harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\frac{\lambda_v^{K_{\ell}}}{\lambda_v^L}=\Bigl\{q_v^{(\dim M_v^{K_{\ell}}-m)/2}\cdot|M_v^{K_{\ell}}|^{-1}\cdot\prod_{i=1}^{m}(q_v^{2i}-1)\Bigr\}\Bigl\{q_v^{(\dim M_v^L-m)/2}\cdot|M_v^L|^{-1}\cdot\prod_{i=1}^{m}(q_v^{2i}-1)\Bigr\}^{-1}\leq q_v^{(n_{v,\nu_v}-1)/2}+1.
\lim_{n\to\infty}\int_{\mathcal{Z}}M\wedge \|\phi\|_{\mathcal{X}}^{2}\, \Theta^{n}(d\phi,dr,dw) = \int_{\mathcal{Z}}M\wedge \|\phi\|_{\mathcal{X}}^{2}\, \bar{\Theta}(d\phi,dr,dw).
\phi(x) = \sum_{l=1}^m \tilde \phi(2^{3 l}x) 2^{-6 l}\cdot \frac 1{l}, & u^o = (u_1^o,u_2^o,0,\cdots,0), & u_1^o=-\partial_2 \phi, u_2^o = \partial_1 \phi.
P^\omega(F_{j,t}^c)=P^\omega(M_{j h(t)}>k j h(t)) \leq P(M_{j h(t)}>k j h(t)) &\leq E[N_{jh(t)}]\:\mathbf{P}_0\bigg(\sup_{0\le s\le j h(t)}|X_s|>kj h(t)\bigg) &= \exp[jh(t)(\beta-k^2/2)(1+o(1))] .
F_{C}( a , T ) = \frac{5 \, \xi^{2}}{3 2 a \pi}- \frac{8 a^{3}\pi^{3}T^{4}\xi^{2}}{1 3 5}- \frac{5 1 2 a^{5}\pi^{5}T^{6}\xi^{2}}{4 7 2 5}+ O ( T^{7}) \, \, .
P_{w,r}^V (V_{\tau_s^R}> x + y| x \leq V_1 \leq x + y/2) \leq\begin{cases}c_7(1 + y^{2/3}x^{-1/3}) e^{-\frac{1}{2}c_6 y^{2/3}x^{-1/3}}~,~ \mbox{if}y \leq x c_7(1 + y^{1/3}) e^{-\frac{1}{2}c_6 y^{1/3}}~,~ \mbox{if}y \geq x.\end{cases}
\mathcal G_{\varphi}=\bigcup_{p\in M}\left\{(X_1, \ldots, X_r, Y_1, \ldots ,Y_s) \in \mathcal F_p (M) \colon X_1, \ldots , X_r \in (D_{\phi})_p, Y_1, \ldots , Y_s \in (D_{\bar\phi})_p\right\},
\begin{aligned}u_{t}=\Delta u -\chi \nabla \cdot (u\nabla w), x \in \mathbb{R}^{N}, t>0, v_{t}=\Delta v - \xi \nabla \cdot (v \nabla w), x \in \mathbb{R}^{N}, t>0, w_{t}=\Delta w - \lambda w + \alpha u +\beta v, x \in \mathbb{R}^{N}, t>0 ,\end{aligned}
\partial_t \int_{\mathbb{R}^3_v}fdv + \nabla_x \cdot \int_{\mathbb{R}^3_v}vfdv =0, &\partial_t \int_{\mathbb{R}^3_v}vfdv + \nabla_x\cdot \int_{\mathbb{R}^3_v}v\otimes vfdv =0, &\partial_t \int_{\mathbb{R}^3_v}\vert v\vert^2fdv + \nabla_x \cdot \int_{\mathbb{R}^3_v}\vert v\vert^2 vfdv =0.
g_{\mathrm{w}}( x ) ={\frac{{\frac{\sqrt{x}}{8 \beta}}- 1}{\sqrt{( x - b^{2}) ( x - a^{2})}}}-{\frac{1}{8 \pi \beta}}\int_{0}^{\infty}d y{\frac{\sqrt{y}}{x + y}}{\frac{1}{\sqrt{( y + b^{2}) ( y + a^{2})}}}.
\alpha_2=0\,\Leftrightarrow\,a_2=0\,\Leftrightarrow\,\ell_t=2\,\ell_b\,\Leftrightarrow\,\beta=\gamma 2\,\alpha_1+3\,\alpha_3=0\,\Leftrightarrow \,a_3=0\,\Leftrightarrow \,\Psi=\frac{3}{2},
\|\hat Q_r(\hat\Sigma-\mu_rI)R_r\|_2 =\sqrt{\sum_{j\in\mathcal{I}_r}(\hat\lambda_j-\mu_r)^2\|(\hat u_j\otimes\hat u_j) R_r\|_2^2}&\leq Cxm_r\mu_r\sqrt{\sum_{j\in\mathcal{I}_r}\|(\hat u_j\otimes\hat u_j) R_r\|_2^2}= Cxm_r\mu_r\|\hat Q_r R_r\|_2.
\sum\limits_{n=1}^{N}\left( \sum\limits_{l=1}^{n}w_{n-l}V_{l}\right) V_n = k^{-1}\sum\limits_{n=1}^{N}\left( \sum\limits_{l=1}^{n}\left(k\chi_{n,l}^{(q)}\right) V_{l}\right) V_n \geq 0, N\geq 1,
\frac{1}{2}\Big( \partial_\mu \varphi^{(1)}\Big)^2 \; + \;\frac{1}{2}\Big( \partial_\mu \varphi^{(2)}\Big)^2 \; + \;\frac{1}{2}\Big( \varphi^{(1)}\Big)^2 \frac{e^2 N}{\pi+gN}\; .
v = \frac{z}{R_{T}}+ \sum_{n = 1}^{\infty}\sin \left( n \frac{z}{R_{T}}\right) \left[ \frac{1}{n}f_{d}\left( n \frac{r}{R_{T}}\right) - \frac{1}{d - 3}\frac{r}{R_{T}}f_{d}^{\prime}\left( n \frac{r}{R_{T}}\right) \right]
\lim_{p\rightarrow N^{-}}\frac{\Lambda_{p}(\Omega)}{\left\vert p-N\right\vert^{p-1}}=\lim_{p\rightarrow N^{-}}\frac{\pi^{\frac{p}{2}}N}{(p-1)^{p-1}}\left( \frac{\Gamma(N/p)\Gamma(1+N-N/p)}{\Gamma(1+N/2)\Gamma(N)}\right)^{\frac{p}{N}}& =\frac{\pi^{\frac{p}{2}}N}{(N-1)^{N-1}}\frac{1}{\Gamma(1+N/2)}=\frac{N\omega_{N}}{(N-1)^{N-1}}.
