harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
3A^2 =|\vec{B}|^2+|\vec{C}|^2+|\vec{D}|^2+|\vec{E}|^2+F^2+G^2+H^2 A\vec{B}=-\vec{C}\times\vec{D}+H\vec{E}\ \ \ \ \AH=\vec{B}\cdot \vec{E}A\vec{C}=-\vec{D}\times\vec{B}+F\vec{E}\ \ \ \ \AF=\vec{C}\cdot \vec{E}A\vec{D}=-\vec{B}\times\vec{C}+G\vec{E}\ \ \ \ \AG=\vec{D}\cdot \vec{E}A\vec{E}=H\vec{B}+F\vec{C}+G\vec{D}.
\langle (x_0, x_1, x_1^\prime, x_2), \sigma^*(y)\rangle = \langle x_1 - x_1^\prime, y \rangle &= \langle \Gamma_0{x}_1 - \Gamma_1{x}_1^\prime, \Gamma{y}\rangle &= \langle (-\Gamma_0{x}_0, \Gamma_1{x}_1, - \Gamma_1{x}_1^\prime, \Gamma_2{x}_2), \delta \Gamma_1{y}\rangle &= \langle \Gamma({x}_0,{x}_1,{x}_1^\prime,{x}_2), \delta \Gamma_1{y}\rangle,
\displaybreak[0] \sum_{[b,c)\in L_s}\int_Y\left(\sum_{i=b}^{c-1}F(y,i)\right)^2d\nu(y) \le\sum_{k=0}^{s-1}\sum_{l=0}^{2^{s-k}-1}\int_Y\left(\sum_{i=2^kl}^{2^k(l+1)-1}F(y,i)\right)^2d\nu(y) \displaybreak[0] \le\sum_{k=0}^{s-1}\sum_{l=0}^{2^{s-k}-1}\frac{4C2^k}{1-q^{-\delta}}=\frac{4Cs2^s}{1-q^{-\delta}}.
{}_{3}F_{2}( \alpha_{1}, \alpha_{2}, \alpha_{3}; \beta_{1}, \beta_{2}; z ) = \sum_{n = 0}^{\infty}\frac{( \alpha_{1})_{n}( \alpha_{2})_{n}( \alpha_{3})_{n}}{( \beta_{1})_{n}( \beta_{2})_{n}}\frac{z^{n}}{n !}
g_{A B}= \left( \begin{array}{c c c}{{- n^{2}( 1 + 2 A )}}&{{a^{2}B_{, i}}}&{{n A_{y}\nonumber}}\\{{a^{2}B_{, j}}}&{{a^{2}\left[ ( 1 + 2{\cal{R}}) \delta_{i j}+ 2 E_{, i j}\right]}}&{{a^{2}B_{y , i}\nonumber}}\\{{n A_{y}}}&{{a^{2}B_{y , i}}}&{{b^{2}( 1 + 2 A_{y y}) \nonumber}}\\ \end{array}\right) \, .
\Big(\Gamma-\liminf_{h\to\infty}f_h\Big)(u)= \sup_{U\in \mathcal{N}(u)}\Big[\liminf_{h\to\infty}\bigl(\inf\{f_h(v):v\in U\}\bigr)\Big]\,, \Big(\Gamma-\limsup_{h\to\infty}f_h\Big)(u)=\sup_{U\in \mathcal{N}(u)}\Big[\limsup_{h\to\infty}\bigl(\inf\{f_h(v):v\in U\}\bigr)\Big]\,,
r_{i}(j)=C_{4}(j) &=\big[\underbrace{B\log_{2}\left(1+\zeta_{m,4}(j)\right)}_{r_{im}(j)=C_{m4}(j)}-\underbrace{B\log_{2}\left(1+\zeta_{e,4}(j)\right)}_{r_{ie}(j)=C_{e4}(j)}\big]^+.
h_E = \frac{p^2_{\xi}}{2m}+\frac{p^2_\eta}{2m}+\frac{p^2_{\phi}}{2m}\left(\frac{1}{\xi^2}+ \frac{1}{\eta^2}\right) - E(\xi^2 + \eta^2) - 2e^2 -\frac{\hbar^2}{8m}\left(\frac{1}{\xi^2}+\frac{1}{\eta^2}\right)
\Gamma^{\alpha}{}_{\beta}= \frac{( - 1 )^{d ( d - 3 ) / 4}}{d !{\sqrt - \gamma}}\epsilon^{i_{1}. . i_{d}}E_{i_{1}}{}^{a_{1}}E_{i_{2}}{}^{a_{2}}\ldots E_{i_{d}}{}^{a_{d}}\Gamma_{a_{1 . . a{_d}}}.
