harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\xi^5 -\big(205X+4300X^2+34000X^3+120000X^4+160000X^6\big)\xi^4 -\big(215X+4475X+35000X^3+122000X^4+160000X^5\big)\xi^3 -\big(85X+1750X^2+13525X^3+46500X^4+60000X^5\big)\xi^2 -\big(15X+305X^2+2325X^3+7875X^4+10000X^5\big)\xi -\big(X+20X^2+150X^3+500X^4+625X^5\big)=0,
4\int_{\mathcal{H}}\int_{\mathcal{H}}\psi(\|x-y\|^{2})d\eta_{t}(x)d\eta_{t}(y) = \int_{\mathcal{H}}\int_{\mathcal{H}}\psi(\|x-y\|^{2})d2\tau_{t}(x)d2\tau_{t}(y) &=\psi(0)( 4t^{2}\mu(\mathcal{H})^{2}+2) - 2 \psi(4t^{2}\|v_{\mu}\|^{2})
\left( \Pi (t_{\nu})-\Pi (t_{\lambda})\right) C_{\lambda \mu}^{\nu}=\sum_{\tilde{\lambda}/d^{\prime}/\lambda}\beta^{|\tilde{\lambda}/d/\lambda |}C_{\tilde{\lambda}\mu}^{\nu}-\sum_{\nu /d^{\prime \prime}/\tilde{\nu}}\beta^{|\nu /d^{\prime \prime}/\tilde{\nu}|}\Pi (t_{\tilde{\nu}})C_{\lambda \mu}^{\tilde{\nu}},
\begin{array}[c]{l}\beta_{0}=\left( \alpha_{0}+\alpha_{1}\right) /2, \beta_{1}=\left( \alpha_{1}+\alpha_{2}\right) /(3\ell^{2}), \beta_{2}=\left( \alpha_{2}+\alpha_{3}\right) /(10\ell^{4}).\end{array}
0\, =-g_\beta(x)\cr =-(x-1)\sum_{l=0}^{r-1}x^l\Bigl[(-1)^{k''(1+l)}\binom\beta{k''(1+l)+1}\frac{x^{r(k''(1+l)+1)}-1}{x^r-1}\cr \kern 3cm +(-1)^{k''(1+l)}\binom\beta{k''(1+l)}\frac{x^{rk''(1+l)}-1}{x^r-1}\Bigr]+1\cr =-(x-1)\sum_{l=0}^{r-1}x^l(-1)^{k''(1+l)}\binom{\tau^4-2}{k''(1+l)+1}+1.
\mathcal{B}_{a2n-a}^{b2n-b}-(\mathcal{B}^{-1})_{a2n-a}^{b2n-b}=(q^{\frac{1}{2}}-q^{-\frac{1}{2}})(\mathcal{M}_{a 2n-a}\mathcal{M}^{b2n-b}-\delta_{a}^{b}\delta_{2n-a}^{2n-b})\hbox{on}V_{2n}.
\omega_\alpha( \textit{\textbf{p}}_\lambda, \textit{\textbf{p}}_\mu)=\max\Big\{\frac{s_\lambda}{s_\mu},\frac{s_\mu}{s_\lambda}\Big\}(1+d_\alpha(\textit{\textbf{p}}_\lambda,\textit{\textbf{p}}_\mu))\,,
\mathfrak{X}_i^{+}(u, \lambda+ \frac{\hbar}{2}\alpha_j)=g_{(\lambda+ \frac{\hbar}{2}\alpha_j, \alpha_i)}(u+z_i)= g_{l+a}(u+z_i)=\frac{\vartheta(u+z_i+l+a)}{\vartheta(u+z_i)\vartheta(l+a)}
(h_1, h_2, h_3)^{[(\sigma_1, \sigma_2, \sigma_3), \gamma]}= \left(h_{1'}^{\sigma_{1'}}, h_{2'}^{\sigma_{2'}}, h_{3'}^{\sigma_{3'}}\right), i' := i^{\gamma^{-1}}\; \forall\; i \in \{1,2,3\}
\frac{\partial}{\partial x^{\nu}}\left( \frac{\partial\mathcal{L}_{0}}{\partial P_{\nu\mu}}\right) =-\frac{4\pi}{c}j^{\mu},\qquad\frac{\partial}{\partial x^{\nu}}\left( \frac{\partial\mathcal{L}_{1}}{\partial P_{\nu\mu}}\right) =\frac{4\pi}{c}j^{\mu}
\, \ \{\rm or \ \as}\ \ \, \ \{\cal{L}}_{BI}= \beta^2 \,\Biggl[ \, \ \ 1 - \sqrt{1 + \frac{1}{2 \beta^2}\, ({\bf B}^2 -{\bf E}^2 ) -\frac{1}{16 \beta^4}\, ({\bf E}\cdot{\bf B})^2}\, \ \ \,\Biggr] \, .
\# \mathcal{N}_A(y) > N_A\left(\frac{[T\log y]+\xi}{T}\right) - N_A\left(\frac{[T\log y]+\xi - 1/9}{T}\right) = N_A\left(\log y\right) - N_A\left(\log y - \frac{1}{9T}\right) =\pi(y;Q,A) - \pi(y e^{-1/(9T)};Q,A) &>\pi(y;Q,A) - \pi(y e^{-1/(9C')};Q,A) &\gg_{C'}\pi(y;Q,A),
-g_{2}'(\ell_{*}) = f_{1}'(\ell_{*}) = \frac{g_{1}(r_{*})-f_{1}(\ell_{*})}{r_{*}- \ell_{*}}& = \frac{[-f_{2}(r_{*}) + g_{2}(\ell_{*})]}{r_{*}- \ell_{*}}= -f_{2}'(r_{*}) = g_{1}'(r_{*}).
