harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\begin{aligned}\kappa \int_0^\infty \partial_y^2 D^\alpha h\cdot \langle y \rangle^{2(l+k)}D^\alpha h\mathrm{d}y \leq -\frac{\kappa}{2}\Vert \langle y\rangle^{l+k}\partial_yD^\alpha h\Vert_{L^2_y}^2+\delta_1\Vert \partial_y (u,h)\Vert_{H^m_0}^2 &+C\delta_1^{-1}E_{u,h}^2(1+E_{u,h}^2).\end{aligned}
G(\vec{X}) : = \dfrac{\alpha}{2}\|\vec{Y}- \vec{X}\|_F^2 + \lambda_0 \sum_{i=1}^{m}s\bigl(\sigma_i(\vec{X});a_0\bigr) + \dfrac{1-\alpha}{2}\|\vec{Y}- \vec{X}\|_F^2 + \lambda_1 \sum_{i=1}^{m}\sum_{j=1}^{n}s(\vec{X}_{ij};a_1),
D_y\tilde{H}(x,y)=\frac{2a_0}{a_0+1}\sum_{k=1}^\infty(-\alpha^{2k-1}D_y\log|\Phi(\Phi(x)+{\bf 2k-1})-\overline{y}| +\alpha^{2k}\log|\Phi(\Phi(x)-{\bf 2k})-y|)-CD_y\log|y|+\frac{a_0-1}{a_0+1}D_y\log|x-y|,
\varphi(y)=\displaystyle\sum_{s ; z_{i+1}\in\partial I(z_i^\prime)}G_{I(y)}(k,z_1) G_{I(z_1^\prime)}(z_1^\prime,z_2)\cdots G_{I(z_s^\prime)}(z_s^\prime,z_{s+1})\varphi(z_{s+1}^\prime),
\begin{array}{ll}(1-\frac{\eta}{\beta})\frac{u_t}{\tilde{u}_y}= \frac{1}{\beta}w_t +\frac{u}{\beta}\frac{\tilde{u}_{yyt}}{(\tilde{u}_y)^2}+ (1-\frac{2\eta}{\beta})u\frac{\tilde{u}_{yt}}{(\tilde{u}_y)^2}.\end{array}
( \partial^{\lambda \dot{\mu}}) = \left[ \begin{array}{c c}{{\partial^{1 \dot{1}}}}&{{\partial^{1 \dot{2}}}}\\{{\partial^{2 \dot{1}}}}&{{\partial^{2 \dot{2}}}}\\ \end{array}\right] = \left[ \begin{array}{c c}{{\partial_{0}+ \partial_{3}}}&{{\partial_{1}+ i \partial_{2}}}\\{{\partial_{1}- i \partial_{2}}}&{{\partial_{0}- \partial_{3}}}\\ \end{array}\right] ,
\gamma_{\operatorname*{N};j}^{\mathbb{B},-}u^{-}=\left\langle \mathbb{B}_{j}^{-}\nabla u^{-},\mathbf{n}_{j}\right\rangle \quad\gamma_{\operatorname*{N};j}^{\mathbb{B},+}u^{+}=\left\langle \mathbb{B}_{j}^{+}\nabla u^{+},-\mathbf{n}_{j}\right\rangle.
m^{p}\;{\mathrm{( i t e r a t e d ~ l i n e ~ i n t e g r a l s ~ i n ~ b o s o n i c ~ p o t e n t i a l s ~ a n d ~ f i e l d s )}}\; ( y - x )_{j}\gamma^{j}\: T^{\mathrm{\scriptsize{r e g}}\: ( n )}( s , l ) \; ,
[\hat{H}_A^{(r)}(m+\frac{r}{\lambda};d)-\delta_{m+\frac{r}{\lambda},0}kd_A]|0\rangle=\hat{E}_\alpha^{(r)}(m+\frac{r-d\cdot\alpha}{\lambda};d)|0\rangle =0{\textrm{\ \when\ \}}m+\frac{r}{\lambda}\geq0
D\phi(x,y,z)=\frac{1}{\phi(x,y,z)}\left[\bar{\theta}|x-y|^{2\bar{\theta}-2}\left( \begin{array}{c}x-y y-x 0 \end{array}\right)+\left( \begin{array}{c}x+y-2z x+y-2z -2x-2y+4z \end{array}\right)\right]
\left< \xi_{i}( \sigma ){\cal O}\left[ \vec{\xi}{}\right] \right>_{\vec{\xi}}= \int_{0}^{1}d \sigma^{\prime}G \left( \sigma - \sigma^{\prime}\right) \left< \frac{\delta{\cal O}\left[ \vec{\xi}{}\right]}{\delta \xi_{i}\left( \sigma^{\prime}\right)}\right>_{\vec{\xi}},
G^{+}\left( x , x^{\prime}\right) = \left| A \right|^{2}G_{E}^{+}\left( x , x^{\prime}\right) + \left| B \right|^{2}G_{E}^{+}\left( x^{\prime}, x \right) + A B^{\ast}G_{E}^{+}\left( x , x_{A}^{\prime}\right) + B A^{\ast}G_{E}^{+}\left( x_{A}, x^{\prime}\right) .
\rho_{t}( \bar{\psi}_{f}, \psi_{f}^{\prime}) = \int D \bar{\psi}_{i}^{\prime}D \psi_{i}J ( \bar{\psi}_{f}, \psi_{f}^{\prime}; t | \bar{\psi}_{i}^{\prime}, \psi_{i}; 0 ) \rho_{0}( \bar{\psi}_{i}^{\prime}, \psi_{i})
b_1\cdot( \delta_m \otimes b_2) \;=\; \delta_m \otimes \alpha_{-m}(b_1)b_2\;, S^n\cdot(\delta_m \otimes b) \;=\; \theta(n,m)\cdot \delta_{m+n}\otimes b \;=\; \delta_{m+n}\otimes \alpha_{-m-n}(\theta(n,m))b\;.
