harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\begin{array}{r c l}{{u_{\lambda , l , m}( r , \theta , \phi )}}&{{=}}&{{N_{\lambda l}\v_{1 , \lambda , l , m}^{( \circ )}=}}\\{{}}&{{=}}&{{N_{\lambda l}\r^{l}\_{2}F_{1}\left({\frac{l + 1 + i \Lambda}{2}},{\frac{l + 1 - i \Lambda}{2}}; l +{\frac{3}{2}}; - r^{2}\right) Y_{l , m}( \theta , \phi ) ,}}\\ \end{array}
\mathcal{I}\left(a, m, 0, \frac{\pi}{2}\right) = \frac{\sqrt{\pi}\Gamma\left(m + \frac{1}{2}\right)}{2(1+a)^{m}\Gamma(m + 1)}\,_{2}F_{1}\left( \frac{1}{2}, m, m + 1, \frac{1}{1+a}\right).
\Phi_i(u)\mathfrak{X}_j^{\pm}(v, \lambda)\Phi_i(u)^{-1}=\frac{\vartheta(u-v\pm a)}{\vartheta(u-v\mp a)}\mathfrak{X}_j^{\pm}(v, \lambda\pm \hbar \alpha_i)\pm \frac{\vartheta(2a)\vartheta(u-v\mp a -\lambda_j)}{\vartheta(\lambda_j) \vartheta(u-v\mp a)}\mathfrak{X}_j^{\pm}(u\mp a, \lambda\pm \hbar \alpha_i).
d_{\Sigma,2}=\max_{\substack{\lambda_1,\lambda_2\in(0,1] \lambda_1(N+\min\{M,K(N-1)\}\leq M \lambda_1(K-1)N+\lambda_2KN\leq M \lambda_1+\lambda_2\leq 1}}KN(\lambda_1+\lambda_2).
\psi_{even, \rho}(x) = \frac{1}{\sqrt{4\pi}}\,|x|^{-\frac{1}{2}+i\rho}, \psi_{odd, \rho}(x) = \frac{{\rm sgn}(x)}{\sqrt{4\pi}}\,|x|^{-\frac{1}{2}+ i\rho}.
\frac{\partial g_{jk}^r}{\partial \xi_k^p}:=\frac{\partial g_{jk}^r}{\partial \xi_k^p}(g_k^{v-1}(z_k,s),h^{v-1}(s)),\,\,\,\,\,\frac{\partial g_{jk}^s}{\partial \xi_k^q}:=\frac{\partial g_{jk}^s}{\partial \xi_k^q}(g_k^{v-1}(z_k,s),h^{v-1}(s))
e(G)\geq f(n,\,\tilde{q}(H))+{\rm{ex}}(\tilde{q}(H)-1,\,\tilde{\mathcal{B}}(H))+(k(H)-1)^2 = e(T_2(n-\tilde{q}(H)+1))+(\tilde{q}(H)-1)(n-\tilde{q}(H)+1)+{\rm{ex}}(\tilde{q}(H)-1,\,\tilde{\mathcal{B}}(H))+(k(H)-1)^2.
\sum_{i}(\frac{1}{r}\frac{d\psi}{dr}\frac{\partial x_{i}}{\partial x_{i}}+x_{i}\frac{d\psi}{dr}\frac{\partial}{\partial x_{i}}\frac{1}{r}+\frac{x_{i}}{r}\frac{\partial}{\partial x_{i}}\frac{d\psi}{dr})
\widetilde{\alpha}_{n,q}(l)=\sum_{i=0}^{2n-2}\textbf{c}_{q,i}^{(n)}l^i\quad\widetilde{\alpha}_{n,q}(l)=\sum_{i=0}^{2n-2}\widetilde{\textbf{c}}_{q,i}^{(n)}l^i,\forall l \geq 2n-1.
\alpha : T T P \to T T P , \quad \alpha \left( p ,{\underline{{p}}}^{\prime}, \dot{\underline{{p}}},{\frac{d{\underline{{p}}}^{\prime}}{d t}}\right) = \left( p , \dot{\underline{{p}}},{\underline{{p}}}^{\prime},{\frac{d \dot{\underline{{p}}}}{d s}}\right) .
\Delta\frac{d}{d\epsilon}\vert_{\epsilon=0}(\bar\Phi_2^{\epsilon(\mu\oplus\nu)})^T\Phi^{\epsilon(\mu\oplus\nu)}_2=y^{-2}\bar\mu\bar\partial\bar\partial(\bar\Phi_+^{0(\mu\oplus\nu)})^T+y^{-2}\mu\partial\partial\Phi^{0(\mu\oplus\nu)}_+)=0,
UU^*\bigl(\Lambda_{\psi}(a)\otimes\Lambda(b)\bigr) =U\bigl((\Lambda_{\psi}\otimes\Lambda)[z(1\otimes b)]\bigr) &=V\bigl((\Lambda_{\psi}\otimes\Lambda)[z(1\otimes b)]\bigr)=E\bigl(\Lambda_{\psi}(a)\otimes\Lambda(b)\bigr).