|W_2| =|T_1\cup(V_2\setminus V_1)\setminus S_2|+|A_3| &\ge|T_1|+|V_2\setminus V_1|-|S_2|+|A_3| &\ge |V_1| +|S_1|-N/2+|(V_2\setminus V_1)| -|S_2|+|A_3|\ge N/4.
\rho\left(M_\alpha(z), M_\beta(w)\right) = \det\left(\mu_{\alpha}(z)\right)^{k+\nu}\det\left(\lambda_\alpha(z)\right)^{-\nu}\det\left(\mu_{\beta}(w)\right)^{k+\nu}\det\left(\lambda_\beta(w)\right)^{-\nu},
\begin{aligned}Q \left( \sqrt{2 \min \left\{\alpha^2 \rho_a, \beta^2 \rho_b \right\}}\right) <P_{abr}& < Q \left( \sqrt{{2 \alpha^2 \rho_a}}\right) + Q\left(\sqrt{{2\beta^2 \rho_b}}\right)\end{aligned}
\partial^{\nu}\Lambda_{\nu \mu}= \varepsilon_{\mu \nu}n^{\nu}\bar{c}^{a}\frac{\delta S}{\delta \phi^{a}}- A_{\mu}^{a}\frac{\delta S}{\delta c^{a}}+ \partial_{\mu}\bar{c}^{a}\frac{\delta S}{\delta b^{a}}+ \mathrm{t o t a l d e r i v .}\quad .
\min_y~ \left\{h_{\nu}(y)+\frac{1}{2t}\|y-x\|^2\right\}=\min_y\min_z~ \left\{h(z)+\frac{1}{2\nu}\|z-y\|^2+\frac{1}{2t}\|y-x\|^2\right\}&=\min_z \left\{h(z)+\frac{1}{2(t+\nu)}\left\|z-x\right\|^2\right\},
h_2=\, a_1^2+a_1^6+b+a_1^4 b+a_1^2 b^2+a_1^6 b^2+b^3+a_1^4 b^3+a_1^2 b^4+a_1^4b^5+a_1^2 b^6+a_1^4 b^7+b^9 &+b^{11}+(a_1^4 +a_1^8 +a_1^4 b^4 )k+(a_1^2 +a_1^6 +a_1^4 b +a_1^8 b +a_1^6 b^4+a_1^4 b^5 +a_1^2 b^8 )k^2\cr +(a_1^4 +a_1^8 +a_1^4 b^2 +a_1^8 b^2 +a_1^4 b^4 +a_1^4 b^6 )k^3+(a_1^6+a_1^8 b^3 +a_1^6 b^4 )k^4+(a_1^8 +a_1^8 b^2 )k^5.
c_{\alpha,\beta}^{\kappa,j}=(-1)^p\frac{\varepsilon_0\varepsilon_1\cdots \varepsilon_{p-1}}{\varepsilon_0\varepsilon_1\cdots\varepsilon_{q-j-1}}\binom{p+j}{j}, j=0,\ldots, q \textrm{and}\kappa=s_\alpha, c_{\alpha,\beta}^{\kappa,j}=(-1)^j\varepsilon_{p-j}\varepsilon_{p-j+1}\cdots\varepsilon_{p-1}\binom{q+j}{j}, j=0,\ldots, p \textrm{and}\kappa=e.
\Phi_{1}^{( 1 )}= P_{e}\; , \; \; \Phi_{2}^{( 1 )}= P_{\chi}\; , \; \; \Phi_{3}^{( 1 )}= P_{\kappa}\; , \; \; \Phi_{4 a}^{( 1 )}= P_{a}+ \imath \psi_{a}\; .
S_{6}^{(m)}({{p}^{r+1}}) \equiv ({{C}_{1}^{(m)}}+{{C}_{5}^{(m)}})S_{6}^{(1)}({{p}^{r}})+({{C}_{2}^{(m)}}+{{C}_{4}^{(m)}})S_{6}^{(2)}({{p}^{r}}) +C_{3}^{(m)}S_{6}^{(3)}(p^r) \pmod{p^{r+1}}.