\Vert g\Vert_{L_{x}^{\infty}L_{\xi}^{2}}=\sup_{x\in{\mathbb{R}^{3}}}|g|_{L_{\xi}^{2}}\,,\quad\Vert g\Vert_{L_{x}^{1}L_{\xi}^{2}}=\int_{{\mathbb{R}^{3}}}|g|_{L_{\xi}^{2}}dx\,.
R_{\le_{alg}(\lambda,\mathbf p)}=\bigoplus_{\substack{(\mu,\mathbf q)\in\Gamma (\mu,\mathbf q)\le_{alg}(\lambda,\mathbf p)}}\mathbb C\xi_{\mu,\mathbf q}\hbox{\and\}R_{<_{alg}(\lambda,\mathbf p)}=\bigoplus_{\substack{(\mu,\mathbf q)\in\Gamma (\mu,\mathbf q)<_{alg}(\lambda,\mathbf p)}}\mathbb C \xi_{\mu,\mathbf q}.
d_{U_1}(v) \le d_{U_i}(v) \le k|M_i| + \frac{k-1}{2}|U_i-M_i| = \frac{k-1}{2}|U_i| + \frac{k+1}{2}|M_i| &< \frac{k-1}{2}\biggl(\frac{n}{r-1}+\delta^{1/5}n\biggr) + \frac{k+1}{2}\cdot\frac{n}{4k^3}&< \frac{k-1/8}{2(r-1)}n,
u^*=\frac{1+b_1}{2}, u_{1}^{*}=\frac{1}{2}\left(b+1-\sqrt{b^{2}-2\left(1+\frac{2 m}{1+a}\right) b+1}\right), u_{2}^{*}=\frac{1}{2}\left(b+1+\sqrt{b^{2}-2\left(1+\frac{2 m}{1+a}\right) b+1}\right).
e^{i\frac{2}{3f}z^{3/2}\theta_\pm^{3/2}}= e^{i\frac{2}{3f}z^{3/2}\mp i \kappa z^{1/2}+i \frac{f\kappa^2}{4}z^{-1/2}}\left(1 + O(z^{-3/2}f^2) \right).
\mu( 1 \otimes e_i \otimes 1 ) = e_i, d_1( 1 \otimes \alpha \otimes 1) = \alpha \otimes e_{t(\alpha)}\otimes 1 - 1 \otimes e_{s(\alpha)}\otimes \alpha, d_{n}(1 \otimes \alpha_1 \cdots \alpha_n \otimes 1) = \alpha_1 \otimes \alpha_2 \cdots \alpha_n \otimes 1 + (-1)^n 1 \otimes \alpha_1 \cdots \alpha_{n-1}\otimes \alpha_n.
H=\frac{b_{11}g_{22}+b_{22}g_{11}-b_{12}g_{21}-b_{21}g_{12}}{2(g_{11}g_{22}-g_{12}g_{21})}=\frac{b_{11}g_{22}+b_{22}g_{11}+2b_{12}g_{12}}{2(g_{11}g_{22}+(g_{12})^2)},
v^{-1}_A\overset{I}{A}= \overset{I}{A}v^{-1}_A, v^{-1}_A \overset{I}{B}= \overset{I}{v}\!^{-1}\overset{I}{B}\overset{I}{A}v^{-1}_A v^{-1}_B \overset{I}{A}= \overset{I}{v}\overset{I}{B}\!^{-1}\overset{I}{A}v^{-1}_B, v^{-1}_B \overset{I}{B}= \overset{I}{B}v^{-1}_B.
Hu_{nk}(r)=(H_{0}+\frac{\hbar}{m_{0}}k\cdot\Pi+\frac{\hbar^{2}k^{2}}{4m_{0}^{2}c^{2}}\nabla V\times p\cdot\overline{\sigma})u_{nk}(r)=E_{n}(k)u_{nk}(r)
\sup_{\theta\in \ell_0[p_n]}r^2(\theta) \le \sup_{\theta\in \ell_0[p_n]}r^2(I^*(\theta),\theta)\le \sigma^2 p_n \log\big(\tfrac{en}{p_n}\big)=O\big(r^2(\ell_0[p_n])\big).
NLP(1,j,0,\neq) = \int_1^T \int B Q^3 B\partial_Z^t \left( (\partial_j^t U^1_0)_{Hi}(\partial_X U^j_{\neq})_{Lo}\right) \,dV\,dt + \int_1^T \int B Q^3 B\partial_Z^t \left( (\partial_j^t U^1_0)_{Lo}(\partial_X U^j_{\neq})_{Hi}\right) \,dV\,dt & = P_{HL}+ P_{LH}.
A !_{\nu}B & \leqslant \{A^{-1}\sharp_{\nu}B^{-1}+ \sum_{k=0}^{n}r_{k}(A^{-1}\sharp_{\frac{m_k}{2^k}}B^{-1}-2 A^{-1}\sharp_{\frac{2m_k+1}{2^{k+1}}}B^{-1}+ A^{-1}\sharp_{\frac{m_k+1}{2^k}}B^{-1})\}^{-1}& \leqslant A \sharp_{\nu}B.
\frac{q_i^{-r}(q_i^{t-r+1}-q_i^{-t+r-1})}{q_i^{t+1}-q_i^{-t-1}}+ \frac{q_i^{t+1-r}(q_i^{r}-q_i^{-r})}{q_i^{t+1}-q_i^{-t-1}}= \frac{q_i^{t-2r+1}-q_i^{-t-1}+q_i^{t+1}-q_i^{t-2r+1}}{q_i^{t+1}-q_i^{-t-1}}=1,
( \xi , \eta ) = \frac{c}{2 \pi}\int ( \xi^{a}\nabla_{a}\nabla_{b}\nabla_{c}\eta^{c}) \, d x^{b}+ \frac{h}{2 \pi}\int ( \xi^{a}D_{a b}\nabla_{c}\eta^{c}) \, d x^{b}- ( \xi \leftrightarrow \eta ) .
