harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
{\cal S}= \left( \begin{matrix}{{1 + 3 \epsilon}}&{{\mu}}&{{0}}&{{0}}\\{{\lambda}}&{{1 + \epsilon}}&{{0}}&{{2 \mu / 9}}\\{{0}}&{{0}}&{{1 - 3 \epsilon}}&{{- \lambda}}\\{{0}}&{{- 6 \lambda}}&{{- \mu}}&{{1 - \epsilon}}\\ \end{matrix}\right)
\left| f_0^{(r-q)}(x) \prod_{s=1}^q f_0^{(s-1)}(x)^{m_s}\right| \leq R^\frac{r-q}{\beta}f_0(x)^\frac{\beta-(r-q)}{\beta}\prod_{s=1}^q R^\frac{(s-1)m_s}{\beta}f_0(x)^\frac{(\beta-s+1)m_s}{\beta}& = R^\frac{r-M_q}{\beta}f_0(x)^\frac{\beta-r+(\beta+1)M_q}{\beta}.
V(t) = \int_{\mathcal{X}^{\sigma}\times \mathbb{R}^d}\left( \int_{\mathcal{X}\times\mathbb{R}^d}g^2(x,t-s)h(y,s-u)p(dx)ds\right) L^{\sigma}(dy,du) &= \int_{\mathcal{X}^{\sigma}\times \mathbb{R}^d}k(y,t-u) L^{\sigma}(dy,du).
L_{\infty,q}(0,\infty):=\cap_{p>0}L_{p,q}(0,\infty), &L_{p,\infty}(0,\infty):=\cap_{q>0}L_{p,q}(0,\infty), &L_{as}(0,\infty):=L_{\infty,\infty}(0,\infty):=\cap_{p>0}L_{p,\infty}(0,\infty).
f(s+1)-f(s) = \left(\frac{s+1}{n-s}+ \frac{i-2s-1}{n-s-1}\right)-\left(\frac{s}{n-s+1}+ \frac{i-2s+1}{n-s}\right) &= \frac{s(n-i-2) - (n-i)(n+1)}{(n-s+1)(n-s)(n-s-1)}=\frac{(n-i)(s-n-1)-2s}{(n-s+1)(n-s)(n-s-1)}.
S_P[\eta, A] = t \sum_x \sum_{\stackrel{\scriptstyle \mu, \nu = 1}{\mu < \nu}}^3A_{\mu \nu}(x) \Big[\eta_{+\mu}(x,\nu) \eta_{-\mu}(x+\hat{\mu},\nu) +\eta_{+\nu}(x,\mu) \eta_{-\nu}(x+\hat{\nu},\mu) \Big],
{\bf \Psi}_{n , k_{3}}^{T}( x , y ) = \left( c_{1}\Psi_{n - 1}( x , y ) , i c_{2}\Psi_{n}( x , y ) , c_{3}\Psi_{n - 1}( x , y ) , i c_{4}\Psi_{n}( x , y ) \right) .
\sum_{i=1}^{\widehat m}\omega_i\leq n^{\frac{1-2\beta}{2\beta+1}}\sum_{i=1}^{\widehat m}\frac{\widehat \Delta_i}{f_0(\widehat x_{i-1})^{\frac{2\beta+3}{2\beta+1}}}\leq 8 n^{\frac{1-2\beta}{2\beta+1}}\int_0^1 f_0(x)^{-\frac{2\beta+3}{2\beta+1}}dx,
F_k(x_1, \dots, x_k) := k! \sum_{\substack{\lambda_1, \dots, \lambda_k \geq 0 \lambda_1 + 2\lambda_2 + \cdots + k \lambda_k = k}}\prod_{j=1}^k \frac{(-s_j(x_1, \dots, x_j))^{\lambda_j}}{\lambda_j! j^{\lambda_j}}, F_0 := 1,
\dot{\epsilon}_{\delta\sigma}=\epsilon_{\delta\sigma}\left[3{\dot{\bar{a}}_R\over\bar{a}_R}(1+w)+2{\dot{\bar{c}}_4\over\bar{c}_4}{{1+\epsilon_{\delta\sigma}}\over{1+\epsilon}}+\left({\dot{\bar{G}}_4\over\bar{G}_4}-4{\dot{\bar{c}}_4\over\bar{c}_4}\right)(1+\epsilon_{\delta\sigma})\right].
C_o = c_{1,1}c_{3,2}+ c_{1,3}c_{1,2}+ c_{1,2}c_{1,3}& = ( w_1 - w_1^2 ) ( - w_2 w_3) + ( - w_1 w_3) ( - w_1 w_2 ) + (- w_1 w_2) ( - w_1 w_3) & = (3 w_1^2 - w_1) w_2 w_3,
P^\epsilon(s_2)-P^\epsilon(s_1)=\bigl(e^{-\frac{s_2-s_1}{\epsilon^2}}-1\bigr)P^\epsilon(s_1)+\int_{s_1}^{s_2}e^{-\frac{s_2-s}{\epsilon^2}}f(q^\epsilon(s))ds+\int_{s_1}^{s_2}e^{-\frac{s_2-s}{\epsilon^2}}\sigma(q^\epsilon(s))d\beta(s).
[L^p(\Omega), H_{per, b.c.}^{2,p}(\Omega)]_{\theta}= [R (\oplus_{i\in I}L^p(\tilde{\Omega}_i)), R (\oplus_{i\in I}H_{b.c.}^{2,p}(\tilde{\Omega}_i))]_{\theta}&= R (\oplus_{i\in I}[L^p(\tilde{\Omega}_i), H_{b.c.}^{2,p}(\tilde{\Omega}_i)]_{\theta}).
-\tfrac{1}{2}\int_0^u f'(s)^2 \, ds =-u+\sqrt 2 c(u+\beta t)^{1/3}-\sqrt 2 c (\beta t)^{1/3}+\tfrac{1}{3}c^2(u+\beta t)^{-1/3}-\tfrac{1}{3}c^2(\beta t)^{-1/3}&\le -u+\sqrt 2 c u^{1/3}+O(t^{-1/3}).
