harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\left (R^{0}|_{S^{1}}f \right)_{p,q}= \alpha(p,q) \frac{\partial f_{p}}{\partial x^{q}}= \frac{1}{2}\left( \frac{\partial f_{p}}{\partial x^{q}}- \frac{\partial f_{q}}{\partial x^{p}}\right) \equiv \frac{1}{\sqrt{2}}\left({\rm curl}\,(f) \right)_{p,q}
{\cal I}( A ) = c_{1}a_{2}^{i j}\partial_{i}\partial_{j}+ ( c_{2}a_{1}^{k}+ c_{3}\partial_{j}a_{2}^{k j}) \partial_{k}+ c_{4}\partial_{i}\partial_{j}a_{2}^{i j}+ c_{5}\partial_{i}a_{1}^{i}+ c_{6}a_{0}
G_{N}( R ) \approx{\frac{\kappa_{5}^{4}| \lambda_{0}|}{4 8 \pi}}\left[ 1 -{\frac{8 1}{2 \kappa_{5}^{8}\lambda_{0}^{4}}}( \xi_{+}^{2}- \xi_{-}^{2})^{2}\right] R^{- \frac{k^{2}}{3}}
u_R(x,t) \le \frac{\| u_0 \|_\infty}{\left( 1-(p-1) \| u_0 \|_\infty^{p-1}\, t \right)^{\frac{1}{p-1}}}\Longrightarrow T_R \ge T:= \frac{1}{(p-1) \| u_0 \|_\infty^{p-1}}\, .
\zeta_{cyl}^{D}(s)=\frac{a^{s-1}}{2\sqrt{\pi}\Gamma\left(\frac{\displaystyle s}{\displaystyle 2}\right)\Gamma\left(\frac{\displaystyle 3-s}{\displaystyle 2}\right)}\sum\limits_{n=-\infty}^{\infty}\int_{0}^{\infty}dy\, y^{1-s}\frac{d}{dy}\ln\left[2y I_n(y) K_n(y)\right].
\max_{1 \leq i \leq M}\|A_i\|_* = \max_{\|v\|_*=1}\max_{1 \leq i \leq M}\|A_iv\|_* &\leq \max_{\|v\|_*=1}\left(\sum_{i=1}^M \|A_iv\|_*^2\right)^{\frac{1}{2}}&=\max_{\|v\|_*=1}\varrho_2(\mathsf{A})\|v\|_* = \varrho_\infty(\mathsf{A})
S_{0}={\frac{1}{2 \pi}}\int d^{2}x \sqrt{- g}\left[ e^{- 2 \phi}( R + 4 ( \nabla \phi )^{2}+ 4 \lambda^{2}) -{\frac{1}{2}}\sum_{i = 1}^{N}( \nabla f_{i})^{2}\right] ,
\sum_{1-l\le q\le n-1-l}{k\choose n-h-q}{n-k\choose q}=\, \sum_{0\le q\le \min(n-1-l,n-h)}{k\choose n-h-q}{n-k\choose q}=\,{n\choose n-h}-\sum_{\min(n-1-l,n-h)+1\le q \le n-h}{k\choose n-h-q}{n-k\choose q}=\,{n\choose h}-\sum_{h\le q\le \max(l,h-1)}{k\choose q-h}{n-k\choose n-q},
\log{\bf E\,}e^{z\xi}=\sum_{r=0}^{\infty}\frac{\Gamma_{r}\left\{\xi\right\}\,z^{r}}{r!}\mbox{and}\Gamma_{r}\left\{\xi \right\}=\left. \frac{d^{r}}{dz^{r}}\log{\bf E\,}e^{z\xi}\right|_{z=0}.
=\frac{|-1 + z_{1,n}|^2 \left(1 - \frac{\pi^2}{a \, n}(2 X^2 -1) \right)\exp\bigl(\frac{2 \pi \, X}{a}\bigr)}{|z_{1,n}|\left|1 + \frac{\pi}{a \, n}e^{i (\psi - \varphi)}\right|-|-1 + z_{1,n}| \,\left(1 - \frac{\pi^2}{2 a \, n}(2 X^2 -1) \right)\exp(\frac{\pi \, X}{a})}
\begin{aligned}\Sigma^2\geq \|{A}e\|_2^2=e^T{\bf G}e=\|e\|_2^2+e^T{\bf (G-I)}e \geq \|e\|_2^2-|e|^T{\bf |G-I|}|e| \geq \|e\|_2^2-M|e|^T{\bf |1-I|}|e| = (1+M)\|e\|_2^2-M\|e\|_1^2 \geq (1+M)\|e\|_2^2-\frac{M}{(\gamma N)^{2/q-2}}\|e\|_{q}^2, \end{aligned}
\begin{array}{ccl}(0~0~-\frac{7}{2}) \rightarrow (0~-\frac{7}{2})(+\frac{7}{2}) \oplus(0~-\frac{5}{2})(+\frac{3}{2}), (0~0~+\frac{3}{2}) \rightarrow (0~+\frac{5}{2})(-\frac{7}{2}) \oplus(0~+\frac{3}{2})(-\frac{3}{2}). \end{array}
\overline\square_B |\phi|^2 = -\sum_a \nabla_{V_a}\nabla_{\bar V_a}|\phi|^2 + \nabla_{H^{0,1}}|\phi|^2 &= \langle \nabla_T^*\nabla_T\phi,\phi\rangle + \langle \phi,\bar\nabla_T^*\bar\nabla_T\phi\rangle -\sum_{a=1}^n\{|\nabla_{\bar V_a}\phi|^2 + |\nabla_{V_a}\phi|^2\}.
w_{ij}(\theta) = b_\alpha N^{\alpha - 1}_{\theta,\alpha}\sigma^{ij},i,j\in\{1,\ldots,d\},~i\leq j,f_{ii}({\bf{x}})=x_i^2,~f_{ij}({\bf{x}})=2x_ix_j,i,j\in\{1,\ldots,d\},~i<j.
r_{}^{(l)}= \nabla F(x^{(l)}) + \sum_{i = 1}^m \lambda^{(l)}_i\nabla g_i(x^{(l)}) + A^T v^{(l)}, \left(r_{}^{(l)}\right)_i = \lambda^{(l)}_i s_i^{(l)}, i=1, \dots, m, \left(r^{(l)}_{}\right)_i = g_i(x^{(l)}) + s_i^{(l)}, i=1, \dots, m, r_{}^{(l)}= Ax^{(l)}- b.
