harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
2 V_1 / 3 \le \mathcal H^n(M - B(p,R_M/2 + \delta)) &\le \mathcal H^n (A_T) &\le C_1(n,D,v) (\mathcal H^{n-1}(B(\Uparrow_p^q, \pi/2+\epsilon))-\mathcal H^{n-1}(B(\Uparrow_p^q, \pi/2-\epsilon))) &\le C_1(n,D,v) \mathcal H^{n-1}(\mathbb S^{n-1}) \theta_n(\epsilon) &\le V_1 /2.
\int_{0}^{\infty}u_{l}^{\lambda_{1}}(r) u_{l}^{\lambda_{2}}(r) dr = \delta(\lambda_{1}-\lambda_{2}) , \int_{0}^{\infty}u_{l}^{\lambda}(r) v_{l}^{\kappa}(r) dr = 0 , \kappa >0 , \int_{0}^{\infty}v_{l}^{\kappa}(r) v_{l}^{\kappa}(r) dr = 0 , \kappa >0
\sum_j 2^j (\nabla p)(2^{-j}x) = \Delta^{-1}\nabla \nabla \cdot ( \sum_j 2^j u(2^{-j}x) \cdot (\nabla u) (2^{-j}x) ) = \Delta^{-1}\nabla \Bigl( \underbrace{\sum_j (\partial u) (2^{-j}x) (\partial u) (2^{-j}x)}_H \Big).
f_{\ell}( k , r ) \equiv w_{\ell}( k r ) g_{\ell}( k , r ) \quad \mathrm{a n d}\quad \phi_{\ell}( k , r ) \equiv \frac{( - k )^{- \nu + \frac{1}{2}}}{2 \nu}\frac{h_{\ell}( k , r )}{w_{\ell}( k r )}\, ,
\begin{cases}\partial_{t}\rho = 0 D \times (t_{2j+1},t_{2j+2}] \partial_{t}u = \sqrt{2}\sum_{k=1}^{n}\sigma_{k,\tau,n,\delta}(\rho,\rho u)\dot{\beta}_{k}D \times (t_{2j+1},t_{2j+2}]. \end{cases}
\mathbb{P}\left[\mathbb{A_{UE}}\right] =2\left(\pi\lambda_{b}\right)^2\int_{0}^{\infty}x_{21}\mathrm{e}^{-\pi\lambda_{b}x_{21}^2}{\min^2}\left(x_{21},\beta^{\frac{\mu-2}{4\mu}}x_{21}^{\frac{2}{\mu}}\right)\mathrm{d}x_{21}\approx \frac{2\, \Gamma\left(\frac{2}{u}\right)}{\mu \left(\pi\lambda_b\right)^{\frac{2}{\mu}-1}\beta^{\frac{1}{\mu}-\frac{1}{2}}},
\int_{0}^{\infty}\ldots \int_{0}^{\infty}\Delta \{R_{1}, . . . , R_{n}\}\prod_{j = 1}^{n}R_{j}^{a}e^{- X_{j}\sqrt{R_{j}}}d R_{j}= 2^{n}\frac{\Delta \{X_{1}^{2}, . . . , X_{n}^{2}\}}{\prod_{j = 1}^{n}X_{j}^{2 ( a + n )}}\prod_{j = 1}^{n}\Gamma \left[ 2 ( j + a ) \right]
q_1(t) = \rho_1 a[\delta^2 t^{2\gamma-\frac12}+ b t^\frac12(\rho_3 + ct)], q_2(t) = \rho_1 a\delta^2 t^{2\gamma-1}- b\rho_2 (\rho_1 + at)(\rho_3 + ct), q_3(t) = -\rho_2(\rho_1 + at)[\delta^2 t^{2\gamma-1}+ b(\rho_3 + ct)], q_4(t) = \rho_3\delta t^{\gamma - 1}[a\rho_1 + \rho_2 (\rho_1 + at)],
\sum_{i=1}^l\lambda_i\nabla f_i(x^0)+\sum_{j=1}^m\mu_j\nabla g_j(x^0)=0, &\sum_{i=1}^l\lambda_i\langle \xi^{*i}, u\rangle+\sum_{j=1}^m\mu_j\langle \zeta^{*j}, u\rangle\geq 0, &\mu=(\mu_1, \mu_2, \ldots, \mu_m)\geqq 0, \mu_j=0,\ \j\notin J(x^0; u), &\lambda=(\lambda_1, \lambda_2, \ldots, \lambda_l)> 0.
\Theta^{}: \begin{pmatrix}r k_u k_s \end{pmatrix}\mapsto \begin{pmatrix*}[l]A_c k_c +g_c \circ K{}{}-k_c \circ (A_c +r) A_u^{-1}k_u \circ (A_c + r) - A_u^{-1}g_u \circ K{}{}A_s k_s\circ (A_c + r)^{-1}+ g_s \circ K{}{}\circ (A_c +r)^{-1}\end{pmatrix*}.
(Y+\mathcal{L}) \cap \mathcal{S}=P_\mathcal{S}(Y) + \mathcal{L}\cap \mathcal{S}, (C + \mathcal{L}^{\perp}) \cap \mathcal{S}= P_\mathcal{S}(C) + \mathcal{L}^{\perp}\cap \mathcal{S}.
\varphi_{i,k}(t)=\sum_{j=1}^{k}\alpha_{i,j}^{k}(t)w_j, \mu_{i,k}(t)=\sum_{j=1}^k\beta_{i,j}^{k}(t)w_j, &q_k(t)=\sum_{j=1}^{k}\gamma_j^k(t) y_j, n_k(t)=1+\sum_{j=1}^{k}\eta_j^k(t) y_j,
(\delta^{2*}_2(\epsilon) \circ \delta^{2*}_0(\epsilon))(m\otimes 1) = \sum_{I,J}\nabla(\xi_J)(\nabla(\xi_I)(m))\otimes \delta^2_2(\Gamma_J)\cdot \delta^2_0(\Gamma_I) \delta^{2*}_1(\epsilon)(m\otimes 1) = \sum_K \nabla(\xi_K)(m)\otimes \delta^2_1(\Gamma_K).