{\cal L}_{g K}= \frac{1}{2}\left[ i \overline{{{\Psi}}}_{L}\left( \partial \! \! \! / + \frac{1}{2}( i{\cal A}\! \! \! / \, \hat{C}+{\cal A}\! \! \! / \: \hat{C}i ) \right) \Psi_{L}+ \overline{{{\Psi}}}_{L}\left( \partial \! \! \! / + \frac{1}{2}( i{\cal A}\! \! \! / \, \hat{C}+{\cal A}\! \! \! / \: \hat{C}i ) \right) \Psi_{L}i \right]
\frac{1}{n+1}\left(\frac{3}{16 n\log^2n}\left(\begin{matrix}1 -i -i -1\end{matrix}\right) + O\left(\frac{1}{n\log^3n}\right)\right) \qquad{}= \frac{3}{16 n^2\log^2n}\left(\begin{matrix}1 -i -i -1\end{matrix}\right) + O\left(\frac{1}{n^2\log^3n}\right),
\left| (\rho-1)a\alpha^n-d_1\cdot \rho^{\ell+m+k}\right| = \left| -(\rho-1)\Pi(n)-(d_1-d_2)\cdot \rho^{m+k}-(d_2-d_3)\cdot \rho^k-d_3 \right| &\leqslant (\rho-1)\cdot \alpha^{-n/2}+(\rho-1)\cdot \rho^{m+k}+(\rho-1)\cdot \rho^k+(\rho-1) &< 3(\rho-1)\cdot \rho^{m+k},
2G_{AB}(q^{A}_{\phantom{A}\mu}q^{B}_{\phantom{B}\nu}+n^{A}n^{B}q_{\mu\nu})/3-G^{A}_{\phantom{A}A}q_{\mu\nu}/6=-2\kappa^2_5[\tau_{\alpha \beta}q^{\alpha}_{\phantom{\alpha}\mu}q^{\beta}_{\phantom{\beta}\nu}-\tau^{\alpha}_{\phantom{\alpha}\alpha}q_{\mu\nu}/4]\delta(y)/3
\frac{\partial\mathcal{U}_i^F}{\partial d_i}\big|_{d_i=d_i^{ne}}=\frac{1}{w_i}\left(\frac{\sum_{j \in \mathcal{S}}\frac{c_j+\lambda_j}{w_j}}{|\mathcal{S}|-1}-\frac{c_i+\lambda_i}{w_i}\right)>0,
\|\alpha_t(au_g) - au_g\|_2^2 = \|f_{c_t, g}au_g-au_g\|_2^2 \le \|a\|^2\|f_{c_t, g}-1\|_2^2 = \|a\|^2\left[2 - 2 \tau(f_{c_t, g})\right] & = 2\|a\|^2\left[1 - \int_X \tau(\omega_x(tb(x, g^{-1}x))) d\mu(x)\right] & = 2\|a\|^2 \int_X \left[1 - e^{-t^2\|b(x, g^{-1}x)\|^2}\right] d\mu(x)\to 0 t \to 0,
(F_i(x_i^k,N\hat v^k_i) -F_i(x^*_i,\bar x^*))^T(x_i^k-x_i^*) \ge -\bar L_iNM \sum_{j=1}^N[W(k)]_{ij}\| v^k_j - y^k\|\cr + (F_i(x^k_i,\bar x^k) - F_i(x^*_i,\bar x^*) )^T(x_i^k-x_i^*).
\xi_k F_{k,m,r}(z) = (4\pi |m|)^{1-k}\biggl\{(1-k)G_{2-k,-m,r}(z) + G_{2-k,-m,r-1}(z)\biggr\}, \xi_k G_{k,m,r}(z) =(4\pi |m|)^{1-k}F_{2-k,-m,r-1}(z),
(|(vM)|_2)^2 \ = \ \left( \sum_{s \in S_1}\left(\frac{1}{3}v(p_s) \right)^2 \right) + \left( \sum_{s \in S_2}\left(\frac{1}{3}v(q_s) \right)^2 \right) &+ \left( \sum_{s \in S_3}\left(\frac{1}{3}v(r_s a) + \frac{1}{3}v(r_s b) \right)^2 \right)+ \left( \sum_{s \in S_4}\left(\frac{1}{3}v(t_s c) \right)^2 \right).
G_{p,q}^{\,m,n}\!\left( \,\begin{matrix}a_1,\cdots , a_p b_1, \cdots , b_q \end{matrix}\; \Big| z \right) := \frac{1}{2 \pi i}\int_L \frac{\prod_{j=1}^m \Gamma(b_j - s) \prod_{j=1}^n \Gamma(1 - a_j +s) z^s}{\prod_{j=m+1}^q \Gamma(1 - b_j + s) \prod_{j=n+1}^p \Gamma(a_j - s)}\mathrm{d}s,
\int_{\Delta^2}|S_1(w_1, z_1) K_2(w_2, z_2)| \frac{|w_2-z_2|^2}{|w-z|^2}dV(z) \leq C_2 \int_\Delta |w_1-z_1|^{-1}dV(z_1) &~~\times \int_\Delta |w_2-z_2|^{-(\frac{3}{2}+ \frac{1}{10})}dV(z_2) \leq C_3
(1+\kappa\nu)(2\kappa)^{\nu}\frac{\Gamma(\frac{1}{2\kappa}+\frac{\nu}{2})}{\Gamma(\frac{1}{2\kappa}-\frac{\nu}{2})}\frac{\alpha\beta^{\nu}}{\Gamma(\nu)}\int_{0}^{x}z^{\alpha\nu-1}exp_{\kappa}(-\beta z^{\alpha})dz
A_r(n,\epsilon) = \{G\in M_r(n): \exists x\in [n]\textnormal{such that for some}l\in[m(r)-2], |N^G_l(x)|\geq \epsilon n\}, A'_r(n,\epsilon) = \{G\in M_r(n) \setminus A_r(n,\epsilon): \exists xy\in{[n]\choose 2}\textnormal{with}d^G(x,y) \in [m(r)-2]\}.
A_{(3,1)}= \begin{pmatrix}Y_{11}Y_{3}Y_{12}0 \end{pmatrix}A_{(3,2)}= \begin{pmatrix}Y_{11}0 Y_{12}Y_{4}\end{pmatrix}A_{(3,3)}= \begin{pmatrix}Y_{11}Y_{5}Y_{12}Y_{5}\end{pmatrix}
a_{i1}=\mu_i\frac{D[q_{4+i}, q_2, q_3]}{D[q_4, q_2, q_3]}; \, \, a_{i2}=\mu_i \frac{D[q_1, q_{4+i}, q_3]}{D[q_1, q_4, q_3]}; \, \, a_{i3}=\mu_i\frac{D[q_1, q_2, q_{4+i}]}{D[q_1, q_2, q_4]};
(4+\dfrac{3ac}{b^2}- \dfrac{6bd}{c^2}-\dfrac{c^2}{bd})= \dfrac{2(n+1)[k^3-(2n+2)k^2+(n^2+3n-5)k-n^2+ 2 n-3]}{(k+1) (k+2 )(n-k+1) (n-k)(n-k-1)}.