\left\|\rho_{p,\ell}(\beta_h) \phi_w((\cdot)^{-1},e)-\int_{X_{p,\ell}}\phi_w\, d\mu_{p,\ell}\right\|_{L^2(Q_{\ge R})}\ll_{w,p,\ell}\left\|\omega_R\right\|_{L^{2s}(X_{p,\ell})}&=\mu_{p,\ell}(\Omega_R)^{1/(2s)}&\le m_\infty(Q_{\ge R})^{1/(2s)}\ll_s R^{-1/s}.
\int_{X}\left|\frac{|f(x)|}{\|f\|_2}-\frac{1}{\sqrt{\mu(X)}}\right|^2\,d\mu(x) =\frac{1}{\|f\|_2^2}\int_{X}|f(x)|^2\,d\mu(x)+\frac{1}{\mu(X)}\int_{X}\,d\mu(x) &\quad-2\frac{1}{\|f\|_2\sqrt{\mu(X)}}\int_{X}|f(x)|\,d\mu(x) &=2\left(1-\frac{1}{\|f\|_2\sqrt{\mu(X)}}\int_{X}|f(x)|\,d\mu(x)\right) &=c_f
\mathcal{E}_w(z)=\left( \begin{array}{ccc}E_w(z) K_w(z) E_{w,+}(z) E_{w,-}(z) K_{w,-}(z) E_{w,-+}(z) \end{array}\right):\mathcal{H}_{w,r}\to\mathcal{B}_{w,r}
{\rm wt}(\Psi_{\Lambda}^{2^{8}}\cdot\Psi_{\Lambda}(\cdot,f_{\Lambda}))=-2^{2}(2^{8}+1)(2^{9}-1),{\rm div}(\Psi_{\Lambda}^{2^{8}}\cdot\Psi_{\Lambda}(\cdot,f_{\Lambda}))=(2^{8}+1){\mathcal D}_{\Lambda}-2^{4}{\mathcal H}_{1}.
\bigg\|\sum_{j=1}^n L_j^2 f \bigg\|_{2}^2+ 2\mathbf{R}_{n}\sum_{j=1}^n \bigg\|\frac{L_j f}{|x|}\bigg\|_{2}^2 = \widetilde{\mathbf{R}}_{n}\bigg\|\frac{P^\perp f}{|x|^2}\bigg\|_2^2 + \sum_{k = 1}^\infty \big(\mu_k( \mu_k + 2\mathbf{R}_{n}) - \widetilde{\mathbf{R}}_{n}\big) \bigg\| \frac{P_kP^\perp f}{|x|^2}\bigg\|_{2}^2
\frac{\delta_{\rm I}^2}{16^{k^2}}\binom{\rho n}2p_{\rm I}\overset{\eqref{eq:ps}}{\ge}\frac{\delta_{\rm I}^2\rho^2\alpha}{16^{k^2}\cdot 6}n^2p \overset{(\ref{eq:dgdr}\,,\ref{eq:delta_I})}{\geq}\frac{b_i^4a_i^{36k^2+8k}}{6^5\cdot2^{109k^4+26k^3+4k^2}}np^2 \overset{\eqref{rec}}{\ge}2b_{i+1}n^2p
\int_\Sigma |(d(\eta_\delta W))^N|^2 = \int_\Sigma|\eta_\delta(d W)^N + (W d\eta_\delta)^N|^2 = \int_{\delta^2\leq |z|\leq \delta}|\eta_\delta(d W)^N + (Wd\eta_\delta)^N|^2+\int_{|z|\geq \delta}|(d W)^N |^2,
\frac{\partial\mathcal{L}(d_i^*)}{\partial d_i^*}\! \! = 1- \frac{\gamma\sum_{j \in \mathcal{I}}\frac{1}{\epsilon_j}}{1+\gamma \sum_{j\in\mathcal{I}}\frac{1}{\epsilon_j}}- \frac{1}{1+\gamma \sum_{j\in\mathcal{I}}\frac{1}{\epsilon_j}}=0\; \forall i\in \mathcal{I}.
\big((\phi_{1}\otimes f_{1})*(\phi_{2}\otimes f_{2})\big)(x,g) = \phi_{1}(x) \int \phi_{2}(s(x,g).h)f_{1}(gh)f_{2}(h^{-1})1_{Q_{s(x,g)}}(h)dh &=\phi_{1}(x) \int \phi_{2}(s(x,gh))f_{1}(gh)f_{2}(h^{-1})1_{Q_{x}}(gh)dh & =\phi_{1}(x) \int \phi_{2}(s(x,u))f_{1}(u)f_{2}(u^{-1}g)1_{Q_{x}}(u)du.
