harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
k_{t}(x,y)=\frac{1}{t^{n+1}}\left.{\frac{\partial \phi}{\partial x_{k}}}\right|_{\frac{x-y}{t}}\bigl(f(x)-f(y)\bigr)+\frac{1}{t^{n}}\phi\left(\frac{x-y}{t}\right)\left.{\frac{\partial f}{\partial x_{k}}}\right|_{y}.
\int\limits_{0}^{\infty}e^{q_1}\, b_1 \, q_1 \, e^{b_1 x}\, e^{-q_1 \, e^{b_1 x}}\, x \, (b_1 - b_2) \, dx = \; e^{q_1}\, q_1 \, (b_1 - b_2) \, \int\limits_{0}^{\infty}b_1 \, e^{b_1 x}\, e^{-q_1 \, e^{b_1 x}}x \; dx.
\left\langle T_ag_t , T_ag_j\right\rangle_{\alpha}= \frac{(1+o(1)) (a+1)(\alpha-\frac{1}{2})}{(a- \alpha +\frac{3}{2})}(j+1)^{a+\frac{1-\alpha}{2}}(t+1)^{-a+\frac{\alpha-3}{2}}.
\begin{array}{l c l}{{\lambda^{( 0 )}= \gamma_{1 , 6}\lambda^{( 0 )}\gamma_{1 , 6}^{- 1}}}&{{,}}&{{\lambda^{( 0 )}= - \gamma_{\Omega R , 6}\lambda^{( 0 )^{T}}\gamma_{\Omega R , 6}^{- 1}}}\\{{\lambda^{( i )}= e^{2 \pi i v_{i}}\gamma_{1 , 6}\lambda^{( i )}\gamma_{1 , 6}^{- 1}}}&{{,}}&{{\lambda^{( i )}= - \gamma_{\Omega R , 6}\lambda^{( i )^{T}}\gamma_{\Omega R , 6}^{- 1}}}\\ \end{array}
f_{\infty}^{(\alpha)}(|p|) = \frac{\Omega_{\alpha}^2}{2\pi}\int_{-\pi}^{\pi}e^{i\kappa p}\left(4\sin^2{\frac{\kappa}{2}}\right)^{\frac{\alpha}{2}}{\rm d}\kappa =\Omega_{\alpha}^2 \frac{2^{\alpha + 1}}{\pi}\int_{0}^{\frac{\pi}{2}}\sin^{\alpha}(\varphi)\cos{(2p\varphi}){\rm d}\varphi
[\begin{matrix}(\begin{matrix}1&0\\ 0&0\end{matrix})^{T}&(\begin{matrix}0&1\\ 0&0\end{matrix})^{T}\\ (\begin{matrix}0&0\\ 1&0\end{matrix})^{T}&(\begin{matrix}0&0\\ 0&1\end{matrix})^{T}\end{matrix}]=[\begin{matrix}1&0&0&0\\ 0&0&1&0\\ 0&1&0&0\\ 0&0&0&1\\ \end{matrix}],
\begin{gathered}W_p(\mathbb{Q}, \mathbb{Q}'):= \Big(\inf_{\pi \in \Pi(\mathbb{Q}, \mathbb{Q}')}\int_{\mathcal{S}_0 \times \mathcal{S}_0}\Vert \mathbf{c}- \mathbf{c}' \Vert_p \pi( \mbox{\upshape d}\mathbf{c}, \mbox{\upshape d}\mathbf{c}')\Big)^{\frac{1}{p}},\end{gathered}
\begin{aligned}C_m(U) =\sup\left\{\int_{U}(\Delta u)^m\wedge\beta_n^{n-m}:u\in QSH_m(\Omega)\cap C(\Omega), -1\leq u<0\right\}&=\sup\left\{\int_{U}(\Delta u)^m\wedge\beta_n^{n-m}:u\in QSH_m(\Omega)\cap C^{\infty}(\Omega), -1\leq u<0\right\}\end{aligned}
D = \left( \begin{array}{cc}v^{-1}\partial_1 u^{-1}\partial_2 u \partial_4 v \partial_3 \end{array}\right) \; , \quad \bar{D}=\hat{V}^{\epsilon \, -1}D = \left( \begin{array}{cc}v \partial_3 -q^{-2}u^{-1}\partial_2 -q^{-2}u \partial_4 v^{-1}\partial_1 \end{array}\right) \; ,
\frac{\partial}{\partial y^j_k}= \frac{\partial}{\partial y^j_{k+1}}+ \sum_l \bigg(\frac{\partial a^k_l}{\partial y^j_k}\bigg) \frac{\partial}{\partial y^l_{k+1}}+ \sum_\tau \bigg(\frac{\partial \beta^k_\tau}{\partial y^j_k}\bigg) \frac{\partial}{\partial \eta^\tau_{k+1}}
\mu_{k}^i\left(B\right) \geq 1\sum\limits_{\theta \in B^c}{\prod\limits_{j=1}^{n}\left( \frac{\mu_0^j(\theta)}{\mu_0^j(\theta^*)}\right)^{\left[A^{k}\right]_{ij}}}\prod\limits_{t=1}^{k}\prod\limits_{j=1}^{n}\left( \frac{\ell^j(s_{t}^j|\theta)}{\ell^j(s_{t}^j|\theta^*)}\right)^{\left[A^{kt}\right]_{ij}}
\partial_{h_1,\dots,h_{2s},k}\phi(x)=\partial_{h_1,\dots,h_{2s},k'}\phi(x)=\partial_{h_1,\dots,h_{2s},k}\phi(x+k') =\partial_{h_1,\dots,h_{2s},k'}\phi(x+k) =v_{1,0,\mathbf{h}}-v_{1,1,\mathbf{h}}-(v_{1,0,\mathbf{h}}-v_{1,1,\mathbf{h}}) =0
\|\widetilde\delta_\Gamma^{-1/2-\varepsilon}\nabla (u - I^k_h u)\|_{L^2(\Omega)}\lesssim (1 + \delta_{0,\varepsilon}|\ln h|^{1/2}) h^{-\varepsilon}\begin{cases}h^s \|u\|_{B^{s+3/2}_{2,1}(\Omega)}, s = 0, h^s \|u\|_{B^{s+3/2}_{2,\infty}(\Omega)}, s \in (0,k), h^s \|u\|_{B^{k+3/2}_{2,1}(\Omega)}, s = k.\end{cases}
\int\limits_{Y_{2s_0}^{(1)}\cap\{s\leq s_0'\}}u_1^f\overline{v_{1s}^g}dsdy' =\int\limits_{Y_{2s_0}^{(2)}\cap\{s\leq s_0'\}}u_2^f\overline{v_{2s}^g}dsdy'.
