harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
T =m\int_{r_1^{1-\delta}}^\infty \mu(E\cap B(x, u)) u^{-m-1}\; du &\leq \int_{r_1^{1-\delta}}^{r_0}m\mu( B(x, u)) u^{-m-1}\; du+\int_{r_0}^\infty m u^{-m-1}\; du &\leq \int_{r_1^{1-\delta}}^{r_0}m u^{s-\epsilon-m-1}\; du+\int_{r_0}^\infty mu^{-m-1}\; du\mbox{(by \eqref{e-8se})}&\leq \frac{m}{m+\epsilon-s}r_1^{-m+\delta m+(1-\delta)(s-\epsilon)}+r_0^{-m}.
\Gamma_{( 0 )}={\frac{1}{( p + 1 ) !{\sqrt - d e t G}}}\epsilon^{i_{1}. . . . . . i_{p}+ 1}\partial_{i_{1}}X^{M_{1}}. . . . . . . \partial_{i_{p + 1}}X^{M_{p + 1}}\Gamma_{M_{1}}^{\prime}. . . . ._{M_{p + 1}}.
\lim_{s\rightarrow t}\int_{\Omega}A\left( \frac{Ru(s)-Ru(t)}{s-t}\right)\mathrm{d}x \leq\lim_{s\rightarrow t}\int_{\Omega}A\left( \frac{u(s)-u(t)}{s-t}\right) \mathrm{d}x & =\int_{\Omega}A\left( \lim_{s\rightarrow t}\frac{u(s)-u(t)}{s-t}\right)\mathrm{d}x=\psi\left( \frac{\mathrm{d}}{\mathrm{d}t}u(t)\right)
T_{r h}\approx \left( \frac{9 0}{\pi^{2}g_{*}}\right)^{\frac{1}{4}}m i n \left( \sqrt{H ( \phi_{e})}, \sqrt{\Gamma}\right) \approx \left( \frac{3 0}{\pi^{2}g_{*}}\right)^{\frac{1}{4}}m i n \left( \Delta , \left( \frac{3}{8 \pi^{2}}\right)^{\frac{1}{4}}p^{\frac{3}{2}}\kappa^{\frac{5}{2 p}}\Delta^{3 - \frac{5 q}{2 p}}\right)
\left| \Psi_{0}[{\psi}_{\mathrm{K}}( y ) ] \right|^{2}= \pi \hbar \exp \left\{- \zeta_{P{\cal K}}( 0 ) \right\}\exp \left\{- \frac{1}{4}\frac{d \zeta_{P{\cal K}}}{d s}( 0 ) \right\}
\lim_{n \to +\infty}J_{n}=\lim_{n \to +\infty}\int_{A_{K_{n}, \sigma_{n}}}(( u-K_{n})_{+})^{\gamma^{*}}dx=\int_{A_{\frac{K}{2}, \frac{R_1}{2}}}(( u-\frac{K}{2})_{+})^{\gamma^{*}}dx.
L_{n}^{(\alpha)}(x)^{2}-L_{n-1}^{(\alpha)}(x)L_{n+1}^{(\alpha)}(x)=\sum_{k=0}^{n-1}\frac{(\begin{matrix}\alpha+n-1\\n-k\end{matrix})}{n(\begin{matrix}n\\k\end{matrix})}L_{k}^{(\alpha-1)}(x)^{2}>0.
\mu(\bigcap_{s\in U}B_s) =\int \prod_s\chi_{B_s}(\{i_j\}_{j\in s})\,d\mu(i_1,\ldots,i_n) =\int \chi_{B_t}(\{i_j\}_{j\in t}) \int \prod_{s\neq t}\chi_{B_s}(\{i_j\}_{j\in s})\,d\mu\,d\mu(\{i_j\}_{j\in[n]\setminus t}).
\Delta f_{def}=\frac{\Delta F_{def}(\vec{R})}{V}=-\frac{k_{B}Tv_{s}\beta(\lambda_{x}^{2}+\lambda_{y}^{2}+\lambda_{z}^{2}-3)}{2}=\frac{k_{B}Tv_{s}\beta}{2}(\lambda_{z}^{2}+\frac{2}{\lambda_{z}}-3)
\sup_{x \in \mathbf{R}}\left| f(x) - \tilde{f}_{N}(x) \right| \leq\| f \| \, \sup_{x \in \mathbf{R}}\left|w(x)\, B_{N}(x; \{a_{i}\}, \mathcal{D}_{d})\right| & = \| f \| \, \sup_{x \in \mathbf{R}}\left|w(x)\, \prod_{j = -N}^{N}\tanh \left( \frac{\pi}{4d}(x - a_{j}) \right) \right|,
[\tau(\lambda )Q(\lambda)\psi]({\vec x})=Tr\left[\int d{\vec t}d{\vec y}\left(\prod_{i=N}^1 L(\lambda|x_i,\partial_{x_i}) R_{\lambda +\beta_i-1}(t_i,x_i|t_{i-1},y_i)\right)\psi({\vec y})\right] ~.
S_2^D(U; \psi_m; \eta): = \big\{\varphi: D^\times_f \to E \, \big| \, \varphi(\delta gu ) = \eta(ad) \psi_m(d) \varphi(g) \textrm{for any}\delta \in D^\times, & g \in D^\times_f, \textrm{and}u\in U \textrm{with}u_p = \big(\begin{smallmatrix}a b c d\end{smallmatrix}\big) \big\}.
\hspace{. 7 5 i n}+ \sum_{j = 1}^{l - 1}\gamma^{( l - j ) / 2}\varphi_{i}( - l + j ) \otimes \gamma^{- l + j}x_{i}^{+}( - j ) \quad{\it m o d}\; U N_{-}\otimes U N_{+}^{2}
\int (\theta(x)-x)^{2}\bar{\mu}_{Q,t}(dx) = \frac{1}{2}\iint((\theta(x)-x)-(\theta(y)-y))^{2}\bar{\mu}_{Q,t}(dx)\bar{\mu}_{Q,t}(dy) \int V'(x)(\theta(x)-x)\bar{\mu}_{Q,t}(dx) =\frac{1}{2}\iint((\theta(x)-x)-(\theta(y)-y))\cot((x-y)/2)\bar{\mu}_{Q,t}(dx)\bar{\mu}_{Q,t}(dy).