- 4 \left( \stackrel{\circ}{P}_{A B C D , a}+ \frac{3}{2}\stackrel{\circ}{P}_{[ C D , a}C_{A B ]}\right) C^{D L}\gamma^{a}\gamma^{0}- 3 \stackrel{\circ}{T}_{[ A B}\delta_{C ]}^{L}\gamma^{a b}= 0 .
c^{S_1}_j = c_j, j \in S_1 = J c^{S_{k}}_j = c^{S_{k-1}}_j - \frac{c^{S_{k-1}}_{\pi_{k-1}}}{w^{S_{k-1}}_{\pi_{k-1}}}\, \left[w^{S_{k-1}}_j - w^{S_{k}}_j\right], j \in S_{k}, 2 \leq k \leq n,
g_{t,u}=\begin{cases}g_0+u\tilde{k}M_0 \setminus B(p_0, 2 t^{1/4}) (1 - \chi_{t}(r) )\phi_{t}^* (g_0+u\tilde{k}) + \chi_{t}(r) t h_{t,u}(1/2) t^{-1/4}< r < 2 t^{-1/4}t h_{t,u}r < (1/2) t^{-1/4}\end{cases}.
\Omega = -{\cal H}\, \mathrm{d}z \wedge \mathrm{d}\bar{z}+ \bar{p}\, \mathrm{d}z \wedge \mathrm{d}\varphi + p \, \mathrm{d}\varphi \wedge \mathrm{d}\bar{z}\quad \mathrm{w i t h}\quad{\cal H}: = p v + \bar{p}\bar{v}-{\cal L}\, .
\sum_{M^h \in M^G}\frac{|x^G \cap M^h|}{|x^G|}\frac{|\{K^g: g \in G, K^g \le M^h \}|}{|K^G|}= |G:M| \frac{|x^G \cap M|}{|x^G|}\frac{|\{K^g: g \in G, K^g \le M \}|}{|K^G|}&= \operatorname{fpr}(x,M^G) \operatorname{fix}(K,M^G).
-\frac{1}{2^{10}}\frac{1}{\pi^4}\big{[}\Omega_5\big{]}^{\frac{1}{2}}\sum_{N=1}^{\infty}\frac{1}{N(N + 3)}C_N^{\frac{3}{2}}(cos{\theta_4^0})\big{[}a^{-2}C_N^{\frac{3}{2}}(cos{\theta_4})\big{]}(a^{-2}L^2)a^3K \frac{1}{\epsilon}+ F. T.
2-6\epsilon+\cdot\cdot\cdot=(\frac{1}{3}-\frac{u}{4}+\cdot\cdot\cdot)(\frac{du}{d\epsilon})^{2}+(\frac{-1}{2}+\frac{u}{3}-\frac{u^{2}}{8}+\cdot\cdot\cdot)\frac{d^{2}u}{d\epsilon^{2}}
0\leq \frac{1}{2}\triangle(|x^{\top}|^{2})=2-2\langle H,H-\xi\rangle+\sum_{i,j,k,l}h_{ij}^{k^{*}}h_{ij}^{l^{*}}(H-\xi)^{k^{*}}(H-\xi)^{l^{*}}, &0=\frac{1}{2}\mathcal{L}(\langle H,\xi\rangle)=\frac{1}{2}(\langle H,\xi\rangle-\sum_{i,j,k,l}h_{ij}^{k^{*}}h_{ij}^{l^{*}}\xi^{k^{*}}(H-\xi)^{l^{*}}),
U_{( 0 , \, r - 1 )}\equiv \frac{1}{2}( Q_{0})_{\alpha_{0}\beta_{0}}( Q_{1})_{\alpha_{1}\beta_{1}}( Q_{2})_{\alpha_{2}\beta_{2}}\cdots ( Q_{r - 1})_{\alpha_{r - 1}\beta_{r - 1}}\epsilon^{\beta_{0}\alpha_{1}}\epsilon^{\beta_{1}\alpha_{2}}\cdots \epsilon^{\beta_{r - 1}\alpha_{0}}.
\|\Delta v_\epsilon\|_{2}^{2}= \mu\gamma_{q}t_\epsilon^{\frac{N(q-2)}{4}-2}\|v_\epsilon\|_{q}^{q}+ \gamma_{4^*}t_\epsilon^{\frac{N(4^*-2)}{4}-2}\|v_\epsilon\|_{4^*}^{4^*}.
\Theta(x+iy,g_t): =y^{3/4}\sum_{h_1,h_2,h_3 \in \mathbb{Z}}e\big(x(h_3^2-4kh_1h_2)\big)e\big(iy(2kt^{-2}h_1^2+2kt^{2}h_2^2+h_3^2)\big) &=\Big(y^{1/4}\sum_{h\in \mathbb{Z}}e\big((x+iy)h^2\big) \Big) \Big(y^{1/2}\sum_{h_1,h_2\in \mathbb{Z}}e\big((-4kxh_1h_2)e(iy(2kt^{-2}h_1^2+2kt^{2}h_2^2)\big) \Big).