\beta_1=f_{11}\eta^1+t_{1\overline{1}}\eta^{\overline{1}}+f_{12}\eta^2+t_{1\overline{2}}\eta^{\overline{2}}+f_{13}\eta^3, \beta_2=f_{21}\eta^1+t_{2\overline{1}}\eta^{\overline{1}}+f_{22}\eta^2+t_{2\overline{2}}\eta^{\overline{2}}+f_{23}\eta^3,
B(x,y,z,t) = 0 \iff x=y z = t \, , B(x,y,z,t) = 1 \iff x = z y=t \, , B(x,y,z,t) = B(x,y,z,w)B(x,w,z,t)\, . B(x,y,z,t) = B(x,t,z,y)^{-1}\, .
\sum_{n=0}^\infty a_n(w;b) z^{n}=\frac{\lambda}{w+\lambda}\left( \mathrm{e}^{b (w+\lambda)(1-p^{-1})-bz}-\mathrm{e}^{-bz p^{-1}}+\left(\mathrm{e}^{-b z}-\mathrm{e}^{-b z p^{-1}}\right)\frac{w+\lambda-z}{z}\right) &-\frac{\lambda}{p}\mathrm{e}^{b(w+\lambda- z )}\tilde{Z}_\downarrow (w, (w+\lambda)p^{-1}; b) .
\int_{\mathcal{X}_n^c}f(x) \int_0^1 \mathrm{B}_{k,n-k}(s) \log u_{x,s}\, ds \, dx &\leq C_{d,f}\int_{\mathcal{X}_n^c}f(x)\biggl\{\log n + \log\biggl(1+ \frac{\|x\|}{\mu_\alpha^{1/\alpha}(f)}\biggr)\biggr\}\, dx = O\bigl(\max\{q_n \log n,q_n^{1-\epsilon}\}\bigr),
\mathrm{S}(\cdot\mid d,\alpha):=\bigg{[}\mathop{\otimes}_{i=1}^{d}\mathcal{N}(0,\varepsilon^{2}d^{2\alpha+1}i^{-(2\alpha+1)})\bigg{]}\otimes\bigg{[}\mathop{\otimes}_{i=d+1}^{\infty}\mathcal{N}(0,0)\bigg{]}.
(1-Q)\sum_{i \geq0}a_i x^i = R + \sum_{i \geq0}(c_1\alpha(a_{i+n-1}) + \dots + c_n\alpha^n(a_i))x^{i+n}&= R + \sum_{i \geq0}a_{n+i}x^{n+i}\textrm{(by assumption)}&= R + \sum_{i \geq n}a_ix^i &= R - \sum_{i=0}^{n-1}a_ix^i + \sum_{i\geq0}a_ix^i
d_{\Gamma_{t+2}}(i,j) =\begin{cases}t+2 \{i,j\}\in \binom{[t]}{2}, t+1 i\in [t] j\in [t+1,t+4], t \{i,j\}\in \{\{t+1,t+4\},\{t+2,t+3\}\}, t-1 \{i,j\}\in \{\{t+1,t+3\},\{t+2,t+4\}\}, 1 \{i,j\}\in \{\{t+1,t+2\},\{t+3,t+4\}\}.\end{cases}
g :=g_0-g_1-g_2+\overline{g}_{n,\alpha}^2 \overline{g}_{m,\alpha}^2 = \Big( g_{n,\alpha}(i_1)\tilde{g}_{m,\alpha}(\ell_{\pi_1}) -\overline{g}_{n,\alpha}\overline{g}_{m,\alpha}\Big) \Big( g_{n,\alpha}(i_2) \tilde{g}_{m,\alpha}(\ell_{\pi_2}) -\overline{g}_{n,\alpha}\overline{g}_{m,\alpha}\Big)\,,
\|T\|_{S_1}\leq \sum_{\lambda\in \Lambda}\|f_\lambda\| \|g_\lambda\| \leq \left(\sum_{\lambda\in \Lambda}\|f_\lambda\|^2\right)^{\frac{1}{2}}\left(\sum_{\lambda\in \Lambda}\|g_\lambda\|^2\right)^\frac{1}{2}= \|\xi^{[N]}\| \|\eta^{[N]}\|.
c= 2u'^2(1-\cos u)+v'^2(2+\cos u)^2, c \kappa= (u''v'-v''u')\sqrt{2}\sqrt{1-\cos u}(2+\cos u)+v'^3\frac{\sin u(2+\cos u)^2}{\sqrt{2}\sqrt{1-\cos u}}&+u'^2v'\frac{(6-3\cos u)\sin u}{\sqrt{2}\sqrt{1-\cos u}}.
\mathcal{K}_{\mathcal{I}_l}\cdot\mathcal{K}_{\mathcal{I}_l}=\mathcal{K}_{\mathcal{I}_l,j,k}\cdot\mathcal{K}_{\mathcal{I}_l,j,k}(1-\nabla_j\cdot\nabla_j-\nabla_k\cdot\nabla_k+(-\nabla_j\cdot\nabla_j)(-\nabla_k\cdot\nabla_k)),
h^1_{n,T^\ast (1)}(\widetilde{\alpha}(y))(\sigma) = \log^0_p(\chi(\gamma_0^{}))\left[\sigma \mapsto \frac{\sigma-1}{\gamma-1}\widetilde{\alpha}(y) - (\sigma-1)\widetilde{\alpha}(b) \right] & = \alpha^\ast \circ h^1_{n,T}.
