harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\Vert x_n-y_i\Vert^2\le\exp\left(-2\theta_k\sum_{j=i}^{n-1}\lambda_j\right)\Vert x_i-y_i\Vert^2+2M^2\sum_{j=i}^{n-1}(\theta_i-\theta_j)\lambda_j+4M^2\sum_{j=i}^{n-1}\lambda_j^2.
\int_{\Omega}|\nabla u |^p = \int_{\Omega}|v'(h(y))|^p |\nabla h (y)|^{p-1}|\nabla h (y)| \,dy &=\int_0^1 \left[ \, \int_{h^{-1}(\{t\}) \cap \Omega}|v'(h(y))|^p |\nabla h (y)|^{p-1}\,d \mathcal{H}^{N-1}(y) \,\right] \,dt.
\rho_{\varepsilon}(x)=\begin{cases}\varepsilon,{\rm if}\x\in\Omega_{\varepsilon}, \frac{M-\varepsilon |\Omega_{\varepsilon}|}{|\Omega\setminus\overline{\Omega}_{\varepsilon}|},{\rm if}\x\in\Omega\setminus\overline{\Omega}_{\varepsilon}, \end{cases}
{\cal S}_{g}= \int d t \sum_{j = 1}^{n}\left[ \Psi_{j}^{\dagger}( i \partial_{t}- A_{0}) \Psi_{j}\right] + \sum_{j = n + 1}^{N}\left[ \Psi_{j}^{\dagger}( - i \partial_{t}+ A_{0}) \Psi_{j}\right] .
\vert \partial_x p_{t-s}G(s,u)\vert = \vert p_{t-s}\left(\partial_x G(s,u)\right)\vert &\leq p_{t-s}\big\vert f'(u+\hat{v}(\cdot +cs))\, u_x + (f'(u+\hat{v}(\cdot +cs))-f'(\hat{v}(\cdot +cs)))\, \hat{v}_x(\cdot+cs)\big\vert &\leq p_{t-s}\big[\eta_3 \, (1+u^2)\vert u_x\vert + \eta_3 \, (\vert u\vert + u^2)\, \hat{v}_x(\cdot+cs)\big]
UL\psi = U(d_A^\ast f + d_B^\ast g) = S(2d_A^\ast d_A - I)(d_A^\ast f + d_B^\ast g) &= S\{d_A^\ast f + (2d_A^\ast T - d_B^\ast) g\}= d_B^\ast f + (2d_B^\ast T - d_A^\ast) g
{\mathfrak q}_0^{Sp(p,q)\times Sp(1)}={\mathfrak l}_0^{Sp(p,q)\times Sp(1)}+{\mathfrak u}_0^{Sp(p,q)\times Sp(1)}\subset{\mathfrak s}{\mathfrak p}(n,{\mathbb C}) \times{\mathfrak s}{\mathfrak p}(1,{\mathbb C}),
\frac{1}{4\pi^2}\int_0^{2\pi}\int_0^{2\pi}\frac{f(r_1e^{i\theta_1},r_2e^{i\theta_2})\,d\theta_1\,d\theta_2}{(r_2e^{i\theta_2})^{\alpha_2}}=a_{(0,\alpha_2)}.
Z_{\mathrm{s c}}( \beta ) = \int_{- \infty}^{\infty}d x_{0}\sum_{j = 1}^{N ( x_{0}, \beta )}\mathrm{e}^{- S_{j}( x_{0}, \beta ) / \hbar}\Delta_{j}^{- 1 / 2}( x_{0}, \beta ) \; ,
\sum_{i>k_n}i^{-1-2\alpha}\leq \int_{k_n}^\infty x^{-1-2\alpha}\, dx + k_n^{-1-2\alpha}= \frac{k_n^{-2\alpha}}{2\alpha}+ k_n^{-1-2\alpha}\leq k_n^{-2\alpha}\frac{1+2\alpha}{2\alpha},
\mathbb K^{(\alpha,\frac{1}{2})}(x,y) = \frac{-1}{x-y}\sum_{j=0}^2 \sum_{i=0}^{2-j}(-1)^j a_{i+j}\left( \vartheta_x^j f(x) \right)\left( \vartheta_y^i g(y) \right)
w^\uparrow_{\{1,2\}}(C_{(1, \{2,3\})})= 0, w^\uparrow_{\{1,2\}}(C_{(2, \{1,3\})})= 0, w^\uparrow_{\{1,2\}}(C_{(3, \{1,2\})})= 1 w^\uparrow_{\{1,2\}}(C_{(\{2,3\}, 1)})= 0, w^\uparrow_{\{1,2\}}(C_{(\{1,3\}, 2)})= 0, w^\uparrow_{\{1,2\}}(C_{(\{1,2\}, 3)})= 1
u_m = v_m \sim \frac{\Lambda_1\Gamma(m/2-1/2)}{(\log(-2-\sqrt{3})s)^{m/2-1/2}}+ \frac{\Lambda_2 \Gamma(m/2-1/2)}{(-\log(-2-\sqrt{3})s)^{m/2-1/2}}+ \frac{\Lambda_3 \Gamma(m/2-1/2)}{(\log(-2+\sqrt{3})s)^{m/2-1/2}}+ \frac{\Lambda_4 \Gamma(m/2-1/2)}{(-\log(-2+\sqrt{3})s)^{m/2-1/2}},
d(m\otimes (x_1\otimes\cdots\otimes x_n)) = \sum_{i=1}^n (-1)^{i+1}\{m,x_i\}\otimes(\alpha(x_1)\otimes\cdots\otimes\hat{\alpha(x_i)}\otimes\cdots\otimes \alpha(x_n))+ &\sum_{i<j}(-1)^{i+j}\beta(m)\otimes([x_i,x_j],\alpha(x_1)\otimes\cdots\otimes\hat{\alpha(x_i)}\otimes\cdots\otimes\hat{\alpha(x_j)}\otimes\cdots\otimes \alpha(x_n))
e^{2\varphi}={{8Q^2_F}\over{\pi^2Q^2_H}}\equiv \lambda^6; \ \Q_H\equiv{{\sqrt{2}}\over{4\pi^2}}\int_{S^3}e^{-4\varphi/3}\star G, \Q_F\equiv{{\sqrt{2}}\over{16\pi}}\int_{S^3}e^{2\varphi/3}\star F.
|S(n)| \leq (V-1)((V-2)^f+(V-2)^{f-1}+...+(V-2)+1) & < (V-1)\cdot 2(V-2)^f &\leq 2(V-1)(V-2)^{\log_V(n)}&=2(V-1)n^{\log_V(V-2)}&=O(n^{\log_V(V-2)}).