\|f^{=d}\|_q \leq \gamma' \rho^{-d}= \gamma \left( 99\cdot2 \, r \sqrt{q}\sqrt{\frac{\log(1 / \gamma)}{d}}\right)^d \leq \gamma \left( 200 r \sqrt{q}\cdot \max \left\{\sqrt{q}, \sqrt{\log(1 / \gamma)}\right\}\right)^d,
\begin{aligned}\sum\limits_{i = 1}^{N}{\left({{n_{i + 1}}-{n_i}}\right)}=n_{N+1}-n_1, &\sum\limits_{i = 1}^{N}{\left({{n_{i + 1}}-{n_i}}\right)\left({2{\tau_{\rm c}}+ 2\mathcal{T}+{n_{i + 1}}+{n_i}}\right)}=\left(n_{N+1}+\tau_{\rm c}+\mathcal{T}\right)^2-\left(n_1+\tau_{\rm c}+\mathcal{T}\right)^2.\end{aligned}
V_1 = U_1 \cap ( V_1 \cup V_2 ) = U_1 \cap ( V_1 \cup V_3 ) = U_1 \cap ( V_1 \cup V_2 \cup V_3 ), V_2 = U_2 \cap ( V_1 \cup V_2 ) = U_2 \cap ( V_2 \cup V_3 ) = U_2 \cap ( V_1 \cup V_2 \cup V_3 ), V_3 = U_3 \cap ( V_1 \cup V_3 ) = U_3 \cap ( V_2 \cup V_3 ) = U_3 \cap ( V_1 \cup V_2 \cup V_3 ).
\Vert T\Vert =\max\{|a_{11}+a_{21}|+|a_{12}+a_{22}|,|a_{11}-a_{21}|+|a_{12}-a_{22}|\}& =|a_{11}+a_{21}|+|a_{12}+a_{22}|=\Vert(a_{11},a_{12},a_{21},a_{22})\Vert_{1},
Z ({\cal M}) = e^{- i \pi ( 1 + b^{1}({\cal M}) )}e^{- i \pi ( \mathrm{d i m}H^{0}+ \mathrm{d i m}H^{1})}e^{- i \pi [ 2 I_{a}- ( 1 - 2 a ) ]}{\tau}_{R}^{- 2}.
(Gf)(x-v_{i}) =T_{w_{x}}^{-1}(T_{\sigma_{x}(i)}^{-1}\cdots T_{\sigma_{x}(i)+d_{i}^{+}(x)-1}^{-1}). \left( \rho(X_{\sigma_{x}(i)+d_{i}^{+}(x)}T_{\sigma_{x}(i)+d_{i}^{+}(x)-1}\cdots T_{\sigma_{x}(i)})f\right)\!(w_{x}x).
\ddot{y}(t) =16(s_c(p-1)+1)E[Q]- 8s_c(p-1) \| \nabla u \|_2^2 &= 8s_c(p-1) \left( \| \nabla Q \|_2^2 - \| \nabla u \|_2^2 \right) = -8 s_c(p-1) \delta(t) <0,
|{\det}_2 ( +A)-{\det}_2 ( +B)|=|g(1)-g(0)|\leq |g(1)-g(0)-g'(0)|+|g'(0)| \leq\frac{1}{2}\sup_{0\leq t\leq 1}|g''(t)|+|g'(0)|\leq\frac{1}{\rho^2}\sup_{|\lambda-1/2|\leq\rho+1/2}|g(\lambda)|+|g'(0)|.
|\{(x,t)\in Q^{(1-\delta)^i}_{\tau^i}: \nabla u\cdot e\le \ell (1-\delta)^i\}|> \mu |Q^{(1-\delta)^i}_{\tau^i}| \quad\mbox{for all}e\in\mathbb S^{n-1}\mbox{and}i=0,\cdots, k,
\eqref{term-2}= \sum_{j,k}\iint |u(t,y)|^2 \psi_R(x-y) (x_j-y_j) \partial_j \partial^2_k (|u(t,x)|^2) dx dy &= - \sum_{j,k}\iint |u(t,y)|^2 \partial_j[\psi_R(x-y) (x_j-y_j)] \partial^2_k (|u(t,x)|^2) dx dy &= \sum_k \iint |u(t,y)|^2 \partial^x_k \left[3\phi_R(x-y) +2(\psi_R-\phi_R)(x-y) \right] \partial_k(|u(t,x)|^2) dx dy,
\Big|\mathcal{S}\Lambda_\iota+\mathcal{T}_\iota\Lambda_\iota^2+O(\Lambda_\iota^3)\Big|\leq N^{{\varepsilon}}\frac{|\mathcal{S}|+ \hat{\Lambda}}{(N\eta)^{\frac13}},\iota=1,2 \Theta(z,{\frac{N^{3\varepsilon}}{(N\eta)^{\frac13}}}\,,{\frac{N^{3\varepsilon}}{\sqrt{N\eta}}})\cap\Omega_3(z)\,.
