harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
F^{\mu \nu}= - \frac{1}{2}\epsilon^{\mu \nu \rho \sigma}\hspace{0 . 8 m m}{^*}\hspace{- 0 . 8 m m}F_{\rho \sigma}= \left( \begin{array}{c c c c}{{0}}&{{- E_{1}/ c}}&{{- E_{2}/ c}}&{{- E_{3}/ c}}\\{{E_{1}/ c}}&{{0}}&{{- B_{3}}}&{{B_{2}}}\\{{E_{2}/ c}}&{{B_{3}}}&{{0}}&{{- B_{1}}}\\{{E_{3}/ c}}&{{- B_{2}}}&{{B_{1}}}&{{0}}\\ \end{array}\right) .
2 [{}^{\ell}Q_{n},{}^{\ell}Q_{m}] = + 2 \omega \tilde{g}\sum_{t = 0}^{n + m - 1}\sum_{j , k}\left( ( \ell_{k}^{+}\ell_{k}^{-})^{t}m_{j k}( \ell_{k}^{+}\ell_{k}^{-})^{n + m - t - 1}- ( \ell_{k}^{+}\ell_{k}^{-})^{t}m_{j k}( \ell_{k}^{+}\ell_{k}^{-})^{n + m - t - 1}\right) = 0 .
F_{1}(x) = \frac{1}{2}(f(x)+\overline{f(x)}) - \frac{1}{2}(f(x)m+\overline{f(x)m})m ; F_{2}(x) =\frac{1}{2}(f(x)n+\overline{f(x)n}) - \frac{1}{2}(f(x)mn+\overline{f(x)mn})m.
\begin{array}{c}\omega={\rm d}{\bf p}\wedge{\rm d}{\bf x}+\sum_{i=1}^N{\rm d}{\bf p}_i\wedge{\rm d}{\bf e}_i,{\cal H}={s}\left[{\bf p}{\bf e}_1+\sum_{i=1}^{N}{k}_{i-1}{\phi}_{i-1.i}+\frac{{k}_{N}}{2c}({\Phi}_{N.N}-c^2)+ \sum_{i,j=1}^Nd_{ij}({\bf e}_i{\bf e}_j-\delta_{ij})\right],\end{array}
u(\lambda, \mu, \tau, \sigma, \eta) := e^{-\frac{\pi i \lambda \mu}{2\eta}}\int_\gamma \Omega_{2\eta}(t; \tau, \sigma) \frac{\theta(t + \lambda; \tau)}{\theta(t - 2\eta; \tau)}\frac{\theta(t + \mu; \sigma)}{\theta(t - 2\eta; \sigma)}dt,
\langle v_{nm}^k,A_{\alpha}^*.v_{n-2k+2l\,m}^{l-1}\rangle= \langle v_{nm}^k,(l-2)v_{n-2k+2l\,m}^{l-2}+ (n-2k+l+1)v_{n-2k+2l\,m}^{l}\rangle\neq0.
h_{a^*\varpi}(a\varpi^j,s;\chi) =\sum_{(n,a^*\varpi)=1}\frac{\chi(n)g_6(a\varpi^j,n)}{N(n)^s}=\sum_{(n,a^*)=1}\frac{\chi(n)g_6(a\varpi^j,n)}{N(n)^s}-\sum_{\substack{(n, a^*)=1 \varpi | n}}\frac{\chi(n)g_6(a\varpi^j,n)}{N(n)^s}.
-\sum_{i=1}^I \frac{1}{x_i}\bigg[J^{}_{i+1/2}- J^{}_{i-1/2}\bigg] = \frac{1}{x_1}J^{}_{1/2}- J^{}_{1+1/2}\left(\frac{1}{x_1}-\frac{1}{x_2}\right) -\cdots- J^{}_{I- 1/2}\left(\frac{1}{x_{I-1}}-\frac{1}{x_I}\right)-\frac{1}{x_I}J^{}_{I+ 1/2}.
u^{\ast}\left( t,x,\mu \right) =-\frac{1}{2}\Pi_{1}^{-1}\left( t\right)\Pi_{3}^{\top}\left( t\right) \left( x-\bar{\mu}\right) -\Pi_{2}^{-1}\left( t\right) \Pi_{4}\left( t\right) \bar{\mu}-\frac{1}{2}\Pi_{2}^{-1}\left( t\right) \Pi_{5}\left( t\right) .