\chi_{\alpha}\equiv \Pi_{\alpha}-\frac{i}{2}\left( \gamma^{0}\right)_{\;\;\alpha}^{\beta}\bar{\psi}_{\beta}\approx 0,\;\bar{\chi}^{\alpha}\equiv \bar{\Pi}^{\alpha}-\frac{i}{2}\left( \gamma^{0}\right)_{\;\;\beta}^{\alpha}\psi^{\beta}\approx 0,
A_{n-k,q}\left( x\right) =\left( \Phi_{n-k}\circ...\circ\Phi_{n}\right)A_{n,q}\left( x\right) =\frac{\left[ n-k\right]_{q}!}{\left[ n\right]_{q}!}D_{q,x}^{k}A_{n,q}\left( x\right) .
= \sum\limits_{s=1}^{n-j+1}{2n\choose 2s+2j-2}{2s+2j-2\choose 2j-1}\frac{4^s-1}{s}B_{2s}B_{2(n-s-j+1)}&= \sum\limits_{s=1}^{n-j+1}{2n\choose 2(s+j-1)}{s+j-1\brack j}B_{2(n-s-j+1)},
T(1,4) =\left[ \begin{array}{ccc}p_{2}p_{1}1\end{array}\right] \mathrm{or}\ \left[ \begin{array}{ccc}p_{3}p_{2}p_{1}\end{array}\right] & \mathrm{or}\ \left[ \begin{array}{ccc}p_{4}p_{3}p_{2}\end{array}\right] \mathrm{or}\ \left[ \begin{array}{ccc}p_{5}p_{4}p_{3}\end{array}\right] .
X_{13}X_{31}=X_{31}X_{13}=1, \frac{qX_{01}X_{13}-q^{-1}X_{13}X_{01}}{q-q^{-1}}=1,\quad\frac{qX_{13}X_{30}-q^{-1}X_{30}X_{13}}{q-q^{-1}}=1, \frac{qX_{23}X_{31}-q^{-1}X_{31}X_{23}}{q-q^{-1}}=1,\quad\frac{qX_{31}X_{12}-q^{-1}X_{12}X_{31}}{q-q^{-1}}=1;
<n>=\frac{1}{e^{\hbar \omega /kT}-1}-\lambda^{2}\frac{\hbar \omega}{kT}\frac{e^{3\hbar \omega /kT}+4e^{2\hbar \omega /kT}+e^{\hbar \omega /kT}}{(e^{\hbar \omega /kT}-1)^{4}}.
x_{i}^{t+1}=\frac{1}{\sigma_{i}^{2}}\eta_{t}\left(\sum_{b\in[m]}A_{bi}\left(r_{b}^{t}+\delta r_{b\rightarrow i}^{t}\right)\right); \delta x_{i\rightarrow a}^{t+1}=\frac{1}{\sigma_{i}^{2}}A_{ai}r_{a}^{t}\eta'_{t}\left(\sum_{b\in[m]}A_{bi}\left(r_{b}^{t}+\delta r_{b\rightarrow i}^{t}\right)\right).
\nonumber \begin{array}{l l}{{S U ( n )^{3}:}}&{{n - 4 + k = 0}}\\{{S U ( 2 )^{2}U ( 1 )_{R}:}}&{{( n + 2 k - 4 ) ( n - 2 ) + ( k - 2 ) k = 0}}\\{{{U ( 1 )_{R}}^{3}:}}&{{n ( n - 1 ) ( n + 2 k - 4 )^{3}+ 2 k n ( k - 2 )^{3}- k ( k - 1 ) n^{3}= 0}}\\ \end{array}
\bigg| \frac{\partial^{p+2}f}{\prod_{j=1}^{p+2}\partial x_{i_j}}(x) - \frac{\partial^{p+2}f}{\prod_{j=1}^{p+2}\partial x_{i_j}}(y) \bigg|\leq \; [h]_{\alpha,p}|x-y|^\alpha \left( A-2\log |x-y| \right), |x-y|\leq 1 \leq \; A\;[h]_{\alpha,p}|x-y| > 1,
d_{V}^{i}:=\left (\begin{array}{cccccc}d^{i, 0}-S^{i, 1}0 0 \cdots 0 0 d^{i, 1}-S^{i, 2}0 \cdots 0 & \cdots \cdots 0 0 0 0 \cdots d^{i, N}\end{array}\right ).
P=\sum_{s_{zN}}\cdot\cdot\cdot\sum_{s_{z2}}\sum_{s_{z1}}\int_{allspace}\cdot\cdot\cdot\int_{allspace}\int_{allspace}|\Psi|^{2}d^{3}r_{1}d^{3}r_{2}\cdot\cdot\cdot d^{3}r_{N}=1
\| a - \sum_{x \in F}\sum_i \i< \xi_{x,i}h_x^{1/2}, a\xi_{x,i}h_x^{1/2}> \| =\sup_{y \in X}\| \| p^y ( a - \sum_{x \in F}\sum_i \i< \xi_{x,i}, a \xi_{x,i}> h_x (y) ) \| \leq \sup_{y \in X}\sum_{x \in F}\| p^y ( a - \sum_i \i< \xi_{x,i}, a \xi_{x,i}> ) \| h_x (y) < \varepsilon.
t \propto \left( \begin{matrix}{\alpha}&{- \xi^{\mathrm{T}}}&{0}\\{\noalign{\vskip 1 m m}\zeta}&{s o ( n - 1 )}&{\xi}\\{\noalign{\vskip 1 m m}0}&{- \zeta^{\mathrm{T}}}&{- \alpha}\\ \end{matrix}\right) .
