harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\| m^{-1}(u) - m^{-1}(v) \| = \left\| \frac{u-v}{\|u\|}+ \frac{v \|v\|-v\|u\|}{\|u\| \cdot \|v\|}\right\| = \left\| \frac{u-v}{\|u\|}+ \frac{v (\|v\|-\|u\|)}{\|u\| \cdot \|v\|}\right\| \leq &\leq \frac{\|u-v\|}{\|u\|}+ \frac{\left| \|v\| - \|u\| \right|}{\|u\|}\leq \frac{2\|u-v\|}{\|u\|}\leq \frac{2}{\beta}\|u-v\|,
\lVert AB-A_0B_0 \rVert^2 =\sum_{i=1}^M N(a_i-a^0_i)^2 &= \sum_{i=1}^{M-1}N(a_i-a^0_i)^2 + N \left( 1-\sum_{i=1}^{M-1}a_i -1 + \sum_{i=1}^{M-1}a^0_i \right)^2 &= \sum_{i=1}^{M-1}N(a_i-a^0_i)^2 + N \left\{\sum_{i=1}^{M-1}(a_i - a^0_i) \right\}^2.
{{\bf{w}}_k^{\star}}= \sqrt{{\mu_k^{\star}}}{{\bf{B}}_k}{\bf{\Sigma}}_k^{- 1}\frac{{{\bf{A}}_k^H{{\bf{e}}_k}}}{{\left\|{{\bf{A}}_k^H{{\bf{e}}_k}}\right\|}}, \ \forall k,
\left\Vert C_{1}^{1/2p}C_{2}^{1/2p}\cdots C_{L}^{1/2p}\right\Vert_{2p}^{2p}=\operatorname{Tr}\left\{\left[ C_{L}^{1/2p}\cdots C_{2}^{1/2p}C_{1}^{1/p}C_{2}^{1/2p}\cdots C_{L}^{1/2p}\right]^{p}\right\}.
\Theta_{\mathrm{D}}(\delta) := \left\{\int_{\mathcal{S}_1 \times \mathcal{S}_2}g\, d \gamma : \gamma \in \Sigma_{\mathrm{D}}(\delta) \right\}\Theta(\delta) := \left\{\int_{\mathcal{S}}f \, d \gamma : \gamma \in \Sigma(\delta) \right\}.
\sum_{i=1}^{m}\omega_{i}u(\xi_{i})=\frac{\Gamma(\gamma)}{\Gamma(\alpha)}\Omega \sum_{i=1}^{m}\omega_{i}\int^{\xi_{i}}_{0}s^{\rho-1}\left(\frac{\xi_{i}^{\rho}-s^{\rho}}{\rho}\right)^{\alpha-1}\left(f(s,u(s),^\rho D^{\alpha,\beta}u(s))-p(s)u(s)\right) ds.
D = \left( \begin{matrix}{\partial \kern - 0 . 5 e m / \otimes{1 \kern - . 4 e m 1}}&{\gamma_{5}\otimes M}\\{\gamma_{5}\otimes M^{\dag}}&{\partial \kern - 0 . 5 e m / \otimes{1 \kern - . 4 e m 1}}\\ \end{matrix}\right)
A\begin{pmatrix}a_1 \vdots a_{\ell-1}\end{pmatrix}=\begin{pmatrix}(C'_1,C'_1) \cdots (C'_1,C'_{\ell-1}) \vdots \vdots (C'_{\ell-1},C'_1) \cdots (C'_{\ell-1},C'_{\ell-1})\end{pmatrix}\begin{pmatrix}a_1 \vdots a_{\ell-1}\end{pmatrix}=\begin{pmatrix}-1 \vdots 0\end{pmatrix}
\phi_\infty(\varphi_{j,\infty}(a)\varphi_{k,\infty}(b)) = \phi_\infty(\varphi_{l,\infty}(a'b') =\phi_l(a'b') = \phi_l(b'\alpha_{l,i\beta}(a')) = \phi_\infty\big(\varphi_{l,\infty}(b')\alpha_{i\beta}(\varphi_{l,\infty}(a'))\big) = \phi_\infty\big(\varphi_{j,\infty}(b)\alpha_{i\beta}(\varphi_{j,\infty}(a))\big).
\bigl\langle \bar{X}_{j}- Y_{j}, \bar{X}_{k}- Y_{k}\bigr\rangle = \bigl\langle \bar{X}_{j}, \bar{X}_{k}\bigr\rangle - \bigl\langle Y_{j}, \bar{X}_{k}\bigr\rangle - \bigl\langle \bar{X}_{j}, Y_{k}\bigr\rangle + \bigl\langle Y_{j}, Y_{k}\bigr\rangle \equiv 0.
\begin{aligned}p_1(D^{g}) = \frac{1}{4\pi^2}\left( \|W^+ +\frac{s_g}{12}I\|^2 + \|R_0^*\|^2 - \|R_0\|^2 - \|W^- + \frac{s_g}{12}I\|^2\right)\operatorname{vol}_g &= \frac{1}{4\pi^2}(\|W^+\|^2 - \|W^-\|^2)\operatorname{vol}_g &= \frac{1}{4\pi^2}\|W^+\|^2\operatorname{vol}_g \end{aligned}
\tilde u_{i,n}: =\frac{\sigma_i}{\sigma_1^2+\cdots + \sigma_d^2}(b_{1,n}+ \tilde b_2) =: b_{1,n}^{\sigma_i}+ \tilde b_2^{\sigma_i}\tilde u_{i}=\frac{\sigma_i}{\sigma_1^2+\cdots + \sigma_d^2}(b_{1}+ \tilde b_2) =; b_1^{\sigma_i}+ b_2^{\sigma_i}.