{\tilde{\cal{E}}}( B ) = \frac{B^{2}}{2}+ \frac{1}{8 \pi^{\frac{3}{2}}}\int_{0}^{\infty}\frac{d s}{s^{\frac{5}{2}}}e^{- m^{2}s}\left[ ( e B s ) \coth ( e B s ) - 1 \right] .
\frac{P'_{k+1}(x_0)}{P_{k+1}(x_0)}- \frac{P'_{k}(x_0)}{P_{k}(x_0)}- \frac{1}{x_0}= \frac{1}{x_0-r_1}-\frac{1}{x_0}+ \sum_{j=2}^{k+1}\frac{1}{x_0-r_j}- \frac{1}{x_0-s_{j-1}}&= \frac{r_1}{x_0(x_0-r_1)}+ \sum_{j=2}^{k+1}\frac{r_j-s_{j-1}}{(x_0-r_j)(x_0-s_{j-1})}.
\vert x-b\vert \leq \max (\vert x-a_i\vert, \vert b-a_i\vert)=\vert x-a_i\vert \leq \vert x-a_j\vert \leq \max (\vert x-a_j\vert, \vert b-a_j\vert)=\vert x-b\vert,
H_{m, n}^{+, 1}(x) = \frac{4 M T}{\pi}\int_{-\infty}^\infty \widehat{k^*}\Bigl( -\frac{M \zeta}{\pi}\Bigr) \cos(x \cosh \zeta) e\Bigl( \frac{T \zeta}{\pi}\Bigr) \, d \zeta,
\begin{aligned}\big( x,\frac{u_1}{v_1},t_2,\dots, t_{n+1}\big), ( k=1) \big( t_1,\dots, t_{k-1},\frac{v_{k-1}}{u_{k-1}},\frac{u_k}{v_k},t_{k+1},\dots, t_{n+1}\big), ( 1<k<n+1) \big( t_1,\dots, t_n,\frac{v_n}{u_n},y). ( k=n+1)\end{aligned}
W(H)= P\left[\begin{array}{ccc}P_\alpha^T HP_{\alpha}P_\alpha^T HP_{\beta}\Theta_{\alpha\gamma}\circ P_\alpha^T HP_{\gamma}[7pt]P_\beta^T HP_{\alpha}W_0(P_\beta^T HP_{\beta}) 0 [7pt] P_\gamma^T HP_{\alpha}\circ{\Theta}_{\gamma \alpha}0 0\end{array}\right] P^T \forall \, H\in{\cal S}^p\, .
-2(p-q) \int_0^1 (\{px \}- \{qx \})\log x \mathrm{d}x = 2(q-p) \left( \int_0^1 \{px \}\log x \mathrm{d}x + \frac{1}{2}- \frac{1}{12}\frac{\log q}{q}+ \mathcal{O}\left(\frac{1}{q}\right) \right) &= \frac{q-p}{q}\left(\left( 1 + 2 \int_0^1 \{px \}\log x \mathrm{d}x \right) q - \frac{1}{6}\log q + \mathcal{O}(1) \right)
\left|\sum_{\substack{(\delta_1,\ldots,\delta_K,j_1,\ldots,j_K, n_1,\ldots,n_K)\in\mathcal{A}_{\{2,4,\ldots,K\}}, \Delta \notin\mathcal{D}}}\frac{1}{2^K}\frac{1}{N^{K/2+1}}\prod_{k=1}^Ka'_{j_k}\right|=o(1)
\Big\|\sum_{n = N_1}^{N_2-1}\big| M_{n+1}f - M_n f \big|\Big\|_{\ell^p}\leq\Big\| \sum_{n = N_1}^{N_2-1}\big| m_{n+1}- m_n \big| \Big\|_{\ell^1}\|f\|_{\ell^p}&\lesssim\frac{\vartheta_B(N_2) - \vartheta_B(N_1)}{\vartheta_N(N_1)}\|f\|_{\ell^p}.
\frac{J^w(\textbf{X}_{minRSSU}^{(n+1)})}{J^w(\textbf{X}_{minRSSU}^{(n)})}=-2J^w(X_{1:n+1}) &=\int_{0}^{+\infty}w(x) f^2_{1:n+1}(x)dx &= (n+1)^2 \int_{0}^{1}w(F^{-1}(u)) (1-u)^{2n}f(F^{-1}(u))du & \geq \frac{(n+1)^2}{2n+1}& \geq 1.
\tilde{H}_{n,k}=n^{-k}\,\prod_{s=0}^{k}(s\,n-1\,)\sum_{m=0}^{\lfloor\frac{k}{2}\rfloor}\frac{1}{m!\,(k-2\,m)!\,n^{m}}\prod_{l=0}^{m}{1\over l\,n -1}\,v^{mn}\,u^{k-2m}
A_\delta = \{|z|^2>|y_1 |^2 + |y_2|^2, \|y_1 - \frac1{\sqrt2}e_1|^2 + |y_2|^2 + |z-\frac1{\sqrt2}e_1|^2>\delta^2\}B_\delta = \{|z|^2<|y_1 |^2 + |y_2|^2, \|y_1 - \frac1{\sqrt2}e_1|^2 + |y_2|^2 + |z-\frac1{\sqrt2}e_1|^2>\delta^2 \}.