\int Vf_1\cdot Jg \,d\mu=\lim \int T^{k_j}f_1\cdot Jg \,d\mu =\lim \int T^{k_j}f_1\otimes g \,d\eta &= \lim \int f_1\otimes S^{-k_j}g \,d\eta=\lim \int J^{\ast}f_1\cdot S^{-k_j}g \,d\nu.
\sup_{x \in [t_i,t_{i+1})}\{\hat{F}_n(x) - F_0(x)\}\leq \rho(\kappa_+)\|\mathbb{F}_n - F_0\|_\infty , \inf_{x \in [t_i,t_{i+1})}\{\hat{F}_n(x) - F_0(x)\}\geq -\rho(\kappa_-)\|\mathbb{F}_n - F_0\|_\infty ,
[x_{i, r+1}^{\pm}, x_{i, s}^{\pm}] - [x_{i, r}^{\pm}, x_{i, s+1}^{\pm}] = \pm (\varepsilon_1 + \varepsilon_2) (x_{i, r}^{\pm}x_{i, s}^{\pm}+ x_{i, s}^{\pm}x_{i, r}^{\pm}),
\int{\prod_{0 \le t \le T}{\omega_{J}\left( t \right)}\exp \left({i J \int_{0}^{T}{d t \left\{{i{\frac{\xi^{*}\dot{\xi}-{\dot{\xi}}^{*}\xi}{1 + \left| \xi \right|^{2}}}+{\frac{1 - \left| \xi \right|^{2}}{1 + \left| \xi \right|^{2}}}}\right\}}}\right)}
\|(1-\nu)AX-\nu XB \|_{2}^{2}\leqslant \underline{K_{t}}^{-1}\| A^{1-\nu}XB^{\nu}\|_{2}^{2}+R_{0}^{2}\|AX-XB\|_{2}^{2}& -\sum_{k=1}^{\infty}r_{k}\|A^{1-\frac{m_{k}}{2^{k}}}XB^{\frac{m_{k}}{2^{k}}}- A^{1-\frac{m_{k}+1}{2^{k}}}XB^{\frac{m_{k}+1}{2^{k}}}\|_{2}^{2}.
\mathrm{div}(H_{q^{0}}) = - \sum_{j}\frac{\partial}{\partial \xi_j}\Big\lbrack q_j(h'(\xi),\xi)\Big\rbrack= - \sum_{j,k}\frac{\partial^2 h}{\partial \xi_k\partial\xi_j}\frac{\partial q_j}{\partial \gamma_k}- \sum_j \frac{\partial q_j}{\partial \xi_j}.
||\omega_i -\omega_0|| =\sup_n \left| \int_{S_R}A_n(\gamma_v)\cdot \left( Id-T\right)(\mu_i)-A_n(\gamma_v)\cdot \left(Id-T\right)(\mu_0) \right| &=\sup_n \left| \int_{S_R}A_n(\gamma_v)\cdot \left(Id-T\right)(\mu_i-\mu_0)\right|.
\tau:=(\nabla^i\nabla_0T_i-\nabla_0T_i\nabla_0T_i)+g^{ij}\partial_j(\nabla_0\nabla_0T_i) \phi:=\frac{1}{2}[\nabla_i\gamma^i-\gamma_i\nabla_0T^i-F^2g^{ij}\partial_i(\psi_j/F)]
\#A_{m,0}\ge \dfrac{\alpha d_{j_{m}}}{\textrm{per}(x_{l})}\Bigl(\dfrac{\alpha d_{j_{m}+1}}{d_{j_{m}}}-1\Bigr)\ge \dfrac{\alpha d_{j_{m}}}{\textrm{per}(x_{l})}\,\cdot\,\dfrac{\alpha d_{j_{m}+1}}{2d_{j_{m}}}=\dfrac{\alpha^{2}d_{j_{m}+1}}{2\textrm{per}(x_{l})}\cdot
\cos \phi^{( 1 )}= \frac{a}{L^{( 1 )}}( \frac{l}{4 d}( a_{l}+ a_{r}) + \frac{w_{1}}{2}) , \hspace{1 c m}\sin \phi^{( 1 )}= \frac{a}{L^{( 1 )}}\frac{\sqrt{3}l}{4 d}( a_{l}- a_{r}) ,
H_{w}e_{(k)}H_{\omega_{(d)}}H_{y}=\sum\limits_{\substack{(\omega',\chi,\varpi')\in B_{k, n}^{*}\times \mathfrak{S}_{2k+1, n}\times B_{k, n}\ell(\omega')+\ell(\chi)+\ell(\varpi')\leq \ell(w)+\ell(\omega_{(d)})+\ell(y)}}r_{\omega'}s_{\varpi'}t^{\chi}H_{\omega'}e_{(k)}H_{\chi}H_{\varpi'}
2^{k(\beta^{2}+q+1)-2\beta^{2}k(k-1)}\int_{[0,1]^{k}}\prod_{j=1}^{k}y_{j}^{\frac{\beta^{2}+q}{2}}(1-y_{j})^{\frac{\beta^{2}+q}{2}}\prod_{1 \leq i < j \leq k}|y_{i}-y_{j}|^{-2\beta^{2}}\,dy_{1}\ldots dy_{k}
\psi(\alpha) =\begin{cases}|\alpha|^{\frac2m}, u^2+v^2 \le1, 1, u^2+v^2>1, \end{cases}\varkappa(\alpha) =\begin{cases}|\alpha|^{-\frac{m-1}m}\sqrt{1-|\alpha|^{\frac2m}}, u^2+v^2\le1, 0, u^2+v^2>1, \end{cases}
\vartheta_1(x+y; \tau) \vartheta_1(x-y; \tau) \vartheta_1(u+v; \tau) \vartheta_1(u-v;\tau) & - \vartheta_1(x+v; \tau) \vartheta_1(x-v; \tau) \vartheta_1(y+u; \tau) \vartheta_1(-y+u;\tau) & = \vartheta_1(y+v; \tau) \vartheta_1(y-v; \tau) \vartheta_1(x+u; \tau) \vartheta_1(x-u;\tau).