\mathbb K^{(1)}_{r=1,\nu_1=\alpha}(x, y) =\mathbb K^{(2)}_{q=0, r=1, \nu_1=\alpha}(x, y)= 4 \left(\frac{y}{x}\right)^{\alpha/2}\mathbb K_{\operatorname{Bessel},\alpha}(4x,4y).
I_1 = \int \frac{d^4k}{(2\pi)^4}\,\frac{d^4l}{(2\pi)^4}\,\frac{d^4q}{(2\pi)^4}\,\frac{2 (q^2 +q_\mu k_\mu)}{k^2 \Big(1+k^{2n}/\Lambda^{2n}\Big)\,l^2\Big(1+l^{2n}/\Lambda^{2n}\Big)\,q^4 (q+k)^4 (q+l)^2}
\begin{psmallmatrix}x(\beta) s(\beta) w(\beta) y(\beta)\end{psmallmatrix}- \bar \mu(z(\beta)) e = (1 - \beta) \Bigl[\begin{psmallmatrix}x^k s^k w^k y^k\end{psmallmatrix}- \mu^k e \Bigr]+ \beta^2 \begin{psmallmatrix}\Delta x \Delta s \Delta w \Delta y\end{psmallmatrix}\, .
\frac{1}{2}\langle\widehat{\Delta x_k}\hat{C}+ \hat{C}\widehat{\Delta x_k}\rangle=k^{ij}\Delta(x_ix_jx_k)- \frac{\hbar^2}{6}k^{ij}\epsilon_{ki}{}^l\epsilon_{lj}{}^m x_m + 2k^{ij}x_i \Delta(x_jx_k)\approx0\,.
\left( (t-s)y\log\bigl((t-s)^{-1}\bigr)\right)^{1/(1+\beta)}\geq\left(2^{-2(n+1)}\,\frac{c}{2}\,2^{-n-1}2n\log2\right)^{1/(1+\beta)}=c_{(\ref{100})}^{-1}2^{-(\bar{\eta}_{\mathrm{c}}+1)n}\,n^{1/(1+\beta)}.
W^{k,p}(\Omega) \hookrightarrow C^{k-1-[n/p],\theta}(\overline{\Omega})\, \ \theta\in[0,\theta_0]\,, \theta_0 =\begin{cases}[n/p]+1-(n/p) , n/p \text{is not an integer}\,, 1 \text{if}n/p \,.\end{cases}
\dot{\alpha}_{0}^{2}= \frac{1}{l^{2}}\left( \frac{f ( t / l )^{\prime}- g ( t / l )^{\prime}}{f ( t / l ) - g ( t / l )}\right)^{2}= \alpha_{1}^{2}- 1 / l^{2}= \frac{\kappa^{2}}{3 l}\rho + \frac{\kappa^{4}\rho^{2}}{3 6}\ .
\frac{d\sigma(s,t)}{dt}= \frac{1}{16 \pi s^2}\,\big\{A_1^2(s, t)- 2i A_1(s, t) A_2(s, t) &+ A_2^2(s, t) + 2A_1(s, t) A_3(s, t) \big\}\;.
0 = ( \phi_{2}^{\prime}+ H_{1}\phi_{1})^{\prime}+ \frac{1}{r^{2}}\phi_{2}( 1 - \phi_{i}^{2}- \phi_{2}^{2}) - ( i m - H_{1}) ( \phi_{1}^{\prime}- H_{1}\phi_{2}) - M^{2}( \phi_{2}+ \sin \beta ) ;
[p,qp,q][a,ba,b] = [p,p,1][1,q,q][a,a,1][1,b,b] &=[p,p,1][a,a,1][1,b,b][1,q,q] &=\begin{cases}[r,r,1][1,s,s] rP= pP\cap aPPs = Pb\cap Pq 0 \end{cases}&=\begin{cases}[r,sr,s] rP= pP\cap aPPs = Pb\cap Pq 0 ,\end{cases}
v_{1}^{1}=y_{1,[2]}^{1,d}-a_{1}^{1,1}(y_{1,[1]}^{1}-y_{1,[1]}^{1,d})-a_{1}^{1,0}(y_{1}^{1}-y_{1}^{1,d}) v_{2}^{1}=y_{2,[4]}^{1,d}-\sum_{\beta=0}^{3}a_{2}^{1,\beta}(y_{2,[\beta]}^{1}-y_{2,[\beta]}^{1,d})
B_1(T(\sigma(a_1)))B_1(a_2) =S(B_1(a_1))B_1(\sigma(a_2))B_1(T(\sigma(a_3)))B_1(a_4) &=S(B_1(a_1))B_1(\sigma(a_2)\circ T(\sigma(a_3)))B_1(a_4) &=S(B_1(a_1))\epsilon(a_2)B_1(a_3)=\epsilon(a)1.