\sum_{k=0}^\infty\int_{A_k}|x|^2|\nabla\varphi|^2\dx^2=\sum_{k=0}^\infty 2^{-2k}\,\int_{A_k}|\nabla\varphi|^2\dx^2\ \cong\ \sum_{k=0}^\infty2^{-2k}\sum_{i,j}\int_{A_k}\nabla\varphi_i\cdot\nabla\varphi_j.
\frac{1}{\rho_{F}(\Omega)}=\frac{1}{\|d_{F}\|_{L^{\infty}(\Omega)}}=\min_{\varphi\in W_{0}^{1,\infty}(\Omega)\setminus\{0\}}\frac{\|F(\nabla \varphi)\|_{L^{\infty}(\Omega)}}{\| \varphi\|_{L^{\infty}(\Omega)}}.
{\cal N}\prod_{0<\tau<\beta}D\mu[M_k(\tau)] ={\cal N}\prod_{0<\tau<\beta}\prod_{\gamma=1}^Nd(M_k(\tau))_{\gamma\gamma}\prod_{\gamma > \nu =1}^N d \mbox{Re}(M_k(\tau))_{\gamma\nu}\;d \mbox{Im}(M_k(\tau))_{\gamma \nu}\; .
\lim_{t \to \infty}\frac{S_j(t)}{M_j(t)}= \lim_{N \to \infty}\frac{S_j(\tau_N)}{M_j(\tau_N)}= \frac{\int_0^{1}(1-p y)^{-1}(1-y) y^{j-1}dy}{\int_0^1 (1-py)^{-1}y^{j-1}dy}= 1-\frac{\int_0^{1}(1-p y)^{-1}y^{j}dy}{\int_0^1 (1-py)^{-1}y^{j-1}dy}=: \varphi_j(p).
P ( t ) \equiv \sum_{L}b_{L}t^{L}= \frac{\prod_{j = 1}^{\kappa}( 1 - t^{2 j})}{\prod_{j = 1}^{N}( 1 - t^{2 j}) \prod_{j = 1}^{\kappa - N}( 1 - t^{2 j})}.
\widehat{V}^{\dagger}(k_{1},k_{2})=e^{i\sqrt{2|\theta^{12}|}\; \left(\kappa\hat{a}+\overline{\kappa}\hat{a}^{\dagger}\right)}= e^{-|\theta^{12}|\kappa\overline{\kappa}}\,e^{i\sqrt{2|\theta^{12}|}\; \overline{\kappa}\hat{a}^{\dagger}}\,e^{i\sqrt{2|\theta^{12}|}\; \kappa \hat{a}}~,
q^{dk}\, \left(\max\limits_{r\in \mathbb F_q^*}\prod\limits_{j=1}^k\left(\sum\limits_{{\bf v}\in S_r}|\widehat{E_j}({\bf v})|^k \right)^{\frac{1}{k}}\right) \lesssim q^{k\alpha+d-1}\left( \prod\limits_{j=1}^k |E_j| \right)^{\frac{1}{\ell}}.
L_{\Omega}(x;z^{\prime},\bar{z}^{\prime})=\frac{\Psi(x;z^{\prime},\bar{z}^{\prime})}{(-r)^{n+m+1}(x;z^{\prime},\bar{z}^{\prime})}+\Phi(x;z^{\prime},\bar{z}^{\prime})\log (-r(x;z^{\prime},\bar{z}^{\prime})).
2 E - E_{c}= \frac{r_{+}l}{2 R}\left[ \left( 2 - \frac{l^{2}}{r_{+}^{2}}k \right) \left( k + \frac{r_{+}^{2}}{l^{2}}\right)^{m - 1}- ( 2 m + 1 ) k + \frac{l^{2}}{r_{+}^{2}}k^{2}\right] .
\begin{cases}\dot{\varrho}_t(x) = \frac{\kappa}{2}\partial_{xx}\varrho_t(x) - \lambda \partial_x \Delta_t(x), \dot{\Delta}_t(x) = \frac{\kappa}{2}\partial_{xx}\Delta_t(x) - \lambda \partial_x\varrho_t(x) -2\gamma \Delta_t(x).\end{cases}
< T_{i}^{k}[ \tilde{g}_{lm}] > = (\frac{\eta}{\tilde{g}})^{\frac{1}{2}}< T_{i}^{k(M)}[ \eta_{lm}] > -{\frac{1}{2880 \pi^2}}\left [{\frac{1}{6}}^{(1)}\tilde{H}_{i}^{k}-^{(3)}\tilde{H}_{i}^{k}\right ]
\sup_{C \in \mathcal{C}}\biggl|\mathbb{P}\biggl(\frac{1}{(n-2)^{1/2}}\sum_{i=3}^n Z_i \in C\biggr) - \mathbb{P}(Z \in C)\biggr| \leq \frac{C_2\mathbb{E}(\|Z_3\|^3)}{(n-2)^{1/2}}.
\bar\partial \tau_i = \frac{1}{2}\sum_{j,k, l=1}^n \left(\bar\partial_l T^{i}_{jk}\right) \phi_j \wedge \phi_k \wedge \bar\phi_l + \frac{1}{2}\sum_{j,k=1}^n T^i_{jk}\bar \partial \phi_j \wedge \phi_k - \frac{1}{2}\sum_{j,k=1}^n T^i_{jk}\phi_j \wedge \partial \phi_k =
\Psi(x,t)=(\frac{1}{x_{0}\sqrt{\pi}})^{1/2}\frac{e^{-x_{0}^{2}p_{0}^{2}/2\hbar^{2}}}{\sqrt{1+i\omega_{0}t}}exp(-\frac{(x-ix_{0}^{2}p_{0}/\hbar)^{2}}{2x_{0}^{2}(1+i\omega_{0}t)}).