\sum_{i=0}^9 a_i t_1^i = a_9\prod_{i=1}^9 (t_1 - q_i) \ll T^{\varepsilon}B_3 |U_1|^6 \left(E_1^{-1/2}((1 + \Upsilon_2^{-1})|U_2|^2 + \Upsilon_2^{-1}T^2 + |U_1U_2|) + E_2^{-1/2}|U_2|^2\right) \ll T^{\frac{17}{2}+\frac{21}{2}b+\varepsilon}
S=\frac 12(^{+}\tau -^{-}\tau )\int_M \varepsilon^{\mu \nu \alpha \beta}\,{\cal R}_{\mu \nu}^{ab}\,{\cal R}_{\alpha \beta}^{cd}\epsilon_{abcd}-\frac i2(^{+}\tau +^{-}\tau )\int_M\varepsilon^{\mu \nu \alpha \beta}\,{\cal R}_{\mu \nu}^{ab}\,{^{*}{\cal R}}_{\alpha \beta}^{cd}\, \epsilon_{abcd},
\left(u\alpha_{1}\right)\left(v\alpha_{3}\right) =\rho^{v_{2}}uv\alpha_{1}\alpha_{3}\ \left(v\alpha_{3}\right)\left(u\alpha_{1}\right) =\rho^{u_{3}}vu\alpha_{1}\alpha_{3}=\rho^{u_{3}+u_{2}v_{3}-u_{3}v_{2}}uv\alpha_{1}\alpha_{3},
d^2f = \sum_{i,k}dx^i \wedge dx^k \frac{\partial^2 f}{\partial x^k \partial x^i}+ \sum_{i, l}dx^i \wedge d\xi^l \frac{\partial^2 f}{\partial \xi^l \partial x^i}& + \sum_{j,m}d\xi^j \wedge dx^m \frac{\partial^2 f}{\partial x^m \partial \xi^j}+ \sum_{j,n}d\xi^j \wedge d\xi^n \frac{\partial^2 f}{\partial \xi^n \partial \xi^j};
O_{\beta \mu_{1}\ldots \mu_{n}}= O_{\beta \mu_{1}\ldots \mu_{n}}^{\mathrm{t w 2}}+ O_{\beta \mu_{1}\ldots \mu_{n}}^{\mathrm{t w 3}}+ O_{\beta \mu_{1}\ldots \mu_{n}}^{\mathrm{t w 4}}+ \ldots ,
\nabla b_k =\nabla u_{j_k}-\nabla g_k=(\nabla u_{j_k}-v_k)+(v_k-\rho_{l_k}v_k)+\rho_{l_k}(v_k-\nabla\tilde{g}_k)-\tilde{g}_k\otimes\nabla\rho_{l_k}&=(\nabla u_{j_k}-v_k)+(1-\rho_{l_k})v_k+\rho_{l_k}\tilde{\sigma}_k-\tilde{g}_k\otimes\nabla\rho_{l_k}
\langle(\psi_{\mathcal{A}, \mathcal{B}})_* (e_S \otimes e_T) , e_S \otimes e_T \rangle_{\mathcal{A},\mathcal{B}}=\frac{\langle \psi_*(j_C)_*(e_S \otimes e_T) , (j_X)_*(e_S \otimes e_T)\rangle}{\prod_{i=1}^s |A_i|}.
\begin{aligned}\mathcal{W}_t - \mathcal{W}_{rr}\leq A_1 \mathcal{W}_r + B (\mathcal{W}+m) +C_1(\mathcal{Z}+n) + E & \leq K_1|\mathcal{W}_r|+ B \mathcal{W}+ C_1 \mathcal{Z}\end{aligned}
h^{\epsilon_1, \epsilon_2}(t, r) = -\epsilon_2 |r_1-r_2| - \epsilon_1\epsilon_2|r_1 - r_3| - \epsilon_1|r_2 - r_3| - \epsilon_1\epsilon_2|t_1+t_2| &+ \epsilon_1\epsilon_2 |t_1 - r_1| + \epsilon_1 |t_1 - r_2 | + |t_1 - r_3| + |t_2 + r_1| + \epsilon_2 |t_2 + r_2| + \epsilon_1\epsilon_2 |t_2 + r_3|.
S_{61}^{[1]}= \sum_{\substack{1\le k\le N 6\mid k}}(-1)^{\frac k2+1}\sum_{m\ge0}a(m)K_k^{[1]}(n,m)\int_{-\frac{1}{k(k+N)}}^\frac{1}{k(k+N)}e^{\frac{2\pi}{k}\left(nz-\frac mz\right)}d\Phi.
\sum_{i=1}^{3}\delta_{1}(\beta_{i}-\beta_{i+1})(\beta_{i}-\delta_{1})^{3}(\beta_{i+1}-\delta_{1})^{3}- \sum_{i=1}^{3}\delta_{2}(\beta_{i}-\beta_{i+1})(\beta_{i}-\delta_{2})^{3}(\beta_{i+1}-\delta_{2})^{3},
I_2 \leq \frac2{\delta^2}\|u^2\|_{L^\infty(L^\infty)}\|\sigma^1-\sigma^2\|_{L^2(L^2)}^2 + C_d \bigg(\sum_{i=1}^2\|u^i\|_{L^\infty(L^\infty)}\bigg) \big\| \nabla \sigma^1\big\|_{L^2(L^2)}^2.
\sum_{l=1}^n \sum_{p=1}^n \sum_{q=1}^n \sum_{r=1}^n \sum_{s=1}^n (b_{pl}b_{li}b_{qj}a_{rk}c_{qr}^s c_{ps}^t - b_{pl}b_{lk}b_{qi}a_{rj}c_{qr}^s c_{sp}^t - b_{pl}b_{lj}b_{qi}a_{rk}c_{qr}^s c_{ps}^t)=0, \forall i,j,k,t : 1 \leq i,j,k,t \leq n.
\sum_{i=1}^{k-t}(s-1)^{(t-r_1+i)}{k-r_1 \choose t+i-r_1}{t+i-r_1-1 \choose t-r_1}=\sum_{i=1}^{k-t}(s-1)^{(t-r_2+i)}{k-r_2 \choose t+i-r_2}{t+i-r_2-1 \choose t-r_2}
\mathsf{M}^G_h(T):=\{((p:\mathfrak{X}\rightarrow T),G,\rho)|\p \omega_{\mathfrak{X}/T}\forall t \in T,\forall k\in \mathbb{N}, \chi(\mathfrak{X}_t,\omega^k_{\mathfrak{X}_t})=h(k)\}/\simeq
f(A) = f(2^K)+\sum_{p\in A}f(p) +\sum_{p\notin A}f(A_{p}) &\le f(2^K) + \sum_{\substack{p>2 p\in A}}f(p) + \sum_{\substack{p>2 p\notin A}}b_pf(p) + \sum_{\substack{p>2 p\notin A}}\sum_{k=1}^{K-1}f((A^k)_{2^k p}),
|(\varphi f)^{(n)}(z)|^2 \leqslant \Big(\sum\limits_{j=0}^n |\varphi^{(n-j)}(z)|^2 |f^{(j)}(z)|^2\Big) \Big(\sum\limits_{j=0}^n{\binom{n}{j}}^2\Big),\,\,\,z\in\mathbb D.