S_1(x) = x_1, S_2(x) = \frac{x_1^2}{2}+x_2, S_3(x) = \frac{x_1^3}{6}+ x_1x_2 + x_3 , S_4(x) = \frac{x_1^4}{24}+ \frac{x_2^2}{2}+ \frac{x_1^2x_2}{2}+ x_1x_3 + x_4.
|\sigma|^2_g=\frac{\i^{n^2}\sigma\wedge \bar{\sigma}}{dV_g}=c_n \frac{(dy_1\wedge\dots \wedge dy_n)\wedge (d\theta_1\wedge\dots \wedge d\theta_n)}{(dx^1\wedge\dots \wedge dx^n)\wedge (d\theta_1\wedge\dots \wedge d\theta_n)}=c_n\frac{dy_1\wedge\dots \wedge dy_n}{dx^1\wedge\dots \wedge dx^n},
|I_{3}(t)| \le \Bigg|\int_{1}^{\infty}t^{-1+\beta/2}x^{-1/2 -1/\beta}w(x^{-1/\beta}, \beta, 1)dx\Bigg| \le C t^{-1+\beta/2}\Bigg|\int_{1}^{\infty}x^{-1/2 - 1/\beta}dx\Bigg| = Ct^{-1+\beta/2}.
\widetilde{\omega}^{+, \iota}_n := \prod_{2 \leq m \leq n, m: \textrm{even}}\Phi_m \left(\frac{1}{1+X}\right) = \prod_{2 \leq m \leq n, m: \textrm{even}}\left( \Phi_m (1+X) \cdot (1+X)^{-p^{m-1}(p-1)}\right) = \left( \prod_{2 \leq m \leq n, m: \textrm{even}}(1+X)^{-p^{m-1}(p-1)}\right) \cdot \widetilde{\omega}^{+}_n ,
\rho\left(q^{1/2}c_{k}(q)\right)=\frac{A(q^{1/2}c_{k}(q);q)}{B'(q^{1/2}c_{k}(q);q)}=\frac{-q^{1/2}\mathcal{S}_{q}(q^{1/2}c_{k}(q))}{\mathcal{C}_{q}'(c_{k}(q))}=-\frac{\mathcal{C}_{q}(qc_{k}(q))}{c_{k}(q)\mathcal{C}_{q}'(c_{k}(q))}
{\rm dim}\;{\rm ker}\; \mathcal{T}={\rm dim}\;{\rm ker}\;(P_{\rm F}-D_aP_{\rm F}D_a-1) ={\rm dim}\;{\rm ker}\; \mathcal{T}' ={\rm dim}\;{\rm ker}\;(\mathcal{P}_{\rm D}-U\mathcal{P}_{\rm D}U-1),
L^1(J)+L^2(J)-L^1(I)-L^2(I) \leq (K^{*}_{11}|K_{21}|+K^{*}_{25}|K_{25}|-1)|\beta_1|-\sum\limits_{i=2,3,4}K^{*}_{2i}|\beta_i| +M\Delta'(\alpha_5,\boldsymbol{\beta}^{*}).
\sum_{\alpha,\beta\in\mathbb{F}_{p^m}}T(\alpha,\beta)^2T(\pi^{\frac{p^k+1}{2}}\alpha,-\pi\beta)=\sum_{\alpha,\beta\in\mathbb{F}_{p^m}}T(\alpha,\beta)T(\pi^{\frac{p^k+1}{2}}\alpha,-\pi \beta)^2.
\begin{aligned}T_{f}\big(A^{{'E}_{0}}_{0}, h^{{'E}_{0}}\big)= T_{f}(A^{C^{\bullet}(W^{u}_{Z_{2}}/W_{Y}^{u}, F)}, h^{C^{\bullet}(W^{u}_{Z_{2}}/W_{Y}^{u}, F)}) &-T_{f}(A^{C^{\bullet}(W^{u}_{Z}, F)}, h^{C^{\bullet}(W^{u}_{Z}, F)})+T_{f}(A^{C^{\bullet}(W^{u}_{Z_{1}}, F)}, h^{C^{\bullet}(W^{u}_{Z_{1}}, F)}), \end{aligned}
\frac{1}{\gamma}\frac{d}{dt}\int_{\Omega}u^{\gamma}(t)+\frac{4(\gamma-1)}{\gamma^2}\int_{\Omega}|\nabla u^{\frac{\gamma}{2}}(t)|^2= (\gamma-1)\chi\int_{\Omega}u^{\gamma-1}(t)\nabla u (t)\cdot \nabla v(t) +\int_{\Omega}u^{\gamma}(t)\Big[a_0(t,\cdot)-a_1(t,\cdot)u(t)-a_2(t,\cdot)\int_{\Omega}u(t) \Big].
\sum_{k=1}^\infty \frac{|\hat\mu(k,k)|^2}{(k+1)^{2\alpha}}\le C\sum_{k=1}^\infty \frac{1}{(2k^2)^{1/\tau}(k+1)^{2\alpha}}\le C\sum_{k=1}^\infty \frac{1}{k^{2\alpha+2/\tau}}.
x_a =\frac{\prod_{q=s_ia_n}^{s_ia_n\oplus(s-1)}\lambda_{n,q}}{\prod_{q=a_j\ominus s}^{a_j\ominus 1}\lambda_{j,q}}x_{a(j,n,a_j\ominus s,(s_ia_n\oplus s)/s_i)}&=\frac{\prod_{u=0}^{s-1}\lambda_{n,u}}{\prod_{u=0}^{s-1}\lambda_{j,u}}x_a &=\frac{\gamma^n}{\gamma^j}x_a
0\! =U\langle A,J\xi\rangle=\langle\bar{\nabla}_U A,J\xi\rangle+\langle A,\bar{\nabla}_U J\xi\rangle=a\langle\nabla_U A,U\rangle+b\langle\nabla_U A,V\rangle+\alpha b\langle A,A\rangle+\alpha a\langle A,\xi\rangle &=-\frac{a}{\alpha-\gamma}\Bigl(A\alpha-\frac{3abc}{4}\Bigr)+b\langle\nabla_U A,V\rangle+\alpha b,
R_{\ell}U_{\ell}= \sqrt{R}_{\ell}(\sqrt{R_{\ell}}U_{\ell}) \in L^{2}_{\rm loc}((0,\infty) \times \mathbb T^d_{\ell})\,, R_{\ell}U_{\ell}\otimes U_{\ell}= \sqrt{R}_{\ell}\, R_{\ell}^{1/4}U_{\ell}\otimes R_{\ell}^{1/4}U_{\ell}\in L^{1}_{\rm loc}((0,\infty) ; L^2(\mathbb T^d_{\ell})).