\langle :X_T\otimes X_T:,\Phi \rangle:= \sum_{j=1}^m\Big( \langle X_T,\phi_j \rangle \langle X_T,\psi_j \rangle - \mathbb{E}(\langle X_T,\phi_j \rangle \langle X_T,\psi_j \rangle)\Big).
{\cal L}_{g f}={\frac{1}{2 \alpha}}( h_{\mu \rho ,}{}^{\mu}-{\frac{1}{2}}h_{, \rho}) ( h^{\nu \rho}{}_{, \nu}-{\frac{1}{2}}h_{,}{}^{\rho}) +{\frac{1}{2 \beta}}\sigma_{, \mu}\sigma_{,}{}^{\mu},
\frac{\partial}{\partial r}(\rho_{g}r^{2}uh_{g})-\frac{\partial}{\partial r}(\lambda_{g}r^{2}\frac{\partial T_{g}}{\partial r})-\frac{\partial}{\partial r}(\sum_{i=1}^{N}\rho_{g}Dh_{i}r^{2}\frac{\partial Y_{i}}{\partial r})=0
I= i R \int^{\rho}_{0}sds \int \frac{d^{4}p}{(2\pi)^{4}}\frac{1}{(p^{2}-s^{2})^{2}}\rightarrow - R \int^{\rho}_{0}sds \int \frac{d^{4}p}{(2\pi)^{4}}\frac{1}{(p^{2}+s^{2})^{2}}.
\rho\circ_{\phi}\Pi^{E}(x) =(_{\phi}\rho^{\alpha}(P_{\alpha}\circ_{\phi}\Pi^{E}(x)))_{\alpha\in\Gamma}=(_{\phi}\rho^{\alpha}(_{\phi}\Pi^{\alpha}(x_{\alpha})))_{\alpha\in\Gamma}&=(x_{\alpha})_{\alpha\in\Gamma}=x.
X_{\alpha}^{\varepsilon}\left( t\right) =M_{\alpha}^{\varepsilon}\left(t\right) +\int_{0}^{t}\left( D_{\alpha}^{\ast}\tilde{V}^{\prime \prime}\right) \left( \hat{\eta}_{s}^{\varepsilon}\left( 0,\varepsilon e_{\alpha}\right) \right) ds,
\frac{\partial A_{i j}}{\partial q^{k}}- \frac{\partial A_{k j}}{\partial q^{i}}= \frac 1 2 \frac{\partial}{\partial \dot{q}^{j}}\left( \frac{\partial B_{i}}{\partial \dot{q}^{k}}- \frac{\partial B_{k}}{\partial \dot{q}^{i}}\right) .
\int_{S^2}(k^{(3)}- h^{(3)}) dS^2 =-\frac{3}{4}\int_{S^2}W_0^2 d S^2-\frac{1}{60}\int_{S^2}|\alpha|^2 dS^2 + \frac{11}{45}\int_{S^2}|\beta|^2 dS^2.
L ~ = ~ i \bar{\psi}^{i}{\partial \! \! \! /}\psi^{i}+ \sigma \bar{\psi}^{i}\psi^{i}+ i \pi \bar{\psi}^{i}\gamma^{5}\psi^{i}- \frac{1}{2 g^{2}}( \sigma^{2}+ \pi^{2})
\theta_1(x) ={\sf c}_1|x|^2{\sf U}_\infty^{p-1}\theta_0 +{\sf h}_1(x) ={a}_1 |x|^\frac{2(p-q)}{1-q}\theta_0 +{\sf h}_1(x) ({a}_1\not=0),{\sf h}_1(x) = |x|^\frac{2(p-q)}{1-q}\theta_0 \sum_{i=1}^{N-1}b_i|x|^\frac{2(p-q)i}{1-q}.
L_0+\tilde{L}_0=\frac{1}{2}k^2+N+\tilde{N}+\frac{1}{2}\Pi_iG^{ij}\Pi_j+\frac{1}{2}L^i(G-BG^{-1}B)_{ij}L^j+L^iB_{ij}G^{jk}\Pi_k.
L= \frac{1}{2h^2}g^{\mu\nu}\partial_\mu \sigma \partial_\nu \sigma-\frac{N}{2\lambda_1 h^2}\sigma^2 +\frac{N}{2\xi_1 h^2}R \sigma^2 -\frac{N}{4! \lambda_2 h^4}\sigma^4 +\bar{\psi}(i\gamma^\mu (x) \nabla_\mu - \sigma) \psi ,
\int_{\{\Psi_1<-t\}\cap U}|\tilde F-F^{1+\delta}|^2e^{-\varphi-\Psi_1}=\int_{\{\Psi_1<-t\}\cap U}|\frac{\tilde F}{F^{1+\delta}}-1|^2e^{-\Psi_1}<+\infty.
\sigma_F:=\sup_{x\in M}\sup_{v,w \in T_xM\backslash\{0\}}\frac{g_v(w,w)}{F(w)^2}=\sup_{x\in M}\sup_{\alpha,\beta \in T^*_xM\backslash\{0\}}\frac{F^*(\beta)}{g^*_\alpha(\beta,\beta)}\in [1,\infty].
\sum_{n=1}^{\infty}\frac{1}{\left(\left(2n-1\right)^{2}+x^{2}\right)^{s}}=\frac{\sqrt{\pi}\,x^{1-2s}}{4\Gamma(s)}\Gamma\left(s-\frac{1}{2}\right)+\frac{2^{\frac{1}{2}-s}\,x^{\frac{1}{2}-s}\,\pi^{s}}{\Gamma(s)}\,\sum_{m=1}^{\infty}(-1)^{m}m^{s-\frac{1}{2}}K_{s-\frac{1}{2}}(\pi xm).