\prod_{i=1}^{k+1}f_i = f_1\cdots f_k f_{k+1}= (h_1 x^{d_2+\cdots +d_k})f_{k+1}= x^{d_2+\cdots +d_k}(h_1f_{k+1}) = x^{d_2+\cdots +d_k}(h x^{d_{k+1}})
\begin{aligned}A_{1}U=C_{1},\VB_{1}=D_{1}, A_{2}X=C_{2},\XB_{2}=D_{2}, A_{3}Y=C_{3},\YB_{3}=D_{3}, A_{4}Z=C_{4},\ZB_{4}=D_{4}, E_1U+VF_1+ E_{2}XF_{2}+E_{3}YF_{3}+E_{4}ZF_{4}=C_c.\end{aligned}
\mathbf{u}_{T}=\Pi_{T}\mathbf{u}=\sum_{\ell=0}^{\infty}\sum_{m\in\iota_{\ell}}\left( v_{\ell}^{m}\nabla_{\Gamma}Y_{\ell}^{m}+w_{\ell}^{m}\mathbf{T}_{\ell}^{m}\right) ,
S_{4g}= \sum_{f \in \mathcal{M}_{\leq 4g}}\frac{d_4(f)}{\sqrt{|f|}}\sum_{\substack{C \in \mathcal{M}_{\leq g}C | f^{\infty}}}\sum_{h \in \mathcal{M}_{2g+1-2d(C)}}\chi_f(h)- q \sum_{f \in \mathcal{M}_{\leq 4g}}\frac{d_4(f)}{\sqrt{|f|}}\sum_{\substack{C \in \mathcal{M}_{\leq g-1}C | f^{\infty}}}\sum_{h \in \mathcal{M}_{2g-1-2d(C)}}\chi_f(h).
\sum_{\substack{N(a) \leq x a \equiv 1 \bmod{(1+i)^3}}}\frac{1}{N(a)}\prod_{\pi | 2a}\left( 1+N(\pi)^{-1}\right)^{-1}=\sum_{\substack{(d) N(d) \leq 2x}}\frac{\mu_{[i]}(d)}{\sigma(d)}\sum_{\substack{a \equiv 1 \bmod{(1+i)^3}\frac{d}{(2,d)}|a,\N(a) \leq x}}\frac{1}{N(a)}.
E_{1}-\widetilde{E}_{1}=UE_{S_{A}}V^{\ast}A^{\dagger}\Phi,E_{3}-\widetilde{E}_{3}=UE_{S_{A}}(I+HH^{\ast}E_{S_{A}})^{-1}, F_{1}-\widetilde{F}_{1}=\Psi A^{\dagger}UF_{S_{A}}V^{\ast},\F_{2}-\widetilde{F}_{2}=-(I+F_{S_{A}}K^{\ast}K)^{-1}F_{S_{A}}V^{\ast}.
\Pi_{odd}(p) = -\frac{4mg^2}{\beta}\sum_{n=- \infty}^{\infty}~\int_{0}^{1}dx \int \frac{d^2l}{(2\pi)^2}[\frac{1 - \frac{1}{2}\{e^{il\tilde p}+ e^{-il\tilde p}\}}{(M_1^2 + ((2n+1)\frac{\pi}{\beta}+i\mu )^2}~] ~~~,
g_{\sigma \sigma}= g_{\tau \tau}= \frac{\widetilde{a}^{2}( r )}{\left( \cosh \tau - \cos \xi \right)^{2}}, \qquad g_{s s}= \frac{\widetilde{a}^{2}( r ) \sin^{2}\xi}{\left( \cosh \tau - \cos \xi \right)^{2}}
\nu_{2}(b_{2}(4n+3))=3, &\nu_{2}(b_{2}(8n+5))=3, &\nu_{2}(b_{2}(16n+i))=3 \;for\; i\in\{6,9,12\}, &\nu_{2}(b_{2}(32n+i))=3 \;for\; i\in\{8,17,26\}, &\nu_{2}(b_{2}(64n+i))=3 \;for\; i\in\{16,33,50\}, &\nu_{2}(b_{2}(128n+i))=3 \;for\; i\in\{32,65,98\}, &\nu_{2}(b_{2}(256n+i))=3 \;for\; i\in\{64,129,194\}, &
h_{2j}(\beta, p , q) \Big |_{\beta = 2 \atop q=0}= c_{2j}\:\: (j \ge 0), h_{2j-1}(\beta, p , q) \Big |_{\beta = 2 \atop q=0}= \tilde{h}_{j}(\beta, p , q) \Big |_{\beta = 2 \atop q=0}= 0 \:\: (j \ge 1) .
\begin{aligned}f(x, u, u_x):=\, c_1\,u_x^3+c_2\,u_x^2\,u+c_3\,u^3+c_4\, u_x^4+c_5\,u_x^3\,u+c_6\,u_x^2\,u^2+c_7 \,u^4+f_{\geq 5}(x, u, u_x),\end{aligned}
U'_{\mathcal X_{t-k}\to y_{t-k}}=\left\{u'\in \mathcal P_{\mathcal X_t}: \big|\{e\in \mathcal A_{\mathcal Y}: \{\beta^{1}_{\mathcal X_1}, \alpha^{1}_{\mathcal X_2},\dots, \beta^{t-k-2}_{\mathcal X_{t-k-1}},u'\}\subseteq e\}\big|\ge \frac{\rho}{2^{t-k-1}}\prod_{j\in [k]\setminus [t-k]}|\mathcal P_{\mathcal X_j}|\right\}.
\begin{cases}l=\dim V = \dim \mathrm{L}\Big(k\cdot \sum_{P\in S}P\Big) \geq ks +1-g l=\dim V =\dim \mathrm{L}\Big(k\cdot \sum_{P\in S}P\Big)= ks+1-g \mathrm{for}\ks\geq 2g-1\end{cases}
C(\varphi_{f,p,q})^{\bullet}:= S^{\bullet}_{X,p}\oplus f_{*}S^{\bullet-1}_{Y,q}d^{\bullet}_{C(\varphi_{X,Y,p,q})}:= \begin{bmatrix}d^{\bullet}_{S_{X,p}}0 \varphi_{f,p,q}^{\bullet}-f_{*}d^{\bullet-1}_{S_{Y,q}}\end{bmatrix}.
dS^2 ={H_7}^{-1/2}[ du dv - \mu^2\sum_{i=1}^4{z^i}^2 du^2 + \sum_{i=1}^4 dz^i dz^i + d\tilde x^2 + d\tilde{y}^2 ] +{H_7}^{1/2}(dr^2 + r^2 d{\theta}^2).