\alpha_0 = 2 q(1-s) + (1-2q+qs) + e^{t(1-q)^2}qs \, , \alpha_1 =- 2 q (1-q)q (1-s) + q^2 (1-2q+qs) \, , \alpha_2 = q^2(1-q)^2 q(1-s) + q^4 (1-2q+qs) \, .
\begin{aligned}L({x_p}{u_i}) =- \sum\limits_{j =1}^n{(X_j^2 + Y_j^2)}({x_p}{u_i}) =- \sum\limits_{j = 1}^n{\left[{2({X_j}{x_p})({X_j}{u_i}) +{x_p}(X_j^2{u_i}) +{x_p}(Y_j^2{u_i})}\right]}={x_p}L{u_i}- 2{X_p}{u_i},\end{aligned}
T^{\mu\nu}(x)=\frac{2}{\sqrt{-g(x)}}\frac{\delta}{\delta g_{\mu\nu}(x)}S_{Maxwell}=\frac{1}{\mu_{0}}(F_{\lambda}^{\mu}(x)F^{\nu\lambda}(x)-\frac{1}{4}g^{\mu\nu}(x)F_{\rho\sigma}(x)F^{\rho\sigma}(x))
M_0 J(k)T_0 T_2=\begin{bmatrix}kI_\mu+o(k) kB_0D_0^{-1}+o(k) kC_0A_0^{-1}+o(k) I_{n-\mu}+o(1)\end{bmatrix},\;\; k\to0,\;\;k\in \overline{\mathbb{C}}^+.
R_{ijkl}= R^B_{ijkl}, & R_{ijkp}= R_{ijpq}= R_{ipqr}= 0, & R_{ipjq}= -f H^f_{ij}h_{pq}, & R_{pqrs}= f^2 \{R^F_{pqrs}- ||{\rm grad}\: f ||^2 (h_{pr}h_{qs}- h_{ps}h_{qr}) \},
\alpha_{a,b}:= M \Big ( f(b)-f(a)- \langle G(a), b-a \rangle \Big ) -\tfrac{1}{2}\|G(a)-G(b)\|^2, & \beta_{a,b}:= \Big \| \tfrac{1}{2}\Big ( G(b)-G(a)+M (x-b) \Big )\Big \|^2, & Z_{a,b}:= \tfrac{1}{2}\Big ( G(a)+G(b)+ M (x-b) \Big ) .
0= 2(1-(\cot (2\vartheta) )^{2}) + (1-(\tan \vartheta )^{2}) +9(1-(\tan (3\vartheta) )^{2}) = (12- 2(\cot (2\vartheta) )^{2}- (\tan \vartheta )^{2}) -9 \frac{(\cot (2\vartheta) +\cot \vartheta )^{2}}{((\cot (2\vartheta) )(\cot \vartheta )-1)^{2}}
X(n)_{s,t}=(1,\int_{s<s_{1}<t}dX(n)_{s_{1}},...,\int_{s<s_{1}<...<s_{\lfloor p\rfloor}<t}dX(n)_{s_{1}}\otimes\cdot\cdot\cdot\otimes dX(n)_{s_{\lfloor p\rfloor}})
\frac{1}{n}\left(\sum^{n}_{k=1}\theta^{\frac{k}{n}}\nu^{b}_{n,k}(a) \omega^{-k}\right)=\frac{1}{n}\left(\sum^{n}_{k=1}\nu^{b}_{n,k}(a) (\theta^{\frac{1}{n}}_{b}\omega^{-1})^{k}\right).
u_{i}^{n+1}=u_{i}^{n}-\frac{\sigma\,\Delta t}{\varepsilon^{2}+\sigma\,\Delta t}u_{i}^{n}-\frac{\Delta t}{2\,\Delta x (\varepsilon^{2}+\Delta t\,\sigma)}\left(p(\tau_{i+1}^{n+1})-p(\tau_{i-1}^{n+1})\right) +\frac{\Delta t\,\lambda}{2 \,\Delta x}\left(\frac{\varepsilon^{2}}{\varepsilon^{2}+\sigma\,\Delta t}\right)(u_{i+1}^{n}-2u_{i}^{n}+u_{i-1}^{n}).
W_\mu({\bf p})\cdot D\left \{V^{-1}(\Lambda_{p}, k)\right \}= D\left \{V^{-1}(\Lambda_{p}, k)\right \}\cdot\left [ W_\mu({\bf k})+\frac{p_\mu+k_\mu}{M(\Delta_0+M)}p_\nu W_\nu ({\bf k})\right ],
\left\Vert \left( f_{v}\right)_{v}\right\Vert_{\ell_{>}^{q(\cdot)}(L^{p\left( \cdot \right)})}:=\inf \left\{\mu >0:\varrho_{\ell^{q(\cdot)}(L^{p(\cdot )})}\left(\frac{1}{\mu}(f_{v})_{v}\right)\leq 1\right\}.
\sum_{j=1}^2 | x_j - x'_j |^{\alpha}\leq 2^{1 - \tfrac{\alpha}{2}}\|x-x'\|^\alpha \leq 2^{1 - \tfrac{\alpha}{2}}A^\alpha B^\alpha \, \vartheta(x,x')^\alpha \leq M_1 \left[ C(0) - C(\vartheta(x,x')) \right]
\begin{aligned}0 \leq p_i^L = u_{i0}(x) \leq \cdots \leq u_{in}(x) = p_i(x), u_{ij}(x) \leq a_{ij}j = 0,\ldots,n, u_{i0}(x) = a_{i0}\leq \cdots \leq a_{in}, \deg(u_{ij}) \leq \tau.\end{aligned}
\sum_{m=1}^\infty \frac{\sin^2(\pi m y)}{m^2 - n^2/y^2}\frac{\sqrt{x + \sqrt{x^2 + m^2}}}{\sqrt{x^2 + m^2}}\approx - \frac{\pi y^{3/2}}{4 n}\frac{\sqrt{\sqrt{n^2 + x^2 y^2}- x y}}{\sqrt{n^2 + x^2 y^2}}.
\int\limits_{0}^{\infty}\frac{\cosh t\sinh^{4}\frac{\alpha}{2}t}{\sinh^{3}t}dt=\frac{\alpha^{2}}{8}\left( \psi \left( \frac{\alpha}{2}\right)+\psi \left( -\frac{\alpha}{2}\right) -\psi \left( \alpha \right) -\psi\left( -\alpha \right) \right) .