{\cal L}_{\mathrm{c}}= -{\frac{3}{2}}\left[ S_{0}\overline{{S}}_{0}\, e^{- K / 3}\right]_{D}+ \left[ S_{0}^{3}( i h + \alpha M^{3}) \right]_{F}+{\frac{1}{4}}\sum_{a = 1}^{2}{\left[ ( S + \beta^{a}T ){\cal W}^{a}{\cal W}^{a}\right]_{F}},
c_1>0, \c_2 = -1, \c_3> 0, \ \d_1>0, \d_2= -1, \d_3\geq0, \ \ \frac{1}{2}\left(\frac{d_3}{c_3}+\frac{d_1}{c_1}\right) < \frac{d_2}{c_2}< \frac{d_1}{c_1}.
\begin{array}{ll}\|w\|_{\mathcal{B}_{0,\ell}^{0,1}}+\frac{1}{\sqrt{\beta + C_{\eta}}}\big\|w|_{y=0}\big\|_{A^0}\leq C(\lambda T)\|\tilde{f}\|_{\mathcal{B}_{0,\ell}^{0,0}}\ .\end{array}
\frac{1}{\lambda_{0}}= \frac{\sigma_{0}^{D-2}}{\lambda_{r}}+ \int_{-\infty}^{\infty}\frac{d\omega}{2\pi}\, \frac{1}{(4\pi)^{\frac{D-1}{2}}}\left\{\omega^{2}+(D-2)\sigma_{0}^{2}\right\}(\omega^{2}+\sigma_{0}^{2})^{\frac{D-5}{2}}\Gamma(-\frac{D-3}{2}) \mbox{tr}\mbox{\boldmath$1$}.
R_{1234}^2 + 2 R_{1342}R_{1423}= x^2-2y(x+y) = \frac 13 [ 2(x-y)^2 + 2(x-y)(x+2y) - (x+2y)^2 ] \ge \frac 13 [ 2(x-y)^2 + 2(x-y)(K_{14}-K_{13}) - (K_{14}-K_{13})^2 ]
D(s, S_f) = L(s, f) + \frac{L(s-1, f)}{s + \frac{k-1}{2}- 1}+ \frac{1}{2\pi i}\int_{(\epsilon)}L(s - z, f) \zeta(z) \frac{\Gamma(z)\Gamma(s + \frac{k-1}{2}- z)}{\Gamma(s + \frac{k-1}{2})}\; dz.
A_{11}+A_{33}=\frac{C^1_{2,1}}{\mu}+\frac{(B^1_{11,3})^2-(C^1_1)^2}{\mu^2};~~A_{11}+A_{ii}=\frac{C^1_{2,1}}{\mu}-\frac{(C^1_1)^2}{\mu^2}, i\geq 4; A_{22}+A_{33}=\frac{C^1_{1,2}}{\mu}+\frac{(B^1_{11,3})^2-(C^1_1)^2}{\mu^2};~~A_{22}+A_{ii}=\frac{C^1_{1,2}}{\mu}-\frac{(C^1_1)^2}{\mu^2}, i\geq 4.
\lvert \phi v(w)-\phi v(x)-\phi v(y)\rvert= \lvert \phi v(x_1y_2)-\phi v(x_1z)-\phi v(z^{-1}y_2)\rvert = \lvert \phi v(x_1y_2)-\phi v(x_1)-\phi v(y_2) &-\phi v(x_1z)+\phi v(x_1)+\phi v(z) &-\phi v(z^{-1}y_2)-\phi v(z)+\phi v(y_2)\rvert \leq 2\lvert v\rvert_S+2\lvert v\rvert_S+2\lvert v\rvert_S < \infty.
\sum_{k=0}^n(-1)^{k+n}\frac{2n}{n+k}\binom{n+k}{n-k}m_k(t)[4(1-\epsilon)]^k = 2(-1)^n \int_0^{\pi}\cos(n \zeta) \nu_t(d(\sin^2(\zeta/2)/(1-\epsilon)) & = 2\int_0^{\pi}\cos(n \zeta) \nu_t(d(\cos^2(\zeta/2)/(1-\epsilon)).
x{\Phi}^\ell={\mathcal{L}}^\ell{\Phi}^\ell ,{\mathcal{L}}^\ell=\left(\begin{array}{ccccc}{a}_0^{(\ell)}\mathbb{I}_p b_1^{(\ell)}{a}_1^{(\ell)}\mathbb{I}_p &b_2^{(\ell)}{a}_2^{(\ell)}\mathbb{I}_p & \ddots \ddots \ddots\end{array}\right).
S_{ij}= g_{ij}+ \sum_{\substack{n \geq 0 d \in H_2(X;\mathbb{Z})}}\frac{1}{n!}\left\langle \phi_i, \tau, \dots, \tau, \frac{\phi_j}{1-qL}\right\rangle^{K\textnormal{th}}_{0,n+2,d}Q^d
\begin{aligned}-dq^i_t =\bigg(\bar{\tilde{b}}_x q^i_t+\bar{\sigma}_x m^i_t+\bar{\tilde{\sigma}}_x \tilde{m}^i_t+\int_{\mathcal{E}}\bar{l}_x n^i_{(t,e)}\nu(de)-\beta q^i_t+\kappa^i_t\bigg)dt &\quad-m^i_tdW_t-\tilde{m}^i_td\xi_t-\int_{\mathcal{E}}n^i_{(t,e)}\tilde{N}(de,dt), \end{aligned}
\mathbf{E}(\mathbf{x}) =\frac{1}{4\pi}\int\rho_e(\mathbf{x}')\frac{\mathbf{r}}{r^3}\,d^3x' -\frac{1}{4\pi}\int\mathbf{J}_m (\mathbf{x}')\,{\bf\times}\,\frac{\mathbf{r}}{r^3}\,d^3x'
\{A_{a,0}\}= [A_{a,0}],\{A_{a,1}\}= [A_{a,1}] +v^{-a-1}[A_{a+1,0}], \{A_{a,2}\}=[A_{a,2}] +v^{-a-1}[A_{a+1,1}] + \gamma_a(2) [A_{a+2,0}].
