s)+1}=\frac{N\lambda-(N-1)}{N\lambda+1}.
[Tu,T_1(v),T_1(w)]+[T_1(u),Tv,T_1(w)]+[T_1(u),T_1(v),Tw] = T\Big(D(T_1(u),T_1(v))w+\theta(T_1(v),T_1(w))u-\theta(T_1(u),T_1(w))v\Big) &+T_1\Big(D(Tu,T_1(v))w+D(T_1(u),Tv)w+\theta(Tv,T_1(w))u &+\theta(T_1(v),Tw)u-\theta(Tu,T_1(w))v-\theta(T_1(u),Tw)v\Big),
E_{o p t}^{( p )}\simeq 0 . 4 2 0 7 9 8 7 - 1 1 . 9 8 \times 1 0^{- 5}/ p + 8 . 9 9 1 \times 1 0^{- 5}/ p^{2}+ \mathrm{O}( p^{- 3}) .
W_y^{QFT}= E_y - E_{\infty}= -\frac{\alpha^2}{L^3}+ \alpha ( \| \lambda_y\|^2 - \|\lambda_{\infty}\|^2) + \|\Phi_{\#}^{\infty}\|^2_{\#}- \|\Phi_{\#}^y\|^2_{\#}&\quad+ O\Big( \frac{\alpha^2}{L^5}\Big) + O\big( \alpha^2 L e^{-L/2}\big) + O(\alpha^4 \log(\alpha^{-1})).
\alpha ((x',v';x,v) \cdot (y',w';y,w)) =(xy, x'w' + v'y' + t v'w' + xw+vy+tvw) \cr =(xy, xw' + tvw' + v'y + tv' w + t v'w' + xw+vy+tvw) \cr\alpha (x',v';x,v) \cdot \alpha (y',w';y,w) =(x,v+v') \cdot (y,w + w') \cr = (xy, x (w+w') + (v+v') y + t (v+v') (w+w')) .
\varphi(hgh'g') =\widetilde\rho(h{}^gh'g_xg'_{x}) &=\rho(h)\rho({}^{g_t}({}^{g_x}h'))\widetilde{\rho}(g_xg'_x) &=\rho(h)\rho({}^{g_x}h')\rho(g_xg'_x) &=\widetilde{\rho}(h{}^{g_x}h'g_xg'_x) &=\widetilde{\rho}(hg_xh'g'_x) &=\widetilde{\rho}(hg_x)\widetilde{\rho}(h'g'_x) &=\varphi(hg)\varphi(h'g'),
\frac{1}{2}\partial_{t}\lVert w\rVert_{L^{2}}^{2}+ \lVert \Lambda^{r_{1}}w\rVert_{L^{2}}^{2}= \int (b\cdot\nabla) j w \leq \lVert \nabla j\rVert_{L^{2}}\lVert b\rVert_{L^{\infty}}\lVert w\rVert_{L^{2}}
G^{( + )}( x , x^{\prime}) = - 2 \pi i \int \frac{d^{n - 1}k}{( 2 \pi )^{n - 1}}e^{2 i k_{\mu}\tilde{x}^{\mu}}\frac{( 1 + f ( k_{0}, \overline{x}) )}{2 | \omega |}
C_{\b c d}^{a}\equiv R_{\b c d}^{a}+{\frac{2}{d - 2}}\left( \delta_{\ [ d}^{a}R_{c ] b}+ g_{b [ c}R_{d ]}^{a}\right) +{\frac{2}{( d - 1 ) ( d - 2 )}}\delta_{\ [ c}^{a}g_{d ] b}R ,
lim_{t\rightarrow\infty,F_{e}\rightarrow0}\frac{1}{t}ln(\frac{p(\overline{J}_{t})=A}{p(\overline{J}_{t})=-A})=lim_{t\rightarrow\infty,F_{e}\rightarrow0}\frac{2A\langle J\rangle_{F_{e}}}{t\sigma_{\overline{J}(t)}^{2}}.
