harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
S^{(1)}\le\sum_{j=1}^{N-2}\|\mu_j\|\sum_{k=j+1}^{N-1}\frac{r_k\,\|\mu_{k}\|}{(r_j+\dots+r_{k})^2}\le\sum_{k=2}^{N-1}\sum_{j=1}^{k-1}\frac{r_j\, r_k\,\|\mu_{k}\|}{(r_j+\dots+r_{k})^2}&\le\sum_{k=2}^{N-1}\|\mu_{k}\|\sum_{j=1}^{k-1}r_k\int_{r_k}^\infty\frac{dx}{x^2}=\sum_{k=2}^{N-1}\|\mu_{k}\|<\|\mu\| \,.
v(t,x)= \int_t^T R_{s-t}\left[e^{-(s-t){A}}G B(s,\cdot)\right](x)\,ds+\int_t^T R_{s-t}\left[e^{-(s-t){A}}\nabla^Gv(s,\cdot) B(s,\cdot) \right](x)\,ds.
\gamma_1 = \begin{pmatrix}- \lambda_q -1 1 0 \end{pmatrix}\ , \ \gamma_2 = \begin{pmatrix}0 1 -1 2 \end{pmatrix}\ , \textrm{and}\gamma_3 = \begin{pmatrix}1 \mu_q 0 1 \end{pmatrix}.
K_0 (t) = -\tilde I_0 (t) \ln \frac t2 + \sum_{m=0}^\infty \frac{t^{2m}}{2^{2m}(m!)^2}\psi (m+1) , \tilde I_0 (t) = \sum_{l=1}^\infty \bigl[(l!)\Gamma (l+1)\bigr]^{-1}2^{-2l}t^{2l}, \psi (j) = 1 + \frac 12 + \cdots \cdots \frac 1j -1 - C
\Omega^{\mathcal{L}eg}_{n-1,w}\cong\bigoplus_{r+s=n-1}H_r(W;{\mathbb Z}_2)\otimes_{{\mathbb Z}_2}\Omega_s\cong\Omega^{\mathcal{L}eg}_{n-1}/K^{\mathcal{L}eg}_{n-1,w}\cong\Omega^{\mathcal{L}eg}_{n-1,s}/K^{\mathcal{L}eg}_{n-1,s,w}.
X_{H}=\frac{\partial H}{\partial p_{\mu}}\frac{\partial}{\partial q^{\mu}}-\frac{\partial H}{\partial q^{\mu}}\frac{\partial}{\partial p_{\mu}}=\frac{dq^{\mu}}{dt}\frac{\partial}{\partial q^{\mu}}+\frac{dp^{\mu}}{dt}\frac{\partial}{\partial p_{\mu}}=\frac{d}{dt}
\Omega_{\bar{1}}^{( 1 )}( x , y ) = \int d^{2}y \Sigma^{( 1 )}( x , y ) L_{i_{1}}^{( 1 )}( y ) = - \kappa \epsilon_{i j}\partial^{i}\dot{A}^{j}- \kappa \partial_{1}\dot{\Phi}^{1}-{\sqrt \kappa}\partial_{2}\dot{\Phi}^{2}= 0 .
U^h(t)= I_h z_R(t), U^h_0= I_h z_0, \Psi^h(s)= \Psi^h_1(s)+\Psi^h_2(s)+\Psi^h_3(s)+\Psi^h_4(s), \Theta^h(s)= -i I_h(z_R(s)\Phi \cdot),
\prod_{i=1}^{n_{1}}\frac{1}{\sqrt{2\pi}\sigma_{1}}exp(-\frac{(x_{i}-\mu_{1})^{2}}{2\sigma_{1}^{2}})\cdot\prod_{i=n_{1}+1}^{n_{1}+n_{2}}\frac{1}{\sqrt{2\pi}\sigma_{2}}exp(-\frac{(x_{i}-\mu_{2})^{2}}{2\sigma_{2}^{2}})
T((x,-1), (y,-1)) = T^{-}(x,y), T((x,+1), (y,+1)) = T^{+}(x,y), T((x,+1),(x,-1)) = T^{+-}(x), T((x,-1),(x,+1)) =T^{-+}(x), T((x,-1),(y,+1)) = 0 \mbox{for $x \neq y$}, T((x,+1),(y,-1)) = 0 \mbox{for $x \neq y$},
\begin{aligned}\frac{S_{(r,s),(k,l)}}{S_{(1,1),(k,l)}}=\chi_{\frac{r-1}{2}}\left(\frac{\pi\lambda^{\prime}}{p}\right)\chi_{\frac{s-1}{2}}\left(\frac{-\pi\lambda^{\prime}}{p'}\right) &=(-1)^{k(s-1)+r(l-1)}\chi_{\frac{r-1}{2}}\left(\frac{\pi kp'}{p}\right)\chi_{\frac{s-1}{2}}\left(-\frac{\pi lp}{p'}\right)\ . \end{aligned}
A^{k_1,k_2,k_3, k_4}=(k_1 \Lambda^{\rho}_{L,{I_{ij}}}, k_2 \Lambda^{\rho}_{R,{I_{ij}}},k_3 \Lambda^{\rho}_{D,{I_{ij}}},k_4 \Lambda^{\rho}_{U,{I_{ij}}}),
\begin{aligned}-\int_{x^*}^{x_n}\int_{x_n}^{x}(t-x_0)\cdots(t-x_n)\dtdx =-\int_{x^*}^{x_n}\int_{t}^{x_n}(t-x_0)\cdots(t-x_n)\dxdt = -\int_{x^*}^{x_n}(t-x_0)\cdots(t-x_n)(x_n-t)\dt = h^{n+3}\int_{\frac{x^*-x_0}{h}}^{\frac{x_n-x_0}{h}}\mu(\mu-1)\cdots(\mu-n+1)(\mu-n)(\mu-\frac{x_n-x_0}{h})\d\mu.\end{aligned}
\|(-\Delta)^{\frac{1}{4}}\mathcal{Q}\|_{L^{2}}^2 = \frac{a_{*}}{2}\iint_{\mathbb{R}^{3}\times\mathbb{R}^{3}}\frac{|\mathcal{Q}(x)|^2|\mathcal{Q}(y)|^{2}}{|x-y|}{\rm d}x{\rm d}y.