X_j(t) = e^{\int_{t^*}^t G_j(v) \: dv}X_j(t^*) + \int_{t^*}^t \mu X_{j-1}(u) e^{\int_u^t G_j(v) \: dv}\: du + e^{\int_{t^*}^t G_j(v) \: dv}Z_j'(t) &= T_{j,1}(t) + T_{j,2}(t) + T_{j,3}(t),
f(\xi)=\int_\Omega\frac{f^{p-1}(\eta)}{|\eta^{-1}\xi|^{Q-\alpha}}d\eta+ \lambda \int_\Omega\frac{f^{p-1}(\eta)}{|\eta^{-1}\xi|^{Q-\alpha-1}}d\eta,\xi\in \overline\Omega,
\left< \xi_{i}( \sigma ){\cal O}\left[ \vec{\xi}{\,}\right] \right>_{\vec{\xi}}= \int_{0}^{1}d \sigma^{\prime}G \left( \sigma - \sigma^{\prime}\right) \left< \frac{\delta{\cal O}\left[ \vec{\xi}{\,}\right]}{\delta \xi_{i}\left( \sigma^{\prime}\right)}\right>_{\vec{\xi}},
\lambda \left\{x \in \mathbb{I}: \left|\frac{\log q_n(x)}{n}- b\right| \geq \varepsilon\right\}= \lambda \left\{x \in \mathbb{I}: \frac{\log q_n(x)}{n}\geq b + \varepsilon\right\}+ \mathrm{P}\left\{x \in \mathbb{I}: \frac{\log q_n(x)}{n}\leq b - \varepsilon\right\},
\sum_{\tilde{C}, \tilde{f}(\tilde{C}) > 0}t^{\tilde{\lambda}_C}P^{\tilde{T}}_t(\tilde{C})= \sum_{C, f(C) > 0}t^{\lambda_C}P^T_t(C) + \sum_{\hat{C}, \hat{f}(\hat{C}) > 0}t^{\hat{\lambda}_C+2}P^{\hat{T}}_t(\hat{C}).
A=\left[\begin{array}{ccccc}0 a_{1}0 0 \cdots a_{1}0 a_{2}0 \cdots 0 a_{2}0 a_{3}\cdots 0 0 a_{3}0 \cdots \vdots \vdots \vdots \vdots \ddots\end{array}\right]\!,\ \a_{k}=v_{k}v_{k+1}\,,\k\in\mathbb{N}.
\frac{f_{r}}{f_{s}}=\frac{1-\frac{|v_{r}|}{|C|}cos(\theta_{C,v_{r}})}{1-\frac{|v_{s}|}{|C|}cos(\theta_{C,v_{s}})}\sqrt{\frac{1-(v_{s}/c)^{2}}{1-(v_{r}/c)^{2}}}
A_{0 , \cdots , p - 1}^{\; \prime}={A_{0 , \cdots , p - 1}}\ ,{Y_{p}}^{\prime}= 2 \pi \alpha^{\prime}\; A_{p}\ , Y_{p + 1 , \cdots , 9}^{\prime}= Y_{p + 1 , \cdots , 9}\ .
\det\left( A_{\backslash\left\{j\right\}}-\lambda_{k}I\right) =\sum_{m=1}^{N}\left( x_{m}\right)_{j}^{2}\lim_{z\rightarrow\lambda_{k}}\frac{\prod_{j=1}^{N}\left( \lambda_{j}-z\right)}{\lambda_{m}-z}& =\left( x_{k}\right)_{j}^{2}\prod_{j=1;j\neq k}^{N}\left( \lambda_{j}-\lambda_{k}\right)
\frac{\left(1-\frac{1}{p}\right)}{\Gamma(k/2)}\left\langle\Psi_2,\textup{res}_p([x])\right\rangle\cdot\left\langle\Psi_1,\partial^{\textup{loc}}_0(d_{\widetilde{\delta}}) \right\rangle\cdot L_p(\frak{X},\kappa,s)=\mathcal{A}\left(\frak{Reg}_{\mathbb{H}_{\mathbf{f}}}(\partial_0(d_{\widetilde\delta}),[x_{\textup{f}}])\right)
V_{vis}(\tilde{q}) = \frac{| \frac{\partial W_{vis}}{\partial \tilde{q}}+ \frac{\partial f_{vis}}{\partial \tilde{q}}\overline{F_{\Phi}}|^2}{\frac{\partial^2 f_{vis}}{\partial \tilde{q}\partial \tilde{q}^\dagger}}- 6{\rm Re}~ W_{vis}(\tilde{q})F_{\Phi}-f_{vis}|F_\Phi|^2.
j^{\mu}= - ( D_{\nu}c^{a}) ( F_{a}^{\mu \nu}+ B_{a}\eta^{\mu \nu}) - \frac{1}{2}{\varphi_{m n}}^{a}c^{m}c^{n}\partial^{\mu}{\bar{c}}_{a}- B_{a}D^{\mu}c^{a}~ ,
\left( \begin{matrix}{{\Gamma_{\phi^{-}}^{1 1}}}&{{\Gamma_{\phi^{-}}^{1 2}}}\\{{\Gamma_{\phi^{-}}^{2 1}}}&{{\Gamma_{\phi^{-}}^{2 2}}}\\ \end{matrix}\right) = M_{C}^{- 1}\left( \begin{matrix}{{\Gamma_{R}^{\phi^{-}}( p )}}&{{0}}\\{{0}}&{{{\Gamma_{R}^{\phi^{-}}}^{*}( p )}}\\ \end{matrix}\right) M_{C}^{- 1}
0 =2f(w,x,y,z) =f(w,x,y,z)+f(x,w,y,z) =(wx,y,z)+x(w,y,z)+(x,y,z)w+(xw,y,z)+w(x,y,z)+(w,y,z)x =([w,x],y,z)+[w,(x,y,z)]+[x, (w,y,z)].
z ( 1 - z ) \frac{d^{2}\Phi_{{l}}}{d z^{2}}+ \left( \frac 9 4 - \frac{1 5}{4}z \right) \frac{d \Phi_{{l}}}{d z}- \frac{3 0 - l ( l + 1 )}{1 6}\Phi_{{l}}= 0
\lim_{x^{+}\rightarrow 0}\, \langle 0|T (\varphi^{\dagger}(x)\sigma_{1}\chi(0) ) | 0 \rangle \neq\langle 0 | \varphi^{\dagger}(x^{+}= 0^{+}, x^{-}, x^{\perp})\sigma_{1}\chi(0) | 0 \rangle \ .