\sigma\sum_{n=0}^\infty\int d^4z_1\dots\int d^4z_n\left(W_{xz_1z_2\dots z_ny}\right)_{AB}\frac{e^{-\frac{(x-z_1)^2}{4\sigma}}}{(4\pi\sigma)^2}\frac{e^{-\frac{(z_1-z_2)^2}{4\sigma}}}{(4\pi\sigma)^2}\dots\frac{e^{-\frac{(z_n-y)^2}{4\sigma}}}{(4\pi\sigma)^2}
\left\langle \hat{f}_v,\, e^{v,w}_x\otimes \bigotimes_{w'\in\partial v\setminus \{w\}}e^{v,w'}_0\right\rangle_{\mathcal{V}_{\partial v}}=0 \left\langle e^{v,w}_x\otimes \bigotimes_{v'\in\partial w\setminus \{v\}}e^{v',w}_0,\,\hat{g}_w\right\rangle_{\mathcal{V}_{\partial w}}=0
SS_{AB}\equiv\frac{(\sum_{ij}Y_{ij}(\overline{Y}_{i\cdot}-\overline{Y}_{\cdot\cdot})(\overline{Y}_{\cdot j}-\overline{Y}_{\cdot\cdot}))^{2}}{\sum_{i}(\overline{Y}_{i\cdot}-\overline{Y}_{\cdot\cdot})^{2}\sum_{j}(\overline{Y}_{\cdot j}-\overline{Y}_{\cdot\cdot})^{2}}
\|\Upsilon_\eta(A)\| \leq \sum_{i=-1}^{k-1}\sum_{\sigma \in X(i) \setminus S_\eta^i(A)}{k+2 \choose i+2}\cdot \|I^\sigma(E(I_\sigma(S_\eta^{i+1}(A)),I_\sigma(S_\eta^{i+1}(A))))\|
g_{\alpha \beta \gamma}:= \int_{\sum_{2}}d^{2}\varphi Y_{\alpha}(\frac{\partial Y_{\beta}}{\partial \varphi^{1}}\frac{\partial Y_{\gamma}}{\partial \varphi^{2}}- \frac{\partial Y_{\gamma}}{\partial \varphi^{1}}\frac{\partial Y_{\beta}}{\partial \varphi^{2}})
\sum_{k=0}^{\infty}a_kx^{s_b(k)}\star \sum_{l=0}^{\infty}b_ly^{s_b(l)}= \sum_{n=0}^{\infty}\sum_{k\leq_b n}a_k b_{n-k}x^{s_b(k)}y^{s_b(n-k)}.
\begin{array}{l c l}{{\Delta X_{\pm ( i , j )}}}&{{=}}&{{X_{\pm ( i , j )}\otimes e^{- h H_{j ; n - 1}/ 2}+ e^{- h H_{i ; n - 1}/ 2}\otimes X_{\pm ( i , j )}+}}\\{{}}&{{}}&{{+ ( 1 - e^{\pm h}) \sum_{k = i + 1}^{j - 1}( X_{\pm ( i , k )}\otimes X_{\pm ( k , j )}) .}}\\ \end{array}
A_{2-n}A_{4-n}A_{6-n}\psi_{n-4}^{(1)}= \Big[A_{2-n}A_{4-n}A_{6-n}\alpha_{n-4}^{(1)}+2(B_{6-n}A_{4-n}A_{2-n}+A_{6-n}B_{4-n}A_{2-n}+A_{6-n}A_{4-n}B_{2-n})\beta_{n-4}^{(1)}\Big]\log r &+A_{2-n}A_{4-n}A_{6-n}\beta_{n-4}^{(1)}\log^2 r+O(r^{n-4}).
0\le a(t)\frac{d}{dt}{\Vert u(t)-v(t) \Vert}^2 = \ 2a(t)(u^{\prime}(t) - v^{\prime}(t), u(t)-v(t)) \le \ 2\sqrt{K}\sqrt{a(t)}\Vert u^{\prime}(t) - v^{\prime}(t) \Vert,
\frac{1}{s_{1}}+\frac{\alpha_{1}+\beta_{1}}{n}=\theta\left(\frac{1}{s_{2}}+\frac{\alpha_{2}+\beta_{2}-1}{n}\right)+(1-\theta)\left(\frac{1}{s_{3}}+\frac{\alpha_{3}+\beta_{3}}{n}\right),
\sum_{\alpha \in \Gamma}{\overline{{\chi}}_{0}}^{\dagger}{\gamma_{\lambda}}^{\dagger}\chi_{\alpha + \lambda}( \chi_{0}) e^{\bar{\alpha}\cdot \lambda \phi ( 1 ) / 3}e^{\bar{\alpha}\cdot \phi ( a ) / \sqrt 3}= \sum_{\alpha \in \Gamma}{\overline{{\chi}}_{0}}^{\dagger}{\gamma_{\lambda}}^{\dagger}\chi_{- \alpha + \lambda}( \chi_{0}) e^{- \bar{\alpha}\cdot \phi ( a ) / \sqrt 3}\, .