\mathfrak{B}_{n,q}\left( x,y\right) ={\displaystyle\sum\limits_{k=0}^{n}}\left[\begin{array}[c]{c}n k\end{array}\right]_{q}\left( \mathfrak{B}_{k,q}\left( x\right) +\frac{\left(-1;q\right)_{k-1}}{2^{k}}\left[ k\right]_{q}x^{k-1}\right)\mathfrak{E}_{n-k,q}\left( y\right) .
K_{\mu}\left( \begin{matrix}{r_{k}}\\{s_{k}}\\ \end{matrix}\right) = \left( \begin{matrix}{r_{k}}\\{s_{k}}\\ \end{matrix}\right) + ( r_{k}q_{\mu}- s_{k}p_{\mu}) \left( \begin{matrix}{p_{\mu}}\\{q_{\mu}}\\ \end{matrix}\right) .
\left({{\dot{a}}\over a}\right)^2={{(h^{\prime}|g_{00}|g-b^2)\left[h^{\prime}g_Sg(C+E)^2-(h^{\prime}|g_{00}|g-b^2)(h^{\prime}g_Sg^3e^{-2\phi}+\ell^2)\right]}\over{4h^{\prime}g_{rr}g_Sg\left[(h^{\prime}|g_{00}|g-b^2)^2g^2e^{-2\phi}+b^2(C+E)^2\right]}}\left({{g^{\prime}}\over g}\right)^2.
|\sum_{b=1}^2k_bj_b^2|\leq\max\{|k_1j_1^2|,|k_2j_2^2|\}\leq\frac{1}{(2-\frac{9}{17})^2}(i+j)^2\leq\frac{2}{(2-\frac{9}{17})^2}(i^2+j^2).
Pr(n_{1},n_{2},...,n_{S}|\theta,J)=\frac{J!\theta^{S}}{1^{\phi_{1}}2^{\phi_{2}}\cdot\cdot\cdot J^{\phi_{J}}\phi_{1}!\phi_{2}!\cdot\cdot\cdot\phi_{J}!\Pi_{k=1}^{J}(\theta+k-1)}
p\left({d_{pF}}\leq \delta \right) = \frac{M\delta^{2nk -{2k^2}}}{k!}\int_{\mathbf{\Omega}_2}\prod\limits_{i = 1}^k{x_i}^{n - 2k}\,\,\, \times \prod\limits_{1 \leq i < j \leq k}{{\left({{x_i}-{x_j}}\right)}^2}d{x_1}\cdots d{x_k}
\Delta^N_{}:=\left(\begin{matrix}\dfrac{\bar{h}_i}{k_{j+1}}\dfrac{\bar{k}_j}{h_{i}}-\left(\bar{k}_j\left(\dfrac{1}{h_i}+\dfrac{1}{h_{i+1}}\right)+ \bar{h}_i\left(\dfrac{1}{k_j}+\dfrac{1}{k_{j+1}}\right)\right) \dfrac{\bar{k}_j}{h_{i+1}}&\dfrac{\bar{h}_i}{k_{j}}\end{matrix}\right).
\tilde{g}(x, y):= (3A-4B)x+\frac{1+a_1}{a_2}(A-2B)y-A\tilde{N}_1-\frac{1+a_1}{a_2}A\tilde{N}_2, \tilde{h}(x, y):= \frac{4B-A}{A}x+\frac{2a_1}{1+a_2}y-\tilde{N}_1.
\rho = i(k+w)z^{1\bar{1}}+ i\overline{(a'+b)}z^{1\bar{2}}+ i(a'+b)z^{2\bar{1}}+ i(u+l)z^{2\bar{2}}r = i(k+u)z^{1\bar{1}}+ i\overline{(a+b')}z^{1\bar{2}}+ i(a+b')z^{2\bar{1}}+i(w+l)z^{2\bar{2}}
{\cal H}_{i n t}( x ) = - e \overline{{\psi}}^{( 0 )}( x ) \beta^{\mu}A_{\mu}^{( 0 )}( x ) [ I + \frac{e}{m}( I - \beta_{0}^{2}) \beta^{\nu}A_{\nu}^{( 0 )}( x ) ] \psi^{( 0 )}( x ) ,
\frac{\alpha}{8}\int_{0}^{2\pi}a_1^2 d\theta + \frac{\beta}{8}\int_{0}^{2\pi}a_2^2 d\theta - \frac{1}{8}\int_{0}^{2\pi}d\theta- \frac{\delta}{8}\int_{0}^{2\pi}a_1^4 d\theta - \frac{\epsilon}{8}\int_{0}^{2\pi}a_2^4 d\theta - \frac{\iota}{8}\int_{0}^{2\pi}a_1^2 a_2^2 d\theta = 0,
A = \begin{pmatrix}\frac{99}{100}\frac{1}{100}0 \ldots 0 0 \frac{98}{100}0 \ldots 0 \ddots \ddots \ddots \ddots \vdots 0 \ldots 0 \frac{1}{100}\frac{1}{100}0 \ldots \ldots 0 \frac{0}{100}\end{pmatrix}, B = (0,\ldots,0, 1, \ldots, 1),
(s)\int_{a}^{b}fg d\mu \leq (s)\int_{a}^{b}\bigg( m2^{1-s}f(a)+\bigg(\frac{x-ma}{b-ma}\bigg)^s[f(b)-mf(a)] \bigg). \qquad\qquad\bigg( m2^{1-s}g(a)+\bigg(\frac{x-ma}{b-ma}\bigg)^s[g(b)-mg(a)]\bigg)d\mu = (s)\int_{a}^{b}p_{1}(x)p_{2}(x)d\mu.