-g_{2}((\nabla \pi_{\ast})(U,\varphi V),\pi_{\ast}\mathcal{C}X) =g_{1}(V,\hat{\nabla}_{U}\phi\mathcal{B}X+\mathcal{T}_{U}\omega\mathcal{B}X)+g_{1}(V,\varphi U)\eta(X)
C+\frac{\sqrt{ab}}{a}D=\frac{\sqrt{ab}}{a}\frac{\alpha}{t^\alpha}\mathrm{e}^{-\frac{\alpha(x+y)}{t^\alpha}}\bigg(\frac yx\bigg)^{\frac{\sqrt{ab}-1}{2}}I_{\sqrt{ab}-1}\bigg(\frac{2\alpha\sqrt{xy}}{t^\alpha}\bigg),
M(0,z)\exp( \int_0^t \Phi_K(s,z)dB(s) -\frac{1}{2}\int_0^t \Phi_K^2(s,z) ds) &=\tilde{p}(0,z) \exp(\int_0^t (b_0(s,z)\pi(s,z)-\frac{a_0(s,z)}{b_0(s,z)})dB(s) &-\frac{1}{2}\int_0^t (b_0(s,z)\pi(s,z)-\frac{a_0(s,z)}{b_0(s,z)})^{2}ds);0\leq t \leq T.
\sum_{k=1}^{\infty}\frac{B_{2k}}{2k(2k-1)z^{2k-1}}= - \frac{1}{2\pi i}\int_{c-i\infty \atop{{\rm Max}[0<c={\rm Re}\,s<1}}^{c+i\infty}\; \frac{z^{-s}\, \zeta(-s)}{\sin(s\pi)}\; ds & + \;\; \frac{1}{2\pi i}\int_{C}\frac{1}{(2\pi z)^{2s-1}}\; \frac{\Gamma(2s-1) \zeta(2s)}{\sin(s \pi)}\; ds \;\;.
f_{1}\star_{\vec{b}}^{n}f_{2}\ (\mathbf{n})=\iint_{S^{2}\times S^{2}}f_{1}(\mathbf{n}_{1})f_{2}(\mathbf{n}_{2})\mathbb{L}_{\vec{b}}^{j}(\mathbf{n}_{1},\mathbf{n}_{2},\mathbf{n})d\mathbf{n}_{1}d\mathbf{n}_{2}
\frac{\partial W}{\partial z}(r,0) =\int_{\{x:\,w(x,0)>w^{\circledast}(r,0)\}}\frac{\partial w}{\partial z}(x,0)\, d\gamma(x)=-d_s\int_{\{x:\,u(x)>u^{\circledast}(r)\}}f(x)\, d\gamma(x) & \geq-d_s\int_{0}^{r}f^{\circledast}(\sigma)\,d\sigma,\quad\hbox{for}~r\in(0,\gamma(\Omega)).
L_k(u) =\left (\begin{array}{cc}u + i S^{(k)}_0 i S_{-}^{(k)}i S_{+}^{(k)}u - i S^{(k)}_0 \end{array}\right )\,, \bar L_k(\bar u) = \left(\begin{array}{cc}\bar u + i \bar S_0^{(k)}i \bar S_{-}^{(k)}i \bar S_{+}^{(k)}\bar u - i \bar S_0^{(k)}\end{array}\right)
\sum_{U^a V^b W^c \in \mathcal{B}_n}U^a V^b W^c X^a Y^b Z^c = (UX+VY+WZ)^n & = \prod_{i \in S_n}\left(U^{2^i}X^{2^i}+V^{2^i}Y^{2^i}+W^{2^i}Z^{2^i}\right).
\frac{\Lambda_{{\rm b}\perp}\left(a=-a_4\right)}{{\cal E}_4}=\frac{\Lambda_{\rm b}\left(a=-a_4\right)}{{\cal E}_4}=\frac{\Gamma_{\rm b}\left(a=-a_4\right)}{{\cal E}_4}=\frac{\Gamma_{\rm{bc}}\left(a=-a_4\right)}{{\cal E}_4}=\frac{\Lambda_{\rm c}\left(a=-a_4\right)}{{\cal E}_4}=0
\eqref{eq:G2-1}=K_i \big[ E_i, [F_i ,F_j]_{q^3}\big]=K_i [\frac{K_i-K_i^{-1}}{q-q^{-1}},F_j]_{q^3}=q^3[3] K_i K_i' F_j = \eqref{eq:G2-1}.
\begin{aligned}|J(u^\circ)-J(u)|= \Bigg| \mathbb{E}^{u^\circ}\left[ \mu\left( \sum_{i=0}^{K-1}L(x_i,u^\circ_i)+\Phi(x_K) \right) \right] \ \ - \mathbb{E}^u\left[ \mu\left( \sum_{i=0}^{K-1}L(x_i,u_i)+\Phi(x_K) \right) \right] \Bigg| \leq e_\Phi (c^Kd_0+s\sum_{i=1}^{K-1}c^i). \end{aligned}
1_{N S}{\frac{\buildrel T_{\alpha}}{\longrightarrow}}( 0_{w}| 1_{N S}) , \ \ \, 5_{N S}{\frac{\buildrel T_{\alpha}}{\longrightarrow}}( 0_{m}| 5_{N S}) , \ \ \, ( p + 1 )_{R}{\frac{\buildrel T_{\alpha}}{\longrightarrow}}( p | p_{R}, ( p + 2 )_{R}) ,
\Phi_i(\mu) = \Psi_{i_1}^{(1)}(\mu_1) \cdot \Psi_{i_2}^{(2)}(\mu_2) \cdot \ldots \cdot \Psi_{i_{q-1}}^{(q-1)}(\mu_{q-1}) \cdot \Psi_{i_q}^{(q)}(\mu_{q}) ,
\mathcal{S}_k:=\sum_{i_k\in \mathbb{Z}^*}\int_{\mathcal{H}}\bigg(\sum_{(i_1,\dots, i_{k-1})\in (\mathbb{Z}^*)^{k-1}}\alpha_{i_1,\dots, i_k}^{(k)}:\mathfrak{Re}\langle \mathfrak{e}_{i_1},\cdot\rangle \dots \mathfrak{Re}\langle \mathfrak{e}_{i_{k-1}},\cdot\rangle :(x)\bigg)^2\,dm(x).