\mathbb{E}^{n+1}=\begin{cases}\mathbb{R}^{n+1}_k , \textrm{if}\mathbb{M}^n_k(c)=\mathbb{S}^n_k, c=+1, \mathbb{R}^{n+1}_{k+1}, \textrm{if}\mathbb{M}_k^n(c)=\mathbb{H}^n_k,c=-1,\end{cases}
\alpha_{w}^{-1}(t_\lambda(z)) = \alpha_{w}^{-1}(z^{d(\lambda)}s_{[\lambda]})= z^{d(\lambda)}\alpha_{w^{-1}}(s_{[\lambda]}) &= z^{d(\lambda)}w^{d(\lambda)}s_{[\lambda]}=(zw)^{d(\lambda)}s_{[\lambda]}=t_\lambda(zw)=t_\lambda(z\cdot w).
\int_\Omega-\det(DF )\psi\,dx = -\int_\Omega \left[( F_1)_{x_1}(F_2)_{x_2}- (F_1)_{x_2}(F_2)_{x_1}\right]\psi \,dx &=\int_\Omega \left[ F_1(F_2)_{x_2}\psi_{x_1}\,dx -F_1( F_2)_{x_1}\psi_{x_2}\right]\,dx
\left(\frac{b-a}{a+b\omega}\right)=\left(\frac{-1}{a+b\omega}\right) \left(\frac{a+b\omega}{a-b}\right)=\left(\frac{-1}{a+b\omega}\right) \left(\frac{a+b\omega+(a-b)\omega}{a-b}\right)=\left(\frac{-1}{a+b\omega}\right) \left(\frac{a}{a-b}\right)\left(\frac{1+\omega}{a-b}\right).
\rho_i(x)=\nabla^2 \eta(x)(E_i,E_i)\leq -\sqrt{a}\sin \sqrt{a}d(x)\frac{-\sqrt{a}\sin \sqrt{a}d_0-\kappa_+\cos \sqrt{a}d_0}{\cos \sqrt{a}d_0-\dfrac{\kappa_+}{\sqrt{a}}\sin \sqrt{a}d_0}.
\tilde{E}_{i}^{2}= E_{i}^{b}R_{2 b}= E_{i}^{b}\hat{E}_{3}^{b}\hat{E}_{3}^{a}\hat{\lambda}^{a}- E_{i}^{b}\hat{\lambda}^{b}\hat{E}_{3}^{a}\hat{E}_{3}^{a}.
|r_{n}|\ = \ \left(A\alpha^{n}-A\alpha^{n-m}\right)|q_{n}|\ \leq \A\alpha^{n}|q_{n}| \ \leq\ \frac{2.4|B||\beta|^{n}}{A\alpha^{n}}\cdot \frac{\alpha^{2m}}{\alpha^{2m}-|\beta|^{m}}\ \leq\ 0.025,
f_{c_{1}}^{\prime}\left( z\right) =\left( \frac{h}{B_{2}}\right)^{\prime}f\left( z\right) +\frac{h}{B_{2}}f^{\prime}\left( z\right) +\left( \frac{B_{1}}{B_{2}}\right)^{\prime}f_{c_{2}}\left( z\right) +\frac{B_{1}}{B_{2}}f_{c_{2}}^{\prime}\left( z\right) .
\lim_{z\rightarrow\lambda,\,\pm Imz>0}\langle R_0(z)f,g\rangle=\pm\frac{\pi i}{2\sqrt{\lambda}}\int_{|\xi|=\sqrt{\lambda}}\widehat{f}(\xi)\overline{\widehat{g}(\xi)}d\sigma+p.v.\int_{\mathbb{R}^n}\frac{\widehat{f}(\xi)\overline{\widehat{g}(\xi)}}{|\xi|^2-\lambda}d\xi
\{X^{I}, \{X^{J}, X^{K}\}\}( - 1 )^{I K}+ \{X^{J}, \{X^{K}, X^{I}\}\}( - 1 )^{J I}+ \{X^{K}, \{X^{I}, X^{J}\}\}( - 1 )^{K J}= 0 ,
\delta_v j^{i_1\ldots i_s}=[D-2+s](v\cdot x) j^{i_1\ldots i_s}+\sum\limits_{p=1}^s(v^{i_p}x_k-x^{i_p}v_k)j^{i_1\ldots i_{p-1}k\; i_{p+1}\ldots i_s}
v_m(x,t)=\frac{1}{2\pi i}\sum_{j=0}^{2}\int_{r-i\infty}^{r+i\infty}e^{st}\frac{\Delta_{j,m}(s)}{\Delta(s)}e^{\lambda_j(s)x}\hat{h}_m(s)ds:=[W_{m}(t)h_m(t)](x)
\int_{\Gamma_1}e^{-m\,|x-y|^{2}/(4\tau)}\,dS(y) = \int_{\Gamma_1}e^{-\,|x-y|^{2}\big/\big[4(\tau/m)\big]}\,dS(y) &\leq \Big(\frac{\tau}{m}\Big)^{(N-1)/2}\,B_1 & \leq \tau^{(N-1)/2}\,B_1.