B^{(\eta)}_{H}(t) := \frac{1}{2\sqrt{2}}\int_{0}^{\infty}\frac{e^{-\eta s}}{s^{1/2+H}}\left([e^{-its}-1]B_{\mathrm{c}}(ds)+[e^{its}-1]\overline{B_{\mathrm{c}}(ds)}\right)
P^{(m)}_{|\frac{p}{N_{0}}\rightarrow \infty}(\gamma_{})=\frac{\left(\frac{\left(\frac{N_{0}\gamma_{}}{p}\right)}{\left(1-\left(\kappa^{2}_{T}(\omega+1)+\omega\right)\gamma_{}\right)}\right)^{n}}{n!}+o\left(\left(\frac{p}{N_{0}}\right)^{-n}\right).
g(\underline{X},\underline{Y})=g(\alpha,\beta,\underline{X},\underline{Y})=\prod_{{{0}}\leq i,j\leq{{r-1}}}(X_i-Y_j)\prod_{{{0}}\leq i<j\leq{{r-1}}}[(X_j-X_i)(Y_j-Y_i)]\enspace.
76(4n-1)^{3}+ 285(3n-1)^{3}> 16(4n-1)^{3}+ 225(3n-1)^{3}+ 60(4n-1)(3n-1)(7n - 2) & = (4(4n-1)+15(3n-1))(4(4n-1)^{2}+15(3n-1)^{2}).
l_{\bar{q}}^{\mu}= \left( k_{g}+ k_{\bar{q}}^{\prime}- k_{\bar{q}}\right)^{\mu}={\frac{\vec{k}_{g_{\! \perp}}^{\, 2}}{2 k_{g}^{+}}}\ \eta^{\mu}\quad \mathrm{a n d}\quad l_{q}^{\mu}= \left( k_{g}+ k_{q}- k_{q}^{\prime}\right)^{\mu}={\frac{\vec{k}_{g_{\! \perp}}^{\, 2}}{2 k_{g}^{+}}}\ \eta^{\mu}
\chi(E_{f}) = \chi(E_{f 1_{Int(A)}}) = \lim_{n}\chi (E_{f\phi_{n}}) = \lim_{n}\int f(g)\phi_{n}(g)dg ~~(\textrm{by~}(2)) = \int f(g)1_{Int(A)}(g)dg = \int f(g)1_{A}(g)dg~~(\textrm{~Since $1_{A}=1_{Int(A)}$ in $L^{\infty}(G)$).}
{\cal H}_{j , j + 1}= \sigma_{j}^{+}\sigma_{j + 1}^{-}+ \sigma_{j}^{-}\sigma_{j + 1}^{+}+ \Gamma \left( \sigma_{j}^{+}\sigma_{j}^{+}+ \sigma_{j}^{-}\sigma_{j + 1}^{-}\right) - \frac{h}{2}\left( \sigma_{j}^{z}+ \sigma_{j + 1}^{z}\right) .
\begin{array}{ll}(|\xi|^2)_t + 2k_2 \xi\nabla\cdot v + 2k_2 v\cdot\nabla\xi + 2 |v|^2 begin{align*}6pt]= -2\gamma k_1 \xi^2\nabla\cdot v - 2k_1 \xi v\cdot\nabla \xi+ \frac{2}{k_1}(\frac{1}{\bar{\varrho}}- \frac{1}{\varrho})\nabla\xi\cdot v.\end{array}
E_{\xi_1^\theta(u)}\biggl[\varphi\biggl(x+u+\sum_{i=2}^{n}Z_i^{\theta[u]}\biggr)\biggr]=E_{\xi_1^\theta(u)}\biggl[\varphi\biggl(x+u+\sum_{i=1}^{n-1}Z_i^{\rho}\biggr)\biggr].
\sum p(5n+4)X^{5n+4}=\frac{\alpha(X^{25})}{\alpha(X^5)^6}(\alpha_0^2\alpha_2^2-3\alpha_0\alpha_1^2\alpha_2+\alpha_1^4)\overset{\eqref{short}}{=}5\frac{\alpha(X^{25})}{\alpha(X^5)^6}\alpha_1^4\overset{\eqref{alpha1}}{=}5X^4\frac{\alpha(X^{25})^5}{\alpha(X^5)^6}.
| J_{a}={\epsilon_{a}}, J_{b}={\epsilon_{b}}, D_{a}= \epsilon_{a}\delta , D_{b}= - \epsilon_{b}\delta \rangle \equiv{\frac{1}{4}}| a_{\epsilon_{a}}b_{\epsilon_{b}}\rangle +{\frac{\delta}{4 \sqrt{3}}}| a_{\epsilon_{a}}b_{\epsilon_{b}}9 \rangle
Z_{\scriptscriptstyle C S}( J ) = e^{- \frac{i \pi}{4}\eta ( 0 , \ast d_{\scriptscriptstyle 1})}\tau_{\scriptscriptstyle R S}( M )^{\scriptscriptstyle 1 / 2}e^{i \langle J , \frac{1}{\ast d_{\scriptscriptstyle 1}}J \rangle}
\left\Vert \beta_{t}\left( g_{t},\cdot \right) \right\Vert \leq M\forall t\in \left[ 0,T\right] \left\Vert \left( \beta_{t}-\beta_{s}\right) \left( g_{t},\cdot \right)\right\Vert_{k}\leq \omega \left( s,t\right)^{\theta -\frac{k}{p}}\forall 0\leq s\leq t\leq Tk=1,2,\dots ,\left[ p\right]
\mathcal{U}=\{\sigma(UV) : U \in \mathcal{U}\}=\{\sigma(VU^{-1}) : U \in \mathcal{U}\}&=\{\sigma(VU^{-1}W) : U \in \mathcal{U}\}=\{\sigma(V^{-1}U) : U \in \mathcal{U}\}.