\sum_{i=1}^{n}{^\beta}\gamma_{in}\frac{\partial F^{j}_{\beta\alpha}}{\partial z_{\beta}^{i}}\biggr|_{z_\beta(\gamma p)}=\sum_{i=1}^{n}{^\alpha}\gamma_{ji}\frac{\partial F^{i}_{\beta\alpha}}{\partial z_{\beta}^{n}}\biggr|_{z_\beta(p)}\;\,\;\,1 \leq j \leq n.
h^2 \|\partial_t \nabla_x (v - \Pi_{h}v)\|_{L_2(Q)}^2 = h^2 \sum_{i=1}^d \|\partial_t \partial_{x_i}(v - \Pi_{h}v)\|_{L_2(Q)}^2 \leq h^2 \sum_{i=1}^d C_s^2 h^{2(s-2)}\|v\|_{H^{s}(Q)}^2 = \left[\sqrt{d}C_s h^{s-1}\|v\|_{H^{s}(Q)}\right]^2.
\sigma_{11}=2C_{1}(\lambda^{2}-\frac{1}{\lambda^{2}})[\sum_{i=1}^{5}i\alpha_{i}\beta^{i-1}I_{1}^{i-1}];\sigma_{22}=0;\sigma_{33}=2C_{1}(1-\frac{1}{\lambda^{2}})[\sum_{i=1}^{5}i\alpha_{i}\beta^{i-1}I_{1}^{i-1}].
E_{1}(f):=-\frac{1}{\pi i}\int_{0}^{2\pi}\frac{f(\beta)\left(\frac{\epsilon(\alpha)}{z(\alpha)}-\frac{\epsilon(\beta)}{z(\beta)}\right)(z(\alpha)z(\beta))^{2}}{(z(\alpha)-z(\beta))^{2}}\partial_{\beta}\left(\frac{\epsilon(\beta)}{z(\beta)}\right)d\beta.
A_{\mu}= \mp \,{\frac{2 i}{g}}\sum_{I = 1}^{K}\, [ \, H_{I}\,{\frac{v_{\mu}^{I}}{\rho_{I}}}+ g \, \kappa \, E_{I \, K + 1}^{\pm}\, R_{\mu}^{I}\, \prod_{I = 1}^{K - 1}\delta ( R^{K}\cdot R^{I}) \, ] .
(\Psi^{-1}(X))^{ij}= X^{i+j, j}d_{\Psi^{-1}(X), n}^{ij}= d_n^{i+j,j}\colon (\Psi^{-1}(X))^{ij}\to (\Psi^{-1}(X))^{i+1-n,j+n};
J^{(N)}(N\!-\!1;1,\ldots,1)=\frac{2\mbox{i}^{1-2N}\pi^{(N-1)/2}}{\prod m_l}\;\Gamma\left(\frac{N\!+\!1}{2}\right)\int\limits_0^{\infty}\ldots \int\limits_0^{\infty}\prod\mbox{d}\alpha_l\left( \sum\limits_{i=1}^{N}\frac{\alpha_i}{m_i}\right) \delta\left(\alpha^T\|c\|\alpha\!-\!1\right) .
(x+i\omega)\sqrt{d_1}u_1^{\mathcal{L}}(x)+\rho_0(x-\alpha_1\rho_0^{-1}-c_0)u_0^{\mathcal{L}}(x)=0, &(x+i\omega)\sqrt{d_{k+1}}u_{k+1}^{\mathcal{L}}(x)+\rho_{k}(x+f_{k}^{-1}\rho_{k}^{-1}s_{k})u_{k}^{\mathcal{L}}(x)+(x-i\omega)\sqrt{d_{k}}u_{k-1}^{\mathcal{L}}(x)=0.
\|\sum_{k > 0}u_{<k}v_k\|_{L^2(Q)}\lesssim \ \|\sum_{k > 0}\chi_{Q}u_{<k}v_k\|_{L^2(Q)}\lesssim \|u\|_{L^\infty}( \sum_k \| \chi_Q v_k\|^2_{L^2(Q)})^\frac12 \lesssim \ \| u\|_{L^\infty}\| \chi_Q S_{>0}(v)\|_{L^2(Q)}.\lesssim \|u\|_{L^\infty}\|v\|_{BMO}.
G_2(q) = \frac{\left(q^6;q^6\right)_\infty}{4\left(q^2;q^2\right)_\infty\left(q^3;q^3\right)_\infty}f(q) + \frac{3\left(q^3;q^3\right)^3_\infty}{4(q;q)_\infty\left(q^2;q^2\right)_\infty\left(q^6;q^6\right)_\infty},
P^{(-)}_{I_{l;i}}V^{(k,q)}_{I_{l;i}}P^{(-)}_{I_{l;i}}\,P^{(-)}_{I_{r;j}}=P^{(-)}_{I_{r;j}}P^{(-)}_{I_{l;i}}V^{(k,q)}_{I_{l;i}}P^{(-)}_{I_{l;i}}P^{(-)}_{I_{r;j}}
Z ( J ) = ( d e t ( - \partial_{\mu}^{2}) )^{- 1}\exp \left[ - \frac{g^{2}}{2}\int d^{4}x \left( J_{i}( P \frac{1}{\partial_{\mu}^{2}}P )_{i k}J_{k}+ J_{i}( ( 1 - P ) \frac{1}{\partial_{0}^{2}}( 1 - P ) )_{i k}J_{k}\right) \right] .