\left\{0\right\}=\mathfrak{g}_{J}^{s+1}\subset\mathfrak{g}_{J}^{s}\subset\mathfrak{g}_J^{s-1}\subseteq\cdots\subseteq\mathfrak{g}_{J}^{k+1}\subseteq\mathfrak{g}_{J}^{k}\subseteq\cdots\subseteq\mathfrak{g}_{J}^{1}{\subset}\mathfrak{g}_{J}^{0}=\mathfrak{g}.
\begin{pmatrix}\beta_{0}\beta_{1}\beta_{2}\vdots \beta_{n}\end{pmatrix}=\begin{pmatrix}N(0,0) N(1,0) \cdots N(n,0) 0 N(1,1) \cdots N(n,1) \vdots \ddots \ddots \vdots 0 \cdots 0 N(n,n)\end{pmatrix}\begin{pmatrix}\alpha_{0}\alpha_{1}\alpha_{2}\vdots \alpha_{n}\end{pmatrix}
P_{h}(i) := 1-S(i)-\frac{(k-h)(2i+\alpha+\beta-1)_2}{(i+k+l+\alpha+\beta)(i+k+\beta-l)(i-n-1)}, begin{align*}1ex] S(i) := \frac{(i-k-l-1)(i+\beta+\alpha+n)(i+l+\alpha-k-1)(2i+\beta+\alpha)}{(2i+\beta+\alpha-2)(i+k+l+\beta+\alpha)(i+k+\beta-l)(i-n-1)}.
\Delta^2 u_1 =\left( R_1^2+R_2^2 \right) \frac{2(u_1+u_2)}{1+|u|^2}+ \frac{4 u_2}{1+|u|^2}|\nabla u|^4 + (R_2-R_1) |\nabla u|^2, \Delta^2 u_2 =\left( R_1^2+R_2^2 \right) \frac{2(u_2-u_1)}{1+|u|^2}- \frac{4 u_1}{1+|u|^2}|\nabla u|^4 - (R_1+R_2) |\nabla u|^2,
\sum_{i=0}^{p-1}i C_{\tau,(p-i-1,b)}- \sum_{i=0}^{p-1}i C_{\tau,(p-i-1,0)}+ p \sum_{l=1}^{b}C_{\sigma,(0,l)}= \sum_{l=1}^{b}\sum_{k=0}^{p-1}\left(C_{\sigma,(0,l)}+ \sum_{i=0}^{k}(C_{\tau,(i,l)}- C_{\tau,(i,l-1)})\right) &= \sum_{l=1}^{b}\sum_{k=0}^{p-1}C_{\sigma,(k+1,l)}= \frac{b}{q}C_\sigma.
\langle VU^*h,h \rangle =\langle U^*h,V^*h \rangle =\sum\limits_{j\in\mathbb{J}}\langle U^*h,f_j\rangle\langle f_j,V^*h\rangle =\sum\limits_{j\in\mathbb{J}}\langle h,Uf_j\rangle\langle Vf_j,h\rangle=\sum\limits_{j\in\mathbb{J}}\langle h,x_j\rangle \langle \tau_j,h\rangle, ~\forall h \in \mathcal{H},
\|y_{\bar u + \theta(u - \bar u)}- \bar y\|_{C(\bar\Omega)}= \|\phi + \theta(y_u - \bar y)\|_{C(\bar\Omega)}&\le (C_2C_{f,K_U}\sqrt{|\Omega|}\|y_u - \bar y\|_{C(\bar\Omega)}+ 1)\|y_u - \bar y\|_{C(\bar\Omega)}.
L(n,k)=(\begin{matrix}n-1\\k-1\end{matrix})\frac{n!}{k!}=(\begin{matrix}n\\k\end{matrix})\frac{(n-1)!}{(k-1)!}=(\begin{matrix}n\\k\end{matrix})(\begin{matrix}n-1\\k-1\end{matrix})(n-k)!
G_1(\varepsilon ;\zeta ):=\widetilde{g}^\varepsilon b(\mathbf{D})(B^0-\zeta \overline{Q_0})^{-1}+g^\varepsilon \bigl(b(\mathbf{D})\widetilde{\Lambda}\bigr)^\varepsilon S_\varepsilon (B^0-\zeta \overline{Q_0})^{-1}.