\frac{}{t}\|u\|^2 -\mu u_{x}(t,1)u(t,1)+\mu u_{x}(t,0)u(t,0)+\mu \|u_x\|^2 = \int_{0}^1f(t,x,u,u_x)ux \leq \int_{0}^1\big(|d_2(t)||u|+M_1u^2+M_2u_xu\big)x := I_2.
g_{ij}(\rho):=\begin{cases}\rho_i \textrm{if}~F_i(\rho)>F_j(\rho),~j\in N(i), \rho_j \textrm{if}~F_i(\rho)<F_j(\rho), ~j\in N(i) , \frac{\rho_i+\rho_j}{2}\textrm{if}~F_i(\rho)=F_j(\rho), ~j\in N(i),\end{cases}
\mu:=\frac{1}{(|\nu_{1}|^{\frac{p}{p-1}}+|\nu_{2}|^{\frac{p}{p-1}})^{\frac{1}{p}}}(\textrm{sgn}(\nu_{1})|\nu_{1}|^{\frac{1}{p-1}},\textrm{sgn}(\nu_{2})|\nu_{2}|^{\frac{1}{p-1}}),
\partial_ip_\gamma =\sum_{\alpha\in\mathbb{Z}_+^n,\,|\alpha|\leq N-k,\,\alpha_i>0}v_{\gamma+\alpha}\frac{y^{\alpha-e_i}}{(\alpha-e_i)!}&=\sum_{\beta\in\mathbb{Z}_+^n,\,|\beta|\leq N-k-1}v_{\gamma+e_i+\beta}\frac{y^{\beta}}{\beta!}=p_{\gamma+e_i}.
\begin{array}{c c c}{\varepsilon^{i}= \left( \begin{array}{c c}{\bar{\varepsilon}_{a}}&{0}\\{0}&{\varepsilon^{a}{}^{t}}\\ \end{array}\right)}&{\rho^{i}= \left( \begin{array}{c c}{0}&{\bar{\rho}^{a}{}^{t}}\\{\rho_{a}}&{0}\\ \end{array}\right)}&{a = 1 , 2 , \cdots , N}\\ \end{array}
|I_3| \leq\int_{\frac{1}{2\beta}\leq|x|\leq\frac{1}{\beta}}\frac{1}{|x|^{n-\beta}}\,dx+\int_{\frac{1}{\beta}\leq|x-z|\leq\frac{2}{\beta}}\frac{1}{|x-z|^{n-\beta}}\,dx &\leq C\frac{1 -2^{-\beta}}{\beta}\beta^{-\beta}+C\frac{2^\beta-1}{\beta}\beta^{-\beta}.
\Gamma^{(k+1)}= \Biggl[\frac{1}T\sum_{t=1}^TE\bigl(z_tf_t'|Z,\theta^{(k)}\bigr) \Biggr] \Biggl[\frac{1}T\sum_{t=1}^TE\bigl(f_tf_t'|Z,\theta^{(k)}\bigr) \Biggr]^{-1},
\mu_R (1 \otimes \mu_R) = \begin{pmatrix}1 1 \end{pmatrix}\begin{pmatrix}1 0 0 0 1 1 \end{pmatrix}= \begin{pmatrix}1 1 \end{pmatrix}\begin{pmatrix}1 1 0 0 0 1 \end{pmatrix}= \mu_R (\mu_R \otimes 1).
\mathbb{H}^s =\{f\in L^2(\mathbb{R}): \|f\|_{\mathbb{H}^s}<\infty\},s\geq 0, \mathbb{E}^{p}_{q}= \{f\in L^2(\mathbb{R}) : \|f\|_{\mathbb{E}^{p}_{q}}<\infty\},p,q>0 \, ,
-\langle R^N(\bar\nabla_{e_j}\big(R^N(V,d\phi(e_j))\tau(\phi)\big),d\phi(e_k))d\phi(e_k),V\rangle =-\langle R^N(V,d\phi(e_k))d\phi(e_k),\bar\nabla_{e_j}\big(R^N(V,d\phi(e_j))\tau(\phi)\big)\rangle
S_{WZ}~=~{1\over 4\pi}\int_0^1 dr \int d^2 x~\epsilon^{\mu\nu}\;{\rm tr}\left( \tilde{g}^{-1}\partial_r \tilde{g}\, \tilde{g}^{-1}\partial_\mu \tilde{g}\, \tilde{g}^{-1}\partial_\nu \tilde{g}\right)~~~.\vspace{2mm}
k_{i}(\Delta)=\sum_{\Upsilon\in \mathcal{T}_{i}(\Delta)}{|\mathbf{T}(\tilde{H}_{i-1}(\Upsilon))|^2}=\sum_{\Upsilon\in \mathcal{T}_{i}(\Delta)}{\mathbf{t}_{i-1}(\Upsilon)^2}.
\mathcal{S}^N(H,H_\tau)=\mathcal{S}^N(\tau):=\sup_{0 \leq \tau' \leq \tau}\left\lbrace \sum_{i=0}^N a^{d(\gamma)}\| \mathcal{D}_i H_\tau \|_i^2 + \sum_{i=0}^{N-1}\| \mathcal{D}_{i+1}H \|_{i+1}^2 + \sum_{i=N}a^{b(\gamma)}\| \mathcal{D}_{i+1}H \|_{i+1}^2 \right\rbrace.
\mathop{\sum_{Q\in\mathcal{D}(E)}}_{Q \subset R}\ell(Q)^{2\beta-(d-n)}\int_{W_Q}U_1(Q)^2\, dx \lesssim\sum_{Q:\,Q \subset R}\mu(Q) \sum_{P:\, Q \subset P \subset R}\Big( \frac{\ell(Q)}{\ell(P)}\Big)^{\beta}\alpha(P)^2 &+ \sum_{Q:\,Q \subset R}\mu(Q) \sum_{P:\, R \subset P}\Big( \frac{\ell(Q)}{\ell(P)}\Big)^{\beta}\alpha(P)^2 = I_1 + I_2.