\int_{\mathbb{R}^n}e^{nu}dx \geq\int_{S\backslash (S_0\cup B_1)}e^{n\left(v(x)+p(x)+q\left(\frac{x}{|x|^2}\right)+\beta \ln(|x|)\right)}dx &\geq C\int_{S\backslash (S_0\cup B_1)}|x|^{n\beta}e^{C_0 n |x|^2}dx &=\infty,
\langle \varphi_{i_{1}}( X_{1}) . . . \varphi_{i_{n}}( X_{n}) \rangle_{\zeta}= ( - i )^{n}\prod_{j = 1}^{n}\frac{\delta}{\delta y_{i_{j}}( X_{j})}{\cal G}[ \vec{y}, \vec{\varphi}] \raisebox{- 2 . 8 5 m m}{\scriptsize{\vec{y}= 0}}.
-A=[\begin{matrix}i&-2-i\\ 2-i&0\end{matrix}]=[\begin{matrix}\overline{-i}&\overline{-2+i}\\ \overline{2+i}&\overline{0}\end{matrix}]=[\begin{matrix}\overline{-i}&\overline{2+i}\\ \overline{-2+i}&\overline{0}\end{matrix}]^{T}=A^{H}
\Phi_g^X(y_0, \ldots, y_m, q) = \sum_{k_0, \ldots, k_m}\sum_{\beta \in H_2(X)}\langle\gamma_0^{k_0}, \ldots, \gamma_m^{k_m}\rangle_{g,\beta}^X \frac{y_0^{k_0}}{k_0!}\cdots \frac{y_m^{k_m}}{k_m!}q^\beta
\leq \;\Bigg[ \int_{\Lambda}\prod_{a=1}^N \prod_{i=1}^{n_a}d^2x_{i}^{(a)}d^2y_{i}^{(a)}\prod_{b=1}^N\Big| A(b,b) \Big|^{\frac{\pi}{\pi+gN}p}\Bigg]^{\frac{1}{p}}
T_\ell = \left[ \Gamma \begin{pmatrix}1 0 0 \ell \end{pmatrix}\Gamma \right] \ell \nmid N, U_\ell = \left[ \Gamma \begin{pmatrix}1 0 0 \ell \end{pmatrix}\Gamma \right] \ell \mid N, \langle a \rangle = [\Gamma \gamma_a \Gamma]
A = - \frac{\kappa^{2}}{4}\, \frac{p_{1 1}^{( 1 )}p_{1 1}^{( 2 )}}{\left( p_{\perp}^{( 1 )}- p_{\perp}^{( 4 )}\right)^{2}}\left( \frac{p_{\perp}^{( 1 )}}{p_{1 1}^{( 1 )}}- \frac{p_{\perp}^{( 2 )}}{p_{1 1}^{( 2 )}}\right)^{4}
{\cal J}\overline{{A}}_{n}^{i}{\cal J}^{- 1}= a_{n}^{i};{\cal J}\overline{{\widetilde A}}_{n}^{\scriptscriptstyle L}{\cal J}^{- 1}= L_{n}^{\scriptscriptstyle T};{\cal J}\overline{{C}}_{n}{\cal J}^{- 1}= c_{n}.
\hat{M_2}(x,\lambda)= 2\lambda M_1(x,\lambda)\Phi(0)+ E_\lambda(-x)\Phi(x)P_\lambda(x) -E_\lambda(x)\int_0^x\Phi(x_1)\Phi'(x_1)dx_1 - \int_0^xE_{\lambda}(x-2x_1)\Phi'(x_1)P_\lambda(x_1)\,dx_1.
\langle X^\mu|P^\mu\rangle=\,\frac{1}{\sqrt{N}}\,\sum^N_{k=1}\,\int_{\varphi^{-1}_k(T_xM)}d^4z_k \exp(\imath\,\vartheta_{kP}(\varphi^{-1}_{k}(x),z_k))\,n_{kP}(\varphi^{-1}_k(x),z_k) &\,\langle X^\mu|\varphi^{-1}_k(x),z_k\rangle.
{\textstyle \sum_{K_2\cup K_{31}\ni j=k}^{\nu}}\,\|x^{j+1}\!-\!x^{j}\|\le\sqrt{2{a}^{-1}}{\textstyle\sum_{K_2\cup K_{31}\ni j=k}^{\nu}}\sqrt{\Phi(x^{\ell(j+1)})-\Phi(x^{j+1})}.
{\rm grad}_h|f|^2=\sum_{i,j=1}^nh^{ij}\frac{\partial|f|^2}{\partial x^i}\frac{\partial}{\partial x^j}+\sum_{i,j=1}^nh^{in+j}\frac{\partial|f|^2}{\partial x^i}\frac{\partial}{\partial x^{n+j}}
\real\frac{z \mathtt{f}'_{a, \nu}(z)}{\mathtt{f}_{a, \nu}(z)}=a^{a/2}-\frac{a^{a/2}}{a\nu-a+1}\real\sum_{n=1}^\infty \frac{2z^2}{\mathtt{j}^2_{\nu, n}-z^2}&\geq a^{a/2}-\frac{a^{a/2}}{a\nu-a+1}\sum_{n=1}^\infty \frac{2|z|^2}{\mathtt{j}^2_{\nu, n}-|z|^2}=\frac{|z| \mathtt{f}'_{a, \nu}(|z|)}{\mathtt{f}_{a, \nu}(|z|)}.
\begin{cases}\dot{x}_{i}^{N}(t)=\frac{1}{N}\underset{1\leq j\leq N:j\neq i}{\sum}m_{j}^{N}(t)\mathrm{sgn}(x_{j}^{N}(t)-x_{i}^{N}(t)), x_{i}(0)=x_{i}^{0,N}, \dot{m}_{i}^{N}(t) = \psi_{i}^{N}(\mathbf{x}_{N}(t),\mathbf{m}_{N}(t)), m_{i}(0)=m_{i}^{0,N}.\end{cases}
\prod_{\substack{1\leq i<j\leq m 0\leq k<\theta}}{\big(t^{N-1}x_i - t^{N-1}x_jq^k\big)^{-1}}\times (q-1)^{-\theta{m \choose 2}}\times\prod_{i=1}^m{\frac{[\theta(N-i+1)-1]_q!}{[\theta N - 1]_q!}}
\mathcal{L}_{\Pi^{\sharp}(d_A f_1 \wedge \cdots \wedge d_A f_{n-1})}g = \rho \Pi^{\sharp}(d_A f_1 \wedge \cdots \wedge d_A f_{n-1}) (g) = \langle \Pi^{\sharp}(d_A f_1 \wedge \cdots \wedge d_A f_{n-1}), d_A g \rangle = \Pi (d_A f_1, \ldots , d_A f_{n-1}, d_A g) = \{f_1, \ldots, f_{n-1}, g \}_{\Pi}.