2e(H) =\sum_{i=1}^n d_G(x_i)-2d_G(x_1)=\sum_{i>1}d_G(x_i)-d_G(x_1) =\sum_{i>1}(\Delta(G)-\mbox{def}_i)-\Delta(G)+\mbox{def}_1>(n-2)\Delta(G)=(|H|-1)\Delta(G).
\frac{2}{(k-1)!}F_{H}\wedge F_{H}\wedge \omega^{k-1}=-\frac{2\phi(\tau)}{(2\pi)^{2}(k-1)!}[(k+1)!(1+\tau)^{k}-k(k-1)(k-1)!(1+\tau)^{k-2}]\omega_{\Sigma_{1}}\wedge\dots\wedge\omega_{\Sigma_{k}}\wedge\frac{\sqrt{-1}dw\wedge d\bar{w}}{\lvert w\rvert^{2}}
\mathbb{E}|\Gamma_{N}^{D}(b)\backslash\hat{\Gamma}_{N}^{D,M}(b)|\leq\ \mathbb{E}|\Gamma_{N}^{D}(b)\backslash D_{N}^{\varepsilon}|+\mathbb{E}|\Gamma_{N}^{D}(b')| &+\sum_{x\in D_{N}^{\varepsilon}}\sum_{k=k_{N}}^{n(x)}\mathbb{P}(h^{D_{N}}(x)-a_{N}\in[b,b'),T_{N,M}^{k}(x)),
\sum_{k = 1}^{r - 1}{}_{r - 1}C_{k}\hat{h}_{i}^{( k )}\hat{H}_{i}^{( r + 1 - k )}= ( r - 1 ) \hat{h}_{i}^{( 1 )}\hat{H}_{i}^{( r )}+ \sum_{k = 2}^{r - 1}{}_{r - 1}C_{k}\hat{h}_{i}^{( k )}\hat{H}_{i}^{( r + 1 - k )}.
s\Gamma^\alpha{}_{\beta \gamma}(u)=\Gamma^\alpha{}_{\beta\gamma}(u)-\left( \begin{array}{cc}0 L^i{}_{jc}C^a{}_{bi}0 \end{array}\right) =\left( \begin{array}{cc}L^i{}_{jk}0 0 C^a{}_{bc}\end{array}\right)
\langle K_{\mu^{n}}-K_{\mu^{m}}, K_{\mu^{n}}- K_{\mu^{m}}\rangle_{\mathcal{H}_{K}}&= \int_{X}\int_{X}K(x,y)[\chi_{\mathcal{C}_{n}}(x) -\chi_{\mathcal{C}_{m}}(x)][\chi_{\mathcal{C}_{n}}(y) - \chi_{\mathcal{C}_{m}}(y)]d\mu(x)d\mu(y) \xrightarrow[m,n \to \infty]{}0
{\bar{\theta}}( x_{k}, y_{k}; x ) \equiv \theta ( x_{k}, y_{k}; x ) - \frac{y_{k}- x_{k}}{L}= \frac{1}{L}\sum_{p \ne 0}\frac{1}{i p}\left( e^{i p y_{k}}- e^{i p x_{k}}\right) e^{- i p x},
\begin{multlined}[t][12.5cm]\widetilde{\Pi}_{\nu_{G},k\nu_{T}}\left(x_{1k},y_{2k}\right) =\frac{d_{\nu_{G}}}{(2\pi)^{d_{T}}}\int_{G}\int_{T}\chi_{\nu_{G}}(g^{-1})t^{-k\nu_{T}}\cdot\Pi\left(\tilde{\mu}^{G}_{g^{-1}}\circ\tilde{\mu}^{T}_{t^{-1}}\left(x\right),p_{0}\cdot x_{2k}\right)dtdg,\end{multlined}
\begin{array}{cc}\left( \bar{C}_{0}\, \eta_{0}\,{\cal A}... \right) \; \, \left( \bar{C}_{1}\, \eta_{0}\, B ... \right) \left( \bar{C}_{1}\, \eta_{1}\,{\cal A}... \right) \; \, \left( \bar{C}_{1}\, \eta_{0}\,{\cal B}... \right) \end{array}
\begin{aligned}K(x,y) = \int_{0}^{\infty}\zeta_{1}(n+y+z)\zeta_{2}(-n-x-z)dz K_{e}(x,y) = W(x,y)-K(x,y) g_{k}= K(x,-k) g_{k,e}= K_{e}(x,-k)\end{aligned}
M(\kappa)^{-1}=\sum_{j=-1}^{\beta-4}\kappa^jC_j+\mathcal O(\kappa^{\beta-3}), C_j =B_j+\sum_{\genfrac{}{}{0pt}{}{j_1\ge 0,j_2\ge 0,j_3\ge 0}{j_1+j_2+j_3=j+1}}B_{j_1}A_{j_2}B_{j_3},
K(\omega):= 1+|z|^3_{L^\infty(0,\tilde{\tau}_M;H_x^{\mathfrak{s}})}+\left|\int_0^\cdot S(\cdot-s)(z(s)\d W(s))\right|_{L^r(0,\tilde{\tau}_M;W_x^{\mathfrak{s},p})}&+\sup_{0\leq t_0\leq \tilde{\tau}_M}\left|\int_0^{t_0}S(\cdot-s)(z(s)\d W(s))\right|_{L^r(0,\tilde{\tau}_M;W_x^{\mathfrak{s},p})},
\partial_t\phi_t(u_1,u_2) = F((\psi_t(u_1,u_2),u_2)),\quad\phi_0(u_1,u_2)=0, \partial_t\psi_t(u_1,u_2) = R((\psi_t(u_1,u_2),u_2)),\quad\psi_0(u_1,u_2)=u_2,
\tiny{\frac{(X(\theta)+Y(\theta))^2}{Z(\theta)}- \theta \frac{(X^0+Y^0)^2}{Z^0}-(1-\theta) \frac{(X^1+Y^1)^2}{Z^1}=-\theta(1-\theta)\frac{[Z^1(X^0+Y^0)-Z^0(X^1+Y^1)]^2}{Z^0Z^1Z(\theta)}.}
q\cdot N_{d}^{\prime}=q^2+q\phi(b)+\frac{\phi(-a)}{q-1}\sum_{n=0}^{q-2}G_{-(d-1)n}G_{nd+\frac{q-1}{2}}G_{-n}G_{\frac{q-1}{2}}T^n\left(\frac{b}{a^d}\right).