\int_{0}^\infty \rho^{-a}u_0(\rho)\rho^{n-1}\,d\rho =\sum_{k=1}^{\infty}k^{n}\int_{k}^{k+k^{-n-1}}\rho^{-a}\,d\rho &\le \sum_{k=1}^{\infty}k^{n}k^{-a}k^{-n-1}=\sum_{k=1}^{\infty}k^{-1-a}<\infty
\frac{d}{dt}\int_{R^6}u_t\,\cdot (x\cdot\nabla u+2u)\phi(\frac{x}{t})\,dx=\int_{R^6}u_t^2(x,t)\phi(\frac{x}{t})\,dx+O\left(\int_{\frac{t}{2}\leq |x|\leq \frac{3}{4}t}|\nabla_{x,t}u|^2+\frac{|u|^2}{|x|^2}\,dx\right).
V_{\mathrm{s t a t}}^{q \overline{{q}}}= - \frac{4}{3}\frac{{\alpha}_{s}}{r}+ \sigma r -{\frac{4}{3}}{\frac{\alpha_{\mathrm{s}}^{2}}{4 \pi}}{\frac{1}{r}}[{\frac{6 6 - 4 N_{f}}{3}}( \ln \mu r + \gamma ) + A ]
\sum_{m=1}^{\infty}\frac{\tau_A(p^{m}) \tau_B(p^{m}) \left( 1+\frac{p^w}{p^{2}(p-1)}-\frac{1}{p-1}\right)}{p^{m(3-\alpha-\beta-w )}}= \frac{\tau_A(p)\tau_B(p)}{p^{3-\alpha-\beta-w}}+ O\left( \frac{1}{p^{1+\varepsilon}}\right).
(\mathrm{id}+ B \circ T) (u) *_B (\mathrm{id}+ B \circ T) (v) &= (Tu) (\mathrm{id}+ B \circ T)(v) + (\mathrm{id}+ B \circ T) (u) (Tv) + H (Tu, Tv) &= T(u) \cdot v + u \cdot T(v) + (Tu) \cdot B (Tv) + B (Tu) \cdot (Tv) + H (Tu, Tv) &= u * v + B ( (Tu) (Tv)) &= u * v + BT (u * v) = (\mathrm{id}+ B \circ T)(u * v).
\Phi_\mathrm u = \left[\begin{array}{cc}1 0 0 -1 \end{array}\right], \Phi_\mathrm i = \left[\begin{array}{cc}0 1 1 0 \end{array}\right], \Phi_\mathrm r = \left[\begin{array}{cc}0 \j -\j 0 \end{array}\right],
s-\frac{S}{\alpha^{1/p}-1}-\alpha^{-\frac{N+1}{p}}\left|\partial_\theta\phi^1(\theta)\right| \ \leq partial_\theta f_{\tau,\theta}^{N+1}(\phi(\theta))\ \ \leq \S+\frac{S}{\alpha^{1/p}-1}+\alpha^{-\frac{N+1}{p}}\left|\partial_\theta\phi^1(\theta)\right|\ .
\Gamma_+^{-1}A^{(r)}C_r - \alpha_r C_3 \Gamma_+^{-1}A^{(r)}=- \frac{1}{2}\frac{\alpha}{1 + \mu \alpha k}C U_3^{-1}Y Y^{\rm T}A^{(r)}U_r^{-1}C, \qquad r \in \{1,2\}.
\langle \tau_t(A) B \rangle - \langle A \rangle \langle B \rangle = \sum_{w, w'=1}^3 a_w b_{w'}\left( \langle \tau_t(\sigma_\xi^w) \sigma_\eta^{w'}\rangle - \langle \sigma_\xi^w \rangle \langle \sigma_\eta^{w'}\rangle \right) \, .
\mbox{\boldmath $E$}= q[\frac{1-v^{2}}{(R-\mbox{\boldmath $R\cdot v$})^{3}}(\mbox{\boldmath $R$}-R\mbox{\boldmath $v$}) +\frac{1}{(R-\mbox{\boldmath $R\cdot v$})^{3}}\mbox{\boldmath $R\times $}\{(\mbox{\boldmath $R$}-R\mbox{\boldmath $v$})\mbox{\boldmath $\times a$}\}]
\displaystyle \int_{\mathfrak{g}(\mathbb{R})}\theta(X)\widehat{\varphi}(X) dX=\int_{\Gamma(\mathfrak{g})}\int_{\mathfrak{g}(\mathbb{R})}D^G(X)^{1/2}\theta(X) \widehat{j}(X,Y)\varphi(Y) dY dX
{\rm tr}' e^{-(\Delta+m^2)t}= \sum_{\delta \neq{\bf 1}}\sum_{p=1}^{\infty}\int_{-\infty}^{\infty}du \left [{{{\sqrt{2}}l(\delta)}\over{(4\pi t)^{3/2}}}\int_{\rho_H}^\infty db{{b e^{-b^2/4t - M^2 t}}\over{{\sqrt{{\rm Cosh}b -{\rm Cosh}\rho_H}}}}\right ] ,
I_t \coloneqq \int_0^t \frac{2\rho (e^{2\rho t}-1)}{2\rho \alpha e^{2\rho s}(e^{2\rho t}-1) + (e^{2\rho(t-s)}-1)(e^{-2\rho(t-s)}-1)}ds & =\int_1^{e^{2\rho t}}\frac{e^{2\rho t}-1}{u \left[2\rho \alpha u(e^{2\rho t}-1) + 2 - \frac{e^{2\rho t}}{u}- e^{-2\rho t}u\right]}du & =\int_1^{e^{2\rho t}}\frac{e^{2\rho t}-1}{u^2 c_t + 2u - e^{2\rho t}}du
\langle\lambda(b_1)\eta_1,\lambda(b_2)\eta_2\rangle =\langle\lambda(c_1)\eta_1,\lambda(c_2)\eta_2\rangle, \langle\rho(b_1)\eta_1,\rho(b_2)\eta_2\rangle =\langle\rho(d_1)\eta_1,\rho(d_2)\eta_2\rangle.