\frac{u(t_{n+1}) -2 u(t_n) + u(t_{n-1})}{\tau^2}= u_{tt}(t_n) + \bar{R}\ , u(t_{n+1}) + 2 u(t_n) + u(t_{n-1}) = 4 u(t_n) + \bar{R}\ , \frac{f'(u(t_{n + 1})) + f'(u(t_{n}))}{2}= f' \left( u \left( t_{n + \frac{1}{2}}\right) \right) + \bar{R}\ ,
\mathbb P\begin{bmatrix}\begin{Vmatrix}{\frac 1{(2k)^l}}\sum_{s\in{\hat S}^l}{\tilde H}_t(xs)\end{Vmatrix}_{L^2}\leq (2^{-l}t^{-\frac n2}+\eta)\end{bmatrix}\geq 1-{\frac{2^{\frac{n+4}2}}{n!\eta}}\exp\begin{pmatrix}{\frac{-k}{16\ln 2}}\end{pmatrix}
\chi_{m}^{( 1 )}( t ) = - \frac{( u_{m}- v_{m})^{2}}{[ ( z_{m}- u_{m}) ( z_{m}- v_{m}) Q_{m}^{2}( t ) ]^{3 / 2}}\, , \quad \phi_{m}^{( 1 )}( t ) = \frac{\vartheta_{m}Q_{m}^{2}( t ) \sqrt{( z_{m}- u_{m}) ( z_{m}- v_{m}) ]}}{( u_{m}- v_{m})}
\mathcal{W}_{x_0}(x)=a(x_0)w_{3/2}\left(\frac{(x-x_0)\cdot \nu_{x_0}}{(\nu_{x_0}\cdot A(x_0)\nu_{x_0})^{1/2}}, \frac{x_{n+1}}{(a^{n+1,n+1}(x_0))^{1/2}}\right).
M_N\leq D^p(1-r^{-p})\Big[\frac1{r^{d/2-p}-1}+\frac1{1-r^{-p}}\Big]\sum_{n\geq 0}a^{pn}\mu(G^a_n) \Big(\frac{\sqrt A r^{d}}{2(r^{d/2}-1)\sqrt{N\mu(G_n^a)}}\Big)^{2p/d}= 2^p \frac{\kappa_{d,p}}{N^{p/d}}\sum_{n\geq 0}a^{pn}[\mu(G^a_n)]^{1-p/d},
{\cal L}= \frac{1}{2 g l_{s}}\mathrm{T r}\; \left[ \dot{X}^{a}\dot{X}^{a}+ \frac{1}{2}[ X^{a}, X^{b}]^{2}+ \theta^{T}( i \dot{\theta}- \Gamma_{a}[ X^{a}, \theta ] ) \right] \, .
a^{i+1}_q = a^i_q + \frac{\alpha}{m}\left( y_i - \hat{y}_i \right) = a^i_q - \frac{\alpha}{m}\left( \sum_{j=1}^m a^i_j - \sum_{j=1}^m a_j \right) , q \in \left\lbrace 1,2,\ldots,m \right\rbrace .
\begin{array}{l}P_{a,k}(r|R_{mk},\theta_{mk}) = \prod\limits_{n=1}^N\mathbb{P}[P_{r,mk}^{new}\geq P_{r,nj}^{new}]= \prod\limits_{n=1}^N\mathbb{P}[\cos(\theta_{nj})\leq \frac{R_{nj}^2+x_{nj}}{2rR_j}]\end{array}
R_{u,v}^{J,q}(q)=(-1)^{\ell(v)-\ell(u)}\left(1-q+\delta_{u^{-1}(i-1),v^{-1}(i-1)}q^{1+a_{v^{-1}(i-1)}(u,v)}\right) begin{align*}5pt] \left(1-q+\delta_{u^{-1}(i),v^{-1}(i)}q^{1+a_{v^{-1}(i)}(u,v)}\right)\prod_{j\in D(u,v)}\left(1-q^{a_j(u,v)}\right).
\mathbf{k}_\lambda =\mathbf{k}_{\mathbf{m},j}=\frac{2\pi}{L}\mathbf{m}, \omega_\lambda= |\mathbf{k}_\lambda| = \frac{2\pi}{L}|\mathbf{m}| \mathbf{e}_\lambda =\mathbf{e}_j(\mathbf{k}_\lambda) .