\sum_{\nu = 0}^{N}\left[ \omega_{\nu}^{2}\eta_{\nu \rho}+ \sum_{k = 1}^{N}\lambda_{k}\left( \eta_{k \rho}\delta_{0 \nu}+ \eta_{0 \rho}\delta_{k \nu}\right) \right] \eta_{\nu \sigma}= \delta_{\rho \sigma}\Omega_{\sigma}^{2}\; .
w_{k}(t,i) = W\left(\frac{\eta_k(t)-\hat{\eta}_k^(t,i)}{\hat{\sigma}(\boldsymbol{\eta}_k(t))}\right) \hat{\eta}_k^(t,i+1) = \frac{\sum_{l\in\mathcal{N}_k\cup\{k\}}w_l(i)\eta_l(t,i)}{\sum_{l\in\mathcal{N}_k\cup\{k\}}w_l(i)}
\int\limits_{0}^{t}D_{\tau}^{q}K(t,\tau )r(\tau )d\tau +r(t)\underset{\tau\rightarrow t-0}{\lim}I_{\tau}^{q-1}K(t,\tau)+D_{0+}^{q}(F(t)-F(0))=D_{0+}^{q}(E(t)-E(0)).
E^{W_\alpha}\left[G_{\delta,M}^2 \right]-E\left[\left(\int_0^1 Z^{(\alpha,0)}(x)\,dx\right)^2\right]\overset{Lem.\,\ref{calc}}{=}E^{W_\alpha}\left[G_{\delta,M}^2 \right]-\left(1+\frac{2}{3(2\alpha+1)}\right)\le \epsilon,
[\dot{\mathrm{B}}^{s_0}_{p_0,q_0}(\mathbb{R}^n),\dot{\mathrm{B}}^{s_1}_{p_1,q_1}(\mathbb{R}^n)]_{\theta}= \dot{\mathrm{B}}^{s}_{p_\theta,q_\theta}(\mathbb{R}^n)
B_2 =\int_{|u-t|>2^{2s}\ell(Q)^{2s}}\frac{|P_s*(\phi\mu)(x,u) - P_s*(\phi\mu)(y,u)|}{|u-t|^{2-\frac1{2s}}}\,du &=\sum_k\int_{\mathcal{A}_k}\frac{|P_s*(\phi\mu)(x,u) - P_s*(\phi\mu)(y,u)|}{|u-t|^{2-\frac1{2s}}}\,du.
{\frac{d H_{2}}{d t}}= \nu \int \sp{1 / a}k \sp 2 H_{k}d k \propto \nu a \sp{- 6 - 4 \Delta_{\psi}}\propto \nu \sp{\frac{3 + \Delta_{\phi}+ \Delta_{\psi}}{\Delta_{\phi}- \Delta_{\psi}}}.
\bar{R}_{\mathrm{MA}}=\frac{I_{\mathrm{GMI}}}{\mathbb{E}_{\mathbf{h}}[\log(1+\|\mathbf{h}\|^2\mathcal{E}_{\mathrm{s}})]},\ \ \bar{E}_{\mathrm{MA}}=\frac{P_{\mathrm{MA}}}{P_{\mathrm{CA}}},
f(tx+\sum_{p \neq q}\tau^{(p)}\theta^{(p)}) + \sum_{i=1}^{N}\tau^{(q)}\theta^{(q)}_i \partial_i f(tx+\sum_{p \neq q}\tau^{(p)}\theta^{(p)}) = f(tx+\sum_{p=1}^{\mathcal{N}}\tau^{(p)}\theta^{(p)}).
(\partial^{*}E)_{\bar{z}}\subset \mathbb{R}^{n-1}\bigg\backslash \left( \bigg\{w \in \mathbb{R}^{n-1}: |w| > r_{\ell}^{\vee}(\bar{z}) \bigg\}\bigg\backslash\{\tau\}\right) &= \overline{B^{n-1}\left( 0, r_{\ell}^{\vee}(\bar{z}) \right)}\cup \{\tau\},
\Psi^{[k+1](\alpha)}=\Psi^{[k](\alpha)}U_{k}^{(\alpha)}, \Psi^{[k](\alpha+1)}=\Psi^{[k](\alpha)}V_{k}^{(\alpha)}, \Psi^{[k-1](\alpha+1)}=\Psi^{[k](\alpha)}W_{k}^{(\alpha)}.
\|g_i - f\| \ge |g_i(s_i) - f(s_i)| = |(g(s_i) - (g(1) + 1)) - f(s_i)| \ge |1 + f(s_i)| - |g(1)-g(s_i)| \ge 2 - \delta - \delta = 2 - 2\delta.
\Delta A := \{a - a' : a, a' \in A, a \ne a'\}, \Delta_+ A := \{a + a' : a, a' \in A, a \ne -a'\}, A-B := \{a-b : a \in A, b \in B, a \ne b\}, A + B := \{a+b: a \in A, b \in B, a \ne -b\}, A + g := \{a + g : a \in A\}.