( D s )_{\beta \gamma}= \delta_{\beta , \gamma}( D s )_{\beta}, \quad ( D s )_{\beta}= - i \sum_{\lambda \in \Delta_{s}, \ \beta \cdot \lambda = 1}\left[ g_{s_{1}}z_{d}( \lambda \cdot q , \xi ) + g_{s_{2}}z_{d}^{( 1 / 2 )}( \lambda \cdot q , \xi ) \right] .
\Lambda_{j}= e^{-\frac{\pi i \alpha}{2}}(4x_{j})^{\beta_{j}}\bigg( \prod_{\substack{k=1 k \neq j}}^{m}\widetilde{T}_{k,j}^{\beta_{k}}\bigg)e^{\sqrt{r}g_{+}(-x_{j})}r^{\frac{\beta_{j}}{2}}c_{-x_{j}}^{\beta_{j}}.
=[[\begin{matrix}\sigma_{xx}-\rho_{w}&\tau_{xz}\\ \tau_{zx}&\sigma_{zz}-\rho_{w}\\ \end{matrix}]+[\begin{matrix}\rho_{w}&0\\ 0&\rho_{w}\\ \end{matrix}]+[\begin{matrix}0&\delta\tau_{xz}\\{\delta\tau}_{zx}&0\\ \end{matrix}]-[\begin{matrix}0&{\delta p}_{w,int}\\{\delta p}_{w,int}&0\\ \end{matrix}]]=
\mbox{Given a sensing matrix}A\in\mathbb{C}^{m\times n}\; \mbox{and magnitudes}|y|\in\mathbb{R}_{\ge 0}^m\;, &\mbox{obtain a signal}\rho\in\mathbb{C}^n\; \mbox{such that}\vert y\vert = \vert A\rho\vert.
[ p_{i}, x_{j}] = \delta_{i j}\, e^{- 2 p_{0}/ \kappa}- \frac{1}{\kappa^{2}}\left( \vec{p}\,{}^{2}\, \delta_{i j}- 2 p_{i}p_{j}\right) , \quad [ p_{0}, x_{i}] = - \frac{2}\kappa \, p_{i}
\mathbb{E}\left[ y[n+1] \right] = \mu \left( 2 - \mu^2 - \frac{1}{\tau}\right) \mathbb{E}\left[ T_{new}[n] \right] - \mu \left( 1-\mu^2-\frac{1}{\tau}+ \frac{\mu^2}{\tau}\right) \mathbb{E}\left[ T_{new}[n] \right] + \frac{\mu^2}{\tau}T
\frac{2}{i}\frac{\delta^{2}a_{D t}}{\delta \phi_{i}\delta \phi_{j}}= \frac{- \sin ( \widehat{\theta}_{t})}{( \cos [ \widehat{\theta}_{t}] - \cos [ \theta_{i}] ) ( \cos [ \widehat{\theta}_{t}] - \cos [ \theta_{j}] )}\left( \frac{1}{2 N^{3}}+ \frac{\delta_{i j}}{N}\right)
J (x^i \frac{\partial}{\partial x^j}-x^j\frac{\partial}{\partial x^i}) =-\frac{1}{24 \pi}\int_{S^2}\epsilon^{ij}_{\,\,\,\,l}\tilde{x}^l * d\left( dA \left[A\tilde{\Delta}A-|\tilde{\nabla}A|^2\right]\right) d\mu_{S^2}.
\int_{X}|\tilde{F}_i-(1-b_{t_0,B}(\Psi))fF^{1+\delta}|^2_he^{v_{t_0,B}(\Psi)-\delta \tilde{M}}c(-v_{t_0,B}(\Psi)) \le (\frac{1}{\delta}c_i(T)e^{-T}+\int_{T}^{t_0+B}c_i(s)e^{-s}ds) \int_{X}\frac{1}{B}\mathbb{I}_{\{-t_0-B<\Psi<-t_0\}}|fF|^2_h,
I_{n}( L , m , d , \vec{\rho}) = \frac{1}{( 2 \pi )^{d - 1}}\int{}d^{d - 1}p \frac{e^{i \vec{p}. \vec{\rho}}}{( \vec{p}^{2}+ ( \frac{n \pi}{L})^{2}+ m^{2})}.