Q(C_i)^2 = \frac{\|\delta_i\|^2}{C_{i,\infty}^2\Bigl(\sqrt{\overline{C_i^2}}+ \overline{C_i}\Bigr)^2}= \frac{\sqrt{\overline{C_i^2}}- \overline{C_i}}{C_{i,\infty}^2\Bigl(\sqrt{\overline{C_i^2}}+ \overline{C_i}\Bigr)}\leq \frac{1}{C_{i,\infty}^2}.
f(x) =\frac{l_k}{(k-1)!}x^{-\lambda}+\frac{b_k}{(\lambda)_k}+\int_0^{\infty}M_1(u)u^{\lambda+k-2}e^{-xu}\, du &=\frac{b_k}{(\lambda)_k}+\int_0^{\infty}\left(M_1(u)u^{k-1}+\frac{l_k}{(k-1)!}\right)e^{-xu}u^{\lambda-1}\, du.
\mathrm{R e}\; \iota_{E}\Omega ={\frac{3}{2}}\left( ( 1 + x_{1}^{2}) \alpha_{0}- 2 x_{1}\; \alpha_{1}+ ( 1 - x_{1}^{2}) \alpha_{2}\right){\frac{d x_{1}\wedge d h}{h}}+ \ldots = 0
-u_f(x) = \int_M G^\omega_M(x,z) \Delta_g u_f d\mu_g(z) + \int_{\partial M}\left(u_f(z)\partial_\nu G^\omega_M(x,z) - G^\omega_M(x,z)\partial_\nu u_f(z)\right) d\mu_h(z) - \int_M u_f(z)_g(F(z)G^\omega_M(x,z)) d\mu_g(z)+\omega^2\int_M u_f(z) G^\omega_M(x,z) d\mu_g(z).
\int_0^1 d\alpha_r \;\cdots \tilde{U}_{| \tau_r = (1-\alpha_1)\cdots(1-\alpha_{r-1}) (\alpha_r - \alpha_r^2) \tau}\;\leq\; \sup_{\alpha_r \in [0,1]}\;\cdots\left( (1-\alpha_1)\cdots (1-\alpha_{r-1}) \tau \right)^{-1}\; g \;\;\; ,
L_{\varepsilon}=\sup_{0<t\leq T_1}t^{\frac{1}{2}-\frac{3}{2p}}||b_{\varepsilon}||_{p}\leq \sup_{0<t\leq T_1}t^{\frac{1}{2}-\frac{3}{2p}}||b_{\varepsilon}-b||_{p}+\sup_{0<t\leq T_1}t^{\frac{1}{2}-\frac{3}{2p}}||b||_{p}&\leq 2\delta\textrm{for}\ \varepsilon\leq \varepsilon_0.
(\alpha_{-1}- \beta_{-1}) \cdot (\alpha_{-2}-\beta_{-2}) = \frac{(3t + 22)t}{2^2 \cdot 3^2}a_2 - \frac{(3t - 10)t}{2^2 \cdot 3^2}a_{-2}- \frac{(3t - 14)t}{2^2 \cdot 3^2}a_3 + \frac{(3t + 2)t}{2^2 \cdot 3^2}a_{-3}&- \frac{2t - 1}{2^2 \cdot 3}v_{(2,3)}+ \frac{t}{2 \cdot 3}a_1 \cdot v_{(2,3)}+ \frac{t}{2}(a_2 \cdot v_{(1,3)}- a_3 \cdot v_{(1,2)}).
\mu_{v\rightsquigarrow w}^*\left(A_{i;w}\right)= \left.F_{i;v\rightsquigarrow w}\right|_{X_{j;v}=\prod_k A_{k;v}^{\epsilon_{jk;v}}}\prod_jA_{j;v}^{g_{ij;v\rightsquigarrow w}}\mu_{v\rightsquigarrow w}^*\left(X_{i;w}\right)= \prod_j F_{j;v\rightsquigarrow w}^{\epsilon_{ij;w}}\prod_j X_{j;v}^{c_{ij;v\rightsquigarrow w}}.
\bigg(\int_0^t \big|\sigma \big(r, X^{\alpha,i}, \varphi^{N,X}_r,\alpha^{i,N}_s \big) \nabla \varphi(X^{\alpha,i}_r) \big|^2 \mathrm{d}r \bigg)_{t \in [0,T]},~i=1, \dots, N.
\big|f^\infty_j*\varphi_t(y)\big| =\bigg|\int_{(2B)^c}\varphi_t(y-z)f_j(z)\,dz\bigg| &\leq C\cdot t^{-n}\int_{(2B)^c\cap\{z:|y-z|\le t\}}\big|f_j(z)\big|\,dz &\leq C\cdot t^{-n}\sum_{\ell=1}^\infty\int_{(2^{\ell+1}B\backslash 2^{\ell}B)\cap\{z:|y-z|\le t\}}\big|f_j(z)\big|\,dz.
\left| \left. a , 1 / 2 \right| n \right\rangle = \frac{1}{g_{\left| \left. a , 1 / 2 \right| n \right\rangle}^{\left| 1 / 2 \right\rangle}}K_{a \frac{1}{2}}\left( i \nu_{n}\right) \left| \frac{1}{2}\right\rangle , \quad \mathrm{w h e r e}\left| \frac{1}{2}\right\rangle \equiv \left| B_{\frac{1}{2}}\right\rangle ,
\mbox{Res}\Big(\frac{J_{R}^{(1)}(s)}{s+x_{1}},s = -x_{1}\Big) = \frac{1}{2}\begin{pmatrix}-(d_{0}+d_{1}d_{0}^{2}) -id_{0}^{2}-i \big( d_{0}^{2}d_{1}^{2}+2d_{0}d_{1}+\frac{3}{4}\big) d_{0}+ d_{1}d_{0}^{2}\end{pmatrix}.