d\Big( T[u],T[v] \Big) \, =\ \Big\| \left(A[u]-\big(F[u]-f \big) \right) \, -\, \left(A[v]-\big(F[v]-f \big) \right) \Big\| =\ \Big\| A[u] -A[v] -\big(F[u] - F[v]\big) \Big\|,
(\partial_t -\Delta) |\nabla H-h^S_{\nu\nu}HV|^2 = -2 |\nabla(\nabla H-h^S_{\nu\nu}H \nu_S^T)|^2 +2 (\nabla_i H - h^S_{\nu \nu}H V_i) \cdot & \Big( \nabla_i(|A|^2 H) -|A|^2 H h^S_{\nu \nu}V_i + h_{ik}h_{kj}\nabla_j H & -H V_i (\partial_t -\Delta) h^S_{\nu\nu}+ O(H^2 +H|\nabla H|) \Big)
-i\sigma_{z}\frac{\omega_{r}}{2}e^{-i\sigma_{z}\omega_{r}t/2}\psi+e^{-i\sigma_{z}\omega_{r}t/2}\frac{\partial\psi}{\partial t}=i(\omega_{1}\sigma_{x}cos\omega_{r}t+\omega_{1}\sigma_{y}sin\omega_{r}t+\omega_{0}\sigma_{z})e^{-i\sigma_{z}\omega_{r}t/2}\psi,
\left\| \sum_{\substack{k_1+k_2 = k |k_1| \gtrsim |k_2|}}^* \frac{\langle k \rangle^{1+s_1}u_{k_1}v_{k_2}}{k^3 - \alpha k_1^3 - \alpha k_2^3}\right\|_{\ell^2_k}
\frac{h}{r^3}\frac{d}{dr}(h r^{3}\frac{dR}{dr}) + (1 + \frac{r_{1}^{2}}{r^2})(1 + \frac{r_{5}^{2}}{r^2}) [\omega^2 - m^{2}+ (\omega \sinh \sigma_{K}- m \cosh \sigma_{K})^{2}\frac{r_{0}^{2}}{r^2}] R = 0
\begin{aligned}\int |u| |\nabla \rho| |\nabla u| |\nabla u_t| dx \le C(1+\|\nabla u_t\|_{L^2}+\|\nabla^3 d\|_{L^2}^2)^{\frac{q+6}{3q}}\|\nabla u_t\|_{L^2}&\le C(\varepsilon)(1+\|\nabla^3 d\|_{L^2}^4)+\varepsilon\|\nabla u_t\|_{L^2}^2.\end{aligned}
\Phi_g\{z|(u_1,\Delta_1)\cdots,(u_n,\Delta_n)|p^{(1)},\cdots,p^{(k)},p\}=[\prod_{i=1}^n exp (u_iW(\Delta_i,z))]\Phi_g\{z||p^{(1)},\cdots,p^{(k)},p\}
x_3 =\lambda^2-a_2-x_2=\frac{(y_2-(t^2+1)^3)^2-a_2x_2^2-x_2^3}{x_2^2}= \frac{(t^2+1)^4x_2+2(t^2+1)^6-2y_2(t^2+1)^3}{x_2^2}.
\nabla_X (Y\wedge Y') =\widetilde{\nabla}_X(Y\wedge Y')+F^{1,l}(X,Y\wedge Y') &=(\widetilde{\nabla}_X Y)\wedge Y'+Y\wedge \widetilde{\nabla}_X Y'+F^{1,l}(X,Y\wedge Y') &=(\nabla_X Y)\wedge Y'+Y\wedge \nabla_X Y'+F^{1,l}(X,Y\wedge Y'),
w_1(t) = b_1 + y_1(t) + \int_0^t\left(-\phi_0(w_1(s)) + \phi_B(w_2(s))\right)ds, w_2(t) = b_2 + y_2(t) + \psi_0(w_1(t)) + \int_0^t\left(-\phi_B(w_2(s))\right)ds \ge 0.
\begin{aligned}\vartheta_3(z;\tau) :=\vartheta_4(z+\tfrac12\pi;\tau), \vartheta_1(z;\tau) :=-ie^{iz+\tfrac14\pi i\tau}\vartheta_4(z+\tfrac12\pi\tau;\tau), \vartheta_2(z;\tau) :=\vartheta_1(z+\tfrac12\pi;\tau). \end{aligned}
R_{\mathrm{b o u n d a r y}}( r ) = \frac{1}{\sqrt{r}}\left( A_{\mathrm{1}}J_{- \lambda}\left( \frac{\omega \ell^{2}}{r}\right) + A_{\mathrm{2}}J_{\lambda}\left( \frac{\omega \ell^{2}}{r}\right) \right) ,
I-M=\begin{pmatrix}1-(1-z^{-1})s^{4}t^{4}(z-1)(s^{2}t^{2}-s^{3}t^{3}) (1-z^{-1})(s^{2}t^{2})(1+st+s^{2}t^{2}+s^{3}t^{3}) 1+(z-1)(s^{4}t^{4}) \end{pmatrix}\ ,
\xi =\lim_{j\to \infty}s_j\int_{0}^{2s_j}\frac{\varphi(t)dt}{(s_j-t)^2+v(s_j)^2}\leq \lim_{j\to \infty}k\int_{0}^{2s_j}\frac{v(s_j)\varphi(t)dt}{(s_j-t)^2+v(s_j)^2}&=\lim_{j\to \infty}k\pi P_{v(s_j)}\varphi(s_j)=0,
T_0= A (y^0\frac{\partial}{\partial y^4}-y^4\frac{\partial}{\partial y^0})- B_i( y^0\frac{\partial}{\partial y^i}+y^i\frac{\partial}{\partial y^0})- C_j (y^4\frac{\partial}{\partial y^j}+y^j\frac{\partial}{\partial y^4})+ D_p \epsilon_{pqr}y^q\frac{\partial}{\partial y^r}.