(\prod_{p}\frac{p\gamma_{c,p}(n)}{(p-1)^{c}})\int_{2\le x_{1}\le\cdot\cdot\cdot\le x_{c}:x_{1}+\cdot\cdot\cdot+x_{c}=n}\frac{dx_{1}\cdot\cdot\cdot dx_{c-1}}{lnx_{1}\cdot\cdot\cdot lnx_{c}},
w(\sqrt{-\Delta})^{\frac{1}{2}}g(\theta)=\frac{1}{(2\pi)^{\frac{n}{2}}}\sum_{k,m}(-1)^k\frac{P^{(k,m)}(\theta)}{i^k}\int_0^\infty g_0^{(k,m)}(r)r^{\frac{1}{2}}w(r)^{\frac{1}{2}}J_{k+\frac{n-2}{2}}(r)\,\mathrm{d}r,
\frac{H_0}{t}\gtrsim\int_t^T\frac{D(T)}{\Big(1+(T-s)D^{3/2}(T)\Big)^{2/3}}\,ds =D(T)^{-1/2}\int_0^{D^{3/2}(T)(T-t)}\;\frac{1}{(1+\sigma)^{2/3}}\,d\sigma \geqslant \frac{D(T)(T-t)}{\Big(1+(T-t)D^{3/2}(T)\Big)^{2/3}}\gtrsim\min\left\{D(T)(T-t),(T-t)^{1/3}\right\},
\left (M \epsilon_{I(0)}+{1\over \sqrt 2}(Q+iP)_{IJ}\gamma^0 \epsilon^J_{(0)}\right)' = e^{i \arg S_0}\left(M \epsilon_{I(0)}+{1\over \sqrt 2}(Q+iP)_{IJ}\gamma^0 \epsilon^J_{(0)}\right) =0 \ ,
i\Pi^{\mu\nu}_{(\beta,\mu)}(0,\overline p) = -\frac{4g^2}{\beta}\sum_{n}\int \frac{d^2k}{(2\pi)^2}\frac{{\mathit{M}}^{\mu\nu}. sin^2(\frac{\tilde p k}{2})}{( (k-p)^2 + m^2 + \omega_{n}^2)(k^2 + m^2 +\omega_{n}^2 )}~~~,
\frac{\partial a^\nu_{3/2, -}(b, s)}{\partial s}= 2b s -\frac{4\nu^2s}{\sqrt{9\nu^2 +4\nu^2s^2 +9b^2/4}}\geqslant 2s\big(b -2\nu^2/\sqrt{9\nu^2 +9b^2/4}\big) \geqslant 0.
h\, Op_h^w(\Gamma) Op_h^w(\Sigma(\xi))\left[ \sum_{|I|=3}m_I^{\Sigma}(d\varphi_I(x))\theta_h(x) v^{\Sigma}_I + \sum_{|I|=3}\widetilde{m}_I^{\Sigma}(d\varphi_I(x))\theta_h(x) v^{\Sigma}_I \right] &+ hOp_h^w(\Gamma)Op_h^w(\Sigma(\xi))R_1(v) \,,
\sum_{i,j=1}^N b_{ij}\frac{\mu_i}{T}\left(\frac{\mu_j}{T}- \frac{\mu_{0,j}}{T_0}\right) \geq \frac{1}{2}\sum_{i,j=1}^N b_{ij}\frac{\mu_i}{T}\frac{\mu_j}{T}-\frac{1}{2}\sum_{i,j=1}^N b_{ij}\frac{\mu_{0,i}}{T_0}\frac{\mu_{0,j}}{T_0},
\sum_{n=0}^{\infty}\epsilon(n) =\int\limits_{0}^{\infty}{dn}\,\epsilon(n) +{1\over 2}\,\epsilon(0) -{1\over 12}\,\epsilon'(0) +{1\over 720}\,\epsilon'''(0) -{1\over 30240}\,\epsilon^{(v)}(0) + \cdots,
I_3 = \int^t_0\int^{t}_s e^{(t-\tau)A}\int^0_{-r}\beta(\tau) A_2 G(\tau-s+\theta)f(s)d\theta d\tau ds &= \int^t_0 e^{(t-\tau)A}\Big(\int^{0}_{-r}\beta(\theta)A_2\int^\tau_0 G(s+\theta-\tau)f(\tau)d\tau d\theta \Big)ds.
T_wc_1=T_s(T_vc_1) =q^{l(v)-1}(q^2c_1+qc_s+qc_t+c_{st})+\sum_{x\in\mathcal{E}\setminus\{1\}}q^{l(v)-l(x)-1}(qc_{tx}+c_{stx}) &=q^{l(w)-1}(qc_1+c_s)+q^{l(w)-2}(qc_t+c_{st})+\sum_{y\in\mathcal{F}\setminus\{1,t\}}q^{l(v)-l(y)}(qc_{y}+c_{sy}) &=\sum_{y\in\mathcal{F}}q^{l(w)-l(y)-1}(qc_{y}+c_{sy})
\sum_{l_1}\frac{\bar{g}_{1l_1}}{2I+1}C_{\sigma_1\sigma'_1}C_{\sigma_2\sigma'_2}\delta_{km_I}\left\{\begin{array}{lll}j_1 j_2 I j_1 j_2 l_1\end{array}\right\}\left(\begin{array}{lll}j_1 j_2 I \sigma'_1 \sigma'_2 m\end{array}\right).