\eta_{X, Y}^\lambda(\mathsf{P}) = \sum_{i,j,k,\ell,m,n}(2c_{j}^{i}-\delta_{j}^{i})\pi(S(X_{(1)}))_{k}^{j}c_{\ell}^{k}\pi(X_{(2)}S(Y_{(1)}))_{m}^{\ell}c_{n}^{m}\pi(Y_{(2)}K_{\lambda})_{i}^{n}.
\lim_{b \uparrow \infty}\frac{\overline{\overline{W}}^{(q+r)}(b)}{I_{-\infty}^{(q,r)}(b)}= \frac{1}{\Phi^2(q+r)}\lim_{b \uparrow \infty}\frac{W^{(q+r)}(b)}{I_{-\infty}^{(q,r)}(b)}.
{f_{\hat{\textbf{p}}_u}}\left(\hat{\textbf{p}}_u \right) = \frac{1}{{2\pi \sqrt{\det \left({\textbf{R}}\right)}}}\exp \left({- \frac{{{{\left({\hat{\textbf{p}}_u -{\textbf{p}}_u}\right)}^T}{{\textbf{R}}^{- 1}}\left({\hat{\textbf{p}}_u -{\textbf{p}}_u}\right)}}{2}}\right)
\phi_{k,n}(y_1,\cdots,y_k)\geq (n/4)^k(2\ln n)^{\frac{k}{2}}\mathbb{P}(\xi^{(n)}(I_{n,k})=0) -(n/4)^k(2\ln n)^{\frac{k}{2}}\sum_{j=1}^k\mathbb{P}(\xi^{(n)}(I_{n,k})=\xi^{(n)}(J_{n,k,j})=0).
u(t,x)+u_x(t,x)=\frac{e^x}{2}\int^{\infty}_x e^y m(t,y)dy\geq 0, u(t,x)-u_x(t,x)=\frac{e^{-x}}{2}\int_{-\infty}^x e^y m(t,y)dy\geq 0, v(t,x)+v_x(t,x)=\frac{e^x}{2}\int^{\infty}_x e^y n(t,y)dy\geq 0, v(t,x)-v_x(t,x)=\frac{e^{-x}}{2}\int_{-\infty}^x e^y n(t,y)dy\geq 0.
\sideset{}{''}\sum_{\substack{\boldsymbol{r},\boldsymbol{t}\in\mathbb{Z}^d\setminus \{\boldsymbol{0}\}r_{j_\beta}v=r_{j_\beta}w}}|c_{\boldsymbol{r}}c_{\boldsymbol{t}}|\ll_d\frac{s^{2d-q}}{N^{2-q/d}}\frac{(v,w)^q}{\sqrt{|vw|^q}},
\phi^{k}_{1}(1) = \left(\frac{4}{5}\right)^{k}+ \frac{1}{2}\left[1-\left(\frac{4}{5}\right)^{k}\right] \qquad \phi^{k}_{2}(0) = \frac{1}{8}\left[1-\left(\frac{3}{5}\right)^{k}\right].
0< \psi(t): = v(t)-x(t) & = v(t)-f(t)-\int_a^t\Big[k(t,s,v(s))+\displaystyle\frac{v(t)-x}{1+(v(t)-x)^2}\Big]\Delta s & < v(t)-f(t)-\int_a^tk(t,s,v(s))\Delta s \leq 0,
|\nabla^k(\varphi_i-U_1)(z)| & \le C(1+|z|)^{-k}\begin{cases}m_i^{-2}, \mbox{if}5\le n\le 7, \max\{\Theta_im_i^{-2}m_i^{\frac{2}{n-4}}(1+|z|)^{-1}, m_i^{-2}\}, \mbox{if}n=8, \max\{\Theta_i m_i^{-2}m_i^{\frac{2(n-8)}{n-4}}(1+|z|)^{8-n}, m_i^{-2}\}, \quad\mbox{if}n\ge 9.\end{cases}
\int_R^\infty \exp_p(\mu|u|^{p'})r^\theta\mathrm dr \leq C_p^p\dfrac{\mu^{p-1}}{\Gamma(p)}\|u\|_{X^{1,p}_\infty}^{p}+C_{pp'}^{pp'}\dfrac{\mu^{p}}{\Gamma(p+1)}\|u\|_{X^{1,p}_\infty}^{pp'}&\quad+\sum_{j=2}^\infty\dfrac{\mu^{p-1+j}}{\Gamma(p+j)}\int_R^{\infty}|u|^{(p-1+j)p'}r^\theta\mathrm dr.
\sum_{k=0}^{\infty}\frac{(\frac12)_k^2}{(1)_k^2}\left(\frac{2-\sqrt 3}{4}\right)^k\sum_{k=0}^{\infty}\frac{(\frac12)_k^2}{(1)_k^2}\left(\frac{2-\sqrt 3}{4}\right)^k \left[ \left(\frac32 + \sqrt 3 \right) k + \frac14 \right] = \frac{1}{\pi},
\int d x \, \sqrt{g}\, V [ \sigma ] = \int d x \, \sqrt{g}\, \frac{N}{2 \lambda_{B}}\sigma^{2}+ \frac{N}{2}\mathrm{T r ~}\, \int_{1 / \Lambda^{2}}^{\infty}\frac{d t}{t}\, e^{- t \sigma^{2}}e^{\not \nabla^{2}t}.