\overline{q}_{ij}^2 \overline{q}_{jk}^2+\overline{q}_{ij}^2\overline{q}_{ik}^2+\overline{q}_{jk}^2 \overline{q}_{ik}^2 = 2 \overline{q}_{ij}\overline{q}_{ik}\overline{q}_{jk}(\overline{q}_{ij}+ \overline{q}_{jk}+\overline{q}_{jk})
\mathbf{G}^{-1}_{N}\approx \sum_{n=0}^{N-1}\left( \mathbf{I}_K - \mathbf{\Theta}\mathbf{G}\right)^n \mathbf{\Theta}& = \prod_{l=0}^{L-1}\left[ \mathbf{I}_K + \left( \mathbf{I}_K - \mathbf{\Theta}\mathbf{G}\right)^{2^l}\right] \mathbf{\Theta},
I_{m-1}(t) = \int_0^\infty \int_0^{\xi t}\left(1- e^{-w}\right)^{m-1}dw \xi^{-m}\pi(d\xi) = \int_0^\infty \left(1- e^{-w}\right)^{m-1}\int_{w/t}^{\infty}\xi^{-m}\pi(d\xi) dw.
{\cal G}^{(A)}_{\alpha_1,\ldots,\alpha_n}(x_1,\ldots,x_n) = \left({A\over\pi}\right)^{(\sum\alpha_i - Q)/b}{F_{\alpha_1,\ldots,\alpha_n}(x_1,\ldots,x_n)\over\Gamma((Q-\sum\alpha_i)/b)}\,.
\begin{array}[c]{rl}\Delta(s)= p(s)\left[\sigma(s,\bar{X}^{t,x;\bar{u}}(s),\bar{Y}^{t,x;\bar{u}}(s),\bar{Z}^{t,x;\bar{u}}(s)+\Delta(s),u)\right. &\left.-\sigma(s,\bar{X}^{t,x;\bar{u}}(s),\bar{Y}^{t,x;\bar{u}}(s),\bar{Z}^{t,x;\bar{u}}(s),\bar{u}(s))\right],\;s\in\lbrack t,T].\end{array}
\alpha \xi^{m}+\alpha \eta^{m}+q_{m}(\xi,\eta)=(\alpha \xi^{m}+\alpha \eta^{m})\left(1+\frac{q_{m}(\xi,\eta)}{\alpha \xi^{m}+\alpha \eta^{m}}\right )\geq \frac{\alpha}{2}( \xi^{m}+ \eta^{m})>c|\eta|.
\begin{aligned}\mathbf{J}(ab-\overline{ab},c,d)+(-1)^{|a||c|+|b||d|}\mathbf{J}(cd-\overline{cd},a,b) &=(-1)^{|a||b|+|a||c|+|a||d|+|b||c|+|b||d|+|c||d|}\mathbf{J}(c,ba,d)+(-1)^{|a||b|+|a||d|+|b||c|+|c||d|}\mathbf{J}(a,dc,b).\end{aligned}
\hspace{15mm}\times\prod_{i = 1}^{p}v_{i}^{a}\prod_{i = 1}^{p - 1}(v_{i}-1)^{b - 1}(v_{i}- z)^{c}\prod_{i = p - k + 1}^{p}(1 - v_{i})^{b - 1}(z -v_{i})^{c}
\Phi(\mbox{sign}(f_1(\mathbf{p})),\dots, \mbox{sign}(f_m(\mathbf{p}))) &=\Phi'\big(f_1(\mathbf{p})<0, -f_1(\mathbf{p})<0, \cdots, f_m(\mathbf{p})<0, -f_m(\mathbf{p})<0\big).
\log\dfrac{|\det D_1(z)|.||f(z)||}{\prod_{i=1}^{q}|(f,\tilde a_i)(z)|}\le\sum_{i=1}^{q}\log^+\biggl(\dfrac{|\det D_i(z)|}{\prod_{j=1,j\ne i}^{q}|(f,\tilde a_j)(z)|}\biggl)+\log^+ \Psi (z)
S^{K\textnormal{th}}\left.\right._\circ \left.\right. P_j^{-1}q^{Q_j \partial_{Q_j}}= \mathcal{A}_j \left.\right._\circ \left.\right. S^{K\textnormal{th}}P_j^{-1}q^{Q_j \partial_{Q_j}}\left.\right._\circ \left.\right. T^{K\textnormal{th}}= T^{K\textnormal{th}}\left.\right._\circ \left.\right. \mathcal{A}_j
\noindent \langle \psi, \varphi_0(0;\widetilde \eta,\tau_0)\rangle = \displaystyle \lim_{k\rightarrow\infty}\langle \psi,\varphi_{\varepsilon_{j_k}}(0;\widehat \eta_{\varepsilon_{j_k}},\tau_{\varepsilon_{j_k}})\rangle \mbox{and}r\|\widetilde \eta\| \leqslant \displaystyle \liminf_{k \to \infty}r\|\widehat \eta_{\varepsilon_{j_k}}\|.
P_1(\xi)\coloneqq\min_{\zeta\in\mathbb{R}}\left\{\gamma'|\zeta|+\frac{1}{2}(\zeta-\xi)^2\right\}=\begin{cases}\gamma'|\xi|-\frac{1}{2}\gamma'^2, |\xi|>\gamma', \frac{1}{2}\xi^2, |\xi|\le\gamma',\end{cases}
c = \frac{2}{K}\frac{1}{K-1}+ \frac{K-2}{K}\frac{2}{K-1}\frac{1}{K-2}+ \frac{K-2}{K}\frac{K-3}{K-1}\frac{2}{K-2}\frac{1}{K-3}+ \ldots = \frac{2}{K}\frac{1}{K-1}\sum_{i=1}^{K-1}1 = \frac{2}{K}.