T_{\alpha \beta}={\frac{1}{4}}\left( \nabla_{\alpha}\tilde{\psi}\gamma_{\beta}\psi + \nabla_{\beta}\tilde{\psi}\gamma_{\alpha}\psi - \tilde{\psi}\gamma_{\alpha}\nabla_{\beta}\psi - \tilde{\psi}\gamma_{\beta}\nabla_{\alpha}\psi \right) ~ .
\left[ \varphi ( x ) , \varphi_{0}^{*}( x^{\prime}) \right]_{t = t^{\prime}}= i \delta ( \mathbf{x}- \mathbf{x}^{\prime}) \left[ \widetilde{\varphi}( x ) , \widetilde{\varphi}_{0}^{*}( x^{\prime}) \right]_{t = t^{\prime}}= - i \delta ( \mathbf{x}- \mathbf{x}^{\prime})
\tilde{u}(x)=\{u(x):k_{x}\frac{\partial^{2}u}{\partial x^{2}}+k_{y}\frac{\partial^{2}u}{\partial y^{2}}+q=0forx\in\Omega,u(x)=u^{*}(x)forx\in\partial\Omega,k_{x}\in k_{x},k_{y}\in k_{y}\}
\tfrac{1}{2}\partial_{t}\big\|\partial_{t}^{k-1}(\bar{\theta}- \theta)(t, \cdot)\big\|_{L^{2}(\Omega)}+a \big\|\nabla \partial_{t}^{k-1}(\bar{\theta}- \theta)(t, \cdot)\big\|_{L^{2}(\Omega)}= 0.
\tilde{\Xi}_n^{c}(z_1,z_2, s)=\sum_{c> 0}\sum_{\substack{a,d, ad \equiv 1 \pmod c}}\frac{c^{-2n}}{|c|^{-4s}}\frac{((c\bar{z_1}+d)(c\bar{z_2}-a))^n}{|(cz_1+d)(cz_2-a)|^{2s}}\frac{((c\bar{z_1}+d)(cz_2-a))^n}{|(cz_1+d)(c\bar{z_2}-a)|^{2s}}.
\widehat{F}_{\mu \nu}=F_{\mu \nu}+\frac{1}{4}\theta^{\rho \sigma}\bigg( 2\left\{F_{\mu \rho},F_{\nu \sigma}\right\}-\left\{A_{\rho},D_{\sigma}F_{\mu \nu}+\partial_{\sigma}F_{\mu \nu}\right\}\bigg) +{\cal O}\left( \theta^{2}\right).
F(s +{{3}\over{2}},1,s + 2,{{3}\over{4}}) ={{\Gamma(s+2)}\over{\Gamma(1) \ \Gamma(s+1)}}\ \int_0^1 (1 - t)^s \ \left(1 -{{3t}\over{4}}\right)^{-s-{{3}\over{2}}}\dt
E_{S_1^*}-E_{\hat{S}_1}=E_{W_{t_2}}+E_{W_{t_2},S_1^*\setminus W_{t_2}}, E_{S_1^*,S_3^*}-E_{\hat{S}_1,\hat{S}_3}=E_{W_{t_2},S_3^*}-d_{{t_2},S_1^*\setminus W_{t_2}}-E_{W_{t_2},S_1^*\setminus W_{t_2}}, E_{S_1^*,V_1}-E_{\hat{S}_1,V_1}=E_{W_{t_2},V_1}E_{S_2^*,V_1}-E_{\hat{S}_2,V_1}={d_{t_2,V_1}}.
(s,\pi)_k=\left(\begin{array}{ccccccccccc}\boxed{s_{11}}s_{12}\cdots s_{1m_1}\boxed{s_{21}}\cdots s_{2m_2}\cdots \boxed{s_{k1}}\cdots s_{km_k}\pi(s_{11}) \pi(s_{12}) \cdots{\boxed{\pi(s_{1m_1)}}}\pi(s_{21}) \cdots \boxed{\pi(s_{2m_2})}\cdots \pi(s_{k1}) \cdots \boxed{\pi(s_{km_k})}\end{array}\right)
e^{--}= d\xi^m e_{m}^{--}(\xi, \eta (\xi)) + d\eta^q (\xi) e_q^{--}(\xi, \eta (\xi)) = d\xi^m (e_{m}^{--}(\xi, \eta (\xi)) + \partial_m \eta^q (\xi) e_q^{--}(\xi, \eta (\xi))
\sigma_{v}^{2}=\frac{1}{2}[(\sigma_{11}-\sigma_{22})^{2}+(\sigma_{22}-\sigma_{33})^{2}+(\sigma_{33}-\sigma_{11})^{2}+6(\sigma_{23}^{2}+\sigma_{31}^{2}+\sigma_{12}^{2})]=\frac{3}{2}s_{ij}s_{ij}
\widetilde{J^{K\textnormal{th}}}(q,Q) := P^{-\ell_q(Q)}J^{K\textnormal{th}}(q,Q) = P^{-\ell_q(Q)}\sum_{d \geq 0}\frac{Q^d}{\left(qP^{-1};q\right)_d^{N+1}}
\tilde{\Theta}^{\pm;1, L}_{\lambda, T}= \Theta^{L_1}_{\lambda, T, \epsilon_k^\pm}\ \tilde{\Phi}^{\pm[\frac{k}{2}|\gamma_1|^2; (\lambda| \gamma_1)]}\ (\tau, -\beta_1, \beta_1 + \gamma^\vee_1),
X_{i}:= \frac{\beta_{i-1}-\beta_{i}}{2}-\frac{(1 +\sigma_{i})}{2}\Big(\frac{1}{2}-\frac{1}{p_0}\Big); Y_{i}:= \frac{\beta_{i-1}}{2}- \big(1+(n-i)(1-\sigma_{i})\big)\Big(\frac{1}{2}-\frac{1}{p_0}\Big).