\vec{p}_{\ell}(z)={}^t\Biggl(P_{\ell}(z),{P_{\ell,1,r-1}(z),\ldots, P_{\ell,1,0}(z)}, \ldots,{P_{\ell,m,r-1}(z),\ldots, P_{\ell,m,0}(z)}\Biggr)\enspace,
\left\vert \dfrac{RT_{i,i_{0}}}{Rl_{i_{0}}t_{i_{0}}}-\dfrac{H_{i}}{K}\right\vert =\left\vert \dfrac{RT_{i,i_{0}}K-H_{i}Rl_{i_{0}}t_{i_{0}}}{KRl_{i_{0}}t_{i_{0}}}\right\vert =\epsilon \pounds
\bigl(\mathcal{S}_{M-1}B^{(f)}_{k}\bigr)(q\,|\,b) = 0,\; k<M-1, \bigl(\mathcal{S}_{M-1}B^{(f)}_{M-1}\bigr)(q\,|\,b) = \bigl(\mathcal{S}_{M-1}B^{(f)}_{M-1}\bigr)(0\,|\,b), \bigl(\mathcal{S}_{M-1}B^{(f)}_{M}\bigr)(q\,|\,b) - \bigl(\mathcal{S}_{M-1}B^{(f)}_{M}\bigr)(0\,|\,b) = -qf(0)\,M! \prod_{j=1}^{M-1}b_j.
\langle M_\phi(U(m,n))\delta_y,\delta_x\rangle = \tilde{\phi}(d(x_1,y_1),...,d(x_N,y_N)) \langle U(m,n)\delta_y,\delta_x\rangle &= \tilde{\phi}(m+n)\langle U(m,n)\delta_y,\delta_x\rangle.
A_m'(x) = \sum_{\substack{t=-m t\neq 0}}^m \sum_{\substack{i \leq m m_0(r_i) \leq m}}\left\vert \sum_{j = \langle m; r_i \rangle + 1}^{\langle m+1 ; r_i \rangle}e(r_i^j t x) \right\vert^2
(o,\delta) * (0_S, a\mapsto 0_A) = \big(o\cdot 0_S,a\mapsto \delta(a) \cdot [0_S, a\mapsto 0_A] + i(o)\cdot 0_A\big) = \big(o\cdot 0_S, a\mapsto \delta(a) \cdot 0_A + 0_A\big) = (0_S,a\mapsto 0_A)
u_{m+2}(\theta_D)=(-\frac{\gamma_{m+1}}{a_1+2-\gamma_{m+1}}) u_{m+1}(\theta_D), &u_{m+2}(\theta')=\frac{\gamma_{m+1}^2(a_1+2-\gamma_{m+1})^{-2}(a_1+1)^{-2m-2}(a_1k+a_1+1)-(a_1+1)}{k a_1}.
h(z;\boldsymbol{\xi})=\frac{\sigma^2}{2\pi}\frac{(1-\xi^{p+1}z^{-1})(1-\xi^{p+2}z^{-1})\cdots(1-\xi^{p+q}z^{-1})}{(1-\xi^1 z^{-1})(1-\xi^2 z^{-1})\cdots(1-\xi^p z^{-1})}
{1\over n!}\left({d^n\over d(\omega^2)^n}\Pi_{\parallel,\bot}\right)\Big{\vert}_{\omega^2=0}={M_{\parallel,\bot}^{1-2n}\over\pi}\int_{0}^{1}dy_{\parallel,\bot}\cdot y_{\parallel,\bot}^{n-1}\cdot \left(\kappa_{\parallel ,\bot}y_{\parallel,\bot}^{-1/2}\right),
(i) \int_{\vert x \vert \leq h}x^2 \, \overline{\nu}^k([0,t] \times \textrm{d}x) \longrightarrow \overline{C}_t, (ii) \overline{B}_t^k \longrightarrow \overline{B}_t, (iii) \overline{\nu}^k \left([0,t] \times \{\vert x \vert > 1 \}\right) \longrightarrow 0
V_{\textsf{GS}}\; = \; \{x \in T_{\textsf{GS}}\; : \; n(x) = 1 \wedge \lambda_1 x^{\star}\notin W_{12}+ \epsilon \}\cup{}\{x \in T_{\textsf{GS}}\; : \; n(x) = 2 \wedge \lambda_2 x^{\star}\notin W_{12}- \epsilon \}\;
d\Phi_{g_{-1}}(o) \begin{pmatrix}\hat w w_3\end{pmatrix}= \begin{pmatrix}I -v 0 1\end{pmatrix}\begin{pmatrix}\hat w w_3\end{pmatrix},\hskip 2ex d\Phi_{g_0}(o)\begin{pmatrix}\hat w w_3\end{pmatrix}= \begin{pmatrix}A^{-1}0 0 \alpha\end{pmatrix}\begin{pmatrix}\hat w w_3\end{pmatrix}
T_{g_1^H}\phi(g_2) =\int_G \phi(y)\> K_{g_1^H}(g_2,dy)= \int_H \phi(g_2 \cdot h(g_1))\> d\omega_H(h) &=\int_H\phi(g_2 ) \cdot \phi( h(g_1))\> d\omega_H(h) = \phi(g_2 ) \cdot \alpha(g_1^H),
\langle u , Z_s - Z_{s'}\rangle = \sum_{k = 0}^{\ell_s}(\sum_{i = k}^{\ell_s}b_{i,\ell_s}- \sum_{i = k}^{\ell_s + 1}b_{i,\ell_s + 1})\langle u , \zeta(V_{s,k}) - \zeta(V_{s,k-1}) \rangle + b_{\ell_s + 1,\ell_s + 1}\langle u, \zeta(V_{s,\ell}) - \zeta(V) \rangle.