{\cal S}_n(\widehat{B}^+, \widehat{B}^-) := \{\pi \in ({\cal S}_n(\widehat{B}^+, \widehat{B}^-))^{(\lambda^n_0 - \lambda_0)-}: \pi(\lambda^n_0 - \lambda_0) = y^n \}\widehat{\cal S}_n(\widehat{B}^+, \widehat{B}^-) := \{\widehat{\pi}\in \widehat{\cal S}_n(\widehat{B}^+, \widehat{B}^-): \sigma_{\widehat{\pi}}\leq (\lambda^n_0 - \lambda_0)\}.
p_{r}=(\begin{matrix}0&0&1&0&0&0&0\\ 0&0&0&0&1&0&0\\ 0&0&0&0&0&1&0\\ 0&0&0&0&0&0&1\\ \end{matrix})(\begin{matrix}0\\ 1\\ 1\\ 0\\ 0\\ 1\\ 1\end{matrix})=(\begin{matrix}1\\ 0\\ 1\\ 1\end{matrix})
\int_{-\infty}^{\infty}y_{1}e^{-\frac{1}{2(1-\rho^{2})}y_{1}^{2}+\frac{\rho y_{2}}{1-\rho^{2}}y_{1}}dy_{1}=\rho\sqrt{2\pi(1-\rho^{2})}y_{2}e^{\frac{\rho^{2}y_{2}^{2}}{2(1-\rho^{2})}}
C [ \Delta , \omega ] = \frac 1 6 \sum_{\pm}\left\{c_{0}+ c_{2}\sinh^{2}( r_{\pm}/ \ell ) + c_{4}\sinh^{4}( r_{\pm}/ \ell ) - \alpha_{0}\log \left( \ell \sinh ( r_{\pm}/ \ell ) \right) \right\}.
\lim_{R \to \infty}\int_{a}^{b}|f(x)- \Psi_R^{\delta}f(x) |^p \omega (x) dx =0,\mbox{where}\displaystyle\Psi_R^{\delta}f = \sum_{n=0}^{\infty}\Big(1-\frac{\chi_n}{R}\Big)^\delta_+ a_n(f) \varphi_n.
\prod_{\alpha = 1}^{3}\left[{\frac{P^{\alpha}! Q^{\alpha}! R^{\alpha}! S^{\alpha}! U^{\alpha}! V^{\alpha}! ( U^{\alpha}! + 2 I^{\alpha}! 1 ) ( 2 I^{\alpha}! + 1 )}{( V^{\alpha}! + 2 I^{\alpha}! + 1 )}}\right]^{1 / 2}\, .
f(\cdot,\omega^{n})=\sum_{(k^1,\ldots,k^{n-1})\in\mathcal{K}_{1,\ldots ,n-1}}\mathcal{A}^{p^{n-1}(\omega^{n})}_{k^1,\ldots,k^{n-1}}(a_{k_1,\ldots k_{n-1}}(\cdot, \omega^{n})),
4 \sin^2\Big(\frac{z}{2}\Big) = (\cos(z)-1)^2 + \sin^2(z) 2\sin^2\Big(\frac{y}{2}\Big) + 2\sin^2\Big(\frac{z}{2}\Big) - 2\sin^2\Big(\frac{z-y}{2}\Big) = (\cos(y)-1)(\cos(z)-1) + \sin(y)\sin(z).
\|A\|_{((\mathbb{C}^2,\|\cdot\|_{\Omega})\rightarrow(\mathbb{C}^2,\|\cdot\|_{\Omega})^*)}=\sup_{(z_1,z_2)\in\Omega}(a_{11}|z_1|^2+2Re(a_{12}z_1\overline{z_2})+a_{22}|z_2|^2).
\Vert\varphi\Vert_{X_m}^2:=\Vert\varphi_{x}\Vert^2_{L^2_\gamma(-\infty, T; H^{m+1}(\mathbb T))}+\Vert\varphi_t\Vert^2_{L^2_\gamma(-\infty, T; H^{m+1}(\mathbb T))}+\Vert\varphi_{tt}\Vert^2_{L^2_\gamma(-\infty, T; H^{m}(\mathbb T))}\,.
\sum_{k=0}^{m}C^{(+)}_k(m) (\tfrac{\beta}{2}-k)_{2k}=(\lambda)_m (\tfrac{\beta}{2})_m, \sum_{k=0}^{m}D^{(1)}_k(m) (\tfrac{\beta}{2}-k+1)_{2k}=(\lambda)_m (\tfrac{\beta}{2})_{m}(\lambda-\beta).
\theta_{0,0}=\frac12 \log v, &\theta_{0,p}=\frac{2^{p-1}(2p-2)!!}{(2p)!}v^{2p},\theta_{0,-p}=(-1)^{p+1}\frac{(2p-1)!}{2^{p+1}(2p)!!}v^{-2p},p\ge 1, &\theta_{1,p}=\frac{2^{p}(2p-1)!!}{(2p+1)!}v^{2p+1},p\ge 0.
V(z)=[1-{z\over{Q_p}}{{1+{{(Q_pb)^{n\over 2}}\over{2(x^2+4Q_pz)^{n\over 2}}}}\over{(1+{{(Q_pb)^{n\over 2}}\over{(x^2+4Q_pz)^{n\over 2}}})^2}}]+{{J^2_{\perp}}\over{2m_fQ^2_p}}{1\over{(1+{{(Q_pb)^{n\over 2}}\over{(x^2+4Q_pz)^{n\over 2}}}})^2},
\lim_{x \rightarrow 0}x \frac{d \hat{G}( x )}{d x}= \lim_{x \rightarrow 0}\frac{1}{\pi^{2}N}\frac{\hat{G}^{3}( x ) \, x^{3}}{m^{2}\left( x^{2}+ m^{2}\right)}\rightarrow 0 ,
H(\phi)= \int_M \log \frac{\omega_\phi^n\wedge \eta}{\omega^n_T\wedge \eta}dv_\phi \mathbb{J}_{-Ric}(\phi)= \frac{n\underline{R}}{(n+1)!}\int_M \phi \sum_{k=0}^n \omega^k_T\wedge \omega_\phi^{n-k}\wedge \eta-\frac{1}{n!}\int_M \phi \sum_{k=0}^{n-1}Ric\wedge \omega^k_T\wedge \omega_\phi^{n-1-k}\wedge \eta
G\left(a,r_0\right)=\lim\limits_{R\rightarrow \infty}\left(\left\langle \infty|S\left(R,r_0\right)|0\right\rangle_a +\int \limits_{r_0}^{R}\left\langle \infty|S\left( R ,r\right) H_{int}S\left( r,r_0\right)|0 \right\rangle_{a}\frac{dr}{r}\right) ,
\frac{\P(X_1>s/t_1)}{\P(X_1>s/t_2)}=\frac{\frac{1}{2}-\frac{s}{2\sqrt{t_1^2+s^2}}}{\frac{1}{2}-\frac{s}{2\sqrt{t_2^2+s^2}}}=\frac{\frac{t_1^2}{\sqrt{t_1^2+s^2}(\sqrt{t_1^2+s^2}+s)}}{\frac{t_2^2}{\sqrt{t_2^2+s^2}(\sqrt{t_2^2+s^2}+s)}}&=\frac{t_1^2}{t_2^2}\cdot\frac{\sqrt{t_2^2+s^2}(\sqrt{t_2^2+s^2}+s)}{\sqrt{t_1^2+s^2}(\sqrt{t_1^2+s^2}+s)}.