\dot{p}_{\mu}\circ \delta q^{\mu}= \frac{1}{2}\dot{q}^{\mu}( \partial_{\alpha}g_{\mu \nu}\delta q^{\alpha}) \dot{q}^{\nu}- \delta q^{\mu}\partial_{\mu}U_{q}+ \frac{1}{2}[ \delta q^{\mu}, \dot{g}_{\mu}]
{\cal U}_{\sigma}({\bf p}) ={\frac{m}{\sqrt{2}}}\left( \begin{matrix}{D^{J}\left( \alpha ({\bf p}) \right) \xi_{\sigma}}\\{D^{J}\left( \alpha^{- 1 \dagger}({\bf p}) \right) \xi_{\sigma}}\\ \end{matrix}\right) \quad ,
(E_1=LK_X^{\frac{1}{2}}\oplus LK_X^{-\frac{1}{2}}\oplus L^{-1}K_X^{\frac{1}{2}}\oplus L^{-1}K_X^{-\frac{1}{2}},\theta_1=\begin{pmatrix}0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0\end{pmatrix}),
d(\phi(\zeta_1,\ldots,\zeta_n) )= \phi^d (\zeta_1,\ldots,\zeta_n) +\sum\limits_{\tau\in \mathcal{S}_n}\gamma_{\tau}\sum\limits_{i=0}^{n-1}\zeta_{\tau(1)}\ldots ,\ldots,d(\zeta_{\tau(j)}),\zeta_{\tau(j+1)}\ldots,\zeta_{\tau(n)}
h_2^2(P) = (v_1+v_2)^{2}+ (v_1+v_3)^{2}+ (v_2+v_3)^{2}+ (v_1+v_4)^{2}+ (v_2+v_4)^{2}+ (v_3+v_4)^{2}- v_1^{2}- v_2^{2}- v_3^{2}- v_4^{2}=(v_1+v_2)^{2}+ (v_1+v_3)^{2}+ (v_2+v_3)^{2}+ 2v_4^2+2v_4(v_1+v_2+v_3) = (v_1+v_2)^{2}+ (v_1+v_3)^{2}+ (v_2+v_3)^{2}+ \tfrac{8}{9}(v_1+v_2+v_3)^{2}.
c'_{\alpha_1}= \max \{c_{\alpha_1+\alpha_2}, c_{\alpha_1}+c_{\alpha_1+\alpha_2}-c_{\alpha_2}\}, c'_{\alpha_1+\alpha_2}= \min \{c_{\alpha_1}, c_{\alpha_2}\}, c'_{\alpha_2}= \max \{c_{\alpha_1+\alpha_2}, c_{\alpha_2}+c_{\alpha_1+\alpha_2}-c_{\alpha_1}\}.
a^{(j)}_{i+1}\beta^{(j)}_{i}:=a^{(j)}_{i+1}\cdot(\beta,-E_i^{(j)*})=a^{(j)}_{i}p^{(j)}_{i}\beta^{(j)}_{i+1}+\Big(\frac{a^{(j)}_{i+1}D^{(j)}_{i}}{p^{(j)}_{i+1}D^{(j)}_{i+1}}-a^{(j)}_{i}p^{(j)}_{i}\Big)\cdot\beta^{\Gamma^{(j)}_i}_{i+1}.
L(z,u_a,D)\equiv\int_0^{\infty}d\hat T\,J(z,u_a,\hat T,D)=L_{02}(z,u_a,D)+g(z,D)G_{Bab}^{1-{D\over 2}}+O\bigl(z^4,G_{Bab}^{2-{D\over 2}}\bigr)
\left(\sigma\circ(\sigma\circ f)\,,\, M\,,\, ((\sigma\circ f)^*h)\circ(f^*h\circ \phi)\,,\,t\right) = \left(f\,,\,M\,,\,(f^*\sigma^*h)\circ f^*h\circ \phi\,,\, t\right) &= \left(f\,,\,M\,,\,f^*(\sigma^*h\circ h)\circ \phi\,,\,t\right) &= \left(f\,,\,M\,,\,f^*(\textnormal{Id}_L)\circ\phi\,,\,t\right), \,\,\,
u_n = \begin{cases}\frac{n (2 j-n+1) (a+c+n-1) (a-c+j-n+1) (-a+c+j-n) (a+c+2 j-n)}{4 (N-2 n+1)^2}begin{align*}1em] -\frac{1}{2}(1-\alpha) j (j+1) (a-c) (a-c+1) (a+c+j-1) (a+c+j) begin{align*}1em] -\frac{1}{2}\alpha j (j+1) (a-c) (a-c+1) (a+c+j-1) (a+c+j) \end{cases}
\begin{array}{ll}f(y):=\alpha~~\operatorname{with}~~y=\sum\limits_{j=1}^r\alpha_jh_j\;\mbox{for all y}\in X_2, g(z^*):=\beta~~\operatorname{with}~~z^*=\sum\limits_{j=1}^r\beta_je_j^*\;\mbox{for all}\;z\in Z^*,\end{array}
\begin{aligned}\frac12\int_{B_{R}^{+}\backslash B_{\rho}^{+}}\nabla\eta^{6}\cdot (ay,b)\omega^{2}dxdy\leq \frac{CR}{R-\rho}\int_{B_{R}^{+}\backslash B_{\rho}^{+}}|\nabla\boldsymbol{w}|^{2}dxdy\leq \frac{CR}{R-\rho}.\end{aligned}
\mathcal{K}_{\mathcal{S}}\coloneqq \left\{(y^{n}_{\mathcal{T}},\tilde{y}^{n}_{\mathcal{T}}) \in \mathcal{Y}^{n}_{\mathcal{T}}\times \mathcal{Y}^{n}_{\mathcal{T}}\ :y^{n}_{\mathcal{S}}=\tilde{y}^{n}_{\mathcal{S}}\ \wedge\y^{n}_{i}\neq \tilde{y}^{n}_{i}, \ \forall i \in \mathcal{S}^{c}\right\}.