\mathcal{G}_i = \frac{1}{\phi(m_{2i}/n)}\sum_{k = m_{2i}}^{m_{2i+1}}[(X_k - x_k^o) - (Y_k - y_k^o)], \mathcal{B}_i = \frac{1}{\phi(m_{2i-1}/n)}\sum_{k = m_{2i-1}}^{m_{2i}}[(X_k - x_k^o) - (Y_k - y_k^o)].
\sum_{n \neq 0}n \left[ N_{n}+ \widetilde{N}_{n}+ \sum_{I = 3}^{8}N_{n}^{( I )}+ \sum_{b = 1}^{8}F_{n}^{( b )}\right] = \frac{\sqrt{f}}{l_{s}}m R_{T}( \tilde{a}_{0}^{\dagger}+ \tilde{a}_{0}) .
\begin{cases}\!h_{1}=\sigma x_{0}+j\sigma y_{0}\!h_{k}=\sigma \left ({\sqrt{1-\mu_{k}^{2}}x_{k}+\mu_{k}x_{0}}\right) \quad\:\: +j\sigma \left ({\sqrt{1-\mu_{k}^{2}}y_{k}+\mu_{k}y_{0}}\right) \:\: k=2,{\cdots},N, \end{cases}
\|\Pi_k\varphi\|^\perp_{\alpha+r,m}= \|\Pi_k\varphi\|_{C^0}+|\Pi_k\varphi|^\perp_{\alpha+r,m}\leq \|\varphi\|_{C^0}+ 1+2C_\mu \lambda_0^{\Lambda k}\leq 2+2C_\mu \lambda_0^{\Lambda k}\leq 4C_\mu \lambda_0^{\Lambda k}.
\phi_{( R )}^{1}= e^{i \alpha_{R}}\phi^{1}\, , \quad \phi_{( R )}^{2}= e^{- i \alpha_{R}}\phi^{2}\, , \quad \Phi_{( R )}= e^{i \alpha_{R}}\phi^{1}+ e^{- i \alpha_{R}}\phi^{2}\, .
\int_{\mathcal{I}_j}|\widehat{d_j}(r,\varphi)|^2 \,dr = \int_{\mathcal{I}_j}|\widehat{h}_j(r\sin(\varphi))|^2 \,dr= |\sin(\varphi)|^{-1}\int_{\sin(\varphi)\mathcal{I}_j}|\widehat{h}_j(r)|^2 \,dr.
\Pi_{\mu\nu\kappa\lambda}(p)=\frac{1}{2}(\overline{\eta}_{\mu\kappa}(p)\overline{\eta}_{\nu\lambda}(p)+\overline{\eta}_{\mu\lambda}(p)\overline{\eta}_{\nu\kappa}(p)-\frac{2}{3}\overline{\eta}_{\mu\nu}(p)\overline{\eta}_{\kappa\lambda}(p))
|S(t,\chi)| \ \le \ \left(\frac{1}{4}+o(1) \right) \frac{\log q(|t|\!+\!1)}{\log \log q(|t|\!+\!1)}\underset{s=\frac12}{\mathrm{ord}}\L(s,\chi) \ \le \ \left(\frac{1}{2}+o(1) \right) \frac{\log q}{\log \log q},
\begin{array}{ll}\left(\frac{1}{2}+\frac{1}{2}\cos\left(\frac{2\pi}{p^2-1}\right)\right)^\ell = \left(1-\frac{\pi^2}{(p^2-1)^2}+O\left(\frac{1}{p^8}\right)\right)^\ell = \mathsf{e}^{-\frac{\pi^2\ell}{(p^2-1)^2}}\left(1+O\left(\frac{\ell}{p^8}\right)\right).\end{array}
R = \sum_{n = 1}^{N}x_{n}^{2}( S_{n}^{2}- T_{n}) = \sum_{\Omega ( n , m , m^{\prime})}\frac{x_{n}^{2}}{( x_{n}^{2}- x_{m}^{2}) ( x_{n}^{2}- x_{m^{\prime}}^{2})}\ ,
\frac{1}{\left(x^{2}+t^{2}\right)^{s}}=\frac{\sqrt{\pi}\,2^{\frac{1}{2}-s}}{\Gamma(s)x^{s-\frac{1}{2}}}\intop_{0}^{\infty}y^{s-\frac{1}{2}}J_{s-\frac{1}{2}}(xy)\,e^{-ty}\,dy,
\mathrm{Pr}\big\{|h^\chi_d|\!\leq\! \sqrt{\frac{\tau_\chi}{P_{BS}}}\big\}\!=\! \int_0^{\sqrt{\frac{\tau_\chi}{P_{BS}}}}f_{|h^\chi_d|}(x)dx \!\approx\! \frac{2}{\Omega_h}(K^\chi_d\!+\!1) \frac{\tau_\chi}{P_{BS}}.
\mathcal{A}_t=\begin{pmatrix}A_{t,11}I_{d\times d}-A_{t,11}A_{t,13}A_{t,13}A_{t,21}-A_{t,21}I_{d\times d}-A_{t,11}^T -A_{t,11}^T 0_{d\times d}0_{d\times d}I_{d\times d}I_{d\times d}-I_{d\times d}I_{d\times d}0_{d\times d}0_{d\times d}\end{pmatrix},
a(\textbf{v},\textbf{w})= (\kappa\nabla_{w}\times\textbf{v},\nabla_{w}\times\textbf{w})_h+s(\textbf{v},\textbf{w}), b(\textbf{v},q)= (\nabla_{w}\cdot (\mu\textbf{v}),q)_h,
\left|f(\mu_1)-f(\mu_2)\right| =\frac{C}{p^*}\left|\int_{B_{F^{*}}}\left(\sum_{i=1}^{n}\left|y^*(y_i)\right|^{p^*}\right)d(\mu_1-\mu_2)(y^*)\right| &\leq \frac{C}{p^*}\int_{B_{F^{*}}}\left(\sum_{i=1}^{n}\left\|y_i\right\|^{p^*}\right)d|\mu_1-\mu_2|(y^*) &=\frac{C}{p^*}\left\|(y_1,\ldots,y_n)\right\|_{\ell_{p^*}^n}^{p^*}\left\|\mu_1-\mu_2\right\|.