0 = \frac{\mu(G_1(\rho)-\rho)}{\rho-1}= \frac{1}{\rho-1}\sum_{k=1}^\infty kp_k (\rho^{k-1}-\rho) = -p_1 + \rho \sum_{k=3}^\infty kp_k \frac{\rho^{k-2}-1}{\rho-1}& = -p_1 + \rho \sum_{k=3}^\infty kp_k (\rho^{k-3}+\rho^{k-4}+\dotsb+1) & \ge -p_1 + \rho \sum_{k=3}^\infty kp_k (k-2)\rho^{k-3}& \ge \sum_{k=1}^\infty k(k-2)p_k\rho^{k-1}
=\frac{1}{2}\sqrt{2}\pi^{-1/2}K(\epsilon)^{-1/2}[\frac{d}{d\epsilon}K(\epsilon)]=\frac{1}{2}\sqrt{2}\pi^{-1/2}K(\epsilon)^{-1/2}\frac{E(\epsilon)-(1-\epsilon^{2})K(\epsilon)}{\epsilon(1-\epsilon^{2})}
\mathrm{e}^{- S_{\mathrm{i n d}}\left( \Sigma \right)}={\frac{1}{Z \left( B_{\mu \nu}\right)}}\ \int{\cal D}B_{\mu \nu}\ \mathrm{e}^{- S_{C}\left( B_{\mu \nu}\right) + i \int_{\Sigma}d \sigma_{\mu \nu}B_{\mu \nu}}
g_4 = \frac{f_1^4}{24}-\frac{f_1^2 f_2}{4}+\frac{f_1 f_3}{3}+\frac{f_2^2}{8}-\frac{f_4}{4}, f_6 = -\frac{f_1^6}{24}+\frac{3 f_1^4 f_2}{8}-\frac{2 f_1^3 f_3}{3}-\frac{3 f_1^2 f_2^2}{8}+\frac{3 f_1^2 f_4}{4}-\frac{f_2^3}{8}+\frac{3 f_2 f_4}{4}+\frac{f_3^2}{3}.
\epsilon_{a , 0}( \beta ) = T \sum_{b = 1}^{P}{\cal I}_{a b}^{( P )}\int_{- \infty}^{\infty}\frac{d \beta^{\prime}}{2 \pi}\frac{P - 1}{\cosh [ ( P - 1 ) ( \beta - \beta^{\prime}) ]}\ln \left( 1 + e^{- \epsilon_{b , 0}( \beta^{\prime}) / T}\right)
^1_{m,1}= \frac{\frac{\alpha_1^2|h_{m,m1}|^2}{L\left(||y_{m,1}||\right)}}{\frac{\alpha_2^2|h_{m,m1}|^2}{L\left(||y_{m,1}||\right)}+^{m,1}_{inter}+\frac{1}{\rho}},
\gamma^i(t, W_t)+h^i(t, W_t) = - (M^i_T -M^i_t) -\int_t^T (\nabla h^i)^* (r, W_r) \mathrm d W_r &+ \gamma^i(T, W_T)+h^i(T, W_T) = \Phi(W_T) - (\tilde M^i_T -\tilde M^i_t),
\frac{\partial E}{\partial\hat{h}_{i}}=\frac{\partial}{\partial\hat{h}_{i}}\sum_{n=-\infty}^{\infty}[x[n]^{2}-2x[n]\sum_{k=0}^{N-1}\hat{h}_{k}s[n-k]+(\sum_{k=0}^{N-1}\hat{h}_{k}s[n-k])^{2}]
\lim_{x\to0}\frac{h(x)}{\exp(-a\theta/x^2) / x^3}= \frac{2\theta}{\operatorname{B}(a,b)}\lim_{x\to0}\frac{\left[ 1-\exp\left(-\frac{\theta}{x^2}\right) \right]^{b-1}}{\operatorname{I}_{1-\exp(-\theta/x^2)}(b,a)}= \frac{2\theta}{\operatorname{B}(a,b)}.
\frac{d f}{d \kappa}\; \; \; = \; \; \; -2 \frac{M_c}{I_c}\left[ 1 + \frac{\delta M}{Mc}- \frac{\delta I}{I_c}\right]\; \; \; = \; \; \;-\frac{2}{3}\; - \; \frac{1}{3}\varepsilon \ln(\varepsilon)\; - \; \varepsilon \; .
\overline{\mu}_1 = \frac{\overline{k}_1}{n}\mu_1 + (1 - \frac{\overline{k}_1}{n}) \delta_0 \mathrm{and}\overline{\mu}_0(A):= \frac{\overline{k}_0}{n}\mu_0 + (1 - \frac{\overline{k}_0}{n}) \delta_0\ .
\nabla_l\nabla_k h_{ij}= \nabla_k\nabla_l h_{ij}+ h_{mj}R_{imlk}+ h_{im}R_{jmlk}&=\nabla_j\nabla_i h_{kl}+ (h_{mj}h_{il}- h_{ml}h_{ij})h_{mk}+ (h_{mj}h_{kl}- h_{ml}h_{kj})h_{mi}.
L_{1}( x(u,z) (N-1,N))^{\ast}=\sum_{m=0}^{\infty}(-1)^{m}\alpha_{m}(u)^{\ast}\overline{z}^{m}\rho( \overline{z}^{-\kappa},\overline{z}^{\kappa}) \sigma C^{^{\prime}\ast}( \overline{x_{N}x_{N-1}})^{-\gamma\kappa}.
\overset{2a}{\Delta}_{m+1}B =-\Big(T^{tt}(\alpha_{-,m+1},\beta_{-,m+1})-T^{tt}(\alpha_{-,m},\beta_{-,m+1})\Big), \overset{2b}{\Delta}_{m+1}B =-\Big(T^{tt}(\alpha_{-,m},\beta_{-,m+1})-T^{tt}(\alpha_{-,m},\beta_{-,m})\Big).
J\bigr[u^{\ast}\bigl] = \mathcal{E}^{G}\left [ \int_0^T c\bigl(s, X_s^{0,x;u^{\ast}}, u_s\bigr) ds + \Psi(X_T^{0,x;u^{\ast}})\Bigl \vert \mathcal{F}_0, \right] \equiv \int_0^T c\bigl(s, X_s^{0,x;u^{\ast}}, u_s\bigr) ds + \Psi(X_T^{0,x;u^{\ast}}),\,\, X_0^{0,x;u^{\ast}}= x.