\Pi_S^{\perp}(\mathcal{A}-\mathrm{I}) \Pi_S [b_3 \partial_{yyy}+b_2 \partial_{yy}+b_1 \partial_y+b_0]=\varepsilon^2\Pi_S^{\perp}[\partial_x (\beta_1\,\partial_x(\alpha_{1, 1}\,\cdot) )]+o(\varepsilon^2)
a^{2}, b^{2}= \frac{1}{2}\left( \frac{\alpha^{2}+ A^{2}}{l^{4}}+ \frac{1}{L^{2}}\pm \sqrt{\left( \frac{\alpha^{2}+ A^{2}}{l^{4}}+ \frac{1}{L^{2}}\right)^{2}- \frac{4 A^{2}\alpha^{2}}{l^{8}}}\right)
c_{h,i,j}(\lambda,\nu) = \frac{2^{-2i-2h}\Gamma(\frac{2\nu+p'+2}{4})\Gamma(\frac{\lambda+\rho+\nu-\rho'}{2}+i)}{h!i!j!\Gamma(\frac{2\nu+p'+2}{4}-j)\Gamma(\frac{p''}{2}+i)\Gamma(\frac{\lambda+\rho+\nu-\rho'}{2})}
a_1\left(\sum_{k=[m_n]+1}^{[m_{n+1}]}|\hat{g}(k)|^2\right)^{1/2}\leq \frac{1}{2^{[m_{n+1}] -[m_n]}}\sum_{\Theta_n \in \Delta_n}M_1(g_{\Theta_n}, s_n)\leq M_1(g_{{\tilde{\Theta}}_n},s_n).
\begin{aligned}c(x',y,t) ={}\sum_{x\in T^{-1}(x')}\frac{1}{\vert T'(x)\vert}(c(x,y,t)-x\frac{\partial}{\partial y}b(x,y,t) & +h(x)\left[\frac{\partial}{\partial y}(y\alpha(y,t))+\frac{1}{2}x^{2}\frac{\partial^{2}}{\partial y^{2}}\alpha(y,t)-\frac{\partial}{\partial t}\alpha(y,t)\right])\end{aligned}
(\beta\otimes \beta)( X\otimes Y - Y\otimes X, P\otimes P) =(\beta\otimes \beta)( X\otimes Y, P\otimes P) -(\beta\otimes \beta)(Y\otimes X, P\otimes P) &=\beta(X,P)\beta(Y,P)-\beta(Y,P)\beta(X,P) &=0.
\int_0^\tau T'(s) g(s) + T(s) g'(s) \d s ={}\int_0^\tau T'(s) \Bigl( g(0) + \int_0^s g'(u) \d u \Bigr) \d s + \int_0^\tau \Bigl( T(0) + \int_0^s T'(u) \d u \Bigr) g'(s) \d s ={}\int_0^\tau T'(s) \d s \, g(0) + T(0) \int_0^\tau g'(s) \d s + \int_0^\tau \int_0^\tau T'(s) g'(u) \d u \d s.
\Delta E_{n}^{( 1 )}[ \chi ] = \frac{\langle V \rangle ( 2 - n )}{( 4 \pi )^{\frac{n}{2}}\Gamma \left( \frac{n}{2}\right)}\int_{0}^{\infty}( \omega ( k ) - m ) k^{n - 3}d k = \frac{\langle V \rangle m^{n - 1}}{( 4 \pi )^{\frac{n + 1}{2}}}\Gamma \Bigl ( \frac{1 - n}{2}\Bigr )
\frac{2 \mathrm{i}\pi^{2}}{\sqrt{- \lambda \left( k_{1 2}^{2}, k_{2 3}^{2}, k_{3 1}^{2}\right)}}\left\{\mathrm{C l}_{2}\left( 2 \theta_{1 2}\right) + \mathrm{C l}_{2}\left( 2 \theta_{2 3}\right) + \mathrm{C l}_{2}\left( 2 \theta_{3 1}\right) \right\}
T_{j,1}(r_{1},\dots ,r_{j-1})(f) : = \widetilde{S}_{j,\mathbf{r}}(f), T_{j,k}(r_{k}, \dots ,r_{j-1})(f) : = (i\pi d_{k-1}+P_{k-1})T_{j,k-1}(r_{k-1}, \dots ,r_{j-1})(f) &= \prod_{i=1}^{k-1}(i\pi d_{i}+P_{i})\widetilde{S}_{j,\mathbf{r}}(f),
\Omega_{j k}^{( \alpha \beta )}= \frac{2 R_{j}^{( \alpha \beta )}R_{k}^{( \alpha \beta )}- ( R^{( \alpha \beta )})^{2}\delta_{j k}}{( R^{( \alpha \beta )})^{4}}, R^{( \alpha \beta )}\equiv R^{( \alpha )}- R^{( \beta )}.
[P^{ij},P^{jk}] = 2 F^{ijk}, [P^{jk},F^{ijk}] = P^{ik}P^{jk}-P^{jk}P^{ij}+2P^{ik}C^j-2P^{ij}C^k, [P^{kl},F^{ijk}] = P^{ik}P^{jl}-P^{il}P^{jk}, [F^{ijk},F^{jkl}] = F^{jkl}P^{ij}-F^{ikl}\big(P^{jk}+2C^j\big)-F^{ijk}P^{jl}, [F^{ijk},F^{klm}] = F^{ilm}P^{jk}-P^{ik}F^{jlm},
~(A\sharp B)\Big[(A!_\lambda B)^{-1}-(A\nabla_\lambda B)^{-1}\Big](A\sharp B)=A\Big[(A!_\lambda B)^{-1}-(A\nabla_\lambda B)^{-1}\Big]B &=B\Big[(A!_\lambda B)^{-1}-(A\nabla_\lambda B)^{-1}\Big]A=\lambda(1-\lambda)(B-A)(A\nabla_\lambda B)^{-1}(B-A).\qquad
\lim_{m\rightarrow\infty}\int_{a_{0}}^{b_{0}}f(\tau)\,p_{m(p+1)+\ell,0}^{2}(\tau)\,\frac{d\sigma_{m(p+1)+\ell,0}(\tau)}{|P_{m(p+1)+\ell,1}(\tau)|}=\frac{1}{\pi}\int_{a_{0}}^{b_{0}}\frac{f(\tau)\,d\tau}{\sqrt{(b_{0}-\tau)(\tau-a_{0})}},
C_n^\lambda(x) =\frac{\Gamma(n+2\lambda)}{\Gamma(n+1)\Gamma(2\lambda)}{}_2F_1\left(-n, n+2\lambda;\lambda+\frac{1}{2};\frac{1}{2}-\frac{x}{2}\right) &=\frac{\Gamma(n+2\lambda)}{\Gamma(n+1)\Gamma(2\lambda)}{}_2F_1\left(-\frac{n}{2}, \frac{n}{2}+\lambda; \lambda+\frac12; 1-x^2\right)
\imath\, \hbar\,\frac{d \hat{u}^i}{d\tau}=\,\left[\widehat{H}_t,\hat{u}^i\right]=\,\imath\, \hbar\,\left[ \sum^N_{k=1}\,{\beta}^{k}(\hat{u})\hat{p}_{k},\hat{u}^i\right]=\,\imath\, \hbar\,{\beta}^{i}( \hat{u}),i=1,...,N.