\sum_{k=0}^{n^{1/6}}Y(m,k) p^k \leq (m-1)! (1+C m^{3/2}n^{-1/2}e^{m^{3/2}n^{-1/2}}) \leq 2(m-1)! C m^{3/2}n^{-1/2}e^{m^{3/2}n^{-1/2}}
h=\begin{bmatrix}1 0 0 -1\end{bmatrix}, e^+=\begin{bmatrix}0 1 0 0\end{bmatrix}, e^-=\begin{bmatrix}0 0 1 0\end{bmatrix}, J^+=\tfrac{1}{2}\begin{bmatrix}-i 1 1 i\end{bmatrix}, J^-=\tfrac{1}{2}\begin{bmatrix}i 1 1 -i\end{bmatrix}, J^0=\begin{bmatrix}0 1 -1 0\end{bmatrix}.
t_{0 n}^{\ell}= ( - 1 )^{n}\left( \frac{( n - 2 \ell - 1 ) !}{n !{( - 2 \ell - 1 ) !}}\right)^{1 / 2}\, e^{- i \ell \psi}\frac{t^{n}}{( 1 - | t |^{2})^{\ell}}\, ,
a^{2r-2s}q^{2(r-s)(d-r-s)}={}_2\phi_1 \biggl( \genfrac{}{}{0pt}{}{q^{2s-2r}, a^{-2}q^{2r+2s-2d}}{0}\,\bigg\vert \, q^{-2}; q^{-2}\biggr).
{\gamma^*_{n-k+1}}=\left( id-t_{n,k}^*\right)\cdot \left( id-t_{n-1,k}^*\right)\cdots\left( id-t_{k+1,k}^*\right),{\delta^*_{n-k+1}}=\left( id-(t_{k}^*)^{2}\, t_{n,k+1}^*\right)\cdot \left( id-(t_{k}^*)^{2}\, t_{n-1,k+1}^*\right)\cdots\left( id-(t_{k}^*)^{2}\, t_{k+1,k+1}^*\right),
V_1 = \frac{1}{\sqrt{2}}\biggl( \begin{matrix}i\zeta 0 0 1 \end{matrix}\biggr) \,, V_2 = \frac{1}{\sqrt{2}}\biggl( \begin{matrix}0 i 1 0\end{matrix}\biggr) \,, |\zeta|=1\,.
\sum_{k=1}^{N_n}\tau^k_n\left\|\frac{u^k_n-u^{k-1}_n}{\tau^k_n}\right\|_{V^*}^q \leq \sum_{k=1}^{N_n}\tau^k_n\left\|f^k_n-A^k_n u^{k-1+\theta}_n-\iota^*\xi^k_n\right\|_{V^*}^q,
\max_{\{B_{k,\ell}\}_{{\mathcal{K}_{\ell}}}:~\sum_{k\in \mathcal{K}_{\ell}}B_{k,\ell}= B_{\ell}}~~-{\sum_{k\in \mathcal{K}_{\ell}}2^{-\frac{B_{k,\ell}}{N_k-1}}P_{k,\ell}\alpha_{k,\ell}N_k}
\partial_t u - \Delta \mu =0{\rm in ~}Q, \mu = -\Delta u +\beta(u)+\pi(u)- f{\rm in~}Q, \partial_{\boldsymbol{\nu}}\mu =0{\rm on~}\Sigma, u_{|_\Gamma}= v{\rm on~}\Sigma, u(0) = u_0{\rm in~}\Omega,
\Big | \sum_{k\neq 0}k|k|^{2s-2}A_{-k}u_k\Big |^2 \leq \Big | \sum_{k\neq 0}|k|^{2s-1}\| A_{-k}\| |u_k| \Big |^2 &\leq \Big (\sum_{k\neq 0}|k|^{2s}\|A_{-k}\|^2\Big ) \cdot \Big ( \sum_{k\neq 0}|k|^{2s-2}|u_k|^2\Big ) &= \|A\|_s^2 \cdot \|u\|_{s-1}^2.
\mathbb{E}\left\{\left|e_{i}^{t}\right|^{2}\right\}\doteq\frac{1}{\sigma_{i}^{2}}\left(\overset{n}{\underset{j=1}{\sum}}\frac{\sigma_{j}^{2}}{m}\mathbb{E}\left\{\left|x_{j}-x_{j}^{t}\right|_{2}^{2}\right\}+\sigma^{2}\right).
\psi(x,t)=(\frac{2a^{2}}{\pi})^{1/4}\frac{e^{i\phi}}{(a^{4}+\frac{4\hbar^{2}t^{2}}{m^{2}})^{1/4}}e^{ik_{0}x}exp[-\frac{(x-\frac{\hbar k_{0}}{m}t)^{2}}{a^{2}+\frac{2i\hbar t}{m}}]
|C(m,n;k)| =|C(0,n-m;k)|^{|\Omega_{m}|}=\Bigl((\tfrac12 + o(1))|G_{n-m}|\Bigr)^{|\Omega_{m}|}(\ref{eqn:BaseCardinality}) = \Bigl(\frac{1}{2^{|\Omega_{m}|}}+o(1)\Bigr)|G_{n-m}|^{|\Omega_{m}|}= \Bigl(\frac{1}{|G_{m}|}+o(1)\Bigr)|G_{n}|.
f(x) = \frac{v_0}{(3r+1)!}x^{3r+1}+ \frac{v_1}{(3r-2)!}x^{3r-2}+ \cdots + \frac{v_i}{(3r+1-3i)!}x^{3r+1-3i}+ \cdots + \frac{v_{r}}{1!}x.
I_1= \sum_{j=1}^{i}\int_{\Lambda_j}u n_j \left[\int_{x_{i+ 1/2}-x_j}^{x_{\alpha_{i,j}-1/2}}+ \sum_{k=\alpha_{i,j}}^I \int_{\Lambda_k}\right] \beta(x_j,v)n(t,v)dv du.