\iint_{\max(|x-y|, |x-z|) > \varepsilon}|K(x,y,z) f(y)g(z)|\,d\mu(y)\,d\mu(z) \lesssim \frac{1}{\varepsilon^{m(1/p_1+1/p_2)}}\|f\|_{L^{p_1}(\mu)}\|g\|_{L^{p_2}(\mu)}.
x'-h-1 =(p^e-p^t)-\Big(1+p^e \ \ \ \ +p+{\sum}_{i=1}^{t-1}(p-1)p^i+(s_{t+1}-2)p^t+{\sum}_{i=t+1}^{e-1}s_{i+1}p^i\Big)\in\Upsilon_x,
\frac{{\partial R}}{{\partial \lambda_j^{n}}}= \frac{{\partial{U^n}}}{{\partial \lambda_j^{n}}}+ \sum\limits_{l = 1}^N \sum\limits_{s = 1}^J{\frac{{\partial{U^l}}}{{\partial{\beta_s}}}\cdot\frac{{\partial{\beta_s}}}{{\partial \lambda_j}}= 0}\ .
Q_1 = \mathrm{P}\left( y<\left(\frac{\epsilon_1}{\rho}\right)^{-\frac{1}{\alpha}}, x>\left(\frac{(1+\epsilon_1)}{\phi_i \left( \rho -\epsilon_1 y^{\alpha}\right)}\right)^{-\frac{1}{\alpha}}\right),
\hat{k}(\xi)=c_{0}c_{0}' \frac{\pi^{2}\csc\left( \pi\left(\frac{1}{2}-\beta+i\xi\right) \right) \csc \left( \pi\left( \frac{1}{2}-\beta-i\xi\right) \right)}{\Gamma(1+2\beta)\Gamma(1-2\beta)}.
q =-\sqrt{\alpha}+s^{-1}\frac{(\alpha-1)\sqrt{\alpha}(v_1+v_2)}{4(v_1\sqrt{\alpha}-v_2)(v_2\sqrt{\alpha}-v_1)}+O(s^{-2}), p =s\frac{v_1\sqrt{\alpha}-v_2}{\sqrt{\alpha}(\sqrt{\alpha}+1)}+\frac{(\sqrt{\alpha}-1)\left[(v_1+2v_2)\sqrt{\alpha}+v_2\right]}{4\sqrt{\alpha}(\sqrt{\alpha}+1)(v_2\sqrt{\alpha}-v_1)}+O(s^{-1}).
\begin{array}{c}\widetilde{\psi}(t,x):= \frac{b^{N/4}}{C_N}\exp\Big[ \Phi(t,x) \Big] \begin{array}{ll}\Phi(t,x):= i \Big( S(t,x)+\frac{1}{2}\langle Q(t) [x-x_c(t)],x-x_c(t)\rangle \Big) + \int_0^t \left(i \langle x_c(s),E(s)\rangle - \frac{[Q(s)]}{2}\right) ds \Big]\,.\end{array}\end{array}
\mu(c) =\gamma(c)\: u^{-1}&=q(c,c\wedge m_\gamma)\iota(c\wedge m_\gamma,m_\gamma)\gamma(m_\gamma)(\gamma(m_\gamma))^{-1}\ref{rmkinvcone}&=q(c,c\wedge m_\gamma)\iota(c\wedge m_\gamma,m_\gamma).
\mathbf F_{\chi}\left(\left(\begin{array}{cc}z\mathbf 1_2 0 0 z\mathbf 1_2\end{array}\right)g \left(\begin{array}{cc}\alpha \beta -\beta \alpha\end{array}\right)\right) = \det(\mathbf k)^{k+1}\mathbf F_{\chi}(g)
\begin{aligned}\exists t\in I\mapsto p(t)\in T^*_{q(t)}M H^1,\p(1)=p_1,\t\in I, \left\lbrace\begin{aligned}0\ \ \ \quad =\partial_u H^\lambda(q(t),p(t),u(t)), (\dot{q}(t),\dot{p}(t)) =\nabla^\omega H^\lambda(q(t),p(t),u(t)).\end{aligned}\right.\qquad\end{aligned}
q^{Q}\prod_{k = 1}^{Q - 1}\frac{q z_{i}- z_{k}}{z_{i}- q z_{k}}= \frac{\left( z_{i}- i \tau \kappa \lambda^{- \frac{1}{2}}q^{\frac{1}{2}}\right) \left( z_{i}+ i \kappa \lambda^{\frac{1}{2}}q^{\frac{1}{2}}\right)}{\left( q^{\frac{1}{2}}z_{i}+ i \tau \kappa \lambda^{- \frac{1}{2}}\right) \left( q^{\frac{1}{2}}z_{i}- i \kappa \lambda^{\frac{1}{2}}\right)}.