H_{n}^{(1)}(z) = \frac{2^{n}(n-1)!}{\pi iz^{n}}\left(1+\mathcal{O}(\frac{1}{n})\right),n \rightarrow +\infty, H_{n}^{(1)\prime}(z) = -\frac{2^{n}n!}{\pi iz^{n+1}}\left(1+\mathcal{O}(\frac{1}{n})\right),n \rightarrow +\infty,
\alpha_{j_1,\ell_1}=\sigma_{j_1}^{2\ell_1}\int_{\mathbb{R}^{2j_1}}D^{\ell_1}f\bigg(\sum_{s=1}^{j_1-1}t_se_s +i\sum_{s=1}^{j_1}t_s^{'}e_s +t_{j_1}e_{j_1}+\zeta\bigg)(e_{j_1},\dots,e_{j_1}) &\times d(\gamma_{\sigma_1}\otimes\gamma_{\sigma_1})(t_1,t_1^{'})\dots d(\gamma_{\sigma_{j_1}}\otimes\gamma_{\sigma_{j_1}})(t_{j_1},t_{j_1}^{'}),
\mathcal{F}_+^{\,\leqslant N}:=\;\bigoplus_{L=0}^N\;\Bigg(\bigoplus_{\substack{n+m=L n\leqslant N_1,\,m\leqslant N_2}}\big( \mathfrak{h}^{(1)}\big)^{\otimes_{\operatorname{sym}}n}\otimes \big(\mathfrak{h}^{(2)}\big)^{\otimes_{\operatorname{sym}}m}\Bigg).
\frac{d^{3}y}{dx^{3}}(J(x,y))+\frac{d^{2}y}{dx^{2}}(2\frac{dJ}{dx}+\frac{\partial J}{\partial x})+\frac{dy}{dx}(\frac{d^{2}J}{dx^{2}}+\frac{d}{dx}(\frac{\partial J}{\partial x}))+F(x,y,\frac{dy}{dx})=0
W(v) =\sum_{z=1}^{2Z-1}W_{z}(v)+W_{0}(v) &=\sum_{z=1}^{2Z-1}V_{z}(v)+\O\left(\frac{vN^{2\gamma-2}}{d\log^7N}\sum_{P<p\leq 2P}(\log p)\sum_{z=1}^{2Z-1}\theta_{z}(p^c)\right)+\O\left(\frac{N^{2\gamma-1}}{d\log^6N}\right)
\int_{-\infty}^{\infty}\delta^{\prime}(x)\varphi(x)dx=\delta(x)\varphi(x)|_{-\infty}^{\infty}-\int_{-\infty}^{\infty}\delta(x)\varphi^{\prime}(x)dx=-\int_{-\infty}^{\infty}\delta(x)\varphi^{\prime}(x)dx=-\varphi^{\prime}(0).
F(\mathbf{\gamma_{6k-1}},\mathbf{\gamma_{6k+1}},\mathbf{\gamma_{6k+2}}):=(\gamma_0,\gamma_1,\ldots,\gamma_{6k-2},\mathbf{\gamma_{6k-1}},\gamma_{6k},\mathbf{\gamma_{6k+1}},\mathbf{\gamma_{6k+2}}).
\Big( T \,{}_{{}_{G_2}}U^{L}_{\eta}\, T^{-1}\Big) \Big( C \mapsto f^C \Big) = \big( T \,{}_{{}_{G_2}}U^{L}_{\eta}\big) \big(f\big) = T \big( \sqrt{\lambda(\eta)}R_\eta f \big) = \Big( \, C \mapsto \sqrt{\lambda(\eta)|_{{}_C}}R_\eta f^C \, \Big),
\begin{array}{c}\dfrac{d\varphi}{dt}-\frac{\epsilon_{0}}{2C_{1}}\varphi \left(h^{-1}\right)^{\prime}\left( 1/\varphi^{\beta}\right) =0,~0<\varphi^{-\beta}\left( 0\right) \leq r_{0}.\end{array}
S_{A,3}= (X_0^2 X_1^2+X_0^2 X_2^2+X_1^2 X_2^2)-2 \cdot (X_0^2 X_1 X_2 +X_0 X_1^2 X_2 + X_0 X_1 X_2^2) & - b_2 \cdot (X_0 X_1 X_2) - b_4 \cdot (X_0 X_1+X_0X_2 + X_1X_2) -b_6 \cdot (X_0+X_1+X_2)-b_8.
[\begin{matrix}1&0&0&1&1&0&0&0\\ 1&0&0&0&0&1&1&0\\ 0&1&0&1&0&1&0&0\\ 0&1&0&0&1&0&1&0\\ 0&0&1&0&1&1&0&0\\ 0&0&1&1&0&0&1&0\\ \end{matrix}]
\int_{\Sigma_r}\alpha_{H}(V^2 \nabla \tau)d \Sigma_r = \int_{\Sigma_r}(\alpha_H) \left [ A\nabla Y^0 + (C_i Y^4 + D_p \epsilon_{pqi}Y^q)\nabla Y^i \right] d \Sigma_r + O(r^6) = \int_{\Sigma_r}(\alpha_H) \left [ A\nabla Y^0 + (C_i (1+ \frac{r^2}{2}) + D_p \epsilon_{pqi}Y^q)\nabla Y^i \right] d \Sigma_r + O(r^6)
Q(t) = \det\begin{bmatrix}s_{-r}s_{-r+1}\ldots s_{-r+K-1}1 s_{-r+1}s_{-r+2}\ldots s_{-r+K}t \vdots \vdots \vdots \vdots \vdots s_{-r+K}s_{-r+K+1}\ldots s_{-r+2K-1}t^{K}\end{bmatrix}.
Q(G) = \left({\begin{array}{*{20}c}{Q(F) + (m - 1)\left({\begin{array}{*{20}c}{I_k}{0^T}0 0 \end{array}}\right)}{\left({\frac{{I_k \otimes \textbf{1}_{m - 1}^T}}{0}}\right)}{(I_k \otimes \textbf{1}_{m - 1}|0)}{I_k \otimes (Q(H_1 ) + I_{m - 1})}\end{array}}\right).
f(\eta) = - \frac{1}{\pi}\frac{1}{\sin \eta}\, \sec^{d-3}\bigg(\frac{\eta}{2}\bigg) \, \frac{d}{d\eta}\int_\eta^\alpha \frac{S(\zeta) \sin \zeta\, d\zeta}{\sqrt{\cos \eta - \cos \zeta}}, 0 \leq \eta \leq \alpha.