( \, w, \alpha \, ) \, \mapsto \, \displaystyle{\frac{1}{N}}\, \displaystyle{\sum_{i=1}^N}\, \ell\left( y_i - \sigma(w^Tx^i + \alpha_i) \, \right) - \gamma \, \displaystyle{\sum_{i=1}^m}\, h_i(w_i),
\Gamma_{\alpha}= \frac{\alpha}{2}\left( - \cos \theta \, V_{\sigma}+ 6 \sin \theta \, V_{s}- \frac{1}{2}\sin 2 \theta \, V_{\varphi}\, e^{\alpha \chi / 2}\right) - \frac{\alpha^{2}}{4}\, \dot{\sigma}^{2}\, \cos^{2}\theta \, - \alpha^{2}\, \dot{\sigma}^{2}\, \sin^{2}\theta \, .
F_2^\delta\circ U_2^\delta \left( \begin{array}{c}z x \end{array}\right) = \left( \begin{array}{c}P(x) z +\dots b_0^\delta(x)+\dots \end{array}\right) & = \left( \begin{array}{c}\left[ \prod_{n=0}^\infty a_1^\delta((b_0^\delta)^{n}(x))\right] z + \dots b_0^\delta (x) +\dots \end{array}\right)
F(0,x_n) = \inf_{u \in \mathcal{S}^N}\left[ \frac{1}{2}\int_0^T |u(s)|_H^2ds + h(Y^{0,u}_{x_n})\right] > \frac{1}{2}\int_0^T |u_n(s)|_H^2ds + h(Y^{0,u_n}_{x_n}) - \frac{1}{n}.
ds^2 = - 4 dx^+ dx^- -\mu^2 \left(\sum_{I=5}^8 x_I^2 \right){dx^+}^2 + \sum_{i=1}^8{dx_i}^2 + 4 \mu x_2 dx_1 dx^+ + 4 \mu x_4 dx_3 dx^+,
\Pi_\Phi \equiv \frac{\delta{\cal L}}{\delta \partial_0 \Phi}= \frac{1}{2}\partial_0 \Phi,{}~~~~\Pi_{\Psi_\pm}\equiv \frac{\delta{\cal L}}{\delta \partial_0 \Psi_\pm}= \frac{1}{2}\bar{\Psi}_\pm,
BR(a)=-f^{-1}(a)=\sum_{k=0}^{\infty}(\begin{matrix}5k\\k\end{matrix})\frac{(-1)^{k+1}a^{4k+1}}{4k+1}=-a+a^{5}-5a^{9}+35a^{13}-285a^{17}+\cdot\cdot\cdot,
st(x,st(\tau,K)) = \{\alpha\in st(\tau,K)|\alpha\cup \{x\}\in st(\tau,K)\}&= \{\alpha\in st(\tau,K)|\alpha\cup \{x\}\cup \tau \in K\}&= \{\alpha\in st(\tau,K)|\alpha\cup \sigma \in K\}&= st(\tau,K)\cap st(\sigma,K) &= st(\sigma,K)
\|f^{s'}(t,\cdot)\|_{L^2}\le \Big(\sum_{\substack{i+j=0 j\le 2}}^{[\frac{s'}{2}]+2}\|\partial^i_x\partial^j_tu(t,\cdot)\|^2_{L^{\infty}}\Big) E^{\partial_1}_{s'}(t, u) \le \|u(t,\cdot)\|^2_{W^{t,\rho,\infty}}E^{\partial_1}_{s'}(t, u)\,,
\begin{aligned}F^{ij}\left(\sqrt{b_{n-1}}(A_1 v - A_2 w) \pm \nabla_\alpha (u - \varphi)\right)_{ij}\geq \, 0 \mbox{in}\omega_1 A \sqrt{b_{n-1}}(A_1 v - A_2 w) \pm \nabla_\alpha (u - \varphi) \leq \, 0 \mbox{on}\partial \omega_1\end{aligned}
|||\partial_2(\dot{H}^+_2\partial_1\hat{\Phi}^{+})(t)|||^2_{s-1,\ast,t}\leq||\dot{H}^+_2\partial_1\hat{\Phi}^{+}||^2_{s,\ast,t}\leq C(K)\Big(||{\mathbf V}||^2_{s,\ast,t}+||\dot{{\mathbf U}}||^2_{W^{1,\infty}_{\ast}(\Omega_t)}||\hat{W}||^2_{s+2,\ast,t}\Big) \leq C(K)\mathcal{M}(t).
g^{(d)}(E)=g_{s}\int\frac{d^{d}k}{(2\pi)^{d}}\delta(E-E_{0}-\frac{\hbar^{2}|k|^{2}}{2m})=g_{s}(\frac{m}{2\pi\hbar^{2}})^{d/2}\frac{(E-E_{0})^{d/2-1}}{\Gamma(d/2)}
0=\int_{\mathcal{H}}\int_{\mathcal{H}}\psi(c+\|x-y\|^{2})d\mu(x)d\mu(y) = \int_{[0, \infty)}\int_{\mathcal{H}}\int_{\mathcal{H}}e^{-r\|x-y\|^{2}}d\mu(x)d\mu(y)d\beta_{c}(r), c>0.
R_{Sylv}\left( i+\tau_2, \, : \right) v = \sum_{k=0}^{\tau_1+\tau_2-1}Y_{k}^{i,2}(x_2^{\ast}) \phi_{k}(x_1^{\ast}) = q_2(x_1^{\ast})\phi_i(x_1^{\ast}) = 0.