\Bigl|\bar{\tau}_\varepsilon^i\cdot\nabla\rho_\eta^\varepsilon-\overline{\nabla_\Gamma g_i}\cdot\overline{\nabla_\Gamma\eta}\Bigr| \leq c\varepsilon\sum_{\xi=\eta,\zeta_0,\zeta_1}\left(|\bar{\xi}|+\Bigl|\overline{\nabla_\Gamma\xi}\Bigr|\right)
\Theta_{p}( x ) = ( 2 \epsilon_{p})^{- 1 / 2}e^{- i \epsilon_{p}t}\varphi_{p}( z ) , \quad \varphi_{p}( z ) = ( 2 \pi )^{- 1 / 2}e^{i p z}, \quad \epsilon_{p}= \sqrt{p^{2}+ m^{2}}, \quad - \infty < p < \infty ,
\tilde{D}_1 := [\gamma/2,\gamma]\times [-\sqrt{1+\alpha}~\gamma^2,\sqrt{1+\alpha}~\gamma^2], \tilde{D}_2 := [N,N+\gamma] \times [\sqrt{1+\alpha}~N^{\frac{\alpha+2}{2}}, \sqrt{1+\alpha}~N^{\frac{\alpha+2}{2}}+\gamma^2].
S ={\frac{1}{2}}({\frac{1}{6}}+ | \xi | ) r_{+}^{2}\sum_{i = 0}^{5}\Delta_{i}M_{i}^{2}\ln M_{i}^{2}+{\frac{1}{4 5}}{\frac{1}{8}}r_{+}^{2}C \sum_{i = 0}^{5}\Delta_{i}\ln M_{i}^{2}~ ~ .
d_{j}^+(\lambda)=\frac{(-1)^{\sum_{\alpha=j}^n(\alpha+1)(\lambda_\alpha-\lambda_{\alpha+1}+\delta_{\alpha j})}}{(\lambda_j+1)j}\bigg(\prod_{\ell=1}^{j-1}\frac{\lambda_j-\lambda_\ell-j+\ell+1}{\lambda_j-\lambda_\ell-j+\ell+[\lambda_j-\lambda_\ell]_2}\bigg)
-1 =u^2+vw &=u^2+v\tfrac{(d-a)u-bv}{c}&=u^2+\tfrac{d-a}{c}uv-\tfrac{b}{c}v^2 &=\Big(u+\tfrac{d-a}{2c}v\Big)^2-\Big(\tfrac{(d-a)^2}{4c^2}+\tfrac{b}{c}\Big)v^2.
\begin{aligned}\Vert f-Qf\Vert=\bigl\Vert\lim_{t\to\infty}e^{itH}\varphi(H^a)(\psi_a(H^a)-P^a)P_a^{{\varepsilon}_a}(t)e^{-itH}f\bigr\Vert\to0.\end{aligned}
\check{d}= (r-p-q)^{r-p-q}\sqrt{\frac{p^p \, q^q \, x^r}{(r-p)^{r-p}\, (r-q)^{r-q}\, (1-x)^{r-p-q}}}, begin{align*}3mm]\check{v}_i = v_i - \frac{1-a-b}{r-p-q}(i = 1, \dots, r-p-q).
\Lambda_{\rm{b4}}\left(a\right)=\sqrt{a}\left({\cal C}_{66}a+{\cal C}_{44}\right)\left[\left({\cal C}_{13}+{\cal C}_{44}\right)\left(e_{15}a+e_{33}\right)-\left({\cal C}_{44}a+{\cal C}_{33}\right)\left(e_{31}+e_{15}\right)\right] ,
S_{T}^{L}=\frac{1}{a_{d,k}(\mathcal{D})\mathcal{L}^{k/2}(T)T^{d-(k\delta)/2}}\left[\int_{\mathcal{D}(T)}F(\xi(\mathbf{x}))d\mathbf{x}-C_{0}^{L}T^{d}|\mathcal{D}|\right],
\int_0^{\infty}dy \,y^{1-s}\frac{d}{dy}t^{2(\rho-1)}=(1-\rho)\,\frac{\Gamma\left(\frac{\displaystyle 3-s}{\displaystyle 2}\right)\Gamma\left(\rho-\frac{\displaystyle 3-s}{\displaystyle 2}\right)}{\Gamma(\rho)},\quad 3-2\mbox{Re}\rho<\mbox{Re}s<3.
M_{g}^*(x):=\max \biggl\{\max_{t=1, \ldots, m}\|g^{(t)}(x) \| \, , \, \sup_{y \in B_x^\circ(r_a(x))}\frac{\|g^{(m)}(y)-g^{(m)}(x) \|}{\|y-x\|^{\beta-m}}\biggr\}
\dot{\tilde{x}}_i = g_1^{ij}(t,\tilde x) \tilde{\xi}_j, & \dot{\tilde{\xi}}_i = - \partial_i g_1^{jk}(t, \tilde x) \tilde{\xi}_j \tilde{\xi}_k + (\tilde{\xi}\cdot \partial g_1^{jk}(t,\tilde{x})\tilde{\xi}_j\tilde{\xi}_k)\tilde{\xi}_i
W_0 (X_0 Y_1 - X_1 Y_0) =\frac{\zeta_0 (X_0 Y_1 - X_1 Y_0)}{\beta_0}= \frac{\zeta_2 (X_1 Y_2 - X_2 Y_1)}{\beta_2}= W_2 (X_1 Y_2 - X_2 Y_1) \;.