(Q_1\times G_j)\cap \left(\bigcup_{i=1}^{s-1}(F_i\times Q_2)\cup \bigcup_{i=1}^{j-1}(Q_1\times G_i)\right) = \bigcup_{i=1}^{s-1}F_i\times G_j\cup \left(Q_1\times (\bigcup_{i=1}^{j-1}G_j\cap G_i)\right).
[B,A](t,x,y) =\left(\frac{\partial b_1(1-t,x,y)}{\partial x}+\frac{\partial b_1(1-t,x,y)}{\partial y}\right)\frac{\partial}{\partial x}+\left(\frac{\partial b_2(1-t,x,y)}{\partial x}+\frac{\partial b_2(1-t,x,y)}{\partial y}\right)\frac{\partial}{\partial y}.
\begin{aligned}|f(x)-f(x')| \leqslant C \Big(\frac{\|x-x'\|}{r}\Big)^\beta \Big\{\frac{1}{V(x,x_0,r+d(x,x_0))}\Big(\frac{r}{r+\boldsymbol{\|x-x_0\|}}\Big)^\gamma \qquad+ \frac{1}{V(x',x_0,r+d(x',x_0))}\Big(\frac{r}{r+\boldsymbol{\|x'-x_0\|}}\Big)^\gamma\Big\}; \end{aligned}
\int_{B_{r}(x_0)}|x|^{-\sigma_i}f_{i}(x,u_{1}(x),{}\ldots,u_{L}(x))\varphi_{r}(x) \,dx ={}u_{i}(x_0) + \sum_{j=1}^{k}\int_{\partial B_{r}(x_0)}u_{ij}(x)\frac{\partial}{\partial n}\Big[(-\Delta)^{k-j}\varphi_{r}(x)\Big]\,ds \leq{}u_{i}(x_0).
b_{10}= c_0 + d_1 t + c_1 + d_2 t - c_2 - d_1 + d_2 + 1, b_2 = (c_0 + d_1 t + c_2 - d_2 - 1)/2, b_3 = (-c_0 - d_1 t + c_2 - d_2 - 1)/2,
\sum_{k=1}^n(-1)^k \frac{k!}{k+1}S(n,k) =\sum_{k=1}^n(-1)^k CT\biggl[\frac{n!}{x^n}\frac{(e^x-1)^k}{k+1}\biggl] &=n! CT \biggl[ \frac1{x^n}\sum_{k=1}^\infty(-1)^k\frac{(e^x-1)^k}{k+1}\biggr] &=n! CT \biggl[ \frac1{x^n}\frac{\ln[1+(e^x-1)]- (e^x-1)}{e^x-1}\biggl] &=n! CT \biggl[ \frac1{x^n}\frac{x}{e^x-1}\biggl] * =B_n.
\pi^{x',y'}_{x,x}(f,g) = \P_{xa_n}\left( \frac{S_{r_n}}{a_n}\in [x',y'], \frac{S_j}{a_n}\in [f,g], j \leq r_n, E_{r_n}\right) \geq \P_{xa_n}\left( \frac{S_{r_n}}{a_n}\in [x',y'], \frac{S_j}{a_n}\in [f,g], j \leq r_n \right) - r_n\P(\xi_1 \geq n).
\|w\|_{V'}:= \sup_{0\ne v\in V}\frac{|\langle w,v\rangle_{V',V}|}{\|v\|_V}= \sup_{0\ne v\in V}\frac{|\langle R_V^{-1}w,v\rangle_{V}|}{\|v\|_V}\le \|R_V^{-1}w\|_V,
\sum_{\kappa\le|k|<2\kappa}|k|^{-4}S_k^{\ll}= \sum_{1\le|j|<\kappa^r}|\gamma_j|^2 \sum_{\kappa\le|k|<2\kappa}|k|^{-4}|k-j|^{2\beta}(U_e^2\xi_{kj}^2+U_f^2\upsilon_{kj}^2).
S \, = \, \int d^{8}z \, \bar{\Phi}\Phi \, - \, \left[ \int d^{6}z \, \left( \frac{1}{2}m \Phi^{2}\, + \, \frac{g}{3 !}\Phi * \Phi * \Phi \right) \, + \, h . c . \right] \, .
W_{\mathrm{e f f}}= S \left\{\ln \left[ \frac{S^{N_{c}- N_{f}}\mathrm{d e t}\left( T \right) \left( 1 - \frac{B T^{N_{f}- N_{c}}\bar{B}}{\mathrm{d e t}( T )}\right)^{\rho}}{\Lambda^{3 N_{c}- N_{f}}}\right] + N_{f}- N_{c}\right\},
P_i(G_k)/P_{i+1}(G_k) \cong \begin{cases}C_2 \times C_2 C_2 \times C_2 \times C_2 \text{if $i \equiv_2 1$ and $i \le k+1$,}C_2 C_2 \times C_2 C_2 . \end{cases}
\sum_{i=0}^{\infty}(x_{k})^{i}>\sum_{j=1}^{\infty}\frac{1}{1+j(p_{1}p_{2}\cdot\cdot\cdot p_{k})}>\frac{1}{1+p_{1}p_{2}\cdot\cdot\cdot p_{k}}\sum_{j=1}^{\infty}\frac{1}{j}=\infty
\left( \begin{array}{c c}{{0}}&{{0}}\\{{j}}&{{p}}\\ \end{array}\right) \ , \ \left( \begin{array}{c c}{{j}}&{{p}}\\{{0}}&{{0}}\\ \end{array}\right) \ , \j , p \in Z \ , \ ( j , p ) \neq ( 0 , 0 ) \ .