\|e^{\tau(t) A}\overline{v}(t) \|_{H^r}^2 + \|e^{\tau(t) A}\widetilde{v}(t) \|_{H^r}^2 + 2 \int_0^t \|A^{r+\frac{1}{2}}e^{\tau(s) A}\overline{v}(s) \|^2 + \|A^{r+\frac{1}{2}}e^{\tau(s) A}\widetilde{v}(s) \|^2 ds \leq \|e^{\tau_0 A}\overline{v}_0\|_{H^r}^2 + \|e^{\tau_0 A}\widetilde{v}_0\|_{H^r}^2.
\begin{array}{r c l}{{\cal A}( \omega ; \omega_{\bot})}&{=}&{\displaystyle \int d x_{0}d x_{3}e^{i ( \omega x_{0}- \sqrt{\omega^{2}- \omega_{\bot}^{2}}x_{3})}\tau ( x_{0}) g ( x_{0}, x_{3}^{2}; \omega_{\bot}) .}\\ \end{array}
\left(\int_{0}^{t}|\psi_{r}| dr \right)^{2}\le \int_{0}^{t}\frac{a}{|z_{r}|^{2}}dr \int_{0}^{t}\frac{a}{|\tilde z_{r}|^{2}}dr = \log \frac{\Im z}{\Im z_{t}}\log \frac{\Im z}{\Im \tilde z_{t}}\le \left( \log (\delta/y) \right)^{2}.
\|g\|^{-2}_{\infty}\|\omega\|^{-2}_2 \int_{\mathcal{W}^+_J}|\widehat{f}_{j}(\xi)|^2 \,d\xi= \int_{\mathcal{I}_j}\int_{\mathcal{A}_J}|\widehat{f}_{j}(r,\varphi)|^2 r \,d\varphi\,dr \lesssim 2^{-2js\alpha}\int_{\mathcal{A}_J}\varepsilon_j^2(\varphi) \,d\varphi\asymp A_J 2^{-2js\alpha}.
\left( \begin{matrix}{a}&{b}\\{c}&{d}\\ \end{matrix}\right) = \left( \begin{matrix}{I - e_{i}}&{e_{i}}\\{e_{i}}&{I - e_{i}}\\ \end{matrix}\right) \mathrm{s . t .}e_{i}= \mathrm{d i a g}( \underbrace{\overbrace{0 , . . . , 0}^{i - 1}, 1 , \overbrace{0 , . . . , 0}^{d - i - 2}}_{d}) .
M_{n}= 4 \pi \int_{0}^{\infty}d r ~ r^{2}\{\frac{F_{\pi}^{2}}{8}[ F^{2}+ 2 \frac{s i n^{2}n F}{n^{2}r^{2}}] + \frac{1}{2 e^{2}}~ \frac{s i n^{2}n F}{r^{2}}~ [ \frac{s i n^{2}n F}{n^{2}r^{2}}+ 2 F^{2}] \}
\|f\|_{L^q_t(W_x^{\alpha,p})}:= \biggl( \int_0^{+\infty}\|f(s)\|_{W^{\alpha,p}}\,ds \biggl)^\frac{1}{q}\,, \|f\|_{L^q_t(C_x^{\theta})}:= \biggl( \int_0^{+\infty}\|f(s)\|_{C^{\theta}}\,ds \biggl)^\frac{1}{q}\,.
\Gamma_{r e g}[ V ] = - i \sum_{i}c_{i}\sum_{n = 2}^{\infty}\frac{1}{n}\int \, d z_{1}\dots d z_{n}v ( z_{1}) \star \bar{D}^{2}D^{2}G ( z_{1}, z_{2}) \star \dots \star v ( z_{n}) \star \bar{D}^{2}D^{2}G ( z_{n}, z_{1}) ,
h(V_1,V_1) = -\frac{\langle V_1,d\xi_pV_1\rangle}{\langle \eta,\xi\rangle}= -\frac{\langle V_1,du^{-1}_{\eta(p)}\circ d\eta_pV_1\rangle}{\langle \eta,\xi\rangle}= -\frac{\lambda_1\langle V_1,du^{-1}_{\eta(p)}V_1\rangle}{\langle\eta,\xi\rangle},
\partial_t({\frac{|\hat{H}|}{|\xi|}})+\mu |\xi||\hat{H}|=Re(\frac{\hat{B}}{|\xi|}\cdot \frac{\bar{\hat{H}}}{|\hat{H}|}+\frac{\hat{G}}{|\xi|}\cdot \frac{\bar{\hat{H}}}{|\hat{H}|}),
|\bar{g}-g|_{g}^{2}= \left(\frac{A-\bar{A}-a^{2}\bar{B}}{A}\right)^{2}+\left(\frac{B-\bar{B}}{B}\right)^{2}+\left(\frac{C-\bar{C}}{C}\right)^{2}+\left(\frac{b^{2}\bar{A}+c^{2}\bar{B}+d^{2}\bar{C}+\bar{D}-D}{D}\right)^{2}+ \frac{2(a\bar{B})^{2}}{AB}+\frac{2(b\bar{A}+ac\bar{B})^{2}}{AD}+\frac{2(c\bar{B})^{2}}{BD}+\frac{2(d\bar{C})^{2}}{CD}.
\begin{cases}x_j^{k+1}=\arg\min\{\mathcal{L}(x_1^{k+1},\dots,x_{j-1}^{k+1}, x_j, x_{j+1}^k,\dots, x_s^k, y^k)+\frac{\beta}{2\alpha_j}\|x_j-x_j^k\|_{2}^2: x_j\in\mathbb{R}^{n_j}\}, j\in\mathbb{N}_s, y^{k+1}=y^k+\beta(\sum_{i=1}^s{A_ix_i^{k+1}-b}). \end{cases}
\psi(u,v) = \psi\biggl(u - \frac{1}{2}(u+v), v - \frac{1}{2}(u+v)\biggr) + \frac{u+v}{2}&= \psi\biggl(\frac{u-v}{2}, \frac{v-u}{2}\bigg) + \frac{u+v}{2}&= \phi(u-v) + \frac{u+v}{2}.