\varrho_{*}\Big ( \sum_{k = 1}^{n}f_{k}d^{\prime}f_{k}^{\prime}\Big ) = \sum_{k = 1}^{n}\big ( r_{U_{\iota}\cap U}^{U}( f_{k}) \# 1 \big ) d \big ( r_{U_{\iota}\cap U}^{U}( f_{k}^{\prime}) \# 1 \big ) ,
\|(d\theta_{t}^{c})^{n}\wedge \theta\wedge \theta_{t}^{c}\| \ge 1-\sum^n_{k=2}\binom nk \|d\theta_{t}^{c}\|^{n-k}\cdot\|d\zeta^k_t\| \cdot \|\theta \| \cdot \|\theta^c\| &-\sum^n_{k=1}\binom nk \|d\theta_{t}^{c}\|^{n-k}\cdot \|d\zeta^k_t\| \cdot \|\theta \| \cdot \|\zeta_t\| &> 1-(2^{n+1}-1)\frac1{2^{n+1}-1}> 0.
\sup_{0\leq y\leq 1}\left\vert\rho^Y_n(y)-u^Y_n(y)\right\vert = O\left(n^{-1/2}\log\log n \right) a.s. \sup_{0\leq y\leq 1}\left\vert u^Y_n(y)\right\vert = O\left(n^{-1/4}\left(\log\log n \right)^{-1/4}\left(\log n\right)^{-1/2}\right) a.s.,
\sum_{\rho \in S_{n}}^{}\chi_{R}( \rho ) D_{R}( \rho^{- 1}) = \frac{n !}{d_{R}}I_{R}, \qquad \mathrm{a n d}\qquad \sum_{\sigma \in S_{n}}^{}D_{R}( \sigma \rho \sigma^{- 1}) = \frac{n !}{d_{R}}\chi_{R}( \rho ) I_{R},
\underbrace{\sum_{({u},{v}) \in [r]\times [s]}\lambda^p_{({u}-1)s +{v}}\cdot c(u^{\ast}; ({u},{v}))}_{}+ \underbrace{\sum_{{v}\in [s]}\rho \cdot c(\overline{u^{\ast}+p}; (u^{\ast},{v}))}_{}= 0~~~p \in \{1,\ldots, r-1\}.
\mathrm{T r}( L^{2}) = 2 ( p_{1}^{2}+ p_{2}^{2}) + 4 m^{2}e^{q_{1}- q_{2}}+ 4 m^{2}e^{- q_{1}- q_{2}}+ 2 m^{2}e^{2 q_{2}}= 4{\cal H}_{A_{3}}^{( 2 )}.
\pi(j_{n_k,\infty}(p_{n_k,\lambda})) = \mu_\psi(\rho(j_{n_k, \infty}(p_{n_k,\lambda}))) = \mu_\psi \circ \mu_{m_k}(p_{n_k,\lambda}) = \mu_\psi(t_{E(\infty)}^0 (\chi_{\psi(Z(\lambda, k))})) = t_F^0(\chi_{Z(\lambda, k)}).
(Q_{2}^{h}x)(t) =\int_{t_{i}}^{t-h}T_{q}(t-s)Bu(s) ds +\int_{t_{i}}^{t-h}T_{q}(t-s)f(s, x(s),(Hx)(s))ds &=T_{q}(h)\int_{t_{i}}^{t-h}T_{q}(t-s-h)Bu(s) ds &+T_{q}(h)\int_{t_{i}}^{t-h}T_{q}(t-s-h)f(s, x(s),(Hx)(s))ds &=T_{q}(h)y_{2}(t,h).
\partial_{\varepsilon, \Phi}f\left(x_{0}\right) =\left\{\phi\left(x\right)=-ax^2 +bx:\left(a+1\right)\left(x_{0}-\frac{b-2}{2\left(a+1\right)}\right)^{2}\leq\varepsilon,a\geq0, b\in \mathbb{R}\right\}, \partial_{\varepsilon, \Psi}g\left(x_{0}\right) =\left\{\psi\left(x\right) = cx:\left(2x_{0}-\frac{c}{4}\right)^{2}\leq\varepsilon, c\in \mathbb{R}\right\}.
\bar{x}^{\ast}:=^{\left( 1/4\right)}\bar{y}=\varepsilon \left \vert\delta \right \vert^{-1/2}\bar{n}+\frac{\left( 2\kappa v+\delta \,\acute{}\, \right)}{2}\left \vert \delta \right \vert^{-3/2}\bar{e},\varepsilon=\operatorname*{sign}\delta.
h_{11,1}^{1^{*}}+h_{22,1}^{1^{*}}=0,h_{11,2}^{1^{*}}+h_{22,2}^{1^{*}}=0,h_{11,1}^{2^{*}}+h_{22,1}^{2^{*}}=0,h_{11,2}^{2^{*}}+h_{22,2}^{2^{*}}=0.
C_1 := \int_0^\infty \bigg|g''(r) + \frac{n-1}{r}g'(r)\bigg|^2 r^{n-1}\, \mathrm{d}r, C_2 := \int_0^\infty \bigg(g''(r) + \frac{n - 1}{r}g'(r)\bigg)\overline{g(r)}r^{n-3}\, \mathrm{d}r, C_3 := \int_0^\infty |g(r)|^2 r^{n-5}\, \mathrm{d}r
\sum_{i=1}^n{p}_i[s+1] \delta{u}_i[s+1] =\sum_{i=1}^n\tilde{p}_i[s] \delta \tilde{u}_i[s],(s=0,\dots,T-1), \sum_{i=1}^n{p}_i[T+1] \delta{u}_i[T+1] =\sum_{i=1}^n{p}_i[T] \delta{u}_i[T].