S_q(x) =\frac{b+q[n]_q}{q(1-q)x^2}+\frac{a(a-[n]_q)}{x^2}+\frac{ab(1+q)-qb[n-1]_q}{qx}&+\frac{a(1-x)}{(1-q)x^3}-\frac{b}{q(1-q)x}+\frac{b^2}{q}.
H(z,w, (\xi_{\alpha})_{1 \leq |\alpha| \leq k+1})=\eta_0(z,w)h_0((\xi_{\alpha})_{1 \leq |\alpha| \leq k+1})+ \cdots + \eta_{s}(z,w)h_{s}((\xi_{\alpha})_{1 \leq |\alpha| \leq k+1})=0
H^{(2)}= - \partial_{\theta}^2 - \cot\theta \partial_{\theta}+ \frac{k^2}{\sin^2 \theta}\biggl(- \partial_{\phi}^2 + \frac{\alpha^2 - 1/4}{\cos^2 \phi}+ \frac{\beta^2 - 1/4}{\sin^2 \phi}- 2 \partial_{\phi}^2 \log P^{(-\alpha-1,\beta-1)}_{m_1}(-\cos2\phi)\biggr),
J \cap K = ((A \setminus \{\bar{a}_n\}) \cap (B \setminus \{\bar{a}_n\}))\cup &\qquad\qquad\cup ((A \setminus \{\bar{a}_n\}) \cap [\bar{b}_n \mapsto \bar{a}_n](H)) &\qquad\qquad\cup ((B \setminus \{\bar{a}_n\}) \cap [\bar{b}_n \mapsto \bar{a}_n](G)) &\qquad\qquad\cup ([\bar{b}_n \mapsto \bar{a}_n](G) \cap [\bar{b}_n \mapsto \bar{a}_n](H))
A_{\alpha , \, \gamma}^{l e p t o n}= \textstyle{\frac{1}{2}}\sqrt{( \eta \,{R_{1}^{( 1 )}})^{2}\, ( \eta \,{R_{2}^{( 1 )}})^{2}+ ( \eta \,{R_{1}^{( 1 )}})^{2}\,{\vec{\cal A}_{\alpha , \, \gamma}}^{2}+ ( \eta{R_{2}^{( 1 )}})^{2}\,{\vec{\cal B}_{\alpha , \, \gamma}}^{2}+ 4{A_{\alpha , \, \gamma}^{q u a r k}}^{2}},
\begin{gathered}R_{V , \epsilon}( z ) = R_\epsilon ( z ) ( I + V R_\epsilon ( z ) \rho )^{-1}( I - V R_\epsilon ( z ) ( 1 - \rho ) ) , \delta < \arg z < 3 \pi/2 - \delta , \ \ | z| \gg 1 , \end{gathered}
\beta_{1}\beta_{2}K^{T}( s_{1}, s_{1}) \equiv \! \int d p_{0}^{\prime}d p_{0}\delta ( p_{0}^{\prime}\! - \! s_{1}) \beta_{1}\beta_{2}K^{T}( p_{0}^{\prime}, p_{0}) \delta ( p_{0}\! - \! s_{1}) .
\|T_m f(\cdot, x)\|_{L^2_\rho L^2_x}^2 =\sum_{k=0}^\infty \big\|m(\tau, k) P_k(\mathcal F_\rho f)(\cdot, x)\big\|_{L^2_\tau L^2_x}^2 \leq \|m(\tau, k)\|_{L^\infty}\sum_{k=0}^\infty \big\|P_k(\mathcal F_\rho f)(\cdot, x)\big\|_{L^2_\tau L^2_x}^2 \lesssim \|f\|_{L^2(\mathbb R^{d+1})}^2.
\Gamma=(\Gamma_{kl})with\Gamma_{kl}=\sum_{j=1}^{J}(\lambda_{j}\wedge\mu_{j})[p_{jk}(\delta_{kl}-p_{jl})+c_{j}^{2}(p_{jk}-\delta_{jk})(p_{jl}-\delta_{jl})]+\alpha_{k}c_{0,k}^{2}\delta_{kl}
\|\tilde{S}\|^2 = \|(A+H)^{-1/2}(A+H)^{1/2}\tilde{S}\|^2 \leq \|(A+H)^{-1/2}\|^2 \|(A+H)^{1/2}\tilde{S}\|^2 = \|(A+H)^{-1}\| \| \tilde{S}^T(A+H)\tilde{S}\| =2\|(A+H)^{-1}\| d_{1}(A+H) \leq 2\|(A+H)^{-1}\| \|A+H\| = 2\kappa(A+H),
\sum_{k=1}^{n}\|\phi_{n,k}- \phi_{k}\|\leq \|v_{n+1}\| \sum_{k=1}^{n}\|v_{n+1}^{-1}\phi_{n,k}- v_{\infty}^{-1}\phi_{k}\|+ \|v_{n+1}\| \|v_{n+1}^{-1}-v_{\infty}^{-1}\| \sum_{k=1}^{n}\| \phi_{k}\|.
\begin{cases}-\partial_{tt}v_n + \Delta v_n = |\nabla v_n|^2 + 2 \nabla u_{n-1}\cdot \nabla v_n v_n|_{t=0}= Q_{N}f_0^\omega~,~ ~ \partial_t v_n |_{t=0}= Q_{N}f_1^\omega ~.\end{cases}
\mathcal{A}_3:=\{(j_1, j_2, j_3)\in\mathbb{Z}^3\setminus\{\textbf{0}\}: j_1+j_2+j_3=0\,\,\mbox{and at least}\,\,2\,\,\mbox{indices among}\,\,j_1, j_2, j_3\,\,\mbox{belong to}\,\,S \}.