\|Q^{-1}x\|_{\mathfrak{l}^s}^2=\sum_{l=1}^N\left(\sum_{j=1}^NQ^{-1}_{l,j}x_j\right)^2(1+l)^{2s}\le (1+N)^{2s}\sum_{l=1}^N\left(\sum_{j=1}^NQ_{l,j}^{-1}x_j\right)^2.
\begin{aligned}H^1(W_F,\widehat{S})\times \{\prescript{L}{}{j}:\prescript{L}{}{S}\rightarrow \prescript{L}{}{G}\}_{\phi_0}/\widehat{S}_0 \rightarrow H^1(W_F,\widehat{G}) (\theta,\theta') \mapsto \mathrm{Ind}_{W_L}^{W_F}(\theta\cdot \theta').\end{aligned}
\hat{S}^m(t)=S_0^m+\int_0^{t\wedge\tau_{m,m}}G_m\left(\hat{S}^m(u)\right)dY(u),t\in[0,T],\quad\tau_{m,l}:=\inf\{t\geq0: \hat{B}^m(t)\geq l\}\wedge T.
- \alpha \beta \frac{3 f^{2}}{6 4 \pi^{2}}\, \gamma_{5}\, / \hspace{- 2 m m}\partial \, \delta^{4}( y ) = ( \alpha^{3}R + \beta^{3}L ) \hat{\Sigma}( y ) \, |_{\stackrel{{\scriptstyle \mathrm{a b n .}}}{{\scriptstyle \mathrm{p a r .}}}}\, \, .
{\cal L}_{e f f}^{( 1 )}={\cal L}_{M C S}+{\cal L}_{D}+ e A^{\mu}( J_{\mu}- G_{\mu}) - \frac{e^{2}}{2 m^{2}}J_{\mu}J^{\mu}+ \frac \lambda 2 ( \partial_{\mu}A^{\mu})^{2}
\forall\omega\in\varOmega,\ \forall t\in\mathbb{R}, H_{X}(t) =\sum_{i=1}^N \alpha_i\bigl\vert\sin(t-\beta_i )\bigr\vert; \forall\omega\in\varOmega,U(X) =2\sum_{i=1}^N\alpha_i; \forall\omega\in\varOmega,A(X) =\frac{1}{2}\sum_{i=1}^N \sum_{j=1}^N\alpha_i\alpha_j\bigl\vert\sin(\beta_i-\beta_j)\bigr\vert.
K_{\nu}\left({\nu z}\right) =\frac{c_{2}\left( \nu \right)}{\left({1+z^{2}}\right)^{1/4}}\exp \left\{{\sum\limits_{s=1}^{n-1}{\left({-1}\right)^{s}\frac{\tilde{{E}}_{s}\left( p\right)}{\nu^{s}}}}\right\}\left\{{e^{-\nu \xi}+\varepsilon_{n,2}\left({\nu ,\xi}\right)}\right\},
\eqref{term-5}=\frac{6(q-1)}{(q+1)\omega_3 R^3}\iiint |\chi_R(y-z)u(t,y)|^2 |\chi_R(x-z)u(t,x)|^{q+1}dxdydz & +\frac{6(q-1)}{q+1}\iint |u(t,y)|^2 (\phi_R-\phi_{q,R})(x-y) |u(t,x)|^{q+1}dxdy & +\frac{4(q-1)}{q+1}\iint |u(t,y)|^2 (\psi_R-\phi_R)(x-y) |u(t,x)|^{q+1}dxdy.
{}_2F_1\bigg[\begin{matrix}\frac14 \frac34 &1\end{matrix}\bigg|\,-\frac13\bigg]_{p-1}\equiv \begin{cases}\big(\frac{\sqrt{-3}}{p}\big)\cdot (a+b\sqrt{-3})\pmod{p^2}, p\equiv1\pmod{4}, -\big(\frac{\sqrt{-3}}{p}\big)\cdot 3(a+b\sqrt{-3})\pmod{p^2}, p\equiv3\pmod{4},\end{cases}
(2j_{3}+1)\sum_{m_{1}m_{2}}(\begin{matrix}j_{1}&j_{2}&j_{3}\\m_{1}&m_{2}&m_{3}\end{matrix})(\begin{matrix}j_{1}&j_{2}&j_{3}^{\prime}\\m_{1}&m_{2}&m_{3}^{\prime}\end{matrix})=\delta_{j_{3},j_{3}^{\prime}}\delta_{m_{3},m_{3}^{\prime}}\{\begin{matrix}j_{1}&j_{2}&j_{3}\end{matrix}\},
\int_{0}^{\infty}K_{G\left( x,b;c\right)}^{L}\left( u\right) du =\frac{\gamma\left( x/b+1,c/b\right)}{\Gamma\left( x/b+1\right)}\text{and}\int_{0}^{\infty}K_{G\left( x,b;c\right)}^{U}\left( u\right) du =\frac{\Gamma\left( x/b+1,c/b\right)}{\Gamma\left( x/b+1\right)}.
\begin{gathered}\frac{d^k}{dx^k}\theta_{p}^{n,k}(x)=\sum_{m=0}^{k-1}r_{m}^{n,k}\frac{d^m}{dx^m}\theta_{p}^{n,k}(x), \frac{d^k}{dx^k}\theta_{v}^{n,k}(x)=\sum_{m=0}^{k-1}r_{m}^{n,k}\frac{d^m}{dx^m}\theta_{v}^{n,k}(x).\end{gathered}
E_{\pi^m_{n_-}}[f(x_n)|Y_n=y_n] = \int f(x_{n}) \frac{g(x_{n},y_{n}) \pi^m_{n_-}(dx_n)}{\int_{\mathcal{X}}g(x_{n},y_{n}) \pi^m_{n_-}(dx_n)}&= \frac{\int f(x_{n}) g(x_{n},y_{n}) \pi^m_{n_-}(dx_n)}{\int_{\mathcal{X}}g(x_{n},y_{n}) \pi^m_{n_-}(dx_n)}