\begin{array}{c}m_d = \min\big\{{2 \over d}- \lambda > 0\;|\; \lambda \in \Lambda_n \big\}= \min\Big\{{2 \over d}-{a \over n}> 0\;|\; a = 0, \ldots, n\Big\}\geq{1 \over dn}\geq{1 \over (2n - 1) n}.\end{array}
\begin{aligned}w_\alpha = u_I^\dagger + (A_I^*A_I)^{-1}A_I^*Av - \alpha (A_I^*A_I)^{-1}\mathop{{\rm sgn}}(w_\alpha) = (A_I^*A_I)^{-1}A_I^*(A_Iu_I^\dagger + Av) - \alpha (A_I^*A_I)^{-1}\mathop{{\rm sgn}}(w_\alpha) = (A_I^*A_I)^{-1}A_I^*y- \alpha (A_I^*A_I)^{-1}\mathop{{\rm sgn}}(w_\alpha), \end{aligned}
L_{-1}^*\otimes S_j \subseteq L_{-1}^*\otimes L_j\subseteq L_{-1}^*\otimes (X_{1, \, j, \, 1}\oplus X_{1, \, j, \, 3}) = L_{-1}^*\otimes X_{1, \, j, \, 1}\oplus L_{-1}^*\otimes X_{1, \, j, \, 3}
d e t ( 1 - h_{2}h_{1}) = a_{N}\int d T d T^{*}d e t_{H^{+}}( 1 - h_{2}T ) d e t_{H^{-}}( 1 + T^{*}h_{2}) \cdot d e t ( 1 + T^{*}T )^{- 2 N - 1},
\exp ( \tau L_{0}) \, c = \left( \begin{array}{c}{{\strut c^{0}+ e^{v}\tau}}\\{{\strut c^{0}\tau + c^{d - 1}+ e^{v}{\frac{\tau^{2}}{2}}}}\\{{\strut - c^{0}\tau + c^{d}- e^{v}{\frac{\tau^{2}}{2}}}}\\ \end{array}\right) \ .
u_n^{(r)}=\begin{cases}0 2\le n\le r-k \frac{1}{(r-k+1)(r+k+2)}n=r-k+1r\ge k+1 \frac{1}{n(n+2k+1)}-\frac{1}{(n-1)(n+2k)}r-k+2\le n\le k+1.\end{cases}
\varphi^*\left(\begin{array}{c c}a b 0 d \end{array}\right) =\left(\begin{array}{c c c}a^{p^{e_2}}0 b^{p^{e_2}}0 1 0 0 0 d^{p^{e_2}}\end{array}\right) ( \, e_2 \geq 0 \, ) .
d(\tilde{S}_n(z_n, \xi),S(z)) = d(- \nabla j(z_n) + \tfrac{1}{t_n}(\bar{z}_n - z_n +t_n g_n^\xi),S(z)) = d(- \nabla j(z_n) - \zeta_{\eta,n}- \zeta_{C,n}, S(z)).
J = \int d^{2}x \epsilon_{i j}x_{i}\Bigl \{2 \mathrm{T r}[ - T^{3}( g^{- 1}\partial_{j}g - ( g^{- 1}\partial_{j}g ) |_{| \vec{x}| \rightarrow \infty}) ] - A_{j}Q^{3}\Bigr \},
\left( \partial_{\alpha}\partial^{\alpha}+M^{2}\right)T^{(\beta\gamma)\nu}=\frac{1}{4}\,J^{(\beta\gamma)\nu}+\frac{1}{4}J^{(\gamma\beta\nu)}+\frac{1}{6M^{2}}\partial_{\alpha}\partial^{\alpha}J^{\left(\gamma\beta\nu\right)},
\begin{aligned}Q(U) =U^3+\big(1-2n^2+\big(1+n\big)^2m^2\big)U^2+ &\big(-n^2(1+n)\big((1-n)+2(n+1)m^2\big)\big)U+\big(n^4(1+n)^2m^2\big) &=(U-n^2)(U^2+(m^2(1+n)^2-n^2+1)U-m^2n^2(1+n)^2)\end{aligned}
\det \left( \begin{array}{c c}{{A}}&{{B}}\\{{C}}&{{D}}\\ \end{array}\right) = \det \left( \begin{array}{c c}{{A - B D^{- 1}C}}&{{B}}\\{{0}}&{{D}}\\ \end{array}\right) = \det ( A - B D^{- 1}C ) \det ( D ) \ ,
p^{\prime}= P_{N}( p ) = \sum_{k = 0}^{k \le N / 2}\left( \begin{array}{c}{{N}}\\{{( N \mathrm{m o d}\, 2 ) + 2 k}}\\ \end{array}\right) \, p^{N - 2 k}\, ( 1 - p )^{2 k}\, \, \, \, \, . \, \, \, \, \,
\frac{W_{\xi^{\prime}, \eta}( i )}{W_{\xi^{\prime}, \eta}( i - 1 )}= \frac{M_{\xi^{\prime}}}{M_{\eta}}\frac{Y_{\eta}- X_{\xi^{\prime}}\omega^{i}}{Y_{\xi^{\prime}}- X_{\eta}\omega^{i}}\qquad \frac{W_{\xi , \eta^{\prime}}( i )}{W_{\xi , \eta^{\prime}}( i - 1 )}= \frac{M_{\xi}}{M_{\eta^{\prime}}}\frac{Y_{\eta^{\prime}}- X_{\xi}\omega^{i}}{Y_{\xi}- X_{\eta^{\prime}}\omega^{i}}
< \alpha | T_{1}| \gamma > = \frac{2}{\left( 2 \pi \right)^{2 \left( \nu - 1 \right)}}\delta \left( p_{1}- q_{1}\right) \frac{e^{- i q_{1}^{o}x}}{\sqrt{2{\omega}_{1}\left( q_{1}^{o}\right)}}\delta \left( p_{2}- q_{2}\right) \times
\tilde A \Big({\rm I}_{N_{S}}-\frac1{1+\sum a_{k}}(a_{j})_{i,j}\Big) ={\rm I}_{N_{S}}+ (a_{j})_{i,j}-\frac1{1+\sum a_{k}}(a_{j})_{i,j}-\frac1{1+\sum a_{k}}(a_{j})_{i,j}^2 &={\rm I}_{N_{S}}+ \Bigg( \frac{a_{j}(1+\sum a_{k})}{1+\sum a_{k}}-\frac{a_{j}}{1+\sum a_{k}}-\frac{a_{j}\sum a_{k}}{1+\sum a_{k}}\Bigg)_{i,j}={\rm I}_{N_{S}}.