\mathbb{E}_{P_0}\frac{P_{T_1}P_{T_2}}{P_0^2}= \exp{(\log \Psi) ( 2 k(m+k - 2t) + 2 (t-k)^2 )}= \exp{(\log \Psi) (2mk + 2k^2 - 4kt + 2k^2 +2 t^2 - 4kt)}= \exp{(\log \Psi) (2mk + 4k^2 - 8kt + 2t^2)}\le \exp{(\log \Psi) ((2m - 4t)k + 2t^2 )},
U:=\{S_{[(k-1)\frac{n}{2^\ell}]},\dots,S_{[(2k-1)\frac{n}{2^{\ell+1}}]}\},\quad\widetilde U:=\{S_{[(2k-1)\frac{n}{2^{\ell+1}}]},\dots,S_{[k\frac{n}{2^\ell}]}\}.
\mathcal{Q}_{t}(x_{t-1}, \xi_{[t-1]}, 1 )=\mathbb{E}_{\xi_t, D_t}\Big[ \mathfrak{Q}_t ( x_{t-1}, \xi_{[t-1]}, 1 , \xi_t , D_t ) | D_{t-1}=1 , \xi_{[t-1]}\Big]
\sum\limits_{l=0}^{K}|\sum\limits_{k=0}^{K}T_i^{k,l}(\pi_{i,k}-\pi'_{i,k})| < \sum\limits_{l=0}^{K}\sum\limits_{k=0}^{K}T_i^{k,l}|\pi_{i,k}-\pi'_{i,k}|.
\big((n+m)!\prod\limits_{j=1}^{n}w_j\big)^{-1}=\big((n+m)!\prod\limits_{j=1}^{m_0}w_j\prod\limits_{j=m_0+1}^{n}w_j\big)^{-1}&\leq(\prod\limits_{j=1}^{m_0}w_j)^{-1}\frac{(n+2m_0)!}{(n+m)!}&\leq(\prod\limits_{j=1}^{m_0}w_j)^{-1}\frac{1}{(n+m)(n+m+2)}.
I(s,f^{(n)},\chi)= \sum_{y_0 \not\equiv 1\bmod{\pi}}\sum_{j=0}^{\infty}q^{-1-j}\int\limits_{O_v^{\times 2}}\chi( ac ( x^{12}[y_0+\pi^{j+1}y]^4\widetilde{f^{(n)}}(x,y_0+\pi^{j+1}y)))\ |dxdy| + \sum_{j=0}^{\infty}q^{-1-j}\int\limits_{O_v^{\times 2}}\mathcal{X}(x^{12}[1+\pi^{j+1}y]^4\widetilde{f^{(n)}}(x,1+\pi^{j+1}y))\ |dxdy|,
a_n-a_m =\frac{n-m}{3}+\frac{2n\sin^{2}\frac{n\pi}{2}}{3\left(1+2\cos^{2}\frac{n\pi}{2}\right)}-\frac{2m\sin^{2}\frac{m\pi}{2}}{3\left(1+2\cos^{2}\frac{m\pi}{2}\right)}&\leq \frac{n-m}{3}+\frac{2n}{3}
A\chi e^{-it_0h^{-1}\tilde{P}_h(0)}\tilde{R}_h(z)\chi B= \;A\chi e^{-t_0X}\tilde{R}_h(z)\chi B &\;\;-h^{-1}\int_0^{t_0}A\chi e^{-ith^{-1}\tilde{P}_h(0)}(W+Q_\infty+q_1) e^{-(t_0-t)X}\tilde{R}_h(z)\chi Bdt.
{\cal L}_{WZ}^{(1)}\,=\,-\,{\frac{1}{2}}\,(\,\dot{\theta}^{2}\,+\,3\,{\theta^{\prime}}^{2}\,)\,-\,\lambda \,(\,\dot{\theta}\,+\,\theta^{\prime}\,)\,-\,{\frac{1}{2}}\,\lambda^{2}\;\;,
\int_0^\infty \eta(t)\int_\Omega \Big(\nabla d\odot\nabla d-\frac12|\nabla d|^2\mathbb I_3\Big): \nabla X+\int_0^\infty \eta(t)\int_\Omega \big\langle \partial_t d+u\cdot\nabla d, X\cdot\nabla d\big\rangle=0,
\begin{array}{c}{\delta_{1}^{^{\prime}2}= s_{2}\left( s_{2}+ 1 \right) \ , \quad \delta_{2}^{^{\prime}2}= m^{2}s_{1}\left( s_{1}+ 3 \right)}\\{s_{1}\geq s_{2}\qquad , \qquad s_{1}, s_{2}= 0 , 1 , 2 , . . .}\\ \end{array}
\sum_{\substack{\mathfrak{u}\subseteq [s] \mathfrak{u}\neq \emptyset}}\gamma_{\mathfrak{u}}\left(1-\left(1-\frac{1}{N}\right)^{|\mathfrak{u}|}\right) \leq \frac{\max(1,\Gamma)\mathrm{e}^{\sum_{i=1}^{\infty}\gamma_i}}{N}\,,
\lim_{r\to 0}\begin{vmatrix}f^+(r) \overline{\widetilde{f}^+(r)}f^-(r) \overline{\widetilde{f}^-(r)}\end{vmatrix}= \begin{vmatrix}\Gamma^+(f) \overline{\Gamma^+(\widetilde f)}\Gamma^-(f) \overline{\Gamma^-(\widetilde f)}\end{vmatrix}.