\int_X|Ric(\tilde\omega_\epsilon)+\tilde\omega_\epsilon|^2_{\tilde\omega_\epsilon}\tilde\omega_\epsilon^n =\int_X|2n\epsilon\omega_{i_0,\epsilon}|^2_{\tilde\omega_\epsilon}\tilde\omega_\epsilon^n &\le 4n^3\int_X\tilde\omega_\epsilon^n &\to 4n^3(2\pi)^n\int_X c_1(K_X)^n &=0
t_{s_{1,g-1}}^{5(g-1)n}[T_{s_{1,1}}, r] = [\mathcal{V}'_1,\mathcal{W}'_1] [\mathcal{V}'_2,\mathcal{W}'_2] \cdots [\mathcal{V}'_{\left[\frac{|5n|}{2}\right]+1}, \mathcal{W}'_{\left[\frac{|5n|}{2}\right]+1}] \cdot t_{d_{g-1}}^{10(g-1)n}[T_d,r],
\P^x\Big(\int_0^\infty f(\xi_s)\,ds<\infty\Big) \geq \P^x\Big(\int_{\tau_0}^\infty f(\xi_s)\,ds<\infty, \tau_0<\infty\Big) &=\P^x\Big(\int_{0}^\infty f(\xi_{s+\tau_0}-\xi_{\tau_0})\,ds<\infty, \tau_0<\infty\Big) &=\P^0\Big(\int_0^\infty f(\xi_s)\,ds<\infty\Big)\P^x(\tau_0<\infty) &>0.
\int_{t_{1}}^{t_{2}}\delta L^{\prime}dt=\int_{t_{1}}^{t_{2}}\sum_{k=1}^{N}(\frac{\partial L}{\partial r_{k}}-\frac{d}{dt}\frac{\partial L}{\partial\dot{r}_{k}}+\sum_{i=1}^{C}\lambda_{i}\frac{\partial f_{i}}{\partial r_{k}})\cdot\delta r_{k}dt=0.
\xi_k':=\frac{1}{\sqrt{2}}\big(\xi_k-\eta_{-k}\big)=\frac{1}{\sqrt{2}}\begin{pmatrix}e^{2\pi k i x}-e^{-2\pi k i x}\end{pmatrix},\eta_k':=\frac{i}{\sqrt{2}}\big(\xi_k+\eta_{-k}\big)= \frac{i}{\sqrt{2}}\begin{pmatrix}e^{2\pi k i x}e^{-2\pi k i x}\end{pmatrix}.
\overline W_t = \frac{p^{\overline P^t_t}e^{\delta t}}{X_0^{-1}+\vartheta\int_0^tp^{\overline P_s^t}e^{\delta s}ds}= \frac1{X_0^{-1}p^{-P_t}e^{-\delta t}+\vartheta\int_0^tp^{-P_{t-s}}e^{-\delta(t-s)}ds}begin{align*}1em] = \Big(X_0^{-1}p^{-P_t}e^{-\delta t}+\vartheta\int_0^tp^{-P_s}e^{-\delta s}ds\Big)^{-1}.
s_1 = \begin{pmatrix}0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0\end{pmatrix}, \ \mbox{and}s_2 = \begin{pmatrix}0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0\end{pmatrix}.
f \left( t \right) = \alpha_{i-1}\, \forall t \in \left[ t_{i-1}+ \tfrac{\epsilon}{2}\tfrac{1}{k}\Delta_{g}, t_{i}- \tfrac{\epsilon}{2}\tfrac{1}{k}\Delta_{g}\right) \forall i \in \left\lbrace 1 , \ldots , k \right\rbrace, f \left( t \right) = \alpha_{k}\, \forall t \in \left[ t_{k}, t_{k+1}\right].
\int_{{\mathbf X}_\infty}\psi(x{}^{\rm t}y)\psi(\frac{1}{2}xz{}^{\rm t}x) P(x) dx ={\rm det}(\frac{z}{\sqrt{-1}})^{-2}\cdot \psi(-\frac{1}{2}yz^{-1}{}^{\rm t}y) P(-yz^{-1}).
\sum_{j,k=1}^2(\tilde{l_p})_{\sigma_j\sigma_k}(0,0)\xi_j\xi_k=\vert\nabla_xl_p(x_0,y_0)\vert\left({\cal A}_{\partial D,x_0}v(\xi)\cdot v(\xi)+{\cal A}_{S,x_0}v(\xi)\cdot v(\xi)\right).
\frac{V_{i}^{\prime \prime}(0)}{(V_{i}^{\prime}(0))^{2}}=2\frac{\left(z_{i}-z_{i+1}\right) -\ell \left( z_{i+1}-z_{i-1}\right)}{\left(z_{i}-z_{i+1}\right) \left( z_{i}-z_{i-1}\right)}
F_g={1\over 2}\epsilon^1_{ij}\epsilon^2_{ij}\int_0^1 dx\left [{(1{-}k_0^2{-}\vec k\cdot \vec p)^2\over x^2}+2{k_0^4{-}(\vec k\cdot \vec p)^2\over x}+ (1{-}k_0^2{+}\vec k\cdot \vec p)^2\right ]x^{\vec k\cdot \vec p+ k_0^2}(1-x)^{-2 k_0^2}\,.
\partial^2_\alpha D(p||p_{\alpha,\mu}) = \frac{1}{\alpha^2}+ \mathbb{E}_p[ (x-\mu)^2 e^{-\alpha(x-\mu)}], \partial_\alpha\partial_\mu D(p||p_{\alpha,\mu}) = -1 + \mathbb{E}_p[e^{-\alpha(x-\mu)}] - \alpha \mathbb{E}_p[(x-\mu)e^{-\alpha(x-\mu)}], \partial^2_\mu D(p||p_{\alpha,\mu}) = \alpha^2 \mathbb{E}_p[ e^{-\alpha(x-\mu)}].
v \sim v_{0}+ \delta v \, , \hspace{0 . 2 c m}\mathrm{w i t h}\hspace{0 . 2 c m}\delta v = \frac{6 \pi m}{\lambda}\, v_{0}\, ( 1 - v_{0}^{2})^{3 / 2}\, e^{- \, 2 \lambda t}< 0 \, .