{\hat{A}}\left[{\frac{T}{2}}({\Omega^{\mu}}_{\nu}+{R^{\mu}}_{\nu \rho \sigma}\eta^{\rho}\eta^{\sigma}) \right] ~ = ~ \sqrt{\det \left[{\frac{\frac{T}{2}({\Omega^{\mu}}_{\nu}+{R^{\mu}}_{\nu \rho \sigma}\eta^{\rho}\eta^{\sigma})}{\sinh [ \frac{T}{2}({\Omega^{\mu}}_{\nu}+{R^{\mu}}_{\nu \rho \sigma}\eta^{\rho}\eta^{\sigma}) ]}}\right]}
s^{(\pi)}_\lambda(X) = [Z^\lambda]\V_\pi(z_1;X) V_\pi(z_2;X) \cdots V_\pi(z_m;X)\cdot 1 \cr = [Z^{\lambda+\delta}]\ \prod_{1\le i<j\le m}(z_i-z_j)\ \prod_{\ell=1}^m\, M(z_\ell;X)\L_\pi(Z) \cr = [s_\lambda(Z)] \M(X,Z)\,L_\pi(Z) \,
(-\triangle_{g})^{\alpha}u(x) =\int_{\mathbb{R}^{d}}g\bigg{(}\frac{|u(x)-u(y)|}{|x-y|^{\alpha}}\bigg{)}\frac{u(x)-u(y)}{|u(x)-u(y)|}\frac{dy}{|x-y|^{d+\alpha}}&= \int_{\mathbb{R}^{d}}g\bigg{(}\frac{u(x)-u(y)}{|x-y|^{\alpha}}\bigg{)}\frac{dy}{|x-y|^{d+\alpha}}\ \ (\g\ ).
i \raise . 1 5 e x \mathrm{~ / ~}\kern - . 5 3 e m \mathrm{~ D ~}= e^{g \gamma_{5}[ \phi ( x ) + f ( x ) ]}U ( x ) i \raise . 1 5 e x \mathrm{~ / ~}\kern - . 5 7 e m \mathrm{~ \partial ~}U^{\dagger}( x ) e^{g \gamma_{5}[ \phi ( x ) + f ( x ) ]},
F_{\alpha}[f](y)=\sqrt{\frac{1-icot(\alpha)}{2\pi}}e^{i\frac{cot(\alpha)}{2}y^{2}}\int_{-\infty}^{\infty}e^{-i(csc(\alpha)yx-\frac{cot(\alpha)}{2}x^{2})}f(x)dx.
R_{1 2}( u - v ) \hat{l}_{1}^{- 1}T_{1}( u ) \hat{l}_{1}\hat{l}_{2}^{- 1}T_{2}( v ) \hat{l}_{2}= \hat{l}_{2}^{- 1}T_{2}( v ) \hat{l}_{2}\hat{l}_{1}^{- 1}T_{1}( u ) \hat{l}_{1}R_{1 2}( u - v ) . \ll{t r a c e 3}
x_i =\#\{j\mid \{R_i,C_j\}\ \mbox{is an edge of $\Delta$}\}&=\#\{j\mid (R_i,C_j)\in B(\Delta)\}, \mbox{and}y_j =\#\{i\mid \{R_i,C_j\}\ \mbox{is an edge of $\Delta$}\}&=\#\{i\mid (R_i,C_j) \in B(\Delta)\}.
n^2\mathbb{E}({\phi_n}_{|R'}) =n^2\left(1-\frac{2}{n}\right)\left(\frac{1}{n^{5/2}}\right)\left|\left(\begin{array}{cc}a (n^{1/2}-1)b c (n^{1/2}-1)d\end{array}\right)\right|_1 &=\left(1-\frac{2}{n}\right)\left|\left(\begin{array}{cc}(1/n^{1/2})a (1-(1/n^{1/2}))b (1/n^{1/2})c (1-(1/n^{1/2}))d\end{array}\right)\right|_1
-\sum_{j=1}^{i+1}\sum_{\xi \in X_{j}^\circ(F)}I(d_{i+1,j}(f))(\xi)+\sum_{j=1}^{i+1}\sum_{\xi \in X_{j}^\circ(F)}I(d_{i+1,j}(\mathcal{F}_{X_{i+1}}(f)))(\xi) &-\kappa d_{i+1,0}(f)(0_{V_0},0,0) +\kappa d_{i+1,0}(\mathcal{F}_{X_{i+1}}(f))(0_{V_0},0,0) & -c_{i+1}(f)+c_{i+1}(\mathcal{F}_{X_{i+1}}(f)).
\ell(\widetilde{U}, r) \smallfrown [\widetilde{T}] = (\ell(\widetilde{U}, r) \smallsmile \pi^* \delta) \smallfrown [\widetilde{U}]= \pi^*\delta \smallfrown (\ell(\widetilde{U}, r) \smallfrown [\widetilde{U}] ).