\frac{\Gamma(s)}{\Gamma(1+\alpha-s) \Gamma(\frac{3}{2}+\alpha -s)}= \Gamma(s) \Gamma(s-\alpha) \Gamma(s-\alpha-\tfrac{1}{2})\frac{\sin \pi (s-\alpha) \sin \pi(s-\alpha-\frac{1}{2})}{\pi^2}& =\Gamma(s) \Gamma(s-\alpha) \Gamma(s-\alpha-\tfrac{1}{2})\frac{1}{(2\pi i)^2}\left(e^{2\pi i(s-\beta)}+ e^{-2\pi i(s-\beta)}\right).
FR_{p,q}= \bigg(\big[-\lfloor p/2\rfloor+1,1\big]\cup\big[\mu,\mu+\lfloor p/2\rfloor\big]\bigg)\times\big[-\lfloor q/2\rfloor+1,\nu+\lfloor q/2\rfloor\big] &\cup\big[-\lfloor p/2\rfloor+1,\mu+\lfloor p/2\rfloor\big]\times\bigg(\big[-\lfloor q/2\rfloor+1,1\big]\cup\big[\nu,\nu+\lfloor q/2\rfloor \big]\bigg).
{\cal I}=-\int_0^1\!\! d\sigma\int_0^\infty \!\!{d\rho\over\rho}{\log(1+\rho)\over(1+\rho)^3}{[\rho(x-z) +(1-\sigma)(x-y)]_+[\rho (y-z) +\sigma (y-x)]_+^2\over [\rho\sigma (x-z)^2 +\rho(1-\sigma )(y-z)^2 +\sigma(1-\sigma) (x-y)^2-i\epsilon]}\ .
H_0 =1, H_{4k}= \frac{(-1)^{k}k^2(2k-1)!^2}{2^{8k^2-4k-2}}\prod_{j=1}^{2k-1}(2j+1)!^4 H_{4k+1}= \frac{(-1)^{k}(2k)!^2}{2^{8k^2}(4k+1)!^2}\prod_{j=1}^{2k}(2j+1)!^4 H_{4k+2}= \frac{(-1)^{k}(2k+1)!^2}{2^{8k^2+4k}}\prod_{j=1}^{2k}(2j+1)!^4 H_{4k+3}=0
D_s X(t)(\omega) = \sigma(s,\omega, X(s)(\omega)) + \int_s^t (D_s b)(r, \omega, X(r)(\omega)) dr + \int_s^t (D_s \sigma)(r, \omega, X(r)(\omega)) dW(r) &+\int_s^t \nabla_x b(r,\omega, X(r)(\omega)) D_s X(r)(\omega) dr + \int_s^t \nabla_x \sigma(r,\omega, X(r)(\omega)) D_s X(r)(\omega) dW(r).
\frac{z^{a}\exp\left( -z\right)}{\Gamma\left( a+1\right)}=\left\{\frac{1+O\left( a^{-1}\right)}{\sqrt{2\pi}}\right\}a^{-1/2}\exp\left\{a\ln\left( \frac{e}{\rho e^{1/\rho}}\right) \right\}& =O\left[ a^{-1/2}\exp\left\{a\ln\left( \frac{e}{\rho e^{1/\rho}}\right)\right\}\right] ,
\mu\widetilde r_\alpha(s) = -\frac{1}{\widetilde f(s^\alpha|\mu)}\frac{d}{ds}{\widetilde f(s^\alpha|\mu)}= -\frac{\alpha s^{\alpha-1}}{\widetilde f(s^\alpha|\mu)}\frac{d}{d s^\alpha}{\widetilde f(s^\alpha|\mu)}=\mu \alpha s^{\alpha-1}\widetilde\rho(s^\alpha)
\left(\int_{\partial M}h^2\right)^{\frac{1}{2}}\leq \frac{c\sqrt{\lambda_1}}{\sqrt{\lambda_1}-\sqrt{\lambda_1-(k-1)c^2}}\left(\int_{\partial M}z^2\right)^{\frac{1}{2}}&=\frac{\sqrt{\lambda_1}}{(k-1)c}(\sqrt{\lambda_1}+\sqrt{\lambda_1-(k-1)c^2})\left(\int_{\partial M}z^2\right)^{\frac{1}{2}}.
R_V(x_{n-1},x_{n-2})\dots R_V(x_1,x_0)\ \dot\varphi^2(x_{n-1}) \star_\omega \dots \star_\omega \dot\varphi^2(x_0):= S(x_{n-1}-x_{n-2})\dots S(x_1,x_{0}) \; r(x_{n-1},\dots , x_0)n\geq 1,
\delta_1(\tau)=\frac{1}{4}+6q+6q^2+\cdots,~~\varepsilon_1(\tau)=\frac{1}{16}-q+7q^2+\cdots, &8\delta_2(\tau)=-1-24q^{\frac{1}{2}}-24q-96q^{\frac{3}{2}}+\cdots,~~\varepsilon_2(\tau)=q^{\frac{1}{2}}+8q+28q^{\frac{3}{2}}+\cdots.
\sum_{n_1 , ... , n_m = 0}^\infty \| T_m U_{n_m}(z_m ,z_m') \cdots T_1 U_{n_1}(z_1 , z_1') \Psi \| < \infty , & \sum_{n_1 , ... , n_m = 0}^\infty \| U_{n_m}(z_m ,z_m') \cdots T_1 U_{n_1}(z_1 , z_1') \Psi \| < \infty .
\chi_k = A\frac{\left(1-i\cot \left( n_1\pi\right) \right)}{\Gamma\left( 1+n_1\right)}\sqrt{\eta}\left( \frac{k\eta}2\right)^{n_1}\left[1-\frac{(k\eta )^2}{2(2+2n_1)}\right] -i B \frac{\csc \left( n_1\pi \right)}{\Gamma\left( 1-n_1\right)}\sqrt{\eta}\left( \frac{k\eta}2\right)^{-n_1}\left[ 1-\frac{(k\eta )^2}{2(2-2n_1)}\right].
||f||^2 =\langle\beta_1g^2+\beta_2g+\beta_3,\beta_1g^2+\beta_2g+\beta_3\rangle &=|\beta_1|^2+|\beta_2|^2+|\beta_3|^2+2\Re\left(\beta_1\overline{\beta_2}g(0)+\beta_2\overline{\beta_3}g(0)+\beta_1\overline{\beta_3}g(0)^2\right) &=(1+2|a|^2-2|a|^4-|a|^6)(1+|a|^2)^2.