{\frac{\partial \qquad}{\partial Q_{I}( t )}}= \int_{0}^{\Delta t}d s \left[{\frac{\partial Q ( s , t )}{\partial Q_{I}( t )}}\right]{\frac{\delta \qquad}{\delta Q ( s , t )}}= \int_{0}^{\Delta t}d s{\frac{s^{I - 1}}{( I - 1 ) !}}{\frac{\delta \qquad}{\delta Q ( s , t )}}\; ,
f_{l}^{m}=\int_{\Omega}f(\theta,\varphi)Y_{l}^{m*}(\theta,\varphi)d\Omega=\int_{0}^{2\pi}d\varphi\int_{0}^{\pi}d\theta sin\theta f(\theta,\varphi)Y_{l}^{m*}(\theta,\varphi).
\Psi^{\prime \prime}+ [ 3{\cal H}- 2 \frac{\varphi^{\prime \prime}}{\varphi^{\prime}}] \Psi^{\prime}+ [ 4{\cal H}^{\prime}- 4{\cal H}\frac{\varphi^{\prime \prime}}{\varphi^{\prime}}] \Psi - \partial_{\alpha}\partial^{\alpha}\Psi = 0 .
\begin{aligned}D_n := 2^{n^2-1}\cdot p^{2(n+3)(n-1)m}\cdot \left(\prod_{j=1}^{n-1}(|\nu_j\nu_{n-j}' p^{2m}|_p^{-1},p^{\ell+m})^{1/2}\right) \cdot (\ell+(n-1)m+1)^{(n^2-1)}\cdot ((n-1)\ell+n)^{\frac{n^3}{2}}.\end{aligned}
QA^{1/2}JA^{1/2}Q^T =\begin{bmatrix}D^{1/2}0 0 D^{1/2}\end{bmatrix}MJM^T\begin{bmatrix}D^{1/2}0 0 D^{1/2}\end{bmatrix}&=\begin{bmatrix}D^{1/2}0 0 D^{1/2}\end{bmatrix}J\begin{bmatrix}D^{1/2}0 0 D^{1/2}\end{bmatrix}=\begin{bmatrix}0 D -D 0\end{bmatrix}.
\begin{array}{rcl}\hat{\lambda}_{(p=1)}[\gamma(1)] = \hat{\lambda}_{(p=1)}[\gamma(0)] +1= \left(M_{(p=1)}\hat{\lambda}_{(p=1)}\right) [\gamma(0)]\, , & M_{(p=1)}= \left(\begin{array}{cc}1 1 0 1 \end{array}\right)=T\, , \end{array}
\mathbf{S}_{H_{1}}=\left(\begin{array}{ccccc}1 0 0 \cdots 0 \frac{1}{2}\frac{1}{2}0 \cdots 0 \vdots \vdots \vdots \ddots \vdots \frac{1}{N}\frac{1}{N}\frac{1}{N}\cdots \frac{1}{N}\end{array}\right).
N'_{W'W'}=\left(|W'|+1-s_{W'}|W'|+\sum_{h(T_V)\ge 1}\frac{m_V |W'|}{m_V|V|+1}+\sum_{h(T_{V'})=0}\frac{|W'|}{|V'|+1}\right) \le 1+|W'|\left(1-s_{W'}+\sum_{h(T_V)\ge 1}\frac{1}{|V|-2}+\sum_{h(T_{V'})=0}\frac{1}{|V'|+1}\right) \le 1+|W'|\left(1-\frac 3 4 s_{W'}\right).
(n+p+q+1)^2(n-\beta+q)(\beta+p)d_{p,q}(\beta) =(n+p+q+1)^2(n-\beta+q)(\beta+p)\sum_{j=0}^nd_{p,q}^{(j)}L_j(\beta) &=\sum_{j=0}^nd_{p,q}^{(j)}(n+p+q+1)^2(n-\beta+q)(\beta+p)L_j(\beta) &=\sum_{j=0}^nd_{p,q}^{(j)}(n+p+q+1)^2(n-j+q)(j+p)L_j(\beta),
\exp \left( - i e \Bigg[ \frac{1}{\sqrt{\pi\!+\!gN}}\Big( A^{(1)}_\mu, \varepsilon_{\nu \mu}\partial_\nu \Phi^{(1)}\Big) +\sum_{I=2}^N \frac{1}{\sqrt{\pi}}\Big( A^{(I)}_\mu, \varepsilon_{\nu \mu}\partial_\nu \Phi^{(I)}\Big) \Bigg]\right) \Bigg\rangle_{\{K^{(I)}\}}\; =
D_{n}^{i}\ :=\ \max_{\tau_{n-1}(i)<k\le\tau_{n}(i)}(S_{k}-S_{\tau_{n-1}(i)})^{-}, H_{n}^{i}\ :=\ \max_{\tau_{n-1}(i)<k\le\tau_{n}(i)}(S_{k}-S_{\tau_{n-1}(i)})^{+}\shortintertext{and}\chi_{n}(i)\ :=\, \tau_{n}(i)-\tau_{n-1}(i),
R^e_n(s_{v_0}) \left(ew_e+\sum_{i=1}^{e-1}(e-i)(w_{e-i}+w_{e+i})\right) &=eR^e_n(s_{v_0})(w_e)+R^e_n(s_{v_0}) \left(\sum_{i=1}^{e-1}(e-i)w_{e-i}\right)+R^e_n(s_{v_0})\left(\sum_{i=1}^{e-1}(e-i)w_{e+i})\right) &= e\sum_{i=0}^ew_{i}+ \sum_{i=1}^{e-1}(e-i)w_{i}+\sum_{i=1}^{e-1}(e-i)w_{e+i}&= p_b^*(v_0)+ p_b^*(v_1).