\lim\limits_{\epsilon\rightarrow 0}I(\mathbf{S}_1+\mathbf{U}_{1,\epsilon},S_2+U_{2,\epsilon};\mathbf{S}_1(\gamma),S_2(\gamma)) &=I(\mathbf{S}_1,S_2;\mathbf{S}_1(\gamma),S_2(\gamma)), &I(\mathbf{S}_1+\mathbf{U}_{1,\epsilon},S_2+U_{2,\epsilon}|\mathbf{S}_1(\gamma),S_2(\gamma))=0,
\tilde{G}=\rho(G') =\left\{{\begin{pmatrix}a 0 0 b_1 A \mathrm{O}_n b_2 \mathrm{O}_n a\,{}^t\! A^{-1}\end{pmatrix}}\Bigg\vert \,a\in \mathbf{R}^*,\, A\in \mathrm{GL}(n,{\mathbf{R}}), \,b_1, b_2\in \mathbf{R}^n\right\}.
\langle\mathcal{Q}_\lambda u,v\rangle_{L^2([a,b],E)}=\int^b_a{\langle Ju'(t),v(t)\rangle dt}=\langle Ju(b),v(b)\rangle-\langle Ju(a),v(a)\rangle+\int^b_a\langle u(t),Jv'(t)\rangle dt.
{\cal W}(t) \; = \; 4 \left({\cal P}\int dt'\frac{\rho(t')}{t-t'}\right)^2-8{\cal P}\int dt'dt''\frac{\rho(t')}{t-t'}\frac{\rho(t'')}{t'-t''}+1-4{\cal P}\int dt'dt''\frac{\rho(t')\rho(t'')}{t'(t'-t'')}\; .
S_2 = \sum_{d_2, \ldots, d_k, d_2', \ldots, d_k'}\lambda_{1, d_2, \ldots, d_k}\lambda_{1, d_2', \ldots, d_k'}\left(\frac{q^n}{\varphi(W) \prod_{j = 2}^k \varphi([d_j,d_j'])}+ O\left(q^{n/2}\right)\right)
F_t(x,t) = \sum_{j=1}^n \prod_{i\neq j}(t-x_i) - (n-1)x_{n+1}t^{n-2}&= x_{n+1}t^{n-1}\sum_{j=1}^n \frac{1}{t-x_j}- (n-1)x_{n+1}t^{n-2}&\geq n x_{n+1}t^{n-2}- (n-1)x_{n+1}t^{n-2}= x_{n+1}t^{n-2}>0.
\left| \sum_{\alpha \beta \gamma \delta}\langle \Phi_{\alpha}\Phi_{\beta}\Phi_{\gamma}\Phi_{\delta}\rangle_{c}\left. \right/ \sum_{\alpha \beta \gamma \delta}\langle \Phi_{\alpha}\Phi_{\beta}\Phi_{\gamma}\Phi_{\delta}\rangle \right| = \left| \langle \Phi^{4}( x ) \rangle_{c}\left. \right/ \langle \Phi^{4}( x ) \rangle \right| \; ( n = 4 ) \; ,
( \Psi_{W W}k ) ( x , y ) \ = \a ( x , y ) \, \left( \begin{matrix}{{1}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{\frac{x y ( q^{2}- 1 )}{q^{2}x^{2}- y^{2}}}}&{{\frac{q ( x^{2}- y^{2})}{q^{2}x^{2}- y^{2}}}}&{{0}}\\{{0}}&{{\frac{q ( x^{2}- y^{2})}{q^{2}x^{2}- y^{2}}}}&{{\frac{x y ( q^{2}- 1 )}{q^{2}x^{2}- y^{2}}}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{1}}\\ \end{matrix}\right) \, k ( y , x ) \ ,
\begin{cases}h''(z) +c(|p|)h'(z)- g(h(z))=0 z\in \left(0,\frac{1}{|p|}\right) h(0)=0,\ \h\left(\frac{1}{|p|}\right)=1 h'\left( \frac{1}{|p|}\right)=h'(0).\end{cases}
\begin{aligned}H_j(x) =(D_j(x))\sp{-1}\cdot A(x)\cdot B_j(x) &=s_{ij}(x)(D_i(x))\sp{-1}\cdot A(x)\cdot s_{ij}(x)B_i(x) &=(s_{ij}(x))\sp2(D_i(x))\sp{-1}\cdot A(x)\cdot B_i(x) &=H_i(x).\end{aligned}
\gamma_{2i}(G_k)/\gamma_{2i+1}(G_k) \cong C_2 \gamma_{2i+1}(G_k)/\gamma_{2i+2}(G_k) \cong\begin{cases}C_2 \times C_2 i\ne 2^{k-1}C_2 i=2^{k-1}.\end{cases}
f_{\boldsymbol{X}^{\ast}}(\boldsymbol{x})=\frac{c_{n}^{\ast}}{\sqrt{|\boldsymbol{\Sigma}|}}\overline{G}_{n}\left\{\frac{1}{2}(\boldsymbol{x}-\boldsymbol{\mu})^{T}\mathbf{\Sigma}^{-1}(\boldsymbol{x}-\boldsymbol{\mu})\right\},~\boldsymbol{x}\in\mathbb{R}^{n}
f(x_1,\ldots,x_k) &=x_1+x_2+x_1x_2+x_2x_3+x_3x_4+\cdots +x_{k-3}x_{k-2}+x_{k-2}x_{k-1}+x_{k-1}+x_k &=x_1+x_2+x_1x_2+x_3(x_2+x_4)+x_5(x_4+x_6)+\cdots+x_{k-2}(x_{k-3}+x_{k-1})+x_{k-1}+x_k &=x_1+x_1x_2+x_3y_4+\cdots+x_{k-2}y_{k-1}+y_2+y_4+\cdots+y_{k-1}+x_k,