K_i^{\mathrm{top}}(\mathrm{Perf}^{\mathrm{gr}}(\mathcal{X}\times \mathbb{C})) =K_i^{\mathrm{top}}(\mathrm{Perf}^{\mathrm{gr}}(\mathcal{X}))= \begin{cases}\bigoplus_{j\in \mathbb{Z}}K(BG)_{\mathbb{Q}}\cdot e_j, i=0, 0, i=1. \end{cases}
t = \frac{b}{\langle \mu, \alpha \rangle}- \sum_{k=1}^{d-1}\frac{\mu_k}{\langle \mu, \alpha \rangle}x_k , t= \frac{c}{\langle \nu, \alpha \rangle}- \sum_{k=1}^{d-1}\frac{\nu_k}{\langle \nu, \alpha \rangle}x_k,
\frac{e^{\tilde{u}_n^{(1)}}-e^{\tilde{u}_n^{(2)}}}{\|\tilde{u}_n^{(1)}-\tilde{u}_n^{(2)}\|_{L^\infty(M)}}=e^{\tilde{u}_n^{(1)}}\zeta_n\left(1+O\left(\frac{1}{\lambda_{n,j}^{(1)}}\right)\right).
E_0(x)=\sum_{x\leq p\leq 2x}\frac{1}{p}\sum_{\substack{0\leq a< p \gcd(n,p-1)=1}}e^{\frac{i2\pi a(\tau^{n}-u)}{p}}\cdot \frac{1}{p}\sum_{\substack{0<\leq b< p gcd(m,p-1)=1}}e^{\frac{i2\pi b(\tau^{m}-v)}{p}}=0.
R \equiv \frac{\sigma(e^{+}e^{-}\rightarrow q\bar{q})}{\sigma(e^{+}e^{-}\rightarrow \mu^{+}\mu^{-})}= 2 \tilde{N}_{c}N_{f}\left(\frac{N_{c}}{\tilde{N}_{c}}\right)^{2}= 2 N_{c}N_{f}\frac{N_{c}}{N_{f}-N_{c}}
\left\|x-\sum_{n=1}^{\lceil \lambda m\rceil}e^*_{\rho_x(n)}(x)e_{\rho_x(n)}\right\| \ \leqslant\ \mathbf C_\ell\left\|x-\sum_{n=1}^{m}e^*_{\rho_x(n)}(x)e_{\rho_x(n)}\right\| \ \leqslant\ \mathbf C_\ell C\sigma^{\mathcal{U}_X, L}_m(x).
\beta = \left( \frac{\beta_{1}- \beta_{0}e^{k_{2}y_{1}}}{1 - e^{( k_{2}- k_{1}) y_{1}}}\right) \; e^{k_{1}( y - y_{1})}\; + \; \left( \frac{\beta_{0}- \beta_{1}e^{- k_{1}y_{1}}}{1 - e^{( k_{2}- k_{1}) y_{1}}}\right) \; e^{k_{2}y}\; ,
\dot{x}(t)=[\begin{matrix}-d_{3}&1&0&0\\ -d_{2}&0&1&0\\ -d_{1}&0&0&1\\ -d_{0}&0&0&0\end{matrix}]x(t)+[\begin{matrix}n_{3}\\n_{2}\\n_{1}\\n_{0}\end{matrix}]u(t)
\mathcal{F}_{D,0}(z,s) := 2^{s-1}\pi^{-\frac{s+1}{2}}|D|^{\frac{s}{2}}L_D(s) B(s) \Gamma\biggl(\frac{s+1}{2}\biggr) P_{\frac{1}{2}, 0}\biggl(z, \frac{s}{2}+ \frac{1}{4}\biggr).
S_{N}f(x_{0}+\frac{L}{2N})=(y_{0}+\frac{c}{2})+\frac{2c}{\pi}(sin(\frac{\pi}{N})+\frac{1}{3}sin(\frac{3\pi}{N})+\cdot\cdot\cdot+\frac{1}{N-1}sin(\frac{(N-1)\pi}{N}))
\mu(r,0,\gamma A+(1-\gamma)B) =(p-q)(1-2\pi_0) &=(p-q)\bigg(1-\frac{2(p^r-q^r)}{2(p^r-q^r)+rpq(p^{r-2}-q^{r-2})}\bigg) &=\frac{rpq(p-q)(p^{r-2}-q^{r-2})}{2(p^r-q^r)+rpq(p^{r-2}-q^{r-2})},
F'(x) = - \frac{1}{x \ln^2(x)}, & F''(x) = \frac{1}{x^2 \ln^2(x)}+ \frac{2}{x^2 \ln^3(x)}, & F'''(x) = - \frac{2}{x^3 \ln^2(x)}- \frac{6}{x^3 \ln^3(x)}- \frac{6}{x^3 \ln^4(x)}.
\frac{g'(t)}{g(t)}=(\ln g(t))' &=\left(\sum_{j=1}^r\ln(e^{a_j t}-1)-r\ln t\right)' &=\sum_{j=1}^r\frac{a_j e^{a_j t}}{e^{a_j t}-1}-\frac{r}{t}&=\frac{\sum_{j=1}^r\prod_{i\ne j}(e^{a_i t}-1)(a_j t e^{a_j t}-e^{a_j t}+1)}{t\prod_{j=1}^r(e^{a_j t}-1)}\,.
\left( \sum_{j=n-s+1}^{n}X_{j}^{\ast},X_{n-s}^{\ast},\sum_{j=1}^{n-s-1}X_{j}^{\ast}\right) /U(n)\overset{D}{\rightarrow}\left( \sum_{k=1}^{s}\Gamma_{k}^{-1/\alpha},\Gamma_{s+1}^{-1/\alpha},\sum_{k=s+2}^{\infty}\Gamma_{k}^{-1/\alpha}\right) .