Y_{2}R_{2 1}^{- 1}d K_{1}R_{2 1}= R_{1 2}d K_{1}R_{2 1}Y_{2}\quad , \quad \tilde{Y}_{2}R_{1 2}d K_{1}^{\dagger}R_{2 1}= R_{1 2}d K_{1}^{\dagger}R_{1 2}^{- 1}\tilde{Y}_{1}\quad ,
d_1( xy^{n_1}x \otimes 1) = x \otimes y^{n_1}x - i_0d_0( x \otimes y^{n_1}x) = x \otimes y^{n_1}x - i_0( 1 \otimes \overline{xy^{n_1}x}) = x \otimes y^{n_1}x - i_0( 1 \otimes xy^{n_1+1}) = x \otimes y^{n_1}x - x \otimes y^{n_1 +1}
\int_\Omega \left(u\log u+\frac{1}{2}|\nabla v|^2+\frac12v^2 \right)dx= \mathcal{F}(u,v)+\int_\Omega u vdx \leq \mathcal{F}(u,v)+\|v\|_{L^\infty(\Omega)}\int_\Omega udx \leq \mathcal{F}(u_0,v_0)+C\Lambda,
\|\mathcal{D}_{J,1}^{m_1}\mathcal{D}_{J,2}^{m_2}\widehat{f}_{j}\|^2_{L^2(\mathcal{W}^+_J)}\lesssim \sum_{\substack{a_1+b_1=m_1 a_2+b_2=m_2}}A_J 2^{-2j(m_1+m_2)s\alpha}2^{-2js\alpha}\lesssim A_J 2^{-2j(m_1+m_2)s\alpha}2^{-2js\alpha}.
M_{\mathrm{CsCl}}=\underset{\epsilon \rightarrow 0+}{\lim}\left( \int_{\epsilon}^{+\infty}\left\lbrace\vartheta_{3}(0\vert i v)^{3}-\vartheta_{4}(0\vert i v)^{3}\right\rbrace dv -\dfrac{2}{\sqrt{\epsilon}}\right)=-2.035 \ 361 \ 508 \ 229 \ \ldots \ .
\hat{H}_{0}= \int_{( \infty )}\frac{d^{3}\vec{k}}{( 2 \pi )^{3}2 \omega ( \vec{k}\, )}\frac{1}{2}\omega ( \vec{k}\, ) \left[ a^{\dagger}( \vec{k}\, ) a ( \vec{k}\, ) + a ( \vec{k}\, ) a^{\dagger}( \vec{k}\, ) \right] \ ,
(t_{g}(b)\cdot\Psi)_{i}(\delta):= g_{i}^{-1}(b)\Psi_{i}(\delta),(t_{g}(u_{\gamma})\Psi)_{i}(\delta):= t_{i}(\gamma^{-1})^{-1}(\Psi_{\gamma^{-1}(i)}(t_{i}(\gamma^{-1})\delta )).
F_{k,\ell,j,m,n}(r)=r^{\kappa_{\ell,jm,n}+\frac{1}{2}}\exp \left(-\frac{\omega r^{2}}{2}\right)L_{k}^{(\kappa_{\ell,j,m,n})}(\omega r^{2}),k=0,1,2... \ .
L_{i j}= \left( \begin{array}{c c c c}{{0}}&{{k_{1}- k_{3}}}&{{- k_{1}+ k_{2}}}&{{k_{3}- k_{2}}}\\{{- k_{1}+ k_{3}}}&{{0}}&{{k_{1}}}&{{- k_{3}}}\\{{k_{1}- k_{2}}}&{{- k_{1}}}&{{0}}&{{k_{2}}}\\{{- k_{3}+ k_{2}}}&{{k_{3}}}&{{- k_{2}}}&{{0}}\\ \end{array}\right) .
\mathbf{v}_A^{\pm}(x,t)=\mathbf{v}_0^{\pm}(x,t)+\varepsilon \mathbf{v}_1^{\pm}(x,t)+\varepsilon^2 \mathbf{v}_2^{\pm}(x,t),p_A^{\pm}(x,t)=p_0^{\pm}(x,t)+\varepsilon p_1^{\pm}(x,t),c_{\pm}^{out}=\pm 1.
\bar{n}_0(t)=[\left(|\alpha_{00}(t)|^2+|\beta_{00}(t)|^2\right)\bar{n}_0+|\beta_{00}(t)|^2+\sum_{k}\left[\left(|\alpha_{0 k}(t)|^2+|\beta_{k 0}(t)|^2\right)\bar{n}_k+|\beta_{k 0}(t)|^2\right]\;.
\Theta_{\alpha}^{+}= \theta_{\alpha}^{i}u_{i}^{+}+ \eta^{+}\tilde{\lambda}_{\alpha}+{\bar{\eta}}^{+}{\bar{\tilde{\lambda}}}_{\alpha}- i \eta^{+}{\bar{\eta}}^{+}{\dot{\theta}}_{\alpha}^{i}u_{i}^{-},
f_{J_{1}n_{1}\sigma_{1}; ~ J_{2}n_{2}\sigma_{2}}^{J_{3}n_{3}\sigma_{3}}= ( - 1 )^{2 J_{1}( J_{2}- n_{2}- 1 )}{\tilde{f}}_{J_{1}n_{1}\sigma_{1}; ~ J_{2}n_{2}\sigma_{2}}^{J_{3}n_{3}\sigma_{3}}.
\begin{array}[t]{r}\dfrac{|k|k|^{2\rho}+\ell|\ell|^{2\rho}+m|m|^{2\rho}|}{(\min (|k|,|\ell|,|m|))^{1/2}}\lesssim |k|^{1/2}\,|\ell|^{\rho}\,|m|^\rho+|\ell|^{1/2}\,|m|^{\rho}\,|k|^\rho+|m|^{1/2}\,|k|^{\rho}\,|\ell|^\rho\,, [5pt]\forall (k,\ell,m)\,;k+\ell+m=0\,,\;k\ell m\neq 0\,,\end{array}
\sum_{j=1}^n \| H^{,j}(s)\|^q_{L^2(\nu_j, \mathbb R^d)}=\sum_{j=1}^n \big(\int_{\mathbb R_0}|H^{,j}(s,z)|^2\nu_j(dz)\Big)^{\frac{q}{2}}& \leq C_0\big(\sum_{j=1}^n \int_{\mathbb R_0}|H^{,j}(s,z)|^2\nu_j(dz)^{\frac{1}{2}}\Big)^q=C_0 \| H(s)\|^q_{L^2(\nu, \mathbb R^d)}.