S_{p,18}^{E-series}=\frac{1}{2}\bigoplus_{s=1}^{p-1}\{|R_{9,s}\oplus2R_{3,s}|^{2}\ominus4|R_{3,s}|^{2}\oplus\bigoplus_{r\in\{1,5,7\}}|R_{r,s}\oplus R_{18-r,s}|^{2}\},
\lim_{K \to -\infty}\frac{\big(x q^{1-K}/\xi, q^{1-K}/(\beta_1 \xi ), q^{1-K}/(\beta_2 \xi ) ;q\big)_{\infty}}{\big(x q^{-\lambda -K}/ \xi, q^{1-K}/(\alpha_1 \xi ), q^{2-K}/(\alpha_2 \xi ) ;q\big)_{\infty}}= 1.
\{f , g \}_{\Lambda}= \omega^{i j}\left( \frac{\partial f}{\partial x^{i}}\frac{\partial_{l}g}{\partial \theta_{j}}- \frac{\partial_{r}f}{\partial \theta_{i}}\frac{\partial g}{\partial x^{j}}\right) + \frac{\partial_{r}f}{\partial \theta^{i}}( \theta^{k}{\partial \omega^{i j}}/{\partial x^{k}}) \frac{\partial_{l}g}{\partial \theta_{j}}
\left(azq;q\right)_{\infty}\sum_{n=0}^{\infty}\frac{q^{n^{2}}S_{n}\left(aq^{-n};q\right)}{\left(azq;q\right)_{n}}\left(-z\right)^{n}=\left(zq/a;q\right)_{\infty}\sum_{n=0}^{\infty}\frac{q^{n^{2}}S_{n}\left(aq^{-n};q\right)}{\left(zq/a;q\right)_{n}}\left(\frac{z}{a}\right)^{n}.
\begin{array}{c}\partial_{\mu,x}\bar{S}^{f(x)}_{x}= \lim_{d^{\mu}x\rightarrow 0}\frac{\mbox{$\bar{S}^{f(x)}_{x+d^{\mu}x}-\bar{S}^{f(x)}_{x}$}}{\mbox{$d^{\mu}x$}}\partial_{\mu,x,f}\bar{S}^{f(x)}_{x}=\lim_{d^{\mu}f(x)\rightarrow 0}\frac{\mbox{$\bar{S}^{f(x) +d^{\mu}f(x)}_{x}- \bar{S}^{f(x)}_{x}$}}{\mbox{$d^{\mu}f(x)$}}.\end{array}
b_z-c_z=qc_{sz}-\sum_{x\in\mathcal{R}}(q_{x,sz}-\mu_{x,sz})c_x+q\sum_{x\in\mathcal{T}_1}q_{x,sz}c_{sx}+q\sum_{x\in\mathcal{T}}q_{x,sz}\Bigl(qc_x+\!\!\sum_{y\in\mathcal{R}(s,x)}\!\!\mu_{y,x}c_y\Bigr).
B^{(2)}= \Omega\bigg( \ln \frac{\mathsf{2 \cdot Num}\cdot 2d}{\xi}\bigg)^{d}; \ \delta^{(2)}= \bigg( \frac{\xi \cdot \vartheta^{1/2d}}{2d \cdot B^{(2)}\cdot \sqrt{2\cdot \mathsf{Num}}\cdot \mathsf{Coeff}^{2/d}}\bigg)^{2d}; \c_{(2)}= \frac{\mathsf{Num}}{\delta^{(2)}\cdot \sqrt{\xi}}.
e(H) \leq \binom{n}{3}-\binom{n-k}{3}&= \frac{1-(1-\beta)^3}{6}n^3+o(n^3) &< \frac{1-(1-\beta_0)^3}{6}n^3+o(n^3) &= \frac{9\beta_0^3n^3}{2}+o(n^3)< \alpha n^3,
\int_GFd\alpha_\gamma(\mu_n) =\int_G\alpha_{\gamma^{-1}}(F) d\mu_n=\langle\rho(\alpha_{\gamma^{-1}}(F))\xi_n,\xi_n\rangle=\langle \pi(u_\gamma)^*\rho(F)\pi(u_\gamma)\xi_n,\xi_n\rangle &=\langle\rho(F)\pi(u_\gamma)\xi_n,\pi(u_\gamma)\xi_n\rangle.
\operatorname{Pr}\Big(|\langle x,\varphi\rangle|\geq\xi\Big)=\operatorname{Pr}\Big(|Z|\geq\xi/\|\Sigma^{1/2}x\|_2\Big)\geq\operatorname{Pr}\Big(|Z|\geq\xi/\sigma_\mathrm{min}\Big)\geq\frac{\sigma_\mathrm{min}}{\xi}\cdot\frac{1}{\sqrt{2\pi}}e^{-\xi^2/2\sigma_\mathrm{min}^2},
D_{\lambda ,x}P_t^\lambda (x,y) =-\frac{4\lambda (\lambda +1)}{\pi}(xy)^\lambda t\int_0^\pi \frac{(\sin \theta )^{2\lambda -1}(x-y\cos \theta)}{((x-y)^2+t^2+2xy(1-\cos \theta ))^{\lambda +2}}d\theta &=:P_t^{\lambda ,1}(x,y)+P_t^{\lambda ,2}(x,y),x,y, t\in (0,\infty ),
F_R(u;\sigma,\rho) =\frac{u^2}{2\sigma^2\sqrt{1{-}\rho^2}}\sum_{k{=}0}^\infty\frac1{4^k(k!)^2}\frac{u^{4k}}{2k+1}\rho^{2k}{}_1F_1\left(2k+1,2k+2, -\frac{u^2}{2\sigma^2(1-\rho^2)}\right),
x_{i,1,k}= 6m-2i+1, \tfrac{1}{2}S -x_{i,1,k}= 2i, x_{i,2,k}= 2i-1, \tfrac{1}{2}S - x_{i,2,k}= 6m-2i+2, x_{i,3,k}= 3m-2i+k+1, \mbox{and}\tfrac{1}{2}S - x_{i,3,k}= 3m+2i-k.