[rsy, y]=1\leq \langle rsy\rangle \cap C_4 &[rsy,s]=r^3sysrsys=y^2=r^2 &[rsy,r]=r^3syr^3rsyr=r^2 &[\langle rsy\rangle,D_8]=\langle r^2\rangle\leq\langle rsy\rangle\cap D_8
c^{2}t_{1}t_{2}-x_{1}x_{2}-y_{1}y_{2}-z_{1}z_{2}>\sqrt{(c^{2}t_{1}^{2}-x_{1}^{2}-y_{1}^{2}-z_{1}^{2})(c^{2}t_{2}^{2}-x_{2}^{2}-y_{2}^{2}-z_{2}^{2})}
|Q_s^n \varphi(v_1) - Q_s^n \varphi(v_2)| \leq \left| \int \varphi( G_n \!\cdot\! v_1) \left[ q_{n}^{s}(v_1,G_n)- q_{n}^{s}(v_2,G_n) \right] \mu(dg_1) \ldots \mu(dg_n)\right| & + \left| \int \left[ \varphi( G_n\! \cdot\! v_1)- \varphi( G_n\! \cdot\! v_2) \right] q_{n}^{s}(v_2,G_n) \mu(dg_1) \ldots \mu(dg_n) \right| & =: I_1+I_2.
\begin{aligned}S_3 = \sum_{I \in \mathcal{D}}\sum_{\alpha \in \Gamma_d}D_I(w,\alpha)^2 \langle w \rangle^{-1}_I \left \langle f w^{\frac{1}{2}}w^{-\frac{1}{2}}\ \right \rangle_{E_{\alpha,I}}^2 \lesssim [w]_{A_2}\| f w^{-\frac{1}{2}}\|^2_{L^2}= [w]_{A_2}\| f \|^2_{L^2(w^{-1})}, \end{aligned}
0 = u_{p,n,e}(p^i) - u_{p,n,e}(p^{i+1}) = e(\nu_p(n) - i) + p^i - e(\nu_p(n) - (i+1)) - p^{i+1}= e + p^i(1-p) , \mbox{\ \equivalently,}i = \log_p\frac{e}{p-1}.
A_3={qQ(r-m)\sin^2\theta\over T[(r-m)^2-m^2\cos^2\theta]},\ \ \A_4=-{qs(1-8{1\over I_0^3}e^{-\tau_0\tau}(1-{2m\over r})^2g_{22})\over 2\sqrt{2}\,T}
\begin{aligned}\frac{d}{dt}\int_{-\infty}^t a(t-s)u(s) \,{\rm d}s = A u(t) + f(\tilde{u}(t))+ g\, \left[r(\tilde u(t),\gamma(t)) + \dot W(t)\right], t \in [0,T] u(t) = u_0(t), t \le 0. \end{aligned}
G_{(1)}^Y(z)_+=z+G^{X_1+X_2}_{X_3}(z)\ ,G_{(1)}^Y(z)_-=t_3z^2+t_2z-W_{(1)}(z)\ , &G_{(2)}^Y(z)_+=z+G^{X_3}_{X_1+X_2}(z)\ ,G_{(2)}^Y(z)_-=t_3z^2/4+t_2z/2-W_{(2)}(z)-W_{(2)}(-z)\ .
W^2_\nu(\mu_s,\mu_t) =\int_XW_2^2(\mu_t^y, \mu_s^y)d\nu(y) = (t-s)^2\int_XW_2^2(\mu_0^y, \mu_1^y)d\nu(y)=(t-s)^2W^2_\nu(\mu_0,\mu_1)
x_1\bar{\otimes}x_2 = (h_{\varphi_1}^{\eta/q}\bar{\otimes}h_{\varphi_2}^{\eta/q})(x_1'\bar{\otimes}x_2')(h_{\varphi_1}^{(1-\eta)/q}\bar{\otimes}h_{\varphi_2}^{(1-\eta)/q})= h_{\varphi_1\bar{\otimes}\varphi_2}^{\eta/q}(x_1'\bar{\otimes}x_2')h_{\varphi_1\bar{\otimes}\varphi_2}^{(1-\eta)/q},
{}_2F_1\bigg[\begin{matrix}\alpha 1-3\alpha &\frac32-2\alpha\end{matrix}\bigg|\,\frac14\bigg]=\frac{16^{-\alpha}}{27^{-\alpha}}\cdot\frac{\Gamma(\frac43)\Gamma(\frac32-2\alpha)}{\Gamma(\frac32)\Gamma(\frac43-2\alpha)}.
\Gamma_1^{(I)}\;=\; \frac{1}{2}{\rm Tr}\ln \left[ \delta(x_1-x_2) - 2 g^2 \int_y \Delta_\varphi(x_1,y){\bar \chi}(y){\mathcal D}^{-1}(y,x_2) \chi(x_2) \right] \;.
F_m(p) := \begin{cases}\min_{v}c \displaystyle \sum_{i=1}^n \sum_{k=1}^m p_i(\phi_{k,i}(x_i) - d_{i,k})\cdot \P(\omega_k) + \sum_{i\neq j}\alpha_{ij}v_{ij}s.t\quad \begin{cases}-v\leq 0 \sum_{j\neq i}v_{ij}- x_{0i}\leq 0,\forall i\in I.\end{cases}\end{cases}
\frac{d^{2}v}{d t^{2}}= \biggl ( \frac{1}{2 v}+ \frac{1}{v - 1}\biggr ) \biggl ( \frac{d v}{d t}\biggr )^{2}- \frac{1}{t}\frac{d v}{d t}- \frac{1}{2}\frac{v ( v + 1 )}{v - 1}.
\begin{aligned}\frac{e^{v(t)}-1}{v(t)}\partial_X V_t(X,Y) =e^{-ty}\frac{1-e^{-tx}}{tx},\frac{e^{v(t)}-1}{v(t)}\partial_Y V_t(X,Y) =\frac{1-e^{-ty}}{ty}.\end{aligned}
\|f_1 f_2\|_{H^k}\leq C\|f_1 f_2\|_{L^2}+C\sum_{j=1}^3\|\nabla \partial_j^{k-1}(f_1 f_2)\|_{L^2}&\leq C\|f_1 f_2\|_{L^2}+C\sum_{j=1}^3\sum_{l=0}^{k-1}\|\nabla (\partial_j^{l}f_1 \partial_j^{k-1-l}f_2)\|_{L^2}.