\psi(t,\vec{x})\mapsto\psi^{cp}(t,\vec{x})=(\begin{matrix}\psi_{L}^{cp}(t,\vec{x})\\ \psi_{R}^{cp}(t,\vec{x})\end{matrix})=\eta_{c}(\begin{matrix}-i\sigma^{2}\psi_{L}^{*}(t,-\vec{x})\\i\sigma^{2}\psi_{R}^{*}(t,-\vec{x})\end{matrix})
E_m=\omega_Q^{\frac{1}{\bar{p}_2}-\frac{1}{\bar{p}_1}}\int_{\mathbb H^n}K(e,y)|y|_h^{-Q/p}dy\|f_j\|_{L_{|x|_h}^pL_\theta^{\bar{p}_2}(\mathbb H^n)}.
\textrm{Det}=0\implies -R_{1c}-R_{1h}R_{3c}-\lambda T'(c)=0 & \implies \mu\frac{K_1}{(1+c)(1+c^2)}\left(\frac{1-b}{1+c}-2c(b+c) \right)+\lambda T'(c)=\frac{\Gamma K}{(K+c)^2}.
{\frac{1}{G}}={\frac{1}{1 2 \pi}}q^{\prime}( 1 ) ={\frac{1}{1 2 \pi}}\left[ \sum_{s}( 1 - 6 \xi_{s}) m_{s}^{2}\ln m_{s}^{2}+ 2 \sum_{d}m_{d}^{2}\ln m_{d}^{2}\right] .
\ddot{q}_{i}=\sum\limits_{j=1,\textrm{}j\neq i}^{n}\frac{m_{j}(q_{j}-\sigma(q_{i}\odot q_{j})q_{i})}{(\sigma-\sigma(q_{i}\odot q_{j})^{2})^{\frac{3}{2}}}-\sigma(\dot{q}_{i}\odot\dot{q}_{i})q_{i},\textrm{}i\in\{1,...,n\}.
f_{A}={\frac{\pi ( e_{1}+ e_{2}) E t e^{- \pi t \epsilon^{2}/ 2}}{2 s i n ( \pi t \epsilon / 2 )}}\, p r o d_{n = 1}^{\infty}{\frac{( 1 - e^{- \pi t n})^{2}}{( 1 - e^{- \pi t ( n + i \epsilon )}) ( 1 - e^{- \pi t ( n - i \epsilon )})}}\ .
\int_\Omega\frac{|u_n-u|^{q}}{|x|^{\alpha}}dx = \int_\Omega\frac{|u_n-u|^{\frac{\alpha}{s}}}{|x|^{\alpha}}|u_n-u|^{q-\frac{\alpha}{s}}dx \leq\Big(\int_\Omega\frac{|u_n-u|^{p}}{|x|^{ps}}dx\Big)^{\frac{\alpha}{ps}}\Big(\int_\Omega |u_n-u|^{\sigma}dx\Big)^{\frac{ps-\alpha}{ps}}, & \leq C\Big(\int_\Omega |u_n-u|^{\sigma}dx\Big)^{\frac{ps-\alpha}{ps}},
F_1= g\cdot\nabla g-f\cdot\nabla f = g\times (\nabla\times g)-f\times (\nabla\times f)+\nabla (\frac{1}{2}|f|^2-\frac{1}{2}|g|^2) \doteq F+\nabla (\frac{1}{2}|f|^2-\frac{1}{2}|g|^2),
\frac{1}{2}\int \prod_{i = 1}^{4}d g_{i}\phi^{2}( g_{1}, \ldots , g_{4}) = \frac{1}{2}\sum_{N_{1}\ldots N_{4}}\sum_{\Lambda}\phi_{\cal A}^{N_{1}\ldots N_{4}\, \Lambda}\phi_{\cal A}^{N_{1}\ldots N_{4}\, \Lambda}.
\xi =q^{-2}\frac{\psi(q)^2 \phi(-q^9)}{\phi(-q) \psi(q^9)^2}&=q^{-2}\frac{\phi(-q^9)^{4}}{\phi(-q^3)^{4}}\left(1+2qw(q^3)+4q^2 w(q^3)^2\right)\left(\frac{1}{w(q^3)}+q\right)^2 &=\frac{\phi(-q^9)^{4}}{\phi(-q^3)^{4}}\left(\frac{1}{q^2 w(q^3)^2}+\frac{4}{q w(q^3)}+9+10q w(q^3)+4q^2 w(q^3)^2\right).
\frac{\partial}{\partial s}\omega_{\psi}^\delta= -{\rm Ric}(\omega_{\psi}^\delta ) +\mu\omega_{\psi}^\delta+(1-\beta)\eta_\delta, & \omega_{\psi}^\delta(\cdot, 0)=\omega^\delta_{\psi_{0, \delta}}=\omega^\delta_{\psi_{s, \delta}}|_{s= 0}=\omega_{\varphi_{t,\delta}}^\delta.