\; = \; \det [ \not \! \partial + i e_{N}\gamma_{0}{\tilde{A}}_{0}+ m ] \; \det \left[ 1 + i e_{N}( \not \! \partial + i e_{N}\gamma_{0}{\tilde{A}}_{0}+ m )^{- 1}\gamma_{0}a_{0}( \vec{x}) \right] \; .
f_{j+1}\bar{e}_{j}=f_{j+1}e_j w_j=e_j f_{j+1}w_j = e_j f_{j+1}\sum_{r\geq0}\frac{(M_{j}-M_{j}^{-1})^r}{r!}z_{j}^{r}&=e_{j}\left(\sum_{r\geq0}\frac{(q^{n+1}M_{j}-q^{-n-1}M_{j}^{-1})^r}{r!}z_{j}^{r}\right) f_{j+1}.
(A^{*m}|A^n|^\frac{2n}{n}A^m)^\frac{m}{m+n}=(A^{*m}A^{*n}A^nA^m)^\frac{m}{m+n}=| A^{m+n}|^\frac{2m}{m+n}=|A|^{2m}=|A^m|^2
\widetilde{\alpha}_{n,q}(l) =\frac{\binom{l-1+q}{l-n-1+q}\binom{l-1}{l-n-1}}{n!(n-1)!(n-q)!l(l-n-1+q)}&=\frac{l^{2n-2}}{n!(n-1)!(n-q)!}\sum_{i=0}^{2n}P_{2n-i}(q) \frac{1}{l^i}\sum_{i=0}^\infty \frac{(n+1-q)^i}{l^i}&=\frac{l^{2n-2}}{n!(n-1)!(n-q)!}\sum_{i=0}^\infty \sum_{j=0}^i P_{2n-i}(q)(n+1-q)^{i-j}\frac{1}{l^i}.
\begin{aligned}\frac{\partial^2R}{\partial z^2}-(p+hD_x)^2R=0, \ -D(x)\leq z\leq0\cr \ R|_{z=0}=1\cr \partial_zR+ih\langle\nabla D(x),(p+hD_x)R\rangle|_{z=-H(x)}=0\cr\end{aligned}
\Delta^{\rm DD}=-n\,\frac{\pi^2}{1440}\left[1+\Big(\frac{\gamma-1}{2}+\ln(2\sqrt\pi)-\frac{\zeta'(4)}{\zeta(4)}-\frac{5}{4}\frac{n+2}{n+8}\Big)\varepsilon\right]+O(\varepsilon^2)\,.
\mathbb{E}[T^{\hat{x}}(G') - T^{\hat{x}}(H) \mid \hat{x}] = \frac{(a-b)}{n}\Big{(}n(\textrm{Unchanged}) - 2n(\textrm{Incorrect, Unchanged}) \Big{)}\times \Big{(}n(\textrm{Changed}) - 2n(\textrm{Incorrect, Changed}) \Big{)},
+ \frac{1}{2!}\int^1_0 d\tau_1 \int_0^1 d\tau_2\left. \frac{\delta^2 W[X]}{\delta X^a(\tau_1) \delta X^b(\tau_2)}\right|_{X=\overline{X}}\xi^a(\tau_1) \xi^b(\tau_2) +{\cal O}(\xi^3)
\Omega^{(r)}_{\lambda}x^{\lambda}=\Omega^{(r-1)}_{\lambda^{*}}T_{\lambda,\lambda^{*}}(x_1,\dots,x_n)x^{\lambda^{*}}=\Omega^{(r-1)}_{\lambda^{*}}T_{\lambda,\lambda^{*}}(1,\dots,1)x^{\lambda},
\frac{e^{2 u_k}- q^{-2s}}{e^{2u_k}q^{-2s}-1}=\prod_{j=1}^{2s}\frac{e^{u_k}q - e^{u_j}}{e^{u_k}- e^{u_j}q},\hspace{10pt}k=1,\ldots,2s.