\mu=\frac{\omega_{p}^{2}r_{B}}{\omega_{H}c}=(\frac{2e^{2}r_{B}}{L_{B}\hbar c})\frac{\nu}{\sqrt{1+\frac{\omega_{p}^{2}}{\omega_{c}^{2}}}}=2\alpha(\frac{r_{B}}{L_{B}})(\frac{1}{\sqrt{1+\frac{\omega_{p}^{2}}{\omega_{c}^{2}}}})\nu
\begin{array}{l c c c}{\mathrm{f l o o r}}&{\mathrm{S U ( 1 / 2 )}}&{\mathrm{S U ( 2 )}}&{\mathrm{f i e l d}}\\{}\\{\mid \mathrm{g r o u n d}>}&{( 0 4 )}&{( 4 )}&{e_{\mu}^{a}}\\{\mid \mathrm{1 s t}>}&{( - 1 3 )}&{( 3 )}&{\Psi_{\mu}}\\ \end{array}
\sigma_{k}(n)=\zeta(k+1)n^{k}[1+\frac{(-1)^{n}}{2^{k+1}}+\frac{2cos\frac{2\pi n}{3}}{3^{k+1}}+\frac{2cos\frac{\pi n}{2}}{4^{k+1}}+\cdot\cdot\cdot]
Q_{1}= \left( \begin{array}{c c}{{0}}&{{\sqrt{H + Z}}}\\{{\sqrt{H + Z}}}&{{0}}\\ \end{array}\right) , \hspace{2 c m}Q_{2}= \left( \begin{array}{c c}{{0}}&{{i \sqrt{H - Z}}}\\{{- i \sqrt{H - Z}}}&{{0}}\\ \end{array}\right)
\lambda_{qrr}[a_{(1\,q)}, a_{(1\,r)}]_c= -3 \lambda_{qrr}\xi^2\chi_{(1\,r)}(g_{q})\lambda_{qqr}a_{(1\,p)}^2g_{qqr}&-3 \lambda_{qrr}\xi^2\lambda_{112}\chi_{(1\,r)}(g_{(2\,q)})a_{(3\,r)}a_{(2\,q)}&+3 \lambda_{qrr}\lambda_{112}\chi_{1}(g_{(3\,p)})\chi_{q}(g_{(2\,r)})\chi_{(1\,r)}(g_{q})a_{(3\,q)}a_{(2\,r)}.
\Omega(\overline{d}) :=\left \{(v,0)^T:v\in \ker T(\overline{d})\right\}, \widehat{\Omega}(\overline{c}) :=\left \{(T_r^{-1}(\overline{d})T(\overline{\widetilde{a}^{-1}})u,u)^T:u\in \ker T(\overline{c})\right \}.
\int \limits_{|a_n|}^{|b_n|}\int \limits_{|a_{n-1}|}^{|b_{n-1}|}\cdots \int \limits_{|a_1|}^{|b_1|}\frac{1}{\sqrt[4s+3]{\sum \limits_{j=1}^{n}x^{4s+3}_j}}dx_1dx_2\cdots dx_n \geq \frac{2\pi \times |\log (\langle a,b \rangle)|\bigg|\prod_{j=1}^{n}|b_j|-|a_j|\bigg|}{||\vec{a}||^{4s+4}+||\vec{b}||^{4s+4}}.
\delta V ( R ) = - \frac{2 ( D - 2 ) \beta^{2}}{M^{4}}\int_{0}^{\infty}d y \, y^{3}\bar{\Omega}\frac{\left( e^{- R y}- e^{- R \bar{\Omega}}\right)^{2}}{\left( 1 - e^{- 2 R y}\right) \left( 1 - e^{- 2 R \bar{\Omega}}\right)}< 0 .
S_{\rm NA}= \int d^{4}x \left[ -{1\over4}{F_{\mu\nu}}^{a}F^{\mu\nu a}+{1\over12}\hat{H}_{\mu\nu\rho}\raisebox{0.98ex}{\mbox{\scriptsize{$a$}}}\hat{H}^{\mu\nu\rho a}-{1\over4}m^{2}\hat{B}_{\mu\nu}\raisebox{0.98ex}{\mbox{\scriptsize{$a$}}}\hat{B}^{\mu\nu a}\right] \,,
H_{nm}^{\{{\bf R}\}}(0,0)\approx\frac{(-1)^{\frac{n+m}2}\epsilon^{-\frac{|n-m|+1}2}}{\sqrt{2\pi}}\Gamma\left(\frac{n+m+1}2\right)R_{11}^{n/2}R_{22}^{m/2}\left(r+\epsilon\right)^{\frac{n+m+1}2}.
\frac{da_1}{dT}=\frac{4(a_1-x_0)}{(a_1-c)(a_1-b_1)(a_1-a_2)(a_1-b_2)}\,,\qquad\frac{db_2}{dT}=\frac{4(b_2-x_0)}{(b_2-c)(b_2-a_1)(b_2-b_1)(b_2-a_2)}\,,
\mathbb P^{(t)}(Z_t=p, T_1\in dt_1, ..., T_n\in dt_n\mid Z_{t-t_n}= y)/ dt_n= y \sum_{k\geq 2}k\mu (k)\times &\mathbb E \Big( \frac{Z_{t_n}^{(1)}(1)Z_{t_n}^{(2)}(k-1)..(Z_{t_n}^{(2)}(k-1)-n+1)}{p(p-1)...(p-n)}, Z_{t_n}^{(0)}(y-1)+Z_{t_n}^{(1)}(1)+Z_{t_n}^{(2)}(k-1)=p, & T_i\in dt_i, i\leq n-1 \Big),
-\Pi^\sharp(\iota_{\Pi^\sharp\beta}d_A\alpha) = -(-1)^{n-k}ab\Pi^\sharp (\iota_{X_k}X^\ast_1\wedge \cdots \wedge X^\ast_n) & = -(-1)^{n-k}(-1)^{k-1}ab\Pi^\sharp (X^\ast_1\wedge \cdots \wedge \widehat{X^\ast_k}\wedge \cdots \wedge X^\ast_n) & = -(-1)^{n-k}(-1)^{k-1}(-1)^{n-k}abX_k = (-1)^kab X_k.