\alpha_k(a,b,c,d,e) =\frac{(1+2a-b-c-d+2k)(a-e+k)}{1+2a-b-c-d-e+k}begin{align*}1mm] \quad+\frac{(1+a-b-c+k)(1+a-b-d+k)(e+k)}{(1+a-b+2k)(1+2a-b-c-d-e+k)}.
{E}^{x,i}\Big[\int_0^{{\tau}^-_0}f(\xi_t)\mathbf{1}_{(J_t =k)}\,dt\Big] = \sum_{j\in E}c_j\int_{y\in[0,\infty)}\int_{z\in[0,x]}U^+_{i,j}(dy)\hat{U}^+_{k,j}(dz) f(x+y -z).
E[X(t)]=E\Big[\int_{-\infty}^te^{{\bf A}(t-s)}{\bf e}dS(s)\Big] =\lim_{d\rightarrow\infty}\sum_{n=0}^{d}\sum_{i=0}^{r-1}e^{{\bf A}t}\int_{(k-n)T+s_{j-2-i}}^{(k-n)T+s_{j-1-i}}e^{-{\bf A}u}{\bf e}E[dS(u)] &+e^{{\bf A}t}\int_{kT+s_{j-1}}^{kT+s}e^{-{\bf A}u}{\bf e}E[dS(u)].
\sum_{p' \not=p}\sqrt{\Delta}\sigma_k^{\ell p'}(x_k^i) Z_{k+1}^{p'}\ \stackrel{{\cal L}}{=}\ \Big( \Delta \sum_{p' \not=p}\big(\sigma_k^{\ell p'}(x_k^i) \big)^2 \Big)^{1/2}Z, Z \sim{\cal N}( 0; 1).
\nu=0 : (1-\ell_{03})p_{20}+ (1-\ell_{02})p_{30}= -i p_{03}p_{20}+i p_{02}p_{30}=0 \nu=1 : (1+\ell_{03})p_{31}+ (1+\ell_{02})p_{21}= -i p_{12}p_{31}+i p_{13}p_{21}=0 \nu=2 : -(1+\ell_{03})p_{02}- (1-\ell_{02})p_{12}= i p_{12}p_{02}-i p_{02}p_{12}=0 \nu=3 : -(1-\ell_{03})p_{13}- (1+\ell_{02})p_{03}= i p_{03}p_{13}-i p_{13}p_{03}=0 \, ,
\nu\otimes\Omega_\nu^T;\bullet_A;\partial^{\#}= \bullet_{TA};\Omega_{\nu}^T;\partial^{\#}= \bullet_{TA};\partial ' = 1_{TA}\otimes \partial ';\nu \otimes 1_A;\bullet_A = \nu \otimes\Omega_\nu^T;1_A\otimes\partial^{\#};\bullet_A
\left|\int_0^{\infty}\psi_tf(\psi)\,\eta\left(\frac{r}{1-t}\right)\,rdr\right|=\left|\int_0^{\infty}r\psi_t\frac{f(\psi)}{r}\,\eta\left(\frac{r}{1-t}\right)\,rdr\right|=o(1-t).
\mathcal{C}\coloneqq\left\{w\in\Sigma_{P}^{*}\middle|\; \left|w\right|\geq m\,\,\mathrm{and}\,\,\forall0\leq k\leq\left|w\right|-m:\,\theta^{k}\left(\left[w\right]\right)\cap A=\emptyset\right\},
\prod_{\substack{\gcd(j_1,j_2,\ldots,j_{m+1},k)=1 j_1,j_2,\ldots,j_{m+1}<k j_1,j_2,\ldots,j_{m+1}\geq 0; k \geq 1}}\left(\frac{1}{1-y^{j_1+j_2+...+j_{m+1}}z^k}\right)^{\frac{1}{k}}
\left|\frac{\alpha_{n,m}}{\beta_{n,m}^x}\right| =\left|\frac{\alpha_{n,p_j}}{\beta_{n,p_j}^x}\right|\cdot \left|\frac{\alpha_{m,p_j}}{\beta_{m,p_j}^x}\right|^{-1}&\leq 1 \cdot \left(\frac{|\mathrm{det}Df_{m,p_{j}}(0)|\cdot |df_{p_{j}, p_{j-1}}(v_{j-1})|^{x+1}}{|df_{m,p_{j-1}}(v_{j-1})|^{x+1}}\right)^{-1}<\left(D^{2^j}\right)^{-1}=D^{-2^j}
M_h^1 [u_h](x) = f_h(x), x \in \cup_{l=1}^N \{\, d_l \, \}\mathcal{M}_h [u_h](x) =0, x \in \Omega_h \setminus \cup_{l=1}^N \{\, d_l \, \}u_h(x) = g(x), x \in \partial \Omega_h.
\mathrm{{T r}}_{L^{2}}\Bigr ( f e^{- t P}\Bigr ) \equiv \int_{M}d x \sqrt{\mathrm{{d e t}}g}\; f ( x ) \mathrm{{T r}}U ( x , x ; t ) \sim ( 4 \pi t )^{- m / 2}\sum_{l = 0}^{\infty}t^{l / 2}A_{l / 2}( f , P ,{\cal B}) .
\operatorname*{l i m}_{m_{1}\rightarrow 0^{+}}\langle ( \bar{\psi}_{1}\psi_{1}\rangle_{A}|_{h = 0}= - \operatorname*{l i m}_{m_{2}\rightarrow 0^{-}}\langle ( \bar{\psi}_{2}\psi_{2}\rangle_{A}|_{h = 0}
\begin{aligned}\frac1{rN}B_{N,n}^{(r)}(x) -\left(\frac{x}{rN}-\frac 1{N(N+1)}\right)B_{N,n-1}^{(r)}(x) &+\sum_{k=0}^{n-2}\binom{n-1}k\frac{B_{N,n-k}}{n-k}B_{N,k}^{(r)}(x)=0.\end{aligned}