\int_\epsilon^M |u'(r)-r^{-1}u(r)|^2\,dr = \int_\epsilon^M |u'(r)|^2 \,dr - M^{-1}|u(M)|^2 + \epsilon^{-1}|u(\epsilon)|^2 & \leq \int_\epsilon^M |u'(r)|^2 \,dr + \epsilon^{-1}|u(\epsilon)|^2 \,.
\int_{\{u_1< u_2\}}\nu(u_1,u_2) =-\int_{\{u_1< u_2\}}\left(\int_{u_1-u_2}^0\chi'(t)dt\right)d\mu &=-\int_{\{u_1< u_2\}}\left(\int_{-\infty}^0\chi'(t)\mathbf{1}_{\{u_1<u_2+t\}}dt\right)d\mu &=-\int_{-\infty}^0\chi'(t)\mu\{u_1<u_2+t\}dt.
\frac{P(z)}{Q(z)}=\displaystyle\sum_{i=1}^{p}\displaystyle\sum_{r=1}^{k_{_{i}}}\frac{a_{ir}}{(z-z_{_{i}})^{r}}+\displaystyle\sum_{j=1}^{q}\displaystyle\sum_{s=1}^{l_{_{j}}}\frac{\beta_{js}z+\gamma_{js}}{\Big(z^{2}-2Re(z_{_{j}})z+|z_{_{j}}|^{2}\big)^{s}}
S_{\mathrm{w o r l d \atop v o l}}= T_{( p )}\int d^{p + 1}\zeta \e^{- \Phi / 2}\sqrt{\vert d e t{\hat{G}}_{\alpha \beta}\vert}+ \mu_{( p )}\int d^{p + 1}\zeta \C_{( p + 1 )}\ ,
|u_r(x)-u_r(z)| = \left|\int_\Omega\left(G_\Omega(x,y)-G_\Omega(z,y)\right)\,[-f(y,u_r(y))+f_r(y)]\;dy\right| & \leq \int_\Omega\left|G_\Omega(x,y)-G_\Omega(z,y)\right|\left(a_1+C\,a_2{\delta(y)}^{p(s-1)}\right)\;dy +\int_\Omega\left|G_\Omega(x,y)-G_\Omega(z,y)\right|\,C\,{\delta(y)}^{-1}\;dy
\Phi^{I}= P^{I}+ \frac{2 i}{\sqrt{2 p^{+}}}\left( ( p^{+}- h P^{-}) \delta^{I J}+ \frac{1}{2}h X^{-}s^{I J}\right) \psi^{J}\approx 0 +{\cal O}( \psi^{3}) ,
S_{\mathrm{b o u n d a r y}}^{G L S M}= \int d x^{0}\left\{i{\frac{\Theta}{2 \pi r}}\sum_{i}( \phi_{i}\widetilde{D}_{0}\overline{{\phi}}_{i}- \overline{{\phi}}_{i}\widetilde{D}_{0}\phi_{i}) \right\}
(a,a^{\prime})[(b,b^{\prime})(c,c^{\prime})] =(a,a^{\prime})(bc,b^{\prime}c^{\prime}), & =\left( a(bc),a^{\prime}(b^{\prime}c^{\prime})\right) , & =\left( (ab)c,(a^{\prime}b^{\prime})c^{\prime}\right) , & =(ab,a^{\prime}b^{\prime})(c,c^{\prime}), & =[(a,a^{\prime})(b,b^{\prime})](c,c^{\prime}).
\mathbb{P}[\Psi^{\phi}(h)\in d\omega_2 \vert \Psi^{\phi}(0)=\omega_1] =\mathbb{P}[^{+}_{\omega \ge \omega_1}(W(\omega)-\phi(\omega-h))\in d\omega_2\vert \Psi^{\phi}(0)=\omega_1] &=\mathbb{P}[^{+}_{\omega \ge \omega_1}(S(\omega)-S(\omega_1)+r^{h}(\omega)) \in d\omega_2 \vert^{+}S(\omega)=\omega_1]
P^*_{\mathrm{CPF}}=[\frac{1-b_m \cdot 2^{2R^*_{\mathrm{CPF}}}}{c_m \cdot 2^{2R^*_{\mathrm{CPF}}}-a_m}]^+ P^*_{r_i}=[\frac{f_i^*\gamma_i^*}{2\beta_i^*\ln 2}-\frac{1}{g_{r_i}}]^+ i = 1,\cdots, M
\rho_i(x)=\nabla^2 \eta(x)(E_i,E_i) \geq -d(x)\frac{\sqrt{a}\sinh \sqrt{a}d_0-\kappa_-\cosh \sqrt{a}d_0}{\cosh \sqrt{a}d_0-\dfrac{\kappa_-}{\sqrt{a}}\sinh \sqrt{a}d_0}&\geq \sqrt{a}d(x)\geq 0.
\frac{1}{2}\frac{d^{2}}{dt^{2}}F(A^{(t)})|_{t=0}\leq\frac{2r^{n+3}e^{-r^{2}}\mathrm{Vol}(S^{n-1})}{n(n+2)(2\pi)^{n}}-\left(\frac{2re^{-r^{2}/2}}{(2\pi)^{n/2}}\int_{B(0,r)}(1-y_{1}^{2})d\gamma_{n}(y)\right).
\begin{array}{l}{( m , - m , 1 , p ) \longleftrightarrow \tilde{\beta}^{\frac{2 k^{\prime}}{\delta}m - \frac{k}{\delta}( p - k^{\prime})},}{( m , - m , 0 , p ) \longleftrightarrow \tilde{\beta}^{\frac{2 k^{\prime}}{\delta}m + \frac{k}{\delta}p}\tilde{\gamma}.}\end{array}
g(x') - g(x) \le c(x',y) - g^c(y) - [c(x,y) - g^c(y)] &=c(x',y) - c(x,y) &=|x'-y|^p - |x-y|^p &\le p (|x-y|^{p-1}\vee |x'-y|^{p-1})|x-x'| &\le C_3 \big [ 1+(|x|\vee |x'|)^{\frac{2p}{\beta}}\big ] |x-x'|,
{\sf L}f(\xi) =\log\left(x^2+(y-1)^2-\frac12\right)+\log\left((x-1)^2+y^2-\frac12\right) &+\log\left((x-2)^2+(y-1)^2-\frac12\right)+\log\left((x-1)^2+(y-2)^2-\frac12\right) &-4\log\left((x-1)^2+(y-1)^2-\frac12\right) &=\log\left(1-\frac{64(x-y)^2(x+y-2)^2}{(2x^2+2y^2-4x-4y+3)^4}\right) \le 0.