T_m(d) = \sum_{Q\in \Gamma\backslash \mathcal Q_{d^2}}\chi_d(Q) \begin{dcases}\int_{C_Q}F_{m,Q}(z,s) y^{-1}|dz| d>0, \int_{C_Q}i \partial_z F_{m,Q}(z,s) \, dz d<0.\end{dcases}
\int |\delta E(m)|^2 dx - \int |m \cdot \delta E(m)|^2 dx = \int \left(|\delta E(w_* + \eta)|^2 - |\delta E(w_*)|^2 \right) dx & - \int \left( |(w_*+\eta) \cdot \delta E(w_*+\eta)|^2 - |w_* \cdot \delta E(w_*)|^2 \right) dx
y_{2}=a_{0}\sum_{r=0}^{\infty}\frac{(\alpha)_{r}(\beta)_{r}}{(1)_{r}^{2}}(lnx+\sum_{k=0}^{r-1}(\frac{1}{\alpha+k}+\frac{1}{\beta+k}-\frac{2}{1+k}))x^{r}.
\sum_{k}\varepsilon(X\triangleright\mathsf{P}_{k}^{j})\varepsilon(Y\triangleright\mathsf{P}_{i}^{k})=\sum_{m,n,o,p}\pi(S(X_{(1)}))_{m}^{j}c_{n}^{m}\pi(X_{(2)}S(Y_{(1)}))_{o}^{n}c_{p}^{o}\pi(Y_{(2)})_{i}^{p}.
\frac{c_{L}}{2}\|z_{m}\|_{H}^2 + h\sum_{n=0}^{m-1}\|B_{1}^{1/2}z_{n+1}\|_{H}^2 + \frac{\omega_{2, 1}}{2}\|v_{m}\|_{V}^2 + \frac{\omega_{2, 1}}{2}h^2\sum_{n=0}^{m-1}\|z_{n+1}\|_{V}^2 \leq C_{3}+ C_{2}h\sum_{n=0}^{m-1}\|v_{n+1}\|_{V}\|z_{n+1}\|_{H}
\phi = \phi_S + \phi_R, \phi_S = \phi_0 + \phi_+ + \phi_-, & \phi_\ast(x) :=\frac{1}{\sqrt{2\pi}}\chi_\ast (x) \int \mathcal{K}_\ast^\#(x,k) \, (\mathcal{F}^\# \phi) (k) \, dk, \ast \in \{0,+,-\}& \phi_R(x) :=\frac{1}{\sqrt{2\pi}}\int \mathcal{K}_R^\#(x,k) \, (\mathcal{F}^\# \phi) (k) \, dk.
A_{m}(\omega,\gamma)=\frac{4}{3\pi}\frac{1}{|m|!}\frac{\gamma^{2}}{1+\gamma^{2}}\sum_{n>v}^{\infty}e^{-(n-v)\alpha(\gamma)}w_{m}(\sqrt{\frac{2\gamma}{\sqrt{1+\gamma^{2}}}(n-v)})
H_{1 / 2}^{R p e r t}\equiv H_{\beta}- H_{1 / 2}+ ( \beta \! - \! 1 / 2 ){\frac{\delta ( x )}{x}}= ( \beta \! - \! 1 / 2 ){\frac{\delta ( x )}{x}}+{\frac{( \beta \! - \! 1 / 2 )^{2}}{x^{2}}}.
f_{n-5}= f_{n-6}+A_{2-n}A_{4-n}A_{6-n}\psi_{n-5}^{(0)}+K_{6-n}\psi_{n-5}^{(0)}= O(r^{n-4})\log r+O(r^{n-4}):=b_{n-4}^{(1)}\log r+O(r^{n-4})+O(r^{n-3})\log r.
\bar{O}( p ) \Gamma_{S}( p ) \bar{O}^{T}( - p ) = \hat{\bar{\Gamma}}_{S}( p ) = \left( \begin{matrix}{{0}}&{{\Gamma_{a}^{\phi_{S}}( p )}}\\{{\Gamma_{r}^{\phi_{S}}( p )}}&{{0}}\\ \end{matrix}\right) .
\big|R_{\theta,0;\lambda}(u)\big| \le\frac{2^{\theta-4}|H_\lambda|^4}{2\pi}\int_{-\infty}^\infty\big|\zeta(4-\theta+it)\Gamma(4+it)\big|\,dt &\ll |H_\lambda|^4=\bigg(\frac{H^2}{1+4\pi^2\lambda^2H^2}\bigg)^2,
E ( x ) = \frac{1}{{\partial}_{-}^{\; \; 2}}{\Delta}( x ; m^{2}) - \frac{x^{-}}{m^{2}- n_{-}{\partial}_{-}^{\; \; 2}}{\delta}( x^{+}) + \frac{2 n_{+}{\partial}_{+}}{( m^{2}- n_{-}{\partial}_{-}^{\; \; 2})^{2}}{\delta}( x^{+}) .
P \bigl( \Delta_n < \mathbf{x}_1^T\mathbf{x}_2 - y_n \bigl(\| \mathbf{x}_1\|^2+\| \mathbf{x}_2\|^2\bigr)/2 \leq0 \bigr) =o(1) \bigl\{\mathcal{L}_{n,y}+p^{-2}\bigr\},