\langle ( n_{j_{1}})_{b}^{j a}| \sin ( \phi_{b}^{j a}/ L ) | ( n_{j_{2}})_{b}^{j a}\rangle = \frac{1}{2 i}\left( \delta_{( n_{j_{1}})_{b}^{j a}, ( n_{j_{2}})_{b}^{j a}+ 1}- \delta_{( n_{j_{1}})_{b}^{j a}, ( n_{j_{2}})_{b}^{j a}- 1}\right)
\begin{aligned}\int \dot{\xi}^2_{(0,1)_\sigma}dy &=\int (-\frac{1}{2}\sqrt{g(y)}+y\dot{\xi}_{(0,1)_\mu})^2dy &=\int [\frac{1}{4}g(y)+y^2\dot{\xi}^2_{(0,1)_\mu}-y\sqrt{g(y)}\dot{\xi}_{(0,1)_\mu}]dy.\end{aligned}
\mathbb{P}_{0}\Biggl(\Biggl|\frac{1}{n}\sum_{k=1}^{n}\bigl|aR_{k}(t)+bI_{k}(t)\bigr|^{2}-\frac{1}{2}\bigl(a^{2}+b^{2}\bigr)\mathbb{E}|D|^{2}\biggr|> \varepsilon \Biggr)\rightarrow 0 \mathbb{P}\mbox{-a.s.}
[0,1] \; = \; f^{-1}\mathbb{R}\; = \; f^{-1}\bigcup_{q \in \mathbb{Z}}(-\infty,q] \; = \; \bigcup_{q \in \mathbb{Z}}f^{-1}(-\infty,q] \; = \; \bigcup_{q \in \mathbb{Z}}L_q
4\overset{0}{H}(X,Y)Z - 2 \overset{1}{H}(X,Y)Z - \overset{2}{H}(X,Y)Z = \overset{g}{W}(X,Y)Z, 2\overset{5}{H}(X,Y)Z - \overset{1}{H}(X,Y)Z + \overset{1}{H}(Y,Z)X = \overset{g}{W}(X,Z)Y,
B_2(m) = \begin{cases}p^{1-3m/2}(-1)^{m+1}\chi_{\psi}(p^m)(1-p^{-1})p^{m}m>0, p^{3m/2}(-1)^{m+1}\chi_{\psi}(p^m)(1-p^{-1})(1+p^{-1}-p^{-m}) m\leq 0. \end{cases}
G \equiv G_{1}+ i G_{2}= \frac{1}{\pi}\oint d \sigma_{i}^{\prime}\ln | \sigma^{\prime}- \sigma | \epsilon_{i j}\partial_{j}G_{1}+ \frac{i}{\pi}\oint d \sigma_{i}^{\prime}\ln | \sigma^{\prime}- \sigma | \partial_{i}G_{1}+ \mathrm{c o n s t .}
X = \{(i,j)u, (i+1,j)u, \ldots, (r,j)u, (r+1,j)v, \ldots \}, Y = \{\ldots, (i-1,j+1)u, (i,j+1)u \}, X^\star = \{(i,j)u,(i+1,j)u, \ldots, (r,j)u\}.
F^\dagger H^\dagger H F (p,q) = \delta(p,q)\!\left(\!\!\!\begin{array}{cccc}3 \! 0 \!\!\!\!2 e^{-ip_1 + ip_2}2 e^{-ip_1 - ip_2}0 \! 3 \!\!\!\! -2 e^{+ip_1 + ip_2}2 e^{+ip_1 - ip_2}2 e^{+ip_1 - ip_2}\! -2 e^{-ip_1 - ip_2}\!\!\!\!3 0 2 e^{+ip_1 + ip_2}\! 2 e^{-ip_1 + ip_2}\!\!\!\! 0 3 \end{array}\!\!\! \right)\!,
R_{(3,5)}(n) = 12\sigma(\frac{n}{3}) - 36\sigma(\frac{n}{9}) + 12\sigma(\frac{n}{5}) -36\sigma(\frac{n}{15}) + 144\, W_{(3,5)}(n) + 1296\, W_{(3,5)}(\frac{n}{3}) - 432\, \biggl(\, W_{(1,5)}(\frac{n}{3}) + W_{(5,9)}(n)\, \biggr).
\int\limits^{+\infty}_{-\infty}\rho(z) \, \overline{W}(\alpha-\beta;x,z)\, W(\alpha - \gamma;y,z)\, \overline{W}(\beta-\gamma;w,z) \,d z =\chi(\alpha,\beta,\gamma)\,W(\alpha-\beta;y,w)\,\overline{W}(\alpha-\gamma;x,w) \,W(\beta-\gamma;x,y),
\log \mu_N\left(P_{N,M}(x,y)\right) = \sum_{m=1}^M \log f_N(x_m,y_m) + \sum_{m=\lfloor wM\rfloor +1}^M \log t_N(y_m) = -M\log Z_N + (w-1)\sum_{m=1}^M \log t_N(y_m) + \sum_{m=\lfloor wM\rfloor +1}^M \log t_N(y_m) = -M \log Z_N + w \sum_{m=1}^M \log t_N(y_m) - \sum_{m=1}^{\lfloor wM\rfloor}\log t_N(y_m)
{\cal L}_{\rm BI}= - \sqrt{-{\rm det}\, \big (\eta_{\mu\nu}+ F_{\mu\nu}+ \bar\chi\Gamma_\mu\partial_\nu\chi +{\textstyle{1\over 4}}\bar\chi \Gamma^a\partial_\mu\chi \bar\chi\Gamma_a\partial_\nu\chi\big )}\,\, .