\int_U|u_k|^2_{h_E,\omega}e^{-\psi}dV_\omega\leq\int_U\langle B_{\psi,\omega}^{-1}f_k,f_k\rangle_{h_E,\omega}e^{-\psi}dV_\omega=\frac{1}{q}\int_U|f_k|^2_{h_E,\omega}e^{-\psi}dV_\omega<+\infty.
0\geq(2-\frac{n-2}{n\gamma})\frac{n-2}{4(n-1)}\mu([g])\left(\int_M u^{\frac{2n\gamma}{n-2}}\right)^{\frac{n-2}{n}}-\sqrt{\frac{n-2}{2(n-1)}}\gamma\int_M \left(|W|^2+\frac{2n}{n-2}u^2\right)^{\frac 12}u^{2\gamma}+\frac{4n\gamma+\frac{1}{\gamma}(n-2)^2-2n(n-2)}{4n(n-1)}R\int_M u^{2\gamma}.
_1 \langle \varepsilon_{1}|_{2}\langle \varepsilon_{2}|_{3}\langle \varepsilon_{3}| \exp [ \sum_{r = 1}^{3}N_{0 0}^{r r}( \varepsilon_{r}^{2}- 2 \sqrt{2}\varepsilon_{r}) + \sum_{r , s = 1}^{3}\sum_{n = 1}^{\infty}N_{0 n}^{r s}\varepsilon_{r}\phi_{n}^{( s )}] ,
\beta\,{F\over L}= e_0\,\beta -\int\limits_{-\infty}^\infty{dp\over 2\pi}\ln\left(1+ e^{-\beta\,\epsilon(p)}\right) = e_0\, \beta +{m\over\pi}\sum_{n=1}^\infty{(-)^n\over n}\,K_1(n\,m\,\beta)
[\begin{matrix}t_{1}&0\\ 0&1/t_{1}\\ \end{matrix}][\begin{matrix}a&b\\c&d\\ \end{matrix}][\begin{matrix}1/t_{1}&0\\ 0&t_{1}\\ \end{matrix}]=[\begin{matrix}at_{1}&bt_{1}\\c/t_{1}&d/t_{1}\\ \end{matrix}][\begin{matrix}1/t_{1}&0\\ 0&t_{1}\\ \end{matrix}]=[\begin{matrix}a&bt_{1}^{2}\\ct_{1}^{-2}&d\\ \end{matrix}]
\frac{1}{1260}=\frac{1}{2}\int_{0}^{1}t^{4}(1-t)^{4}dt<\int_{0}^{1}\frac{t^{4}(1-t)^{4}}{1+t^{2}}dt<\int_{0}^{1}t^{4}(1-t)^{4}dt=\frac{1}{630}.
eT_{\mu\nu}= e\Theta_{\mu\nu}+ \frac{1}{2}g_{\mu\nu}\left({\tilde{\overline\Psi}}_{L}\frac{\delta{S}^-}{\delta{\tilde{\overline\Psi}}_{L}}+{\tilde\Psi}_{L}\frac{\delta{S}^-}{\delta{\tilde\Psi}_{L_0}}\right).
\Gamma^{2}= - i \lim_{\theta \rightarrow 2 i \pi / 3}( \theta - \frac{2 \pi i}{3}) S ( \theta ) = 2 \sqrt{3}\frac{( c + 1 ) ( 1 + 2 c )}{( c - 1 ) ( 1 - 2 c )}\: ,
E_iG^{\mathrm{low}}(b).v_{r\rho}= [ \varphi_i(b)+r+1]_iG_{r\rho}^{\mathrm{low}}(\pi_{r\rho}(\tilde{e}_ib)) &+\sum_{\substack{\tilde{b}\in B(\infty) \varepsilon_i(b)-1<\varepsilon_i(\tilde{b})}}\widehat{E}_{\pi_{r\rho}(b), \pi_{r\rho}(\hat{b})}^{(1), i}G_{r\rho}^{\mathrm{low}}(\pi_{r\rho}(\tilde{b})),
\,{\, m_{s}(y_{1}) \,}\widehat{\mathrm M}_{s}({\mathrm d}y_{1}{\mathrm d}y_{2}) \, =\,{\,m_{s}(y_{2})\,}\mathrm M_{s}({\mathrm d}y_{1}{\mathrm d}y_{2}) \, ; 0 \le s \le T \, , \, \, (y_{1}, y_{2}) \in \mathbb R^{2}\, .