\alpha'\tilde{p}' ={\gamma\alpha^{(1)}}\tilde{p}^{(1)}+{(1 - \gamma) \alpha^{(2)}}\tilde{p}^{(2)}{}\le \gamma\left( \frac{B_k}{T_f}+\alpha^{(1)}E_k \right) +(1 - \gamma) \left( \frac{B_k}{T_f}+\alpha^{(2)}E_k \right)= \frac{B_k}{T_f}+ \alpha'E_k,
{\rm Hilb}\big(3T^{(0)}(K_{n_1,\ldots,n_r})^{ab},t\big) =\sum_{{\boldsymbol{k}}=(k_1,\ldots,k_r)\atop 0 < k_j \le n_j}(-t)^{|{\boldsymbol{n}}|-|{\boldsymbol{k}}|}\prod_{j=1}^{r}{n_j \brace k_j}\prod_{j=1}^{|{\boldsymbol{k}}| -1}(1+j t),
{}^{\Psi}B_{p}^{(\alpha,\beta)}(x,y) =2\int_{0}^{\frac{\pi}{2}}\cos^{2x-1}(\theta) \sin^{2y-1}(\theta){}_{0}\Psi_{1}\left( \alpha,\beta;-p \sec^{2}(\theta) \csc^{2}(\theta)\right)d\theta,{}^{\Psi}B_{p}^{(\alpha,\beta)}(x,y) =\int_{0}^{\infty}\frac{u^{x-1}}{(1+u)^{x+y}}{}_{0}\Psi_{1}\left( \alpha,\beta;-2p-p\left( u+\frac{1}{u}\right) \right)du.
S_1 =\sum_{i=1}^k(-1)^{k-i}\sum_{\substack{2\leq j\leq k j\equiv k\ ({\rm mod}\p)}}(j-1)\frac{(k-2)!}{(k-j)!}p(k-j,i-1) q^{i-1}&=(-1)^{k-1}(k-2)! \sum_{\substack{2\leq j\leq k j\equiv k\ ({\rm mod}\p)}}(j-1)(-1)^{\frac{k-j}{p}}\binom{q/p}{(k-j)/p}& =(-1)^{k-1}(k-2)! \sum_{0\leq \ell\leq \lfloor\frac{k-2}{p}\rfloor}(k-1-p\ell)(-1)^{\ell}\binom{q/p}{\ell}
\alpha y(t) = F(t)+\int_{t}^{\infty}\frac{y(s)}{\log(\lambda_{0}(s))}\left(\frac{1}{1+s-t}+\frac{1}{(\lambda_{0}(t)^{1-\alpha}+s-t)(1+s-t)^{3}}\right) ds, t \geq T_{0}
\overline{g(w,\mu)}g(z,\lambda)+ \overline{f(w,\mu)}f(z,\lambda)+\langle v_{z,\lambda},v_{w,\mu}\rangle_{\cal{H}_M}= \left\langle\begin{pmatrix}g(z,\lambda) f(z,\lambda) v_{z,\lambda}\end{pmatrix},\begin{pmatrix}g(w,\mu) f(w,\mu) v_{w,\mu}\end{pmatrix}\right\rangle_{\mathbb{C}^2\oplus\cal{H}_M},
V_{int}\approx\phi(0)\int_{V}\rho(r)d^{3}r-\sum_{i=1}^{3}F_{i}\int_{V}\rho(r)r_{i}d^{3}r\equiv q\phi(0)-\sum_{i=1}^{3}\mu_{i}F_{i}=q\phi(0)-\mu\cdot F,
S = \int d^{2}x \hspace{1 m m}\mathrm{t r}\left( - \frac{1}{2}F_{\mu \nu}F^{\mu \nu}+ D_{\mu}\phi D^{\mu}\phi + i \bar{\Psi}\gamma^{\mu}D_{\mu}\Psi - i g \phi \bar{\Psi}\gamma_{5}\Psi \right) ,
E_{F}= \int_{x_{1}}^{x_{2}}\left[{\frac{1}{2}}( \nabla \phi )^{2}+{\frac{1}{2}}( \nabla \psi_{1})^{2}+{\frac{1}{2}}( \nabla \psi_{2})^{2}+ V ( \phi , \psi_{1}, \psi_{2}) \right] d x \ .
\left[ \frac \delta{\delta \Phi \left( y\right)},w\left( x\right) \right]=-d_{{\rm W}}\left( \Phi \right) \delta \left( x-y\right) \frac \delta{\delta \Phi \left( x\right)}\,\,,\;\;\;\Phi =\left( b^a,\,h^a\right) \,\,,
E_{0}(r)=\frac{1}{4\pi\epsilon_{0}}\int_{\Omega}\rho(r^{\prime})\frac{r-r^{\prime}}{\|r-r^{\prime}\|^{3}}dr^{\prime}\equiv\frac{1}{4\pi\epsilon_{0}}\int_{\Omega}e(r,r^{\prime})dr^{\prime}
\begin{array}{cc}\lbrack\widehat{p}_{k},\widehat{x}_{l}]=-i\delta_{kl}, \lbrack\widehat{p}_{0},\widehat{x}_{0}]= i \lbrack\widehat{p}_{k},\widehat{x}_{0}]=-{\frac{i}{\kappa}}\widehat{p}_{k}, \lbrack\widehat{p}_{0},\widehat{x}_{l}]=0, \lbrack\widehat{p}_{\mu},\widehat{\lambda}_{\rho}^{\lambda}]=0, \end{array}
\frac{1}{2 \pi}c^{( f )}: e^{\mp i 2 \sqrt{\pi}\sqrt{\frac{\pi}{\pi + g N}}U_{1 f}\Phi^{( 1 )}( x )}:_{M^{( 1 )}}\; \prod_{I = 2}^{N}: e^{\mp i 2 \sqrt{\pi}U_{I f}\Phi^{( I )}( x )}:_{M^{( I )}}e^{\pm i \frac{\theta}{N}}\; .