T= \begin{pmatrix}1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 \zeta_7^3 0 0 0 0 0 0 0 \zeta_7^5 0 0 0 0 0 0 0 \zeta_7^6 \end{pmatrix},
\mbox{var}\left( \#\mathcal{P}_{\Omega}\cap [0, \textbf{x}] \right) = N |[0,\textbf{x}]| - N^2\sum_{i=1}^{N}{\left( |\Omega_i \cap [0,\textbf{x}]| \right)^2}&\leq N |[0,\textbf{x}]| - N|[0,\textbf{x}]|^2.
\gamma(2) =-{\displaystyle\int\limits_{0}^{1}}\frac{(x-1)^{2}(2x^{3}-3x^{2}+2)}{x^{4}-2x^{3}+x^{2}-1}dx \ =2-\frac{1}{\sqrt{5}}\ln\left( \frac{\sqrt{5}-1}{\sqrt{5}+1}\right)-\frac{\pi}{\sqrt{3}}=0.6166095767...,
\begin{aligned}\frac1{rN}y -\left(\frac{x}{rN}-\frac 1{N(N+1)}\right)\frac1n y' &+\sum_{k=0}^{n-2}\binom{n-1}k\frac{B_{N,n-k}}{n-k}\frac{k!}{n!}y^{(n-k)}=0\end{aligned}
r^{L_0}= V(g_3^{-1}\tilde g_1)r_1^{L_0}V(g_3^{-1}\tilde g_1)^*V(g_3^{-1}\tilde g_2)r_2^{L_0}V(g_3^{-1}\tilde g_2)^* V(g_3\tilde g_3).
\Lambda^{\varepsilon,\ell}_{\vec{\mathbb{P}}}(f_{1},f_{2},f_{3}):=\sum_{\vec{P}\in\vec{\mathbb{P}}}\frac{|C^{\varepsilon}_{Q_{\vec{P}}}|}{|I_{\vec{P}}|^{\frac{1}{2}}}|\langle f_{1},\Phi^{1}_{\vec{P}_{1}}\rangle||\langle f_{2},\Phi^{2}_{\vec{P}_{2}}\rangle||\langle f_{3},\Phi^{3,\ell}_{\vec{P}_{3}}\rangle|,
\frac{(Q'(z)-Q'(w))\Im(zw^{-1})}{|z-w|^{2}}\ge\left(\beta-\frac{1}{2\log 2}\right)\frac{\mathrm{Arg}(zw^{-1})\Im(zw^{-1})}{|z-w|^{2}}=\left(\beta-\frac{1}{2\log 2}\right)\frac{\mathrm{Arg}(zw^{-1})\Im(zw^{-1})}{|1-zw^{-1}|^{2}}.
\|v_{n,T}(0)-\phi_n\|_{\dot{H}^1}=\left\|[e^{-itT\Delta}\chi_nP_ne^{iT\Delta}-1]\phi\right\|_{\dot{H}^1}=\|[\chi_nP_n-1]e^{iT\Delta}\phi\|_{\dot{H}^1}\to0
\mathcal{A}_{th}(0,\infty )=\mathcal{N}=\mathcal{A}_{0}(1,\infty),\,\,\,\,\,\,\,\,\,\,\mathcal{A}_{th}(-\infty ,0)=\mathcal{N}^{\prime}\cap \mathcal{M=A}_{0}(0,1)
(1-\tau_i)\phi(f) = \phi(f-f\circ\tau_i) =\langle M_fv,v\rangle - \langle M_{f\circ\tau_i}v,v\rangle =\langle M_fv,v\rangle - \langle M_f\tau_iv,\tau_iv\rangle =\langle M_fv,v-\tau_iv\rangle + \langle M_f(v-\tau_iv),\tau_iv\rangle.