L_1'(x, g(x), g'(x))h_{1, \Delta}(x) +L_2'(x, g(x), g'(x))h_{2, \Delta}(x)+ L_3'(x, g(x), g'(x))h_{2, \Delta}'(x) & + L \operatorname{tr}h_{3, \Delta}(x)+ L\beta^\nu\left(h_{2, \Delta}(x), x \right) = 0.
\mathbf{E}Y\, \leq\,c_\beta\,c_{(\ref{6.3})}\,c^{-1-\beta}\int_{0}^{t}\mathrm{d}ss^{-1/2}\,(t-s)^{-1}\log^{-1-q(1+\beta)}\bigl((t-s)^{-1}\bigr)\, begin{align*}1pt] =:\,c_{(\ref{6.4})}\,c^{-1-\beta}
R\Big(t-\frac12\Big) =-n!^{A-15}2^{-3}(2t+n-1)\frac{\Gamma(t-\frac12)^{A+3}\Gamma(2t+4n)^3\Gamma(2n-2t+2)^3}{\Gamma(t+n+\frac12)^{A+3}}\cdot\frac{\sin(2\pi t)^3}{\pi^3}.
\prod\limits_{\delta|N}\delta'^{r_\delta}=(p_1p_2)^rp_2^{-p_1r}p_1^{-p_2r}=p_1^{r(1-p_2)}p_2^{r(1-p_1)}=p_1^{\frac{-24}{p_1-1}}p_2^{\frac{-24}{p_2-1}}.
S_{\frak M}(y,z) =\sum_{u\le y}\left|\mathcal{I}_1(u,z)+\mathcal{I}_2(u,z)+\mathcal{I}_3(u,z)\right|^2 &\ll \sum_{u\le y}\left(\left|\mathcal{I}_1(u,z)\right|^2+\left|\mathcal{I}_2(u,z)\right|^2+\left|\mathcal{I}_3(u,z)\right|^2\right)
\frac{d}{dt}V_{m}^{(1)}(t) =(-1)^{m}(-A)\left(\sum_{j=0}^{m}\begin{pmatrix}1+(m-1) 1+(j-1)\end{pmatrix}\frac{(-tA)^{j}}{j!}e^{-tA}v_0\right), \frac{d}{dt}V_{m}^{(2)}(t) =(-1)^{m}(-A)\left(\sum_{k=0}^{m-1}\begin{pmatrix}1+(m-1) 1+k\end{pmatrix}\frac{(-tA)^{k}}{k!}e^{-tA}u_1\right).
\| D \Delta^{-1}\nabla \times \omega^B(t) \|_{L_x^{\infty}(|x|> \sqrt{\epsilon})}= \| \mathcal R_{ij}( \chi_{|y|<\frac 12 \sqrt{\epsilon}}\omega^B(t,y) ) \|_{L_x^{\infty}(|x|>\sqrt{\epsilon})}\lesssim \epsilon^{-C}\| \omega^B(t) \|_{1}\lesssim \epsilon^{-C}\| \frac{\omega_0}r \|_{1}<\epsilon,
\phi(x)_g = \phi(x)_g z_g=\phi(x)_g\alpha_g(z_{g^{-1}})=w^*\lim_\omega \phi(x)_g\alpha_g(f_\omega) &= w^*\lim_\omega \theta(E_M(x\theta^{-1}(f_\omega)\sigma(z_{g^{-1}}g^{-1})))=\theta (E_M(x\theta^{-1}(z_{g^{-1}})\sigma(z_{g^{-1}}g^{-1}))) &=\theta(E_M(x\sigma(z_{g^{-1}}g^{-1}))),\ \ \ \x\in W^*(X_0),\ \ \ \g\in G,
Z [ T , ( e_{i}, \vec{x}_{i}) ] = \frac{\int [ d A_{0}( \vec{x}) ] \mathrm{e}^{- S_{\mathrm{e f f}}[ A_{0}]}\prod_{j}\mathrm{e}^{i e_{j}A_{0}( \vec{x}_{j}) / T}}{\int [ d A_{0}( \vec{x}) ] \mathrm{e}^{- S_{\mathrm{e f f}}[ A_{0}]}}
P\left(T_n< q_n^*(1-\alpha)\right) \leq P\left(T_n< q_n^Z(1-\alpha+\varepsilon_n)\right)+P(\mathcal{E}_n^c) &\leq P\left(\max_{\theta\in\mathcal{G}_n}|Z_n(\theta)|< q_n^Z(1-\alpha+\varepsilon_n)\right)+2\varepsilon_n=1-\alpha+3\varepsilon_n.
\left( 2 \kappa \, \operatorname{s i n h}{\frac{p_{0}}{2 \kappa}}\right)^{2}- \overrightarrow{p}^{2}e^{{\frac{p_{0}}{\kappa}}}= \left( 2 \kappa \, \operatorname{s i n h}{\frac{m}{2 \kappa}}\right)^{2}
V(\rho)=E\left [1 - \frac{1 +\frac{\rho_*^4}{2 (\rho^2-l^2)^2}}{\left (1+\frac{\rho_*^4}{(\rho^2-l^2)^2}\right )^2}\right ] + \frac{L^2}{2m_p\rho^2}\frac{1}{\left (1+\frac{\rho_*^4}{(\rho^2-l^2)^2}\right)^2},
\sum_{j=1}^{2}\epsilon_{\mu}^{j}(p)\epsilon_{\nu}^{j*}(p)==\sum_{j==1}^{2}\epsilon_{\mu}^{j*}(p)\epsilon_{\nu}^{j}(p)=\delta_{\mu\nu}-\frac{p_{\mu}p_{\nu}}{E^{2}}.