\delta S_{g}= \frac{i}{4 8}[ \int ( H -{\bar{H}}) F^{{\mathbf i}{\mathbf j}}F_{{\mathbf i}{\mathbf j}}+ \frac{3}{1 0}( H -{\bar{H}})_{{\mathbf i}{\mathbf j}{\mathbf k}{\mathbf l}}F^{{\mathbf i}{\mathbf j}}F^{{\mathbf k}{\mathbf l}}]
\int_{\Omega_{T}}\varphi_{n}d\mu +\int_{\Omega}\varphi_{n}(0)d\sigma \geq \mu^{+}(K)+(\sigma^{+}\otimes \delta_{\{t=0\}})(K)-(|\mu |+|\sigma|\otimes \delta_{\{t=0\}})(O\backslash K) & \geq \mu^{+}(K)+(\sigma^{+}\otimes \delta_{\{t=0\}})(K)-\varepsilon .
S^{env}_{s,t}= -\int_{s}^t \beta(r,q_r) \partial_{p_k}H (r,x_r) \circ d(p_r)_k &+\int_{s}^t \beta(r,x_r) \partial_{p_k}H (r,x_r)(- \nabla_q H+\tilde F)_k(r,x_r) -\nabla_p\cdot \tilde F(r,x_r)dr.
{\, \lower 0 . 9 p t \mathrm{\hrule ~ \mathrm{\vrule ~ h e i g h t ~ 0 . 2 ~ c m \hspace{~ 0 . 2 ~ c m}~ \vrule ~ h e i g h t ~ 0 . 2 c m}\hrule}\,}F_{\alpha \beta}- 2 ( \partial_{\beta}F_{\alpha \mu , \mu}- \partial_{\alpha}F_{\beta \mu , \mu}) = 0 \quad ,
\mathfrak{R}_{w}(q,\beta)= (1, 6, 21, 36, 51, 48, 26)_{\beta}+q \beta (6, 36, 126, 216, 306, 288, 156)_{\beta}\hphantom{\mathfrak{R}_{w}(q,\beta)=}{}+q^2 \beta^3 (20, 125, 242, 403, 460, 289)_{\beta}+q^3 \beta^5 (6, 46, 114, 204, 170)_{\beta}.
\int_{3R^N(t)\leq |\mu-x|}\frac{\rho^N(x,t)}{|\mu-x|^2}\,dx \leq \frac94\int_{2R^N(t)\leq |\mu-x|}\frac{f_0^N(x,v)}{|\mu-x|^2}\,dxdv\leq & C\int_{2R^N(t)\leq |\mu-x|}\frac{g(|x|)}{|\mu-x|^2}\,dx.
-\operatorname{div}^{\Gamma_0}\frac{\delta l^\mathcal{L}}{\delta\Gamma}=-\operatorname{div}^{\Gamma}\frac{\delta l^\mathcal{L}}{\delta\Gamma}+\operatorname{Tr}\left(\operatorname{ad}^*_{\sigma^2}\frac{\delta l^\mathcal{L}}{\delta\Gamma}\right).
H_{3}= 2 \mu \delta_{J_{1}, J_{2}+ J_{3}}{\frac{\sqrt{J_{1}J_{2}J_{3}}}{N}}J_{1}\Big ({\frac{J_{3}}{J_{2}}}\Big )^{\frac{1}{2}}{\frac{\sin^{2}( \pi n y )}{\pi^{2}n^{2}}}\Pi_{2}^{\prime J_{1}}( q_{n}) \bar{\Pi}_{1 , 0}^{\prime J_{2}}O_{0 , 1}^{\prime J_{3}}.
\mathcal{L}_{\star}^f \left( \tilde{\varrho}\right)= \left[ - \left( \kappa + \mathrm{i}\omega \right) a^\dagger a + \left( \alpha_0 a^\dagger - \overline{\alpha_0}\, a \right), \tilde{\varrho}\right] +2 \kappa \, a\, \tilde{\varrho}\, a^\dagger
Z_j = \big(A_{1},A_{j},\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots,A_{j}\,,\,(0),\ldots,(0),(1),(0),\ldots,(0),(0)\big)\j\in[m], Z_{m+1}= \big(A_{2},A_{1},\ldots,A_{m},B_1,B_1,B_2,\ldots,B_{s-m-1},(0),\ldots,(0),(0),(0),\ldots,(0),(1)\big),
\sum_{N^-<p<N^+}\frac{H(D_N(p))}{p}=\frac{1}{N}\sum_{N^-<p<N^+}H(D_N(p))\left(1+O\left(\frac{1}{\sqrt{N}}\right)\right) =\frac{1}{N}\sum_{N^-<p<N^+}H(D_N(p))+\frac{1}{N^{\frac{3}{2}}}\sum_{N^-<p<N^+}|H(D_N(p))|
\overline X^{1,N}_{t+1}= \Psi\left(t, \overline X^{1,N}_t, \mu_t,\widetilde \Upsilon^{1,N}(t,\widetilde X^{1,N}, \widetilde \mu^{1,N}),\xi^1_{t+1}\right),\mathbb{P}_N.
q_1(z) =\frac{q^{n+\gamma}}{(1-q^\gamma) (1-zqb)}\left[(q^{-n-\gamma}-1)(1-zqb)+\frac{b(q^n-1)(aq^n-1)}{(abq^{2n}-1)}\right] \q_0(z) =\frac{q^{\gamma}(q^{n}-1)(bq^n-1)}{(1-zqb)(1-q^\gamma)(abq^{2n}-1)}.