G_{l+\nu}(r'',r';T)=-\frac{2a}{\hbar}\sqrt{r' r''}\int_{0}^{\infty}\frac{dT}{T}\,I_{l+\nu}\left( \frac{2 a \, r'' r'}{iT}\right)\,\exp \left\{i\left[\frac{(E+ i 0^{+})}{\hbar}T+a \left( r'^{2}+r''^{2}\right)\frac{1}{T}\right]\right\}\; ,
V_{T}( \sigma_{1}, \tau_{1}; k^{( 1 )}) V_{T}( \sigma_{2}, \tau_{2}; k^{( 2 )}) = | z_{1}- z_{2}|^{\alpha^{\prime}k_{\mu}^{( 1 )}g^{\mu \nu}k_{\nu}^{( 2 )}}\times : V_{T}( \sigma_{1}, \tau_{1}; k^{( 1 )}) V_{T}( \sigma_{2}, \tau_{2}; k^{( 2 )}) : ~ ,
s^{(1)}_m(y) := \sum_{k=0}^{m}{m\choose k}(\lambda-\tfrac{\beta}{2}+k+1)_{m-k}(\tfrac{\beta}{2}+k+1)_{m-k-1}\times \quad\quad\times\big[k(\lambda+\beta+2m)+\tfrac{\beta}{2}(\lambda-\beta-2m)\big] R^{(1)}_k(y) =: \sum_{k=0}^m D^{(1)}_k(m)R^{(1)}_k(y)
I_{1}(t)=\int_{0}^{\infty}\int_{\mathbb{R}^{d}}t^{\beta - \beta - 1 - \beta d/2}x^{-1-1/\beta-d/2}|y|\exp\{-|y|^{2}/4axt^{\beta}\}w(x^{-1/\beta}, \beta, 1)dy dx = C \int_{0}^{\infty}t^{-1 + \beta/2}x^{-1/2-1/\beta}w(x^{-1/\beta}, \beta, 1)dx.
H_{n m}^{\{{\bf R}\}}( 0 , 0 ) = \frac{\mu_{n m}! | n - m | ! ( - 1 )^{\frac{m + n}2}}{2^{\frac{| m - n |}2}\left( \frac{| m - n |}2 \right) !}\left[ R_{1 1}^{n}R_{2 2}^{m}\left( r^{2}- 1 \right)^{\mu_{m n}}\right]^{\frac 1 2}C_{\mu_{m n}}^{\frac{| m - n | + 1}2}\left( \frac r{\sqrt{r^{2}- 1}}\right) .
\mathfrak{i}_{\mathrm{p}}\left( t\right) :=\underset{(\eta,l^{-1})\rightarrow (0,0)}{\lim}\left\{\left( \eta^{2}l^{d}\right)^{-1}\mathfrak{I}_{\mathrm{p}}^{(\omega ,\eta \mathbf{A}_{l})}\left( t\right)\right\}=\int\nolimits_{t_{0}}^{t}\int\nolimits_{t_{0}}^{s_{1}}\mathbf{X}_{\infty}(s_{1},s_{2})\ \mathrm{d}s_{2}\mathrm{d}s_{1}\ .
\int_{0}^{\infty}d m \frac{m^{2}e^{- \epsilon_{m}m}}{e^{- \beta \Omega_{H}( 1 - x ) m}- 1}= - \frac{2}{\epsilon_{m}^{3}}- \frac{2 \zeta ( 3 )}{[ \beta \Omega_{H}( 1 - x ) ]^{3}},
\begin{cases}\mathbb{H}(A_t,Z_t, W_t, (\partial_t \mathbb{G}^{u,v}, \mathbb{G}, \partial_x \mathbb{G}, \partial_{xx}\mathbb{G})(A_t;Z_t, W_t) ) \mathbb{G}(A_T;Z_T, W_T) = m(A_T),~ (A_T,Z_T, W_T) \in \Lambda_T \times \hat{\Lambda}_T.\end{cases}
\varphi^3 (-q) \varphi (-q^5) = E_{\chi_0,\chi_1}(-q) -2E_{\chi_0,\chi_1}(q^2) -4 E_{\chi_0,\chi_1}(q^4) & + 5E_{\chi_1,\chi_0}(-q) + 10E_{\chi_1,\chi_0}(q^2) -20E_{\chi_1,\chi_0}(q^4).
g^2 \frac{1}{\hbar^2 f(k^2) k^2 - m^2}\int \frac{d^4 l}{(2 \pi)^4}\tilde{\Omega}(l) \frac{1}{\hbar^2 f[(k-l)^2] (k-l)^2 - m^2}\tilde{\Omega}(k - k_i-l) + \dots
\begin{aligned}1_{c(G_2)}=\frac 16(\phi_{(1,0)}-\phi_{(1,3)}'-\phi_{(1,3)}''+\phi_{(1,6)}+\phi_{(2,1)}-\phi_{(2,2)}), 1_{c(G_2)^2}=\frac 16(\phi_{(1,0)}+\phi_{(1,3)}'+\phi_{(1,3)}''+\phi_{(1,6)}-\phi_{(2,1)}-\phi_{(2,2)}), 1_{c(G_2)^3}=\frac 1{12}(\phi_{(1,0)}-\phi_{(1,3)}'-\phi_{(1,3)}''+\phi_{(1,6)}-2\phi_{(2,1)}+2\phi_{(2,2)}). \end{aligned}
lim_{\lambda_{B}\rightarrow0}[\frac{N_{A0}\lambda_{A}}{\lambda_{B}-\lambda_{A}}(e^{-\lambda_{A}t}-e^{-\lambda_{B}t})]=\frac{N_{A0}\lambda_{A}}{0-\lambda_{A}}(e^{-\lambda_{A}t}-1)=N_{A0}(1-e^{-\lambda_{A}t}),
(Z_1 \widehat{\phi})_j(x)=\widehat{\phi}_{j-a}(x-\frac{1}{m}),~~(Z_2 \widehat{\phi})_j(x)= e^{-2\pi i j/m}e^{2\pi i x/(n-m\theta)}\widehat{\phi}_{j}(x).
{\cal F}_{\mu \nu}= - \frac{i}{2}T r \left\{\hat{\Phi}\partial_{\mu}\hat{\Phi}\partial_{\nu}\hat{\Phi}\right\}+ T r \left\{\partial_{\mu}( \hat{\Phi}A_{\nu}) - \partial_{\nu}( \hat{\Phi}A_{\mu}) \right\}\ .
c_a\cdot c_\alpha:=c_a c_\delta\cdot \zeta_{2d}^{\alpha_0+\cdots+\alpha_n}\prod_{i=1}^{\frac{n}{2}+1}\left(\frac{a_{{2i-2}}^{d-1}-(\zeta_{2d}^{\alpha_{2i-2}})^{d-1}}{a_{{2i-2}}-\zeta_{2d}^{\alpha_{2i-2}}}\right).