E(\theta_i \mid \delta_i=1,{\bf y}_n, \phi)=\frac{\int_{}^{}\!\, \theta_i d_i(\theta_i,\phi) N(\theta_i; m_i, \phi v_i) d \theta_i}{\int_{}^{}\!\, d_i(\theta_i,\phi) N(\theta_i; m_i, \phi v_i) d \theta_i},
\mathbb{P}\left\{g^{(i)}_t V^{(i)}_t\leq \bar{t}\right\}= \mathbb{P}\left\{\frac{V^{(i)}_t}{N_i n_x}- 1 \leq \frac{4\alpha \|\Sigma^{(i)}_t\|}{(1-2\alpha)^2\rho^{(i)}\| \Sigma^{(i)}_t\|+ 4\sqrt{n_x}}\right\},
\overline{c}_{X^{\ast}}(g_1, g_2) =m_{X^{\ast}}(g_1g_2)^{-1}m_{X^{\ast}}(g_1) m_{X^{\ast}}(g_2) \widetilde{c}_{{X^{\ast}}}(g_1,g_2) &=[\widetilde{s}(g_1g_2)m_{X^{\ast}}(g_1g_2)^{-1}] [\widetilde{s}(g_1)m_{X^{\ast}}(g_1)^{-1}]^{-1}[\widetilde{s}(g_2)m_{X^{\ast}}(g_2)^{-1}]^{-1}.
\bar{\cal R}^{\frac{1}{2}\,z\,\omega ,\frac{1}{2}\,z\,\omega}\,\sim \,\left( \begin{array}{cccc}t 0 0 0 0 s^{-1}t-t^{-1}0 0 0 s 0 0 0 0 -t^{-1}\end{array}\right)\,.
({\cal F}_{\mu \nu}^{( 0 )})_{k l m n}={\frac{1}{h}}\left[ ({\cal A}_{\nu})_{k l m + 1 n + 1}- ({\cal A}_{\nu})_{k l m n + 1}- ({\cal A}_{\mu})_{k l m + 1 n + 1}+ ({\cal A}_{\mu})_{k l m + 1 n}\right] .
H_{MF}=\sum_{i}[h_{i}^{(0)}-zt(\psi^{*}\hat{b}_{i}+\psi\hat{b}_{i}^{\dagger})]withh_{i}^{(0)}=-\mu\hat{n}_{i}+\frac{U}{2}\hat{n}_{i}(\hat{n}_{i}-1)+zt\psi^{*}\psi
[E_x,E_y](y) = E_x(x^{p-1}) = \sum_{k=1}^{p-1}{{p-1}\choose k}x^{p-1-k}(y^{p-1})^{(k-1)}&= -\sum_{k=1}^{p-1}(k-1)!\,x^{p-1-k}y^{p-k}= - \sum_{n=1}^{p-1}(p-1-n)! \,x^{n-1}y^{n}.
\partial_{\bar q}\mathcal F^Z_g(q, \bar q) = \frac{1}{2}C^{qq}_{\bar q}\Big( D_{q}^2 \mathcal F^Z_{g-1}(q, \bar q) +\sum_{g_1+g_2=g}D_{q}\mathcal F^Z_{g_1}(q, \bar q) D_{q}\mathcal F^Z_{g_2}(q, \bar q) \Big),
i\dot{\gamma}\left( t\right) =u\left( t\right) \bar{\delta}\left(t\right) +v\left( t\right) \gamma\left( t\right) i\dot{\delta}\left( t\right) =u\left( t\right) \bar{\gamma}\left(t\right) +v\left( t\right) \delta\left( t\right) .
\mathcal A(\rho^*,m^*) =\sup_{S\in \mathbb H^1_R}\mathcal L(\rho^*,m^*,S)\ge \inf_{\rho,m}\sup_{S\in \mathbb H^1_R}\mathcal L(\rho,m,S) =\mathcal A(\rho^{*,R},m^{*,R})+R\mathcal E(\rho^{*,R},m^{*,R}) \ge \mathcal A(\rho^{*,R},m^{*,R}).
G_{n_1,n_2}(\tau) = \zeta(n_1,n_2)+\sum_{\substack{k_1+k_2=n_1+n_2 k_1,k_2\ge2}}\big( \delta_{n_1,k_1}+{\tt b}_{n_1,n_2}^{k_1}\big) \zeta(k_1)g_{k_2}(q) + g_{n_1,n_2}(q),
\int_{\mathbb{R}^{2n}}e^{-\pi \langle x, \mathcal{Q}x\rangle}\prod_{j=1,2}f_j(x_j)^{c_j}\, dx\le \big( \frac{\sqrt{1-e^{-2s}}}{(2\pi)^{\frac12(c_1+c_2) -1}}\big)^n\prod_{j=1,2}\big( \int_{\mathbb{R}^n}f_j\, dx_j \big)^{c_j}
\|\tilde{B}T(t,A)-\tilde{B}T(t_{0},A)\|_{q}=\|\tilde{B}T(t_{0}/2,A)(T(t-t_{0}/2,A)-T(t_{0}/2,A))\|_{q}\leq\|\tilde{B}T(t_{0}/2,A)\|_{q}\|T(t-t_{0}/2,A)-T(t_{0}/2,A)\|_{\infty}.
Q( \theta, \varphi, u) = - \frac{1}{4\pi}\int d^2{\bar{\cal S}}\int\limits_{0}^{\infty}d{\bar r}(\log{\bar r}) \frac{\partial}{\partial{\bar r}}\Bigl[{\bar r}^3{\cal Y}({\bar r},{\bar \theta},{\bar \varphi},{\bar u})\Bigl|_{{\bar u}= f^{*}}\Bigr]+\; O[ X\times\Re ] \; ,\quad
\lambda_{\mathrm{max}}(\widehat{\mathbf{Y}}) = \|\widehat{\mathbf{Y}}\|_2 \leq \|\mathbf{Y}\|_2 + \|\mathbf{Y}- \widehat{\mathbf{Y}}\|_2 \leq \lambda_{\mathrm{max}}(\mathbf{Y}) + 0.5\, \lambda_{\mathrm{min}}(\mathbf{Y}) < 2\,\lambda_{\mathrm{max}}(\mathbf{Y}).