I[\mu] = \iint f(\theta - \phi) d \mu(\theta) d\mu (\phi) = \sum_{k \neq 0}c_k(f) \iint e^{i (\theta - \phi)}d \mu(\theta) d \mu(\phi) = \sum_{k \neq 0}c_k(f) | c_k(\mu) |^2 \geq 0.
\| \big(A^{(k, i)}\big)^n e^{(k, i)}_{p}\|^{2}= \sum_{r=0}^{n}\prod_{s=1}^{r}\sum_{j_{s}=s-1}^{j_{s-1}+1}\left\{\frac{[i+(p-r+1+j_{s})k]!}{[i+(p-r+j_{s})k]!}\right\}^{2}\frac{[i+(p+n-2r)k]!}{(i+pk)!},
W^{(\kappa)}(y)=(1-y)^{\alpha+\kappa}y^{\beta+\kappa}\, \Psi_0(y) \,T^{(\kappa)}\, \Psi_0(y)^*, D^{(\kappa)}=\Psi_0^{-1}\,\Omega^{(\kappa)}\, \Psi_0,
2\alpha n > e(\widetilde{H}) \ge e(\widetilde{H}-X)+ \left( \sum_{x \in X}d_{\widetilde{H}}(x) \right) -2\ge e(H''-X)-1 +|X|(3 \gamma n/ 5 -2)-2 & \ge e(H'')-\sum_{x \in X}d_{H''}(x)+|X|(3 \gamma n/ 5 -2)-3 & =e(H'')-\sum_{x \in X}(d_{H''}(x)-3 \gamma n/ 5 +2)-3
\sup_{v \in L^\infty(\Omega )}\frac{\int_\Omega uv\,dx}{\|v\|_{L^{\widetilde A}(\Omega )}}= \sup_{\varphi \in C^\infty_0(\Omega )}\frac{\int_\Omega u\varphi\,dx}{\|\varphi\|_{L^{\widetilde A}(\Omega )}}.
T_+(z)=T_-(z)\begin{pmatrix}0 -1 1 0\end{pmatrix}, z\in \Sigma_2, &T_+(z)=T_-(z)\begin{pmatrix}1 0 -e^{\pi i\alpha}1\end{pmatrix}, z\in \Sigma_1, &T_+(z)=T_-(z)\begin{pmatrix}1 0 -e^{-\pi i\alpha}1\end{pmatrix}, z\in \Sigma_3.
\bar R_{SD}\to \frac{\bar \gamma}{2 \ln(2)}\sum_{k=1}^{2M}\frac{1}{k}, \textrm{as}\bar \gamma\to 0 &\bar R_{\rm conv}\to \frac{\bar \gamma}{4 \ln(2)}\sum_{k=1}^{M}\frac{1}{k}, \textrm{as}\bar \gamma\to 0.
S_{A B}={\frac{i}{2}}e^{K_{V}/ 2}~ ( \sigma_{x})_{A}{}^{C}~ \epsilon_{B C}~ P_{I}^{x}( q ) ~ X^{I}( z ) ={\frac{i}{2}}( \sigma_{x})_{A}{}^{C}~ \epsilon_{B C}~ P_{I}^{x}( q ) ~ L^{I}\, .
\int_B e^{-i\varphi}q(x)u_1u_2dx = \int e^{-i\varphi}(A+\nabla\vartheta)\cdot(A_1+A_2)u_1u_2dx\cr +\int_B ie^{-i\varphi}(A+\nabla\vartheta)\cdot(u_1\nabla u_2-u_2\nabla u_1)dx\cr +\int_B e^{-i\varphi}(A+\nabla\vartheta)\cdot\nabla\varphi u_1u_2dx+\mathcal{R}\cr:= \mathcal{J}_1+\mathcal{J}_2+\mathcal{J}_3+\mathcal{R},
\varrho_{c}= \frac{3}{\kappa}H_{0}^{2}= 1 . 9 h^{2}\times 1 0^{- 2 6}\frac{\mathrm{k g}}{\mathrm{m}^{3}}= 8 1 h^{2}( \mathrm{m e V})^{4}= 2 . 4 h^{2}\times 1 0^{- 1 2 0}m_{P l}^{4}.
F_{m n}^{a}= - 4 \rho^{2}\frac{\overline{{\eta}}_{m n}^{a}}{( x^{2}+ \rho^{2})^{2}}+ 8 \rho^{2}\frac{( \overline{{\eta}}_{m k}^{a}x_{n}- \overline{{\eta}}_{n k}^{a}x_{m}) x_{k}}{x^{2}( x^{2}+ \rho^{2})^{2}}~ .
\int F\cdot \psi \,d\mu = \int g(z) \cdot \int \tilde{\psi}(z+w)f(w) \,dm(w) \,dm(z) = \int \int f(w)g(z-w) \,dm(w)\, \tilde{\psi}(z)\, dm(z) = \int f*g(z) \cdot \tilde{\psi}(z) \,dm(z) = \int (f*g)\circ \pi \cdot \psi \,d\mu,
U_{\beta}(\omega_{j,n}) ~=~ \frac{P'_{\beta}(\omega_{j,n})}{f'_{\beta}(\omega_{j,n})}\neq 0 \mbox{and}U_{\beta}(\frac{1}{\beta}) ~=~ \frac{P'_{\beta}(\frac{1}{\beta})}{f'_{\beta}(\frac{1}{\beta})}\neq 0.
\displaystyle\limsup_{h\to 0+}\dfrac{\sup_{t\in [0, 1]}\sup_{|s-t|\leq h}|\widetilde Z^H(t)- \widetilde Z^H(s)|}{h^{H}(\log{1/h})^{\frac 1 2 +\frac{1}\alpha + \epsilon}}=0\;\;a.s.
err_1(t)= \int_0^t \sum_{(e,\tilde{e})}\,2C^n_{\tilde{e},e}\,\big(\phi_s(x^e_1)-\phi_s(x^{\tilde{e}}_1)\big)\,\big(u^n_{s-}(x^{\tilde{e}}_1) - u^n_{s-}(x^e_1)\big)\,ds.