\int\limits_{M}\;\mu\;(2(P^{jk}{}_{,i}D_{\mu j}^{l}+P^{mk}{}_{,ij}A_{\mu m}P^{jl})-D_{\mu i}^{m}P^{kl}{}_{,m}+(\epsilon^{\rho\nu}D_{\rho}X^{i})\epsilon_{\mu\nu}P^{kl}{}_{,ji})C_{l}C_{k}=0.
g(\beta)=1+\frac{\psi(1/\beta)}{\beta}-\frac{\sum_{i=1}^{N}|x_{i}-\mu|^{\beta}log|x_{i}-\mu|}{\sum_{i=1}^{N}|x_{i}-\mu|^{\beta}}+\frac{log(\frac{\beta}{N}\sum_{i=1}^{N}|x_{i}-\mu|^{\beta})}{\beta},
I_{1,\ast}(t)+\Vert\varphi(t)\Vert^2_{L^2(\mathbb R)}\le C_3\int_0^t(I_{1,\ast}(s)+\Vert\varphi(s)\Vert^2_{L^2(\mathbb R)})ds+C_3\Vert{\mathbf F}\Vert^2_{H^1_\ast(\Omega_t)}\,.
\frac{1}{(2\pi i)^2}\int_{-\frac{1}{2}-i\infty}^{-\frac{1}{2}+ i \infty}ds \int_{\Sigma}dt\prod_{j=0}^{r-q}\frac{\Gamma(s+1+\nu_j)}{\Gamma(t+1+\nu_j)}\frac{\sin \pi s}{\sin \pi t}\frac{R(t)}{R(s)}\frac{x^t y^{-s-1}}{s-t}.
u_n = \sum_{k=1}^{N_D}U_k \otimes s_{n_k,n}, w_n = P_C \otimes s_{{n_0},n}+ u_n, t_n = \sum_{l=1}^{N_C}T_l \otimes s_{n_l,n}, z_n = P_D \otimes s_{{n_0},n}+ t_n.
\scriptsize{{\rm SINR}_{r_{p}d}=\frac{\sum\limits_{k=1}^K (\textbf{w}_{r_pd}^k)^H \rho_{r_pd}^k \textbf{w}_{r_pd}^k}{\sum\limits_{k=1}^K \sum\limits_{\substack{l=1 l\neq p}}^L (\textbf{w}_{r_ld}^k)^H \rho_{r_ld}^k \textbf{w}_{r_ld}^k + \sigma^2(\textbf{w}_{r_pd}^k)^H(\textbf{w}_{r_pd}^k)},}
H=P=\left(\begin{array}{ccc}\ast \ast \ast 0 \ast \ast 0 0 \ast \end{array}\right),U=\left(\begin{array}{ccc}1 0 0 \ast 1 0 0 0 1 \end{array}\right),V=\left(\begin{array}{ccc}1 0 0 0 1 0 \ast \ast 1 \end{array}\right),W=\left(\begin{array}{ccc}1 0 \ast 0 1 \ast 0 0 1 \end{array}\right),
\lambda^{'}=(\overbrace{\alpha_{1},\cdots,\alpha_{1}}^{M_{1}},\overbrace{\alpha_{2},\cdots,\alpha_{2}}^{M_{2}},\cdots,\overbrace{\alpha_{s},\cdots,\alpha_{s}}^{M_{s}},\overbrace{-\alpha_{s},\cdots,-\alpha_{s}}^{N_{s}},\cdots,\overbrace{-\alpha_{1},\cdots,-\alpha_{1}}^{N_{1}})
S[B,\tilde{B},F] = \int d^{5}x \left\{{\textstyle\frac{1}{2\cdot 3!}}H^2 -{\textstyle\frac{1}{4}}m^2 B^2+{\textstyle\frac{1}{4}}\epsilon \partial\tilde{B}(F+mB) \right\}\, .
\sum_j\phi(m_j^*)pw_h\phi(\alpha_{h^{-1}}(m_j)) =\sum_j\phi(m_j^*\sigma(p)z_hh\alpha_{h^{-1}}(m_j)) &=\phi(\sum_jm_j^*m_j\sigma(p)z_hh)=pw_h\ne 0,
\psi_- = \xi_{1,-\alpha^+}\,\xi_{-1,-\alpha^-}\prod_{r=1}^R \xi_{\tau_r,-\alpha^+_r}\; \xi_{\bar{\tau}_r,-\alpha^-_r}\;, \psi_+ = \eta_{1,\alpha^+}\;\;\, \eta_{-1,\alpha^-}\;\,\prod_{r=1}^R \eta_{\tau_r,\alpha^+_r}\;\, \eta_{\bar{\tau}_r,\alpha^-_r}\;.
\int_{\mathbb{R}^{d}}(h,U^{t})_{\mathbb{R}^{N}}dx =\lambda_{h}(U^{t})=\langle\psi(h),U^{t}\rangle_{s}=\lim_{j\rightarrow\infty}\langle\psi(h),U^{j}(t)\rangle_{s}&=\lim_{j\rightarrow\infty}\lambda_{h}(U^{j}(t))=\lim_{j\rightarrow\infty}\int_{\mathbb{R}^{d}}(h,U^{j}(t))_{\mathbb{R}^{N}}dx=\int_{\mathbb{R}^{d}}(h,U(t))_{\mathbb{R}^{N}}dx,
[\![X,Y]\!]= [X,Y]_K+K^{-1}T_K(X,Y) = [KX,Y]+[X,KY]-K[X,Y] &+K^{-1}([KX,KY]-K[KX,Y]-K[X,KY]+K^2[X,Y]) = K^{-1}[KX,KY]
[z^n][y^k]T(y, z) =\frac1k\binom{2k-2}{k-1}[z^{n-k+1}](1-z)^{1-k}= \frac1k\binom{2k-2}{k-1}\binom{1-k}{n-k+1}(-1)^{n-k+1}=\frac1k\binom{2k-2}{k-1}\binom{n-1}{n-k+1}.