m_H \partial_{m_H}\rightarrow m_H \tilde{\partial}_{m_H}\; \; , \; \; e \partial_e \rightarrow e \tilde{\partial}_e \; \; , \; \; f_{\xi}\partial_{\xi}+ (\chi \partial_{\xi}+ \chi_A \partial_{\xi_A}) f_{\xi}\partial_{\chi}\rightarrow \tilde{\partial}_{\xi}\; \; \; ,
\sum_{i \neq j}{N \choose 2}^{-1}\left( X^k_t(i) (1 - X^k_t(j)) + X^k_t(j)(1 - X^k_t(i)) \right) &= \sum_{i \neq j}\frac{X^k_t(i) (1 - X^k_t(j))}{N (N - 1)}&= \frac{N}{N - 1}M^k_t (1 - M^k_t) - \sum_i \frac{X^k_t(i)^2}{N(N - 1)}&\geq M^k_t (1 - M^k_t) - \frac{1}{N - 1}.
f(d\cdot n,m)=\tilde{a}_{v,0}^{(m,d)}-\sum\limits_{l\mid m}\sum\limits_{k=0}^{\tilde{d}-2}\dfrac{l^k}{(\tilde{d}-2-k)!}\cdot\tilde{p}_{k}^{\,(\tilde{d},l)}\cdot \tilde{a}_{v-\tilde{d},k}^{(m,d)},
\bar{\mathcal{F}}_2 (x_1 , x_2) = \kappa(x_1 x_2) \exp{\left\{\frac{[(1+q^4)(y_1 + y_2) + 4q^2 \sqrt{y_1 y_2}\Delta_{L-1}]}{(q^2 -1)^2 (y_1 + y_2)}\log{\zeta(x_1 , x_2)}\right\}}\; ,
\|Op_h((1-\chi)(h^{\beta}\xi))v \|_{H^{s'}_h}^2 = \int (1+ |h\xi|^2)^{s'}|(1-\chi)(h^{\beta}h\xi)|^2 |\hat{v}(\xi)|^2 d\xi & = \int (1+ |h\xi|^2)^s (1+ |h\xi|^2)^{s'-s}|(1-\chi)(h^{\beta}h\xi)|^2 |\hat{v}(\xi)|^2 d\xi & \le C h^{2\beta(s-s')}\|v\|^2_{H^s_h}\, ,
\sum_{j=0}^m v^{(m-j)(m-j+1)}\frac{[a+m-j]!}{[m-j]!}\frac{\prod_{u=1}^j [a+m+1-u] \cdot (1 -v^2)^j}{\prod_{k=1}^{\lfloor \frac{j}2\rfloor}(1-v^{4k})}= \frac{[a+m]!}{[m]!}.
L=\left(\begin{array}{ccc}\sqrt{\frac{\theta q}{2R}}(P-q) 0 \frac{\theta}{2}\sqrt{P+(1-2a)q}0 \sqrt\frac{q}{a}(P-q) \sqrt{aq}\theta -\sqrt{\frac{\theta q}{2R}}(P-q) 0 \frac{\theta}{2}\sqrt{P+(1-2a)q}\end{array}\right),
\begin{array}{c}\displaystyle\vert p-z\vert-\vert p-x^*\vert=\frac{\vert\xi\vert^2-2(p-x^*)\cdot\xi}{\vert p-z\vert+\vert p-x^*\vert}, \displaystyle\vert z-x\vert-\vert x^*-x\vert=\frac{\vert\xi\vert^2+2(x^*-x)\cdot\xi}{\vert z-x\vert+\vert x^*-x\vert}.\end{array}
\int_{x_{1}^{0}}^{x_{1}^{1}}\cdot\cdot\cdot\int_{x_{r}^{0}}^{x_{r}^{1}}\Gamma^{(\lambda)}(x)_{nm}^{*}\Gamma^{(\mu)}(x)_{n^{\prime}m^{\prime}}\omega(x)dx_{1}\cdot\cdot\cdot dx_{r}=\delta_{\lambda\mu}\delta_{nn^{\prime}}\delta_{mm^{\prime}}\frac{|G|}{l_{\lambda}},
Hf(x) = \int h_H(x,t)\,f(x-t)\,dt = \iint \eta_H(t,\nu)\,e^{2\pi i\nu (x-t)}\,f(x-t)\,d\nu\,dt = \int \sigma_H(x,\xi)\, e^{2\pi i x \xi}\widehat f(\xi)\,d\xi,
\chi_{a,\Delta}(x) = \begin{cases}1, \mbox{if}x\in \left[ \frac aq + \Delta, \frac{a+1}q -\Delta\right], 0, \mbox{if}x\in [0,1] \setminus \left[ \frac aq - \Delta, \frac{a+1}q +\Delta\right]. \end{cases}
\langle Jr(\xi a)^\ast r(\eta) J x,y\rangle = \langle r(\xi a) y^\ast , r(\eta) x^\ast\rangle = \langle y^\ast\xi a, x^\ast\eta \rangle = \langle J(x^\ast\eta), J(y^\ast\xi a)\rangle &= \langle \eta x, a^\ast\xi y\rangle = \langle \ell(\xi)^\ast \ell(a\eta) x,y\rangle \; ,
\sum_{n_1,\cdots,n_7\geq1}\frac{1}{(n_1+\cdots+n_7)^5(n_2+\cdots+n_7)^2n_3(n_4)^2(n_5+n_6+n_7)^2n_6n_7}=~ 8\zeta(5,2,1,2,2,1,1) + 16\zeta(5,2,1,3,1,1,1) + 2\zeta(5,2,1,2,1,1,2) + 4\zeta(5,2,1,2,1,2,1) + 48\zeta(5,2,2,2,1,1,1) +~ 28\zeta(5,2,2,1,2,1,1) + 8\zeta(5,2,2,1,1,1,2) + 16\zeta(5,2,2,1,1,2,1) + 40\zeta(5,2,3,1,1,1,1) .