\Vert l_{j k}\Vert_{0 \le j , k \le 2}= \left( \begin{array}{c c c}{- \frac{1}{3}}&{\frac{2}{3}}&{\frac{1}{3}}\\{\frac{2}{3}}&{- \frac{1}{3}}&{- \frac{2}{3}}\\{\frac{1}{3}}&{- \frac{2}{3}}&{- \frac{1}{3}}\\ \end{array}\right) .
|\partial \hat{U}_i \cap Q^k_i(q)|_{\cal H}\le C|\partial \hat{W}\cap Q^{2k}_i(q)|_{\cal H}& \ \ \ +C\epsilon^{-1}\big(\gamma(\hat{W}\cap Q^{2k}_i(q)) + \delta_4(\hat{W}\cap Q^{2k}_i(q))\big).
\big\langle a_1,a_2\big\rangle_0:=\lim_{t \to 0}\big\langle a_1,a_2\big\rangle_t= \big\langle b_1,b_2\big\rangle_{A^*_-}- \big\langle x_Z^{-1}c_1, x_Z^{-1}c_2\big\rangle_{T^*_Z}.
\lim \limits_{r\rightarrow0}\sup \limits_{x\in{\mathbb{R}^{n}}}\varphi_{1}(x,r)^{-1}\, \left \Vert f\right \Vert_{L_{p}\left( B\left( x,r\right)\right)}=0\lim \limits_{r\rightarrow0}\sup \limits_{x\in{\mathbb{R}^{n}}}\varphi_{2}(x,r)^{-1}\, \left \Vert T_{\Omega,\alpha}f\right \Vert_{L_{q}\left( B\left( x,r\right) \right)}=0.
\omega_{\mathbb{A}}\left[\begin{pmatrix}0 T S 0\end{pmatrix}\right]\geq\frac{1}{2}\sqrt{\langle\big(S^{\sharp_A}(S^{\sharp_A})^{\sharp_A}+ (T^{\sharp_A})^{\sharp_A}T^{\sharp_A}\big)x\mid x\rangle_A+ 2c_A(TS)}.
1-\psi (1-z) = 1-b(1-z)-\int_{[0, \infty)}\big( 1-e^{-t(1-z)}\big){\: \rm d}\nu (t) = bz+\int_{[0, \infty)}e^{-t}\sum_{n\geq 1}\frac{t^n z^n}{n!}{\: \rm d}\nu(t) &= bz+\sum_{n\geq 1}\frac{z^n}{n!}\int_{[0, \infty)}e^{-t}t^n{\: \rm d}\nu(t) &=\sum_{n\geq 1}c(\psi ,n)\, z^n
\textrm{sign}\,M_n \left( - \frac{1}{4}\right) = \textrm{sign}\, \left( - \frac{1}{4}\right)\textrm{sign}\, \left\{2M_{n-1}\left( - \frac{1}{4}\right) - \frac{1}{2}\, M_{n-1}' \left( - \frac{1}{4}\right) \right\}= (-1)^n\, ,
s_{\xi}^K(\mathbf{y}|\mathbf{x})=\sum_{k=1}^{K}\frac{\boldsymbol{\pi}_k}{(2 \pi)^{q/2}(\boldsymbol{\Sigma}_k)^{1/2}}\exp \left\{-\frac{(\mathbf{y}-\boldsymbol{\beta}_{k}\mathbf{x}){^\top}\boldsymbol{\Sigma}_{k}^{-1}(\mathbf{y}-\boldsymbol{\beta}_{k}\mathbf{x})}{2}\right\},
\begin{cases}\Delta_g\hat{u}_i^k+\sum\limits_{j\in I}k_{ij}^{\mathbf{c}^{t}}\rho^k_{j}\left(\frac{\hat h^k_{j}e^{\hat{u}^k_j}}{\int_{M}\hat{h}^k_{j}e^{\hat{u}^k_j}\mathrm{d}V_{g}}-\dfrac{1}{|M|}\right)=0 \ \M, \ \ \forall \i\in I, \hat{u}_1^k+2\sum\limits_{i\in I\setminus\{1,n+1\}}\hat{u}_i^k+\hat{u}_{n+1}^k\equiv0,\end{cases}
\left( \begin{array}{l l}{\xi_{1 1}}&{\xi_{1 2}}\\{\xi_{2 1}}&{\xi_{2 2}}\\ \end{array}\right) \rightarrow \left( \begin{array}{l}{\xi_{1 1}}\\{\xi_{2 1}}\\{\xi_{1 2}}\\{\xi_{2 2}}\\ \end{array}\right)
\frac{\partial f_{0}}{\partial y_{j_{1}\ldots j_{k}}^{\sigma}}-d_{i}f_{\sigma}^{i,j_{1}\ldots j_{k}}-f_{\sigma}^{j_{k},j_{1}\ldots j_{k-1}}=0\qquad\mathrm{Sym}(j_{1}\ldots j_{k}),\,\,k\leq s, \frac{\partial f_{0}}{\partial y_{j_{1}\ldots j_{s+1}}^{\sigma}}-f_{\sigma}^{j_{s+1},j_{1}\ldots j_{s}}=0\qquad\mathrm{Sym}(j_{1}\ldots j_{s+1}).
dx x^{-1}=(k+1)yz^{k+1}\frac{dz}{z}\mod z^{k+2}\mathfrak{g}(\mathcal{O})\frac{dz}{z},{\rm Ad}(x)A=(A+[yz^{k+1},A])\frac{dz}{z}\mod z^{k+2}\mathfrak{g}(\mathcal{O})\frac{dz}{z}.