\|Sf\|^2=\big\|\sum_{i\in{I_n},j\in{J_m}}c_{ij}\Lambda_{i+1}^*e_j\big\|^2 =\big\|\sum_{i\in{I_n},j\in{J_m}}r_{ij}\Lambda_{i+1}^*e_j\big\|^2 &\leq B_{\Lambda}\sum_{i\in{I_n},j\in{J_m}}|r_{ij}|^2.
H = \int d^{3}x \left( \frac 1 2 \pi_{i a}\pi_{i}^{a}+ \frac 1 4 F_{i j}^{a}F_{a}^{i j}- A_{0}^{a}D_{i}\pi_{a}^{i}+ \frac 1 2 M^{2}A_{\mu}^{a}A_{a}^{\mu}\right) ,
\sum_{n \geq 0}^{\infty}{}_3F_2(-n, \frac{m+2}{2}, \frac{m+3}{2}, \frac{3}{2}, 2; \epsilon)(w\cos^2(\theta/2))^{n+1}= \frac{w\cos^2(\theta/2)}{(1-w\cos^2(\theta/2))}&{}_3F_2\left(1, \frac{m+2}{2}, \frac{m+3}{2}, \frac{3}{2}, 2; -\frac{\epsilon w\cos^2(\theta/2)}{1-w\cos^2(\theta/2)}\right).
v_A^{-1}\triangleright \varphi = \varphi^{v^{-1}}= \varphi(v^{-1}?), v_B^{-1}\triangleright \varphi = \mu^l(v)^{-1}\left(\mu^l\!\left(g^{-1}v\,?\right) \varphi^v\right)^{v^{-1}}
\begin{aligned}L_{8}(q) = -1 + \sum_{\substack{n \geq 0 -n \leq j \leq n}}q^{6n^2 + 2n - 2j^2 - 2j}+ q^{6n^2 + 10n + 4 - 2j^2 -2j}& + \sum_{\substack{n \geq 0 -n-1 \leq j \leq n}}q^{6n^2 + 16n + 11 - 2j^2}+ q^{6n^2 + 8n + 3 -2j^2}\end{aligned}
I m \, \Pi_{2}^{\mathrm{v a c}}= \frac{3 \lambda g^{2}}{1 6 \pi^{3}}\left\{\frac{1}{\epsilon}+ 4 - 2 \gamma_{E}+ 2 \ln \left( \frac{4 \pi \mu^{2}}{P^{2}}\right) \right\}\, .
\sum_{l=1}^{2k-2}\biggl|\tilde{a}^{n}_{l}-\sum_{j=0}^{l}\frac{\alpha_{l-j}}{j!}\tau^j\biggr|+\biggl|\tilde{a}^{n}_{2k-1}-\sum_{j=1}^{2k-1}\frac{\alpha_{2k-1-j}}{j!}\tau^j\biggr| &+\sum_{j=1}^{2k-2}\biggl|\tilde{b}^{+,n}_{j}-\sum_{l=0}^{2k-2}\frac{\alpha_{2k-2-l}}{(l+j+1)!}\tau^{l+j+1}\biggr|\geq\epsilon>0
I_{-\infty}:= \{i :[\mkern-.4mu[y_{i-}^{\textsc{g}}, y_{i+}^{\textsc{g}}]\mkern-.4mu]\mkern-2.5mu=\mkern-2.5mu [-\infty, y_{i+}^{\textsc{g}}]\},\;\;\;\;I_{+\infty}:=\{i : [\mkern-.4mu[y_{i-}^{\textsc{g}}, y_{i+}^{\textsc{g}}]\mkern-.4mu]\mkern-2.5mu=\mkern-2.5mu [y_{i-}^{\textsc{g}},+\infty]\}.
x(t)= C(t,0)(x_0-g(x))+S(t,0)(y_0-h(x))+\int_0^tS(t,s)f(s)ds+\int_0^tS(t,s)Bu(s)ds &+\sum_{0<t_i<t}C(t,t_i)I_i(u(t_i))+\sum_{0<t_i<t}S(t,t_i)J_i(u(t_i)), t\in I.
\int_\Omega \mathbb{P}[\mu]\nabla p\cdot \nabla \phi\, \d x = \sum_{i\in \mathcal{I}}G_i\int_0^1 \partial_{s_i}\phi \,\d s_i = \sum_{i\in \mathcal{I}}G_i(\phi(x_i^-)-\phi(x_i^+)) = \sum_{j\in \mathcal{V}}S_j\phi(x_j),
\|fg\|_{L^{p,r}}\leq C \|f\|_{L^{p_0,r_0}}\|g\|_{L^{p_1,r_1}}, \frac1p=\frac{1}{p_0}+\frac{1}{p_1}, ~~\frac1r=\frac{1}{r_0}+\frac{1}{r_1}.