\aligned\sum_{k_1+k_2=k+1}\sum_{\beta_1+\beta_2=\beta}\sum_{i=1}^{k-k_2+1}&(-1)^*{\frak m}_{k_1,\beta_1}(h_1,\ldots,{\frak m}_{k_2,\beta_2}(h_i,\ldots,h_{i+k_2-1}),\ldots,h_{k}) = 0\endaligned
\oint_{F}Q_{n,2}(z)P_n(z)\biggl\{h_{n+m}(z)\frac{\tau_n^2(z)}{q_{n}(z)}\int_E\frac{q_{n}^2(\zeta)\tau_n^2(\zeta)}{z-\zeta}\frac{d{\mathfrak m}_n(\zeta)}{P_n(\zeta)}f(z)\,\biggr\}\,dz=0
F[arcsin(x),k]=\frac{2}{1+\sqrt{1-k^{2}}}F[arcsin(\frac{(1+\sqrt{1-k^{2}})x}{1+\sqrt{1-k^{2}x^{2}}}),\frac{1-\sqrt{1-k^{2}}}{1+\sqrt{1-k^{2}}}]
\frac{\theta(z;q)+\theta(-z;q)}2 =\sum_{k=-\infty}^\infty q^{2k\cdot(2k-1)/2}z^{2k}=\theta(-q z^2;q^4), \frac{\theta(z;q)-\theta(-z;q)}2 =-\sum_{k=-\infty}^\infty q^{(2k+1)\cdot 2k/2}z^{2k+1}=-z\,\theta(-q^3z^2;q^4).
\Gamma_{\mu}^{( \mathrm{f )}}= \int{\frac{d^{3}k}{( 2 \pi )^{3}}}G_{\alpha \beta}( k ){\frac{( 2 p - k )_{\alpha}}{( p - k )^{2}+ M^{2}}}\Sigma ( q - k ) \Gamma_{\beta \mu}( k , q - k , - q ) ,
\left(\mathbf{H}\mathbf{D}\mathbf{V}\mathbf{D}^H\mathbf{H}^H+\sigma^2_n\mathbf{I}_M\right)^{-1}=\;\frac{1}{\sigma_n^2}\mathbf{I}_M-\frac{1}{\sigma_n^4}\mathbf{H}\left(\mathbf{E}^{-1}+\frac{1}{\sigma^2_n}\mathbf{H}^H\mathbf{H}\right)^{-1}\mathbf{H}^H\;,
h^\infty_{\mu \nu}(x) = - \frac{2 x^\alpha x^\beta}{r^2}\int_r^\infty dr' \, r' \, R^L_{\alpha \mu , \beta \nu}(r' \hat{x}) + \frac{2 x^\alpha x^\beta}{r^3}\int_r^\infty dr' \,{r'}^2 R^L_{\alpha \mu , \beta \nu}(r' \hat{x})
\mathrm{~ \frac{1}{2}~}( S_{m}, S_{m})^{a}+ V_{m}^{a}S_{m}= i \hbar \Delta^{a}S_{m}, \qquad \mathrm{~ \frac{1}{2}~}\{S_{m}, S_{m}\}_{\alpha}+ V_{\alpha}S_{m}= i \hbar \Delta_{\alpha}S_{m};
\sup_{k \geq 0}\tilde \phi (\mathcal{A}_0^k, \mathbf{Z}_{k+n}^{k+2n-1}):= \sup_{k\geq 0}\sup_{f \in \mathcal{F}^n}\big\| \mathbb{E}\bigl( f(\mathbf{Z}_{k+n}^{k+2n-1}) \big| \mathcal A_0^k \big) - \mathbb{E}f(\mathbf{Z}_{k+n}^{k+2n-1}) \big\|_{\infty}\leq c \cdot e^{- b n}
J_{\varepsilon}(u)=\frac{1}{2}\int_{M}a(x)\left(|\nabla_{g}u|^{2}+h(x)u^{2}\right)d\mu_{g}-\frac{1}{2_{m}^{*}-\varepsilon}\int_{M}a(x)\left(u^{+}\right)^{2_{m}^{*}-\varepsilon}d\mu_{g}.
\frac{1}{\pi\varepsilon^2}\int\limits_{B(\zeta_0,\varepsilon)\cap{\Bbb B}^2}Q(z)\,dm(z)\leqslant\frac{3}{\pi\varepsilon}\int\limits_{1-\varepsilon}^1\beta(r)\,dr\leqslant\frac{6}{\pi}:=c\,, 0<\varepsilon<\varepsilon_0
\frac{\lambda_2^{K_{\ell}}}{\lambda_2^L}=(2^{(\dim M_2^{K_{\ell}}-m+1)/2}\cdot\frac{\displaystyle{\prod_{i=1}^{m-1}}(2^{2i}-1)}{|M_2^{K_{\ell}}|})(2^{(\dim M_2^L-m)/2}\cdot\frac{\displaystyle{\prod_{i=1}^{m}}(2^{2i}-1)}{|M_2^L|})^{-1}\leq 2^{1/2}\cdot \frac{2^{(n_{2,\nu_2}-1)/2}+1}{2^{n+1}-1}
\begin{aligned}|f_n(x,y)| \leq n^{-1/2}||y|^{\ell_\mu}-|x|^{\ell_\mu}|\cdot |\sigma(y)|\le 4 |x-y| \cdot\frac{1+|y|^{\ell_\sigma+1}}{1+|y|}\leq c\cdot (1+|y|^{\ell_\sigma})\cdot |x-y|.\end{aligned}
\left( Q_{\widetilde{\eta}^{-1}}- (D^2 f L_1^{-1}P)_{\widetilde{\eta}^{-1}}\right)(\nabla \chi ) =\left( Q_{\widetilde{\eta}^{-1}}- (D^2f L_1^{-1}P)_{\widetilde{\eta}^{-1}}\right)(-\nabla p ).