\max_{\left( u,m,d\right) \in \overline{K_{1}^{2}\left( M_{2}\right)}\times \overline{K_{2}\left( M_{3}\right)}}\left\{\max_{\overline{Q}_{T}}\left[\max_{\left\vert \beta \right\vert \leq 4}\left\vert D^{\beta}F_{i}\left(X_{i}\right) \right\vert \right] \leq M_{4},i=1,2\right\},
\log h_{i}=\lim_{w\rightarrow w_{i}}\left(\log|w-w_{i}|^{2}+\frac{2e^{-\varphi(w)/2}}{|w-w_{i}|}\right),i=1,\dots,n-1, \intertext{and}\log h_{n}=\lim_{w\rightarrow\infty}\left(\log|w|^{2}-\frac{2e^{-\varphi(w)/2}}{|w|}\right),
e_A:= \frac{\partial}{\partial y^A}-(g_u)^{ij}\left( c_{iA}+\frac{\partial u^B}{\partial x^i}h_{BA}\right)\left( \frac{\partial}{\partial x^j}+\frac{\partial u^C}{\partial x^j}\frac{\partial}{\partial y^C}\right),\ 1\leq A\leq k
|I| =|\int_{\frac{1}{|y|}\leq|x|<\frac{1}{\beta},~|x-z|\leq\frac{1}{\beta}}(\frac{x}{|x|^{n+1-\beta}}-\frac{x-z}{|x-z|^{n+1-\beta}})e^{2\pi ix\cdot y}\,dx| &\le C(\frac{2^\beta-1}{\beta}\beta^{-\beta}+\frac{1-2^{-\beta}}{\beta}\beta^{-\beta}+\frac{\beta^{-\beta}}{1-\beta})
(2\pi)^3{\hat G}({\hat \xi}{\hat \eta};{\hat \xi}'{\hat \eta}') = \sum_{123}D^{-1}({\hat q}_{12}) \int d{{\hat q}_{12}}'' K({\hat q}_{12},{{\hat q}_{12}}''){\hat G}({\hat \xi}''{\hat \eta}'';{\hat \xi}'{\hat \eta}')
\begin{cases}|\sigma_y(t)|\to R\ \,\ \,u(\sigma_y(t))\to 0\ \,\ \,t\to t_y^-,\ \ |\sigma_y(t)-z|\to 0 \ \,\ \,u(\sigma_y(t))\to u(z)\ \,\ \, t\to t_y^+. \end{cases}
h_{\mu\nu}(t,{\bf x})= -8 \pi{\cal G}\frac{1}{(2\pi r)^{(D-2)/2}}\partial_{t}^{(\frac{D-4}{2})}\left[ \int d^{D-1}{\bf x'}S_{\mu \nu}(t-|{\bf x-x'}|,{\bf x'})\right]\,,
\prod_{i=1}^n |x_i|^{\frac{p-2}{n+1}}=\left(\sqrt[n]{\prod_{i=1}^n |x_i|^p}\right)^{\frac{n}{n+1}\cdot\frac{p-2}{p}}\leq \left(\frac{1}{n}\left(\sum_{i=1}^n |x|_i^p\right)\right)^{\frac{n}{n+1}\cdot\frac{p-2}{p}}=n^{-\frac{n}{n+1}\cdot\frac{p-2}{p}}
S = \sum_{n = 0}^{\infty}\int d^{6}x \ [ \dot{A}_{a b}^{( n )}E_{a b}^{( n )}- \frac{( - 1 )^{n}}{4}( E_{a b}^{( n )}E_{a b}^{( n )}+ B_{a b}^{( n )}B_{a b}^{( n )}) +{\Lambda}_{a b}^{( n + 1 )}{\Psi}_{a b}^{( n + 1 )}- 2 A_{t a}^{( n )}{\cal G}_{a}^{( n )}] ,
\log G(z\,|\,\tau) = \int\limits_0^\infty \frac{dt}{t}\Bigl[\frac{1-z}{e^{t\tau}-1}+(1-z)e^{-t\tau}+(z^2-z)\frac{e^{-t\tau}}{2\tau}+\frac{1-e^{-t(z-1)}}{(e^t-1)(1-e^{-t\tau})}\Bigr].