\left(\int_\Omega|u|^{2p/(p+1)}\,dx\right)^{(p+1)/p}-\left(\int_\Omega|u_0|^{2p/(p+1)}\,dx\right)^{(p+1)/p}\le\int_\Omega Vu^2\,dx-\int_\Omega U_0\,u_0^2\,dx,
\sum_{i=0}^{2\ell}\binom{2\ell}{i}\, i\,K_k(i)K_j(i)=2^{2\ell}\, \binom{2\ell}{k}^{-1}\begin{cases}-\frac{k+1}{2}, j=k+1, \ell, j=k, -\frac{2\ell-k+1}{2}, j=k-1, 0, ,\end{cases}
K_\mu(x;p,N)={\Gamma(N-x+1)\Gamma(N-\mu+1)\over\Gamma(N+1)\Gamma(N-\mu-x+1)}\sum_{j=0}^x{(-\mu)_j(-x)_j\over j!(N-\mu-x+1)_j}\left(1-{1\over p}\right)^{j}.
\frac{dP}{dr}=-\frac{GM(r)\rho(r)}{r^{2}}(1+\frac{P(r)}{\rho(r)c^{2}})(1+\frac{4\pi r^{3}P(r)}{M(r)c^{2}})(1-\frac{2GM(r)}{rc^{2}})^{-1}
{\cal Z}_{\Psi}~ = ~ \int [ d q^{i}d p_{i}] [ d \lambda^{a}d \pi_{a}] [ d{\cal C}^{a}d{\overline{{{\cal P}}}}_{a}] [ d{\cal P}^{a}d{\overline{{{\cal C}}}}_{a}] ~ e^{i S_{\Psi}}~ ,
D_t\Delta u = (-\Delta +D_j\widetilde{b}^j+\widetilde{b}^jD_j+\widetilde{c})\Delta u+2(D_j\nabla \widetilde{b}^j+\nabla \widetilde{b}^jD_j+\nabla \widetilde{c})\nabla u &+(D_j\Delta \widetilde{b}^j+\Delta \widetilde{b}^jD_j+\Delta \widetilde{c})u-i\Delta f.
\lambda_1(x) = \frac{1}{2}\left(-x_1-x_2-x_3+2\right), \lambda_2(x) = \frac{1}{2}\left(x_1+x_2-x_3\right), \lambda_3(x) = \frac{1}{2}\left(x_1-x_2+x_3\right), \lambda_4(x) = \frac{1}{2}\left(-x_1+x_2+x_3\right).
\begin{cases}\left(s\tau-\frac{\lambda}{c_{1}^{2}}\mu\right)\gamma\left(\frac{\lambda}{c_{1}}\right)=s\left(t+\tau\right) \left(\lambda\tau-s\mu\right)\gamma\left(\frac{\lambda}{c_{1}}\right)=\lambda\left(t+\tau\right)-s\left(\left\Vert x\right\Vert +\mu\right)\end{cases}
((\overline{D_x}+\overline{D_y})\circ\det) (k) = (\overline{D_x}+\overline{D_y})(\det(k)) \overset{}{=}\overline{D_x}(\det(k)) + \overline{D_y}(\det(k)) = (\overline{D_x}\circ\det)(k) + (\overline{D_y}\circ\det)(k) \xrightarrow{\alpha_x+\alpha_y}D_x(k)+D_y(k) \overset{}{=}(D_x+D_y)(k)
k_{2n}(X) =-\sum_{m_1+2m_2+\dots n m_n=n}\frac{(2n)!(m_1+\dots+m_n-1)!}{m_1!m_2!\dots m_n!}\times &\prod_{1\leq j\leq n}\left(-\frac{\sigma^{2j}\Gamma(1/\alpha)^j\Gamma(\frac{2j+1}{\alpha})}{\Gamma(3/\alpha)^j\Gamma(1/\alpha)(2j)!}\right)^{m_j},
f(\Phi_{p,q}) =\left(\frac{p'(X_{1})}{p(X_{1})}+2\psi_{1}(X_{1})\right)\cdot\Phi_{p,q}+\Phi_{p,q}\cdot\left(\frac{q'(X_{1})}{q(X_{1})}-2\psi_{1}(-X_{1})\right) =\frac{d}{dX_{1}}\log\left(p(X_{1})\Gamma_{1}(X_{1})^{2}\right)\cdot\Phi_{p,q}+\Phi_{p,q}\cdot\frac{d}{dX_{1}}\log\left(q(X_{1})\Gamma_{1}(-X_{1})^{2}\right).
T=\frac{1}{1-N^{-1}}\sum_{\ell=0}^\infty N^\ell T_\ell, \quad\mbox{where}T_\ell f(g_1):=\int_{X^\ell(g_1)}(f(g_1)-f(g_2))\mathrm{d}m_\mathcal{G}(g_2).
{\cal L}_{b u l k}= M^{3}\left\{- \frac{1}{2}K ( \chi ) ( \partial_{y}\chi )^{2}- V ( \chi ) + \sum_{\vec{n}}\frac{a_{\vec{n}}( \chi )}{M^{| n |}}( \partial \chi )^{n_{1}}. . . ( \partial^{N}\chi )^{n_{N}}\right\},
\displaystyle \int_{- \infty}^{\infty}\frac{d x}{2 \pi}\varphi_{\pm}^{,}( x ){\cal Q}_{\pm}( x ) = \pm \left( \frac{\pi}{1 2}- \frac{{\cal Q}_{\pm}^{2}( \mp \infty )}{8 \pi ( 1 - \gamma / \pi )}\right) \: .