\hat G_{*,*}= \frac{1}{n}\sum_{i=1}^n G_{i,i}\hat G_{k,*}= \frac{1}{n}\sum_{i=1}^n G_{k,i}\hat G_{*,k}= \frac{1}{n}\sum_{i=1}^n G_{i,k}\hat G = \frac{1}{n^2}\sum_{i=1}^n \sum_{j=1}^n G_{i,j}
\begin{aligned}\frac{(-1)^r}{r!}\frac{\mathrm{\partial}^r}{\mathrm{\partial}\alpha^r}Z^{\star}_{I}(\{k_i\}_{i=1}^{n};(\alpha,\beta))=\sum_{\begin{subarray}{c}r_1+\cdots+r_{n-1}= r r_i\in\mathbb{Z}_{\ge0}\end{subarray}}Z^{\star}_{I}(\{k_i,\{1\}^{r_i}\}_{i=1}^{n-1},k_n;(\alpha,\beta))\end{aligned}
\prod_{j=0}^{k-1}\frac{\Gamma(\frac{j}{4}+\frac{1}{2})}{\Gamma(\frac{j}{4}+\frac{5}{4})}= \frac{\Gamma(\frac{k+1}{4})\Gamma(3/4)}{\pi 2^{-2k+\frac{1}{2}}k!}.
\sum_{n=0}^{\infty}\textup{PD}_\textup{t}(864n+216)q^n \equiv \dfrac{f_3^2}{f_{6}}\cdot f_2^3 \equiv \dfrac{f_3^2}{f_{6}}\left(f_6a(q^6)-3q^2f_{18}^3\right)~(\textup{mod}~4),
(\phi_{\mu,\nu}(g)P)(\begin{matrix}z_{1}\\z_{2}\\ \end{matrix})=[\phi_{\mu,\nu}(\begin{matrix}a&b\\c&d\\ \end{matrix})P](\begin{matrix}z_{1}\\z_{2}\\ \end{matrix})=P((\begin{matrix}a&b\\c&d\\ \end{matrix})^{-1}(\begin{matrix}z_{1}\\z_{2}\\ \end{matrix})),P\in P_{\mu,\nu}^{2}.
\nabla_x\phi^+(x,\xi)-\xi=\lim_{t\to\infty}\int_0^t(\nabla_xI^L_{a\rho})\left(\tau,q(\tau,t,y(0,t,x,\xi),\xi)\right)d\tau, &q(\tau,t,y(0,t,x,\xi),\xi)=q(\tau,0,x,\eta(t,0,x,\xi)).
\frac{u_{-}-v_{+}+h}{u_{-}-v_{+}}f_{1}^{+}(u)k_{1}^{+}(u)f_{1}^{-}(v)k_{1}^{-}(v)=\frac{u_{+}-v_{-}+h}{u_{+}-v_{-}}f_{1}^{-}(v)k_{1}^{-}(v)f_{1}^{+}(u)k_{1}^{+}(u).
- \frac{k}{2 \pi}g^{- 1}\partial_{-}g \equiv \widetilde{a}^{I}G^{I}\equiv \left( \widetilde{J}^{3}\frac{\sigma^{3}}{2}+ \widetilde{J}^{+}\sigma^{+}+ \widetilde{J}^{-}\sigma^{-}\right) + \left( \frac{d - 1}{2 C_{\rho}}\widetilde{B}^{a}T^{a}\right) + \left( \widetilde{F}_{+ \alpha}R^{+ \alpha}+ \widetilde{F}_{- \alpha}R^{- \alpha}\right)
F \rightarrow F + \frac{\Gamma ( - 1 / 2 )}{( 4 \pi )^{3 / 2}R}[ \int_{B^{3}}( \frac{\lambda}{2}\hat{\phi}^{2})^{3 / 2}d v + \frac{2}{3 R}\int_{S^{2}}( \frac{\lambda}{2}\hat{\phi}^{2})^{1 / 2}d s ]
K ( \alpha_{i},{\bar{\alpha}}_{i}) \approx \sum_{i}^{n}\alpha_{i}\bar{\alpha}_{i}+{\frac{1}{4}}\sum_{i}^{n}\alpha_{i}^{2}\bar{\alpha}_{i}^{2}+ \sum_{i < j}^{n}( \alpha_{i}\bar{\alpha}_{i}\alpha_{j}\bar{\alpha}_{j}-{\frac{1}{4}}\alpha_{i}^{2}\bar{\alpha}_{j}^{2}-{\frac{1}{4}}\alpha_{j}^{2}\bar{\alpha}_{i}^{2}) ,
\Phi_\varepsilon(t,x,y):=u^\varepsilon(t,x)-v_\varepsilon(t,y)-\varphi(t,x,y)\quad &\Phi_{\varepsilon,\delta}(t,x,s,y):=u^{\varepsilon,\delta}(t,x)-v_{\varepsilon,\delta}(s,y)-\varphi(t,x,y)-\frac{|t-s|^2}{\delta}.
H_{k,\tau}:=4\left[1+\tau\beta_{k,\max}(2-\beta_{k,\min})\right],\mbox{}C:=\max_{j\in [m]}c^j,\mbox{}C_g:=\min_{j\in[m]}C_g^j,\mbox{}G_\tau:=\frac{C C_g^2\tau}{(\tau-1)},\mbox{}A_{k,\tau}:=\frac{\mathsf{B}_k}{G_\tau}.
\delta_{\alpha+\beta,n-1}+ \sum_{\substack{\ell\ge \alpha,m\ge \beta, \ell+m=n}}{\ell\brack\alpha}_r{m\brack \beta}_s = \sum_{k=0}^n{n\brack k}_{r+s}\left({k-\beta \brack \alpha}_{r}+{k-\alpha\brack \beta}_{s}\right).
F^2_{n,\underline{k}}(g)(x):=C^{n+1}\int_{\mathbb{R}^{n+2}}g(x-\rho)\prod_{l=1}^{n+1}P'_{k_l}h(x-\rho-y_l) &\times 2^{2k_{n+1}}\chi'_0(2^{k_{n+1}}y_{n+1})\prod_{l=1}^{n}K_{k_l}(y_l,\rho)\,d\rho d\underline{y}.