1-\overline{\cal{F}_{F(\mu)}(w)}\cal{F}_{F(\lambda)}(z)= \overline{\gamma(\mu,w)}(1-\overline{w}z)\gamma(\lambda,z)+\eta(\mu,w)^*(I-F(\mu)^*F(\lambda))\eta(\lambda,z) = (1-\overline{w}z)N_F(z,\lambda,w,\mu)+(1-\overline{\mu}\lambda)M_F(z,\lambda,w,\mu)
\sup_{x_1,\dots,x_{d_n}\in\mathbb{R}}\left|P\left(\bigcap_{k=1}^{d_n}\left\{F_{n,k}\leq x_k\right\}\right)-P\left(\bigcap_{k=1}^{d_n}\left\{Z_{n,k}\leq x_k\right\}\right)\right|\to0.
f_{5}^*(x_c) - f_{7}^*(x_b) = \frac{1}{6}j_{x}(x_i) + \frac{\Delta x^{2}}{144}\left(8\sigma_{5}\sigma_{8}-3\right)\frac{\partial^{2}j_{x}}{\partial y^{2}}(x_{i}) +{\rm{O}}(\Delta x^3) \,
\zeta: =-\tfrac{1}{2}\left(\overline{f}^1_{11}-\overline{f}^2_{12}+\overline{f}^2_{21}-\epsilon t^1_{2\overline{2}}\right)\eta^{\overline{1}}+\tfrac{1}{2}\left(f^1_{11}-f^2_{12}+f^2_{21}-\epsilon\overline{t}^1_{2\overline{2}}\right)\eta^1.
\frac{(u_{+}-v_{-})^{2}}{(u_{+}-v_{-})^{2}-h^{2}}k_{1}^{-}(u)k_{n}^{+}(v)=\frac{(u_{-}-v_{+})^{2}}{(u_{-}-v_{+})^{2}-h^{2}}k_{n}^{+}(v)k_{1}^{-}(u)
\sum_{n = 1}^{\infty}\, f \left( n \right) = - \frac{1}{2}f \left( 0 \right) + \int_{0}^{\infty}\, d x \, f \left( x \right) + i \int_{0}^{\infty}\, d x \, \frac{f \left( i x \right) - f \left( - i x \right)}{e^{2 \pi x}- 1}.
[a^{(r)\mu}_n, a^{(s)\nu\dagger}_m]=\eta^{\mu\nu}\delta_{nm}\delta^{rs},\quad\{c^{(r)}_n,b^{(s)\dagger}_m\}= \{b^{(r)}_n,c^{(s)\dagger}_m\}=\delta_{nm}\delta^{rs},\quad (n,m\ge 1) ,
\begin{aligned}Var(\theta_{\gamma}) =Var(\bar{X}-\gamma(\bar{Y}-E(\bar{Y}))) & =Var(\bar{X})+Var(\gamma(\bar{Y}-E(\bar{Y})))-2Cov(\bar{X},\gamma(\bar{Y}-E(\bar{Y}))) & =Var(\bar{X})+\gamma^2 Var(\bar{Y})-2 \gamma Cov(\bar{X},\bar{Y})\end{aligned}
\rho\left(\frac{\partial v}{\partial t}+u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial r}\right)+\frac{\partial P}{\partial r}=\mu\left(\frac{\partial^{2}v}{\partial x^{2}}+\frac{\partial^{2}v}{\partial r^{2}}+\frac{1}{r}\frac{\partial v}{\partial r}-\frac{v}{r^{2}}\right),
\left. \begin{array}{l l}\eta_E = x_1(z_3\!-\!z_2)+x_2(z_1\!-\!z_3)+x_3(z_2\!-\!z_1) \eta_1 =x_2(z_E\!-\!z_3)+x_3(z_2\!-\!z_E)+x_E(z_3\!-\!z_2) \eta_2 =x_3(z_E\!-\!z_1)+x_1(z_3\!-\!z_E)+x_E(z_1\!-\!z_3) \eta_3 = x_1(z_E\!-\!z_2)+x_2(z_1\!-\!z_E)+x_E(z_2\!-\!z_1).\end{array}\right.
\lim_{t\to 0^+}\frac{\sin\big(\frac{(s+d)\sqrt{k}}{2}\big)}{\sin(s\sqrt{k})}= 1 , \lim_{t\to 0^+}\frac{2\sin\big(\frac{(s-d)\sqrt{k}}{2}\big)}{\sin(t\sqrt{k})}= \lim_{t\to 0^+}\frac{s-d}{t}, \lim_{t\to 0^+}\frac{\cos(s\sqrt{k})\sin\big( \frac{t\sqrt{k}}{2}\big)}{\sin(s\sqrt{k})\cos\big( \frac{t\sqrt{k}}{2}\big)}= 0
A_1= \prod_{\substack{p>2}}\left(1+\sum_{\alpha\ge1}f_t(p^\alpha)\right)= \prod_{\substack{p>2}}\left(1-\sum_{\alpha\ge1}\frac{p^t-1}{p^{\alpha(t+1)-1}|\Gamma(p^\alpha)|(p-1)}\right).
\sum_{\ell=1}^L n_\ell \le \frac{L-1}{-\varepsilon C}\left[\ln\left(\frac{2\varphi(\alpha)}{1 + 2\varphi(\alpha)}\right)+ \ln\left(\frac{\beta}{3}\right)+ \frac{1}{L-1}\ln\left( \frac{\varepsilon}{F(x^{(1)}) - \eta^1}\right)\right] + L.