\delta_0(r) = \delta(ra_0) = \left(r \pi_hh^{-1}\right)' h= \left( u^{k+1-\alpha}s\pi_{v}v^{-1}\right)' u^\alpha v & = (k+1-\alpha) u^k u's\pi_{v}+ u^{k+1}v\, \left(s\pi_{v}v^{-1}\right)'.
\left(b_in_i-v_p\left(n_i!\right)\right)- \left(b_lp^{i-l}n_i-v_p\left((p^{i-l}n_i)!\right)\right) &=n_i(b_i-b_lp^{i-l})+v_p\left((p^{i-l}n_i)!\right)-v_p\left(n_i!\right) &=n_i(b_i-b_lp^{i-l}+\frac{p^{i-l}-1}{p-1}) &=n_ip^i\left(p^{-i}\left(b_i-\frac{1}{p-1}\right)-p^{-l}\left(b_l-\frac{1}{p-1}\right)\right) &\geq0.
[\begin{matrix}y_{1}\\y_{2}\\y_{3}\\y_{4}\\y_{5}\\y_{6}\\y_{7}\end{matrix}]=[\begin{matrix}1&0&0\\ 1&0&0\\ 1&0&0\\ 1&1&0\\ 1&1&0\\ 1&0&1\\ 1&0&1\end{matrix}][\begin{matrix}\mu\\ \tau_{2}\\ \tau_{3}\end{matrix}]+[\begin{matrix}\epsilon_{1}\\ \epsilon_{2}\\ \epsilon_{3}\\ \epsilon_{4}\\ \epsilon_{5}\\ \epsilon_{6}\\ \epsilon_{7}\end{matrix}]
\begin{array}{rl}(4,4,6)= 6.(2,0,1)+1.\left(0,4,0\right), (4,4,6)= 6.(2,0,1)+3.\left(0,4,0\right), (4,4,6)= 6.(2,0,1)+5.\left(0,4,0\right), (4,4,6)= 6.(2,0,1)+7.\left(0,4,0\right).\end{array}
\Omega_{s}= - \mathrm{d}x^{a}\wedge \mathrm{d}p_{a}+ \Omega_{\cal L}\qquad \Omega_{\cal L}= - 2 i s \frac{\mathrm{d}z \wedge \mathrm{d}\bar{z}}{( 1 - z \bar{z})^{2}}\, ,
J_{3,3}=3T_{x^2y}T_{xy^2}-T_{xy}T_{x^2}T_{y^2}+T_{xy}^3+T_{xy}T_x^2T_{y^2}-5T_{xy}^2T_xT_y-3T_{x^3}T_{y^3}&+2T_{xy}T_{xy^2}T_x+T_{x^2}T_xT_{y^3}-3T_{x^2}T_{xy^2}T_y+2T_{x^2y}T_{xy}T_y+3T_{x^2}T_xT_{y^2}T_y &+T_{x^3}T_{y^2}T_y+T_{x^2}T_{xy}T_y^2-T_x^3T_{y^2}T_y+2T_x^2T_{xy}T_y^2-T_{x^2}T_xT_y^3-3T_xT_{x^2y}T_{y^2}
\langle \widehat{u}g, \widehat{u}f_0 \rangle_{L^2 (\mathbb{R}^2)}+ \langle \widehat{v}g, \widehat{v}f_0 \rangle_{L^2 (\mathbb{R}^2)}= F^{(\alpha)}[f_0] \langle g,f_0 \rangle_{L^2 (\mathbb{R}^2)},
\int_{E_f(\lambda)}\Delta \log \left(\frac 1{1-|z|^2}\right) \frac{dxdy}{4\pi}=\int_{\partial E_f(\lambda)}\frac{\partial}{\partial n}\log \left(\frac 1{1-|z|^2}\right) \frac{ds}{4\pi}&= \int_{\partial E_f(\lambda)}\frac 1{1-|z|^2}\frac{\partial}{\partial n}|z|^2 \frac{ds}{4\pi},
dq_t^\prime= \frac{1}{m}(p_t^\prime-\psi(q_t^\prime))dt, d(p_t^\prime)_i= \left(-\frac{1}{m}\gamma(t^*)((p_t^\prime)_i-\psi_i(q_t^\prime))-\partial_{q^i}V(t^*,q_t^\prime)+\tilde F(t^*,q_t^\prime)\right. &\left.+\frac{1}{m}\partial_{q^i}\psi_k(q_t^\prime)\delta^{kj}((p_t^\prime)_j-\psi_j(q_t^\prime))\right)dt+\sigma(t^*,q_t^\prime)\delta_{i\rho}dW^\rho_t
\mathbf a_2 = [0,\; \ldots,\; 0,\; p(S_{m-1})C''_1,\; p(S_{m})C''_2,\; p(S_{m+1})C''_3,\; p(S_{m+2})C''_4,\; 0,\; \ldots,\; 0]^T, \mathbf a_1 = [0,\; \ldots,\; 0,\; w(S_{m-1})C'_1,\; w(S_{m})C'_2,\; w(S_{m+1})C'_3,\; w(S_{m+2})C'_4,\; 0,\; \ldots,\; 0]^T.
\mbox{Prod}\left(\mathbf{x}^{\top^{\left(m-1\right)}},\mathbf{x}^{\top^{\left(m-2\right)}},\cdots,\mathbf{x}^{\top^{j}},\cdots,\mathbf{x}^{\top^{1}},\mathbf{x}^{\top^{0}}\right)=\mbox{Prod}\left(\mathbf{y}^{\top^{\left(m-1\right)}},\mathbf{y}^{\top^{\left(m-2\right)}},\cdots,\mathbf{y}^{\top^{j}},\cdots,\mathbf{y}^{\top^{1}},\mathbf{y}^{\top^{0}}\right).