1 = |L_{N, 1}| + |L_{N, 2}| +\cdots + |L_{N, 2N+1}| &\leq (1+ 1+ \cdots +1)^{1/q}\cdot (|L_{N, 1}|^p + |L_{N, 2}|^p +\cdots + |L_{N, 2N+1}|^p)^{1/p}&\leq (2N+1)^{\frac{p-1}{p}}\cdot (|L_{N, 1}|^p + |L_{N, 2}|^p +\cdots + |L_{N, 2N+1}|^p)^{1/p}
\begin{array}{c c c c c c c}{{}}&{{}}&{{}}&{{1}}&{{}}&{{}}&{{}}\\{{}}&{{}}&{{0 \!}}&{{}}&{{\! 0}}&{{}}&{{}}\\{{}}&{{0 \!}}&{{}}&{{h^{1 , 1}}}&{{}}&{{\! 0}}&{{}}\\{{1}}&{{}}&{{h^{1 , 2}\!}}&{{}}&{{\! h^{1 , 2}}}&{{}}&{{1}}\\{{}}&{{0 \!}}&{{}}&{{h^{1 , 1}}}&{{}}&{{\! 0}}&{{}}\\{{}}&{{}}&{{0 \!}}&{{}}&{{\! 0}}&{{}}&{{}}\\{{}}&{{}}&{{}}&{{1}}&{{}}&{{}}&{{}}\\ \end{array}\ ,
|I_{A}| \le C \int_{|y|^{\alpha}t^{-\beta}}^{\infty}\xi^{-1-1/\beta}w(\xi^{-1/\beta}, \beta, 1)t^{-d\beta/\alpha}\xi^{-d/\alpha}d\xi = C \int_{|y|^{\alpha}t^{-\beta}}^{\infty}\xi^{-1-1/\beta}t^{-\beta d/\alpha}\xi^{-d/\alpha}d\xi \le C t^{-\beta d/\alpha}(|y|^{\alpha}t^{-\beta})^{-1/\beta - d/\alpha}\le C |y|^{-\alpha/\beta - d}t,
\varphi_{xy}(t)(x + ty) = (x + ty)^{[p]^n}= x^{[p]^n}+ t^{p^n}y^{[p]^n}+ \sum_{i=1}^{p^n-1}t^i s_i(x,y) =\lambda(x)x + t^{p^n}\lambda(y)y + \sum_{i=1}^{p^n-1}t^i s_i(x,y) ,
H_{p}(\mathbb{T}^{\infty},X) = \Big\{f \in L_{p}(\mathbb{T}^{\infty},X) \,:\,\, \hat{f}(\alpha) = 0 \, , \, \,\, \,\forall \alpha \in \mathbb{Z}^{(\mathbb{N})}\setminus \mathbb{N}_{0}^{(\mathbb{N})}\Big\}
\phi ( \tilde{\tau}) = \frac{1}{2 i}\left\{\frac{\pi \tilde{\tau}}{\omega_{2}}\frac{\theta_{1}^{\prime}}{\theta_{1}}( \frac{\pi a}{2 \omega_{2}}) + \log \mid \frac{\theta_{1}( \frac{\pi ( \tilde{\tau}- a )}{2 \omega_{2}})}{\theta_{1}( \frac{\pi ( \tilde{\tau}+ a )}{2 \omega_{2}})}\mid \right\},
\|\eta\overline{u}^{\alpha+1}\|_{m^\star}^m\le \frac{C}{(R_1-R_2)^m}\int_{\Omega_{R_2}}\overline{u}^{m(\alpha+1)}+\frac{CG(R_1,R_2)}{m\alpha+1-Cm\epsilon^{m'}}\int_{\Omega_{R_2}}\overline{u}^{m(\alpha+1)}+C\left(\frac{\|f\|_q}{(m\alpha+1-Cm\epsilon^{m'})k^{m-1}}\right)^{N/(mq-N)}\int_{\Omega_{R_2}}\overline{u}^{m(\alpha+1)}.
lim_{t\rightarrow\infty}f_{zyy}(x)=lim_{t\rightarrow\infty}\sum_{i=1}^{N}w_{i}\varphi_{zyy}(x-c_{i})=(\sum_{i=1}^{N}w_{i})3b_{2}(b_{2}^{2}+b_{1}^{2})/(b_{1}^{2}+b_{2}^{2}+b_{3}^{2})^{3/2}=0.
\kappa_{n}=(m-1) \frac{\sum \limits_{i=1}^{m-1}\kappa_i}{m-1}= \sum \limits_{i=1}^{m-1}\kappa_i=\sum \limits_{i=1}^{m-1}\sum \limits_{j=1}^{m-1}\sum \limits_{k=1}^{m-1}\tau_{ij}\tau_{ik}l_{jk}= \sum \limits_{i=1}^{m-1}\sum \limits_{j=1}^{m-1}h^{ij}l_{ij},
u=u_{+}\sqcup u_{-}\colon U_{X}=U_{X,+}\sqcup U_{X,-}=R_{\mathbb{C}/\mathbb{R}}\mathbb{G}_{m,\mathbb{C}}\sqcup R_{\mathbb{C}/\mathbb{R}}\mathbb{G}_{m,\mathbb{C}}\to\mathbb{S}^{1}\sqcup\mathbb{G}_{m,\mathbb{R}}=Z_{+}\sqcup Z_{-}=Z
\begin{array}{r c l}{{{\bf{Q}}_{-{\frac{1}{k}}}^{-}}}&{{=}}&{{[{\bf{Q}}_{-{\frac{1}{k}}}^{+}]^{+}}}\\{{{\bf{Q}}_{1 -{\frac{1}{k}}}^{-}}}&{{=}}&{{[{\bf{Q}}_{1 -{\frac{1}{k}}}^{+}]^{+}}}\\ \end{array}
\begin{array}{l l}{\hat{\gamma}^{\alpha}= \gamma^{\alpha}\otimes 1 \! \! 1_{3 2}}{\alpha = 0 , 1 , 2 , 3}{\hat{\gamma}^{a}= \gamma_{\parallel}\otimes \gamma^{a}}{a = 4 , 5 , . . . , 1 3}\end{array}\hspace{5 . 5 c m}
\deg_{x_n}(a_{i_0}(x) v_{i_0, n+1-i_0}(x)) \qquad{}=\deg_{x_n}(a_{i_0}(x))+1 >\deg_{x_n}(a_{i}(x) v_{i,n+1-i_0}(x)),i\ne i_0,i\in I,
\int_A 1_{B}\left( T_{t_2}(\eta,x)(\omega)\right) \mathbb{P}(d\omega) = \int_A \int_{\Omega}1_{B}\left( \left( T_{t_2}^{t_1}(T_{t_1}(\eta,x)(\omega')) \right)(\omega) \right) \mathbb{P}(d\omega) \mathbb{P}(d\omega')
\begin{pmatrix}m_{11}0 0 0 m_{22}0 0 0 0\end{pmatrix},\quad\begin{pmatrix}m_{11}m_{12}0 -m_{12}m_{11}0 0 0 0\end{pmatrix},\quad\begin{pmatrix}m_{11}0 0 1 m_{11}0 0 0 0\end{pmatrix}.