A\circ B =(\mathcal{R}A+i\mathcal{I}A)\circ(\mathcal{R}B+i\mathcal{I}B) &=(\mathcal{R}A\circ\mathcal{R}B-\mathcal{I}A\circ\mathcal{I}B)+i(\mathcal{R}A\circ\mathcal{I}B+\mathcal{I}A\circ\mathcal{R}B).
|f|^p_p \leq C |f|^{(6-p)/2}_2 |\nabla f|^{(3p-6)/2}_2, begin{align*}8pt] |g|_\infty\leq C |g|^{q(r-3)/(3r+q(r-3))}_q |\nabla g|^{3r/(3r+q(r-3))}_r.
|\alpha_-(t)| \lesssim e^{e_0 t}\sum_{n=[t]}^\infty \Bigg| \int_{n}^{n+1}e^{-e_0 s}B(R, \mathcal{Y}_+)(s) ds \Bigg| \lesssim e^{e_0 t}\sum_{n=[t]}^\infty e^{-(e_0 + c_1) n}\lesssim e^{-c_1 t}.
x\mapsto x, y\mapsto x y\Longrightarrow\mathcal{I}_{1,\beta_1,\beta_2,\beta_3}\!\!=\!\!3\!\!\int_{[0,1]^2}\! \hat{f}_{\mathbf{t}}(x,y)^s x^{3\beta_1+4\beta_2+13\beta_3+1+26s}y^{\beta_3+1}q(xy) \frac{dx}{x}\frac{dy}{y}
\int \mathbb{B}(x) |u|^{r(x)}\leq 2^{r_+}\int \mathbb{B}(x) \left( |u-v|^{r(x)}+ |v|^{r(x)}\right)\leq c(r^+)\int \mathbb{B}(x) \left( |u-v|^{r_+}+ 1 + |v|^{r(x)}\right) = c(r^+)\left(\int \mathbb{B}(x) |u-v|^{r_+}+ \int \mathbb{B}(x) |v|^{r(x)}+ \int \mathbb{B}(x) \right).
\begin{aligned}h_0(r) =c_*\int_r^1 s^{1-n}\,E_+(s)\,ds, h_1(r) =\int_r^1 s^{1-n}\,E_+(s)\int_0^s \rho^{n-1}\,E_-(\rho)\,B[v](\rho)\,d\rho\,ds, \end{aligned}
{\bf M}_{\Lambda}^{R L}= \frac{1}{2}\theta_{\Lambda}\Lambda - 2 , \quad{\bf M}_{\Lambda}^{R i}= - \frac{1}{2 \sqrt{2}}\hat{\beta}\theta_{\Lambda}\gamma^{i}\theta_{\Lambda}, \quad{\bf M}_{\Lambda}^{L i}= \frac{1}{2 \sqrt{2}\hat{\beta}}\Lambda \gamma^{i}\Lambda ,
\sum\limits_{\substack{q\in \mathbb{N}Q/\sqrt{2}<q\le Q}}\sum\limits_{\substack{r \bmod{q}(r,q)=1}}\left|\sum\limits_{\substack{n\in \mathbb{Z}[i] \mathcal{N}(n)\le N}}a_n \cdot e\left(\mbox{\rm Tr}\left(\frac{nr}{2q}\right)\right)\right|^2= \Sigma\left(Q^2,N;\mathbb{N}\right).
\vert \, \sum_{j \in \tau_{a}}k_{0 , j}\, \vert \leq 2 ( m - \eta ) \quad \mathrm{~ f o r ~ a l l ~}\quad \tau_{a}\subseteq \{1 , \ldots , n \}\quad , \quad k_{0 , n}= - \sum_{j = 1}^{n - 1}k_{0 , j}
0=\Delta_{\mathbb{S}^{n-1}}Y(\theta) + \mu Y(\theta) \mathbb{S}^{n-1}, 0=(p-1) |v'|^{p-2}f'' + \frac{(p-1)(n-1)}{r}|v'|^{p-2}f' -\frac{\mu}{r^2}|v'|^{p-2}f +(p-1)(p-2)|v'|^{p-4}v'v'' f' +\alpha v^{p^*-2}f [0,\infty).
\left( \begin{array}{c}{{\psi_{1}}}\\{{\psi_{2}}}\\ \end{array}\right) \longrightarrow \left( \begin{array}{c}{{\widetilde \psi_{1}}}\\{{\widetilde \psi_{2}}}\\ \end{array}\right) = \left( \begin{array}{c}{{a}}\\{{c}}\\ \end{array}\begin{array}{c c}{{b}}\\{{d}}\\ \end{array}\right) \left( \begin{array}{c}{{\psi_{1}}}\\{{\psi_{2}}}\\ \end{array}\right) ,
\begin{aligned}W(\xi, \xi) = |\xi'|^2 W\Big(\frac{\xi'}{|\xi'|}, \frac{\xi'}{|\xi'|}\Big) + (1 - |\xi'|^2) W(\nu, \nu) &\leq |\xi'|^2 W(\xi, \xi) + (1 - |\xi'|^2) W(\nu, \nu),\end{aligned}
\Psi_{m,2N}(a,b,q) - G_{m,3N}(a,b,q) = a^{\delta_{2,m}}b^{\delta_{1,m}}q^{3N+m}+\dots, \Psi_{m,2N+1}(a,b,q) - G_{m,3N+3-m}(a,b,q) = a^{\delta_{1,m}}b^{\delta_{2,m}}q^{(3(N+2)-1)\delta_{1,m}+ (3(N+1)-2)\delta_{2,m}}+\dots,
\hat{G}_{1,\varphi}(x) = \frac{x - i}{x + i}{G}_{1,\varphi}(x),\;\; \varepsilon \hat{N}_{\varphi}(x) := \hat{M}_{0,\varphi}(x) = \varepsilon \frac{x - i}{x + i}{N}_{\varphi}(x),
0\equiv\nabla_{X}S_{33}-\nabla_{X}S_{01}=X(S_{33}-S_{01}) -2S_{a3}\nabla_{X}m_{3}^{a}+S_{a1}\nabla_{X}k^{a}+S_{0a}\nabla_{X}l^{a}=3S_{13}\nabla_{X}k_{3},