\frac{1}{[ v \theta / \pi ]_{H_{\scriptscriptstyle 0}}^{\frac{1}{2}}}\frac{1}{[ \det ( \theta / \pi ) ]_{H^{\mu}\phi_{\scriptscriptstyle 0}}^{\frac{1}{2}}}\frac{1}{[ \det ( \theta / 4 \pi ) ]_{H_{\scriptscriptstyle 0}B^{\mu}}^{\frac{1}{2}}},
R:= \begin{pmatrix}0 +i M_3^* -i M_2^* 0 -iM_3^* 0 +iM_1^* 0 +iM_2^* -i M_1^* 0 0 0 0 0 0\end{pmatrix}, S:= \begin{pmatrix}0 +i C_3^* -i C_2^* 0 -iC_3^* 0 +iC_1^* 0 +iC_2^* -i C_1^* 0 0 0 0 0 0\end{pmatrix}
\frac{d}{dt}E(\partial_t^\mu v) + \langle \partial_t^{\mu+1}v, B_\lambda \partial_t^{\mu+1}v \rangle = - \sum_{1 \leq \nu \leq \mu}\binom{\mu}{\nu}\langle \partial_t^{\mu+1}v, \partial_t^\nu B_\lambda \partial_t^{\mu-\nu +1}v \rangle + \langle \partial_t^{\mu+1}v, \partial_t^\mu F_\lambda \rangle ,
\bigl\lbrack A ,\lbrack B,C \bigr\rbrack\rbrack_{\mu_1 ... \mu_{d-2}}+(cycl. perm.)=\partial_{\lbrack\mu_1}\epsilon_{\mu_2 ... \mu_{d-2}\rbrack\nu\lambda\sigma}\xi^\nu(A) \xi^\lambda (B) \xi^\sigma (C)
L ({\bf \dot{R}},{\bf \ddot{R}},{\mathrm{\boldmath \rho}}_{b},{\mathrm{\boldmath \dot{\rho}}}_{b}) = \Gamma^{- 1}F ({\mathrm{\boldmath \alpha}}_{b},{\mathrm{\boldmath \beta}}_{b},{\mathrm{\boldmath \gamma}}) , b = \overline{{1 , N - 1}};
\tilde{\mathbf{P}}_0^*-\breve{\mathbf{P}}_0^i+\nu^* \mathbf{1}\mathbf{1}^\top \tilde{\mathbf{P}}_0^* =\mathbf{0}\tilde{\mathbf{Q}}_0^*-\breve{\mathbf{Q}}_0^i+\nu^*\mathbf{1}\mathbf{1}^\top \tilde{\mathbf{Q}}_0 =\mathbf{0}.
k_H\left(\frac{n}{N},\frac{x}{N}\right)=C_H N^{H-\frac{1}{2}}x^{\frac{1}{2}-H}\int\limits_{\frac{x}{N}}^{\frac{n}{N}}s^{H-\frac{1}{2}}\left(s-\frac{x}{N}\right)^{H-\frac{3}{2}}ds.
\frac{e^{t(|\xi|^2-|\eta|^2-|\xi-\eta|^2)}-1}{|\xi|^2-|\eta|^2-|\xi-\eta|^2}-\frac{e^{t(|\xi|^2-|\eta|^2)}-1}{|\xi|^2-|\eta|^2}= \sum_{k=1}^\infty\frac{t^{k+1}(|\xi|^2-|\eta|^2-|\xi-\eta|^2)^{k}}{(k+1)!}-\sum_{k=1}^\infty\frac{t^{k+1}(|\xi|^2-|\eta|^2)^{k}}{(k+1)!}.
\widehat{Q}_{B}\widehat{\Phi}( 0 ) = \oint \frac{d z}{2 \pi i}j_{B}( z ) \cdot t \xi c e^{- \phi}( 0 ) \otimes \sigma_{3}\sigma_{1}= - i t \left( \frac{1}{2}\xi c \partial c e^{- \phi}+ \eta e^{\phi}\right) ( 0 ) \otimes \sigma_{2},
k_{j}^{2+2\alpha}\left(\frac{|A_{j}|}{|A_{j}\setminus A_{j+1}|}\right)^{2}\leq \left(\frac{\gamma \rho}{\beta}\right)^2 \frac{1}{|A_{j}\setminus A_{j+1}|}\iint_{A_{j}\setminus A_{j+1}}|\nabla (n-\mu_{+}+k_{j})_{+}|^2 \,dx\,dt &\leq C_1 (\alpha, \theta, \gamma, \beta)k_{j}^{2+2\alpha}\frac{|\Omega_{\tau}|}{|A_j \setminus A_{j+1}|}.
e(H)\leq\begin{cases}\left(1-\frac{1}{c}\right)\frac{v(H)^2}{2}v(H) \leq n\sqrt{\frac{c}{c-1}}\,, \frac{n^2}{2}n\sqrt{\frac{c}{c-1}}< v(H) \leq \frac{5n}{4}\,, \frac{n}{2}\left(v(H)-\frac{n}{4}\right) \frac{5n}{4}< v(H)\,.\end{cases}
M_2 = T^2 -2D \le \left(1+ \alpha_1 t+ \alpha_2 t^2 + \alpha_3 t^3 \right)^2 - 2 \left( \beta_1 t +\beta_2 t^2 + \beta_3 t^3 + \beta_4 t^4 \right) \le 1+ \gamma_1 t + \gamma_2 t^2 + \gamma_3 t^3+ \gamma_4 t^4 + \gamma_5 t^5+ \gamma_6 t^6 \, ,
\tilde{S}_{Q1f}^{pot}(B_i ) = g\left[4\sum_{j<k}\frac{1}{N^2}\mbox{Tr}\, B_j^2 \mbox{Tr}\, B_k^2 \left( \sum_{i}U_{ij}^2 U_{ik}^2 \right)+\sum_{j}\frac{1}{N}\mbox{Tr}\, B_j^4 \left( \sum_{i}U_{ij}^4 \right) \right]
\check{H}_{k-1,x}=-\displaystyle\frac{1}{w}\sum_{l=0}^{k-1}\check{H}_{l-1,x}\check{G}_{k-1-l}-\sum_{l=0}^{k+1}\left(2\check{H}_{l-1}\check{G}_{k-l}+\frac{2v}{w}\check{F}_{l-1}\check{H}_{k-l}\right),
\vartheta_{m}(w ) = \sum_{\gamma \in M_{k}}\mathcal{Z}^{m+ \gamma}t^{\varphi(m+\gamma)}= \sum_{\gamma \in M_{k}}\exp \pi\sqrt{-1}(2\langle w, m+ \gamma\rangle+ \frac{\log t}{2 \pi \sqrt{-1}}Z(m+\gamma, m+\gamma)),
B_\ell\left(\sum_{j \neq 1}f_\ell^{(1,j)}+ (n-2)v, \; \cdots,\; \sum_{j \neq n}f_\ell^{(n,j)}+ (n-2)v\right)=\prod_{i}\mathcal{B}_\ell\left( \sum_{j \neq i}f_\ell^{(i,j)}+(n-2)v\right).