-\frac{n-1}{2n}\log\left(p-1\right) +\frac{n-1}{2n(p-1)}\left[-(n+p-1)\Big(\log(n) +\frac{p-1}{n}\Big) + np\log \left(n\right)\right]+C &=-\frac{n-1}{2n}\log\left(p-1\right) +\frac{n-1}{2n(p-1)}\left[(n-1)(p-1)\log(n) -(n+p-1)\frac{p-1}{n}\right]+ C &=-\frac{n-1}{2n}\log\left(p-1\right) + \frac{(n-1)}{2n}\left[(n-1)\log(n) -\frac{n+p-1}{n}\right]+C
{\cal L}_{P}= - \frac 1 2 G_{\mu \nu}^{\dagger}G^{\mu \nu}- M^{2}\phi_{\mu}^{\dagger}\phi^{\mu}+ \frac{\theta}{2}\varepsilon_{\mu \nu \rho}A^{\mu}\partial^{\nu}A^{\rho}+ \frac{\lambda}{2}( \partial_{\mu}A^{\mu})^{2},
|F_1\cup F_2| =|F_1|+|F_2|=r_1(F_1)+k(F_1)+r_1(F_2)+k(F_2) &\geq r_1(F_1\cup F_2)+k(F_1)+k(F_2) &\geq r_1(F_1\cup F_2)+k(F_1\cup F_2)+1 &\geq g_1(F_1\cup F_2)+1,
-\left[\cos(\kappa_j x) + \frac{\xi}{\kappa_j}\sin(\kappa_j |x|)\right]\sin(\kappa_j t) =-\frac{\xi L}{2}\left[ \frac{\kappa_j t}{Z_j^2}+ 2\sum_{n=1}^\infty \left(\frac{\kappa_j}{k_n}\right) \frac{\cos(k_n x)\sin(k_n t)}{Z_j^2-(\pi n)^2}\right]
\sum_{x \in Y_1, z \in X \setminus Y_1}2\mathfrak{Re}\langle K(x,z)\xi(z) | \xi(x)\rangle &= \sum_{ycab \in Y_1, z \in X \setminus Y_1}2\mathfrak{Re}\langle K(ycab,yc)K(yc,z)\xi(z) | \xi(ycab)\rangle &= \sum_{ycab \in Y_1, z \in X \setminus Y_1}2\mathfrak{Re}\langle V_z\xi(z) | V_{yc}K(yc,ycab)\xi(ycab)\rangle.
\psi_{P,a}\left( \mathbf{O}\left( A,Y;\mathcal{G}\right);\mathcal{G}\right) =\chi_{P,a,eff}^{1}(\mathbf{V};\mathcal{G})=q_{\mathcal{G}}\left( \mathbf{W};P\right) +h_{\mathcal{G}}\left( A,\mathbf{O,M,}Y;P\right) P\in\mathcal{M(G)}.
\alpha^2 = \frac{1}{4}(y_1^2\pi^2+y_2^2p+2y_1y_2\pi\sqrt p) = \frac{1}{4}\pi(y_1^2\pi+y_2^2\pi'+2y\sqrt p),\ \p=\pi\pi'\ \y=y_1y_2. = \frac{1}{4}\pi(2x+2y\sqrt p),\ \ 2x=y_1^2\pi+y_2^2\pi'. = \pi\epsilon_{p},\ \ \epsilon_{p}=\frac{1}{2}(x+y\sqrt p).
R_{\qquad kl}^{nm}=\delta_{\quad k}^{n}\delta_{\quad l}^{m}+\epsilon^{nm}\epsilon_{kl}+i\eta \epsilon^{nm}\epsilon_{kl}+\frac{1}{2}\epsilon^{nm}b_{kl}+\frac{1}{2}b^{nm}\epsilon_{kl}+o\left( \eta^{2}\right) .
\frac{S_{1}R_{2}R_{4}(S_{3}+R_{3})+S_{2}S_{3}(S_{1}+R_{1})(S_{4}+R_{4})}{S_{1}R_{4}(S_{3}(S_{2}+R_{2})+R_{2}R_{3})+S_{2}S_{3}S_{4}(S_{1}+R_{1})}
\beta^{( a )}( \tau ) ={2 \pi i \large \frac{{\int_{- 1}^{1}}d x \sqrt{{\frac{x \Theta_{3}^{4}+ \Theta_{2}^{4}- \Theta_{4}^{4}}{x^{2}- 1}}}}{( \Theta_{2}^{8}- \Theta_{2}^{4}\Theta_{3}^{4}) \left[ \large{\int_{- 1}^{1}}d x \sqrt{{\frac{1}{( x^{2}- 1 ) ( x \Theta_{3}^{4}+ \Theta_{2}^{4}- \Theta_{4}^{4})}}}\right]^{3}}}.
\biggl|\sum_{q_1, q_2}Q(x,y,q)(x_{q_1q_2}-y_{q_1q_2})\biggr| = \biggl|\sum_{q_1, q_2}g(q_1)h(q_2)(x_{q_1q_2}-y_{q_1q_2})\biggr| &\le N\|x-y\|_{\textup{op}}.
\begin{array}{ccl}A_{1,2}= \displaystyle\frac{\mu_M}{(1-r)\gamma}M_{1,2}, F_{1,2}= \displaystyle\frac{(\gamma+\mu_{A,1}+\mu_{A,2}A_{1,2})A_{1,2}}{\phi}, M_{1}= \displaystyle\frac{M_T^*}{\alpha_{+}}, M_{2}= \displaystyle\frac{M_T^*}{\alpha_{-}}.\end{array}
S\left[\left(m+\frac{\delta}{2}\right)iw+q \right] = \frac{(1+\gamma_3)/2}{[ -(1/2) \delta+iq_3+ \left|{\bf q}\right|^2 / 2m ] + \Sigma_{+}(q_3,\left|{\bf q}\right|)}\, + \, O\left(\frac{\ln m}{m}\right)
f(s) =\begin{cases}s >0\,: \frac{1}{2}s w_{0}(s) + \frac{1}{2}\int_{0}^{s}(\sigma w_{1})(\sigma) \, d \sigma s< 0\,: \frac{1}{2}s w_{0}(-s) + \frac{1}{2}\int_{0}^{-s}(\sigma w_{1}) (\sigma) \, d \sigma.\end{cases}
a_{12}^{2134}=a_{12}^{2143}=a_{12}^{2314}=a_{12}^{2341}=a_{12}^{2413}=a_{12}^{2431}= a_{12}^{3214}=a_{12}^{3241}=a_{12}^{3421}=a_{12}^{4213}=a_{12}^{4231}=a_{12}^{4321}
\hat{f}_\theta(k) = e^{-d \theta/2}\hat{f}(e^{-\theta}k ) = e^{-d \theta/2}\tilde{Q}_{\sigma,a}(e^{-\theta}k ) e^{-\frac{\pi}{\sigma}e^{-2\theta}|k|^2}e^{- e^{-\theta}k \cdot u}.