q_{\sigma_{1}\sigma_{2}}\left( x,t\right) =q\left( x,t\right)-2\partial_{x}^{2}\log\det\left\{I+\mathbf{K}_{12}\left( x,t\right)\right\}& =q\left( x,t\right) -2\partial_{x}^{2}\log\det\left\{I+\mathbf{K}_{21}\left( x,t\right) \right\}=q_{\sigma_{2}\sigma_{1}}\left(x,t\right) .
\int d^2w\frac{1}{[z_1-w]^\alpha[z_2-w]^\beta [z_3-w]^\gamma}=\frac{\pi a(\alpha,\beta,\gamma)}{[z_2-z_1]^{1-\gamma}[z_1-z_3]^{1-\beta}[z_3-z_2]^{1-\alpha}}\,,
{\rm{det}}(\mathbf{M}) ={\rm{det}}(\mathbf{M}') = \prod_{0 \leq k < (m+1)n}M'_{k, k}= \prod_{0 \leq k < (m+1)n}\lambda_{q+1}^f \prod_{1 \leq i < q+1}(\lambda_{q+1}-\lambda_i)^{m+1}.
\sum_{n = 0}^{\infty}e ( X^{[ n ]}) ( t \tilde{t})^{- n}q^{n}= \frac{1}{( ( t \tilde{t})^{- 1}q )_{\infty}( t \tilde{t}^{- 1}q )_{\infty}( q )_{\infty}^{2 0}( t^{- 1}\tilde{t}q )_{\infty}( t \tilde{t}q )_{\infty}}
exp(\mathbf{z}^{-1})|\lambda^{sp}\rangle=(\sum^k_{i_1=0}e_{i_1}Y_{-\lambda_1+i_1})\cdots (\sum^k_{i_{l(\lambda)}=0}e_{i_{l(\lambda)}}Y_{-\lambda_{l(\lambda)}+i_{l(\lambda)}})|0\rangle.
\begin{multlined}[t][12.5cm]\mathfrak{T}\left(T_{\nu_{G},k\nu_{T}}[f]\right)\,=\,\frac{d_{\nu_{G}}^{2}}{2^{d_{T}-1}\pi^{d_{T}-1}}\left(\frac{\|\nu_{T}\|k}{\pi}\right)^{d_{M}-d_{P}+1}\cdot \cdot\int_{X_{0,\nu_{T}}}\frac{f(\pi(x))\|\Phi_{T}(\pi(x))\|^{-d_{M}+d_{P}-2}}{\mathcal{D}(\pi(x))}dV_{X}(x)+ \cdots,\end{multlined}
{\cal Q}(d\vec{V})=\left[\begin{matrix}{\cal Q}_1(d\vec{V}){\cal Q}_2(d\vec{V})\end{matrix}\right]\qquad{\cal Z}\left[\begin{matrix}\psi |\sigma|\end{matrix}\right]=\left[\begin{matrix}-{\cal K}-|\sigma|{\cal K}_{01}|\sigma|{\cal K}_{10}\end{matrix}\right],
\int_{- \infty}^{\infty}d \tau \left[ A \frac{v^{6}}{r^{1 1}}+ B \frac{v^{4}( v \cdot r )^{2}}{r^{1 3}}+ C \frac{v^{2}( v \cdot r )^{4}}{r^{1 5}}+ D \frac{( v \cdot r )^{6}}{r^{1 7}}\right] .
x_{{}_D}\mapsto \big( V_{x_{{}_D}}{f}^{{}^{{}^{x_{{}_D}}}},{\widetilde{g}}^{{}^{{}^{x_{{}_D}}}}\big)_{x_{{}_D}}= \big({\widetilde{f}}^{{}^{{}^{x_{{}_D}}}},{\widetilde{g}}^{{}^{{}^{x_{{}_D}}}}\big)_{x_{{}_D}}\,\,
\begin{array}{l}w_{2i+1}=x^{-1}_ix_{i+1},~~i=0,1,2,\cdots,N-1, w_{2j}=y_j,~~j=1,2,\cdots,N-1, w_{2N}=\prod^{N-1}_{l=1}y^{-1}_l,\end{array}
E [ f^{( 0 )}+ f^{( 1 )}] = E [ f^{( 0 )}] + \frac{\delta E}{\delta f}[ f^{( 0 )}] \cdot f^{( 1 )}+ \frac{1}{2}\frac{\delta^{2}E}{\delta f^{2}}[ f^{( 0 )}] \cdot ( f^{( 1 )})^{2}+ \cdots .