{\displaystyle S(h)^{-c/2}{n \choose{c/2}}}\Big(\dfrac{\sum_{j=1}^{n}a_j}{n}\Big)^{c/2}\leq \displaystyle \sum_{1\leq i_1 < \cdots < i_{c/2}\leq n}a_{i_1}a_{i_2}\cdots a_{i_{c/2}}\leq{\displaystyle{n \choose{c/2}}}\Big(\dfrac{\sum_{j=1}^{n}a_j}{n}\Big)^{c/2}\:.
|W(x,y) - W(x_{0},y)| \leq V(y') \Bigg({1\over{\big( 1 +{{d(x_{0},y)}\over{\rho(y')}}\big)^{k}}}\bigg|{1\over{d(x,y)^{n}}}-{1\over{d(x_{0},y)^{n}}}\bigg| + & +{1\over{d(x,y)^{n}}}\bigg|{1\over{\big( 1 +{{d(x,y)}\over{\rho(y')}}\big)^{k}}}-{1\over{\big( 1 +{{d(x_{0},y)}\over{\rho(y')}}\big)^{k}}}\bigg| \Bigg) = A+B.
\left|T_ns(x) - s(x) \right| \le \int_0^1 |s(t) - s(x)| \cal K_n(x,t) dt \le \widetilde{\mathcal{L}}\int_0^1 |t - x| \cal K_n(x,t) dt & \le \widetilde{\mathcal{L}}\left( \int_0^1 |t - x|^2 \cal K_n(x,t) dt \right)^{\frac{1}{2}}\left( \int_0^1 \cal K_n(x,t) dt \right)^{\frac{1}{2}}= O\left( \frac{1}{\sqrt{n}}\right), n \to \infty,
F^{\perp}:(X_0+Y_0) + (X_0+Y_0) = \big((X_0+Y_0)\circ F\big)^{\perp}+ (X_0+Y_0) = \left(x_1^{a_1}\cdots x_n^{a_n}+ y_0^{b_0-1}y_1^{b_1}\cdots y_m^{b_m}\right)^{\perp}+ (X_0+Y_0) = \left(x_1^{a_1}\cdots x_n^{a_n}+ y_0^{b_0-1}y_1^{b_1}\cdots y_m^{b_m}\right)^{\perp}+(Y_0),
P_{0}= 1 , \quad P_{1}= - \tau , \quad P_{2}= \frac{1}{2}\, \kappa^{2}, \quad P_{3}= \frac{1}{2}\, \kappa^{2}\tau , \quad P_{4}= \frac{1}{2}\, \kappa^{( 1 ) \, 2}+ \frac{1}{2}\, \kappa^{2}\tau^{2}- \frac{1}{8}\, \kappa^{4}, \quad \ldots ,
\beta(F,A_1,A_2)\cdot_{\overline{M_{i+1}}}\delta_j = m_1m_2 F\cdot \delta_jj\neq 0,1,i \beta(F,A_1,A_2)\cdot_{\overline{M_{i+1}}}\delta_0 = m_1m_2F\cdot \delta_0+\sum_{\ell=1}^{2}\Big(m_{2-\ell}(m_\ell(2h-2)-(2g(A_\ell)-2)+A_\ell\cdot A_\ell)-A_1\cdot A_2\Big).
\sum_{k=1}^\infty e^{-\eta k}(1 + pk)^q\ \le int_{1/2}^\infty e^{-\eta x}(1 + px)^q \, \mathrm{d}x \ = frac{e^{\eta/p}}{p}\int_{1 + p/2}^\infty e^{-\eta t/p}\, t^q \,\mathrm{d}t.
\lim_{n\to\infty}\tau_{\Lambda^{(n)}\times\Delta^{(n)},N^{(n)}}\big( A(\Lambda^{(n)}\times\Delta^{(n)},N^{(n)};f_1)\dotsm A(\Lambda^{(n)}\times\Delta^{(n)},N^{(n)};f_k)\big) &\quad=\tau(A(f_1)\dotsm A(f_k)).
G_{d,r}(\textbf{t},\epsilon):=\mathbb{E}_{S_{r}}[\prod_{\ell=1}^{\infty}(1+\epsilon^{\ell}t_{d\ell})^{X_{\ell}}] = \exp\left( \sum_{\ell}\epsilon^{\ell}\frac{t_{d\ell}}{\ell}\right).
ds^2=\frac{1}{1-\xi^0\xi^0}d\xi^0d\xi^0 -(1-\xi^0\xi^0)\left[(1-\rho^2)^{-2}d\rho^2 +(1-\rho^2)^{-1}\rho^2d{\bf u}d{\bf u}^{\prime}\right] ~,