{\cal Z}= \sum_{k = 0}^{+ \infty}( - 1 )^{k}\exp ( - \frac{\pi k}{g_{n c}^{2}\theta}) \frac{1}{2 \pi i}\int_{{\cal C}_{0}}\frac{d z}{z^{k + 1}}\exp \left[ z \Phi ( - z ; \frac{3}{2}; 1 ) \frac{A_{n c}}{\sqrt{2 \pi g_{n c}^{2}\theta^{3}}}\right] ,
\|\widetilde{Tf}(z)-\widetilde{Tf}(e^{i\theta_0})\| =\|x^*(f)\tilde{h}(z)-x^*(f)\tilde{h}(e^{i\theta_0})\| & \leq \|x^*(f)\|\|\tilde{h}(z)-\tilde{h}(e^{i\theta_0})\| & \leq \|x^*\|\|\tilde{h}(z)-\tilde{h}(e^{i\theta_0})\| &< \|x^*\|\dfrac{\epsilon}{\|x^*\|}& =\epsilon,
\nabla^{4}w:=\frac{\partial^{2}}{\partial x_{\alpha}\partial x_{\alpha}}[\frac{\partial^{2}w}{\partial x_{\beta}\partial x_{\beta}}]=\frac{\partial^{4}w}{\partial x_{1}^{4}}+\frac{\partial^{4}w}{\partial x_{2}^{4}}+2\frac{\partial^{4}w}{\partial x_{1}^{2}\partial x_{2}^{2}}.
\begin{aligned}\bar H^6-3\bar H^4S+3\bar H^2S^2-3S^2+2S &\geq \dfrac{(\bar H^2-S)^2}4\bigl[(5-\dfrac{7}{80})S-(4-\dfrac3{8})\bar H^2+\dfrac5{8}\bigl ].\end{aligned}
D(w)=\partial(w),\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, D(sw)=w',\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, D(w')=0.
|\Phi^{e_j}_{t}(x,t_{n-1})|^2 = |x|^2+\left|\int_{t_{j-1}}^t v_{j-1}^{N}(\Phi_s^e(x,t_{j-1}))\right|^2+ 2|x|\cdot\left|\int_{t_{n-1}}^t v_{n-1}^{N}(\Phi_s^{e_j}(x,t_{n-1}))\right| &\leq |x|^2+(V_{max}\Delta t^N)^2+2V_{max}\Delta t^N|x|,
0<\lim_{t \to \infty}L(t) =\begin{cases}\big(\frac{\sigma^2d}{2a}\big)^{\frac{c}{a-\frac{\sigma^2d}{2}}}, \mbox{if}\ \Delta \neq 0, e^{-\frac{c}{a}}, \quad\quad\quad\ \mbox{if}\ \Delta=0 .\end{cases}
{\cal L}={\cal L_{0}}+ \frac{1}{2}( D_{\mu}\pi )^{2}+ \frac{g_{4}^{( 1 )}}{2}( D_{\mu}\pi \cdot D^{\mu}\pi )^{2}+ \frac{g_{4}^{( 2 )}}{2}( D_{\mu}\pi \cdot D_{\nu}\pi ) ( D_{\mu}\pi \cdot D_{\nu}\pi ) + \cdots
\left|\frac{v_1^{\sigma_0}a_{\sigma_0}e^{2t_nr}x+v_2^{\sigma_0}}{\|v^{\sigma_0}\|}\right| = \left|a_{\sigma_0}(\rho(\alpha\beta)^n)^{-1}x\frac{v_1^{\sigma_0}}{\|v^{\sigma_0}\|}+\frac{v_2^{\sigma_0}}{\|v^{\sigma_0}\|}\right|
0 \geq \nabla_k\nabla_k \|\nabla w\|^2=2\left(\langle \nabla_k\nabla_k\nabla w,\nabla w\rangle + \|\nabla_k\nabla w\|^2\right) &\geq 2\langle \nabla_k\nabla_k\nabla w,\nabla w\rangle =2\|\nabla w\| \langle \nabla_k\nabla_k\nabla w,e_n\rangle.
\sum_{i=1}^{4}M_{i}(r_{i}cos\phi_{i}-\frac{r_{i}^{2}}{2l_{i}}cos(2\phi_{i}))=0;\sum_{i=2}^{4}M_{i}a_{i}(r_{i}cos\phi_{i}-\frac{r_{i}^{2}}{2l_{i}}cos(2\phi_{i}))=0
U_{0,k+1,n-1}(a)=1-V_{0,k,n-1}(a)=1-\int_{a}^a U_{0,k,n-1}(t)/d_{0,k,n-1}dt=1-0=1, &U_{i,k+1,n-1}^{(i)}=\frac{U_{i-1,k,n-1}^{(i-1)}(a)}{d_{i-1,k,n-1}}-\frac{U_{i,k,n-1}^{(i-1)}(a)}{d_{i,k,n-1}}=\frac{U_{i-1,k,n-1}^{(i-1)}(a)}{d_{i-1,k,n-1}}\neq 0,
G_{lm,l^{\prime}m^{\prime}}\left( \mathbf{R}\right) =\frac{1}{\left(2\pi\right)^{3}}\int d^{3}k\mathbf{\}\exp\left( i\mathbf{kR}\right)\hat{G}_{lm,l^{\prime}m^{\prime}}\left( \mathbf{k}\right) .
H_{x}=[\frac{m\pi}{a}(Ae^{-jk_{x\epsilon}x}+Be^{jk_{x\epsilon}x})+j\frac{k_{x\epsilon}k_{z}}{\omega\mu}(Ce^{-jk_{x\epsilon}x}-De^{jk_{x\epsilon}x})]cos(\frac{m\pi}{a}y)e^{-jk_{z}z}(36)
\Big ( \sum_{m \ge 1}\sup_{|a|_U =1}| \nabla_k \nabla_{a}^G P_t[\Phi_m](x)|^2 \Big)^{1/2}\, \le \frac{c}{t^{\frac{4 - 3\alpha}{2}}}|k|_K \, \| \Phi\|_{\big(C_b(H,J) , C^1_K(H,J) \big)_{\alpha , \infty}}.
\big\| t|f^2_{\alpha a}|+(t+r)|f^3_{\alpha a}|\big\|^2_{L^2(r\leq \langle t \rangle/2)}&\lesssim\sum_{\tiny\begin{matrix}\beta+\gamma =\alpha b+c=a\end{matrix}}\big\| \langle t \rangle |\nabla U^{(\beta,b)}||\nabla U^{(\gamma,c)}| \big\|^2_{L^2(r\leq \langle t \rangle/2)}.