\phi_1\ast_f\phi_2(a_1\cdots a_n)=\sum\limits_{(\pi^1,\pi^2)\in ANC_n^S}\alpha_{\pi^1,\pi^2}\Big(\prod_{\pi_j\in\pi^1}\phi_1(a_{\pi_j})\Big)\Big(\prod_{\pi_k\in\pi^2}\phi_2(a_{\pi_k})\Big).
\delta{\widetilde\varphi}= - \frac{{\rm e}^{\varphi_0}\,(m-\chi_0\,n)^2-{\rm e}^{-\varphi_0}n^2}{2\,{\widetilde\Delta}_{m,n}}\,\frac{{\widetilde Q}_1}{r^6}~~~,~~~\delta{\widetilde\chi}= \frac{{\rm e}^{-\varphi_0}\,(m-\chi_0\,n)\,n}{{\widetilde\Delta}_{m,n}}\,\frac{{\widetilde Q}_1}{r^6}~~,
\dfrac{\partial}{\partial z^{h}}= \dfrac{\partial\widetilde{z}^k}{\partial z^h}\dfrac{\partial}{\partial\widetilde{z}k}+\dfrac{\partial M_\beta^\alpha}{\partial z^h}u^\beta\dfrac{\partial}{\partial \widetilde{u}^\alpha}, \dfrac{\partial}{\partial u^\beta}= M_\beta^\alpha\dfrac{\partial}{\partial\widetilde{u}^\alpha},
(1-G_u)\sum_{1\leq k\leq n}w_ku_{k1}= -\frac{h'}{2}Au_1-\frac{g'\gamma^1}{2}A+G_{uu}u_1^2w_1+G_{ux_1}w_1u_1 &+\sum_{1\leq k\leq n}w_kG_{ux_k}u_1+\sum_{1\leq k\leq n}w_kG_{x_kx_1},
VA_{s}' =yA_{d-1}'A_{s}'+UA_{s}'=\left(\bigoplus_{i=1}^{d+s}y^iA_{d+s-i}\right)+A_{s}U+yA_{s-1}U+\dots+y^sU &=\bigoplus_{i=1}^{d+s}y^iA_{d+s-i}\oplus UA_{s}
\psi_{\mathrm{r o t}}( x ) = e^{i \gamma_{5}\chi}\psi_{\mathrm{i n}}( x ) = \left( \begin{matrix}{\nu}&{i \mu}\\{i \mu}&{\nu}\\ \end{matrix}\right) \psi_{\mathrm{i n}}( x ) = \left( \begin{matrix}{\nu + i \mu \lambda}\\{i \mu + \lambda \nu}\\ \end{matrix}\right) e^{i k x},
h\circ f_1 =(Ax^d, \; Bx^{dm}(y^d+\sum_{j=2}^da_j(x)y^{d-j})+C(x^d) ) f_2\circ h =(A^d x^d, \; (Bx^my+C(x))^d+\sum_{j=2}^db_j(Ax) (Bx^my+C(x))^{d-j}) .
R_1+R_2+R_3 \leq \max_{\alpha\in[0,1]}\left\{\mathcal{C}\left(\tfrac{\bar{\alpha}P_1}{1+\alpha P_1}\right) + \mathcal{C}(a\alpha P_1)\right\}+ \mathcal{C}(P_2) &= \mathcal{C}(aP_1)+\mathcal{C}(P_2).
n_{\mathrm{M}}:=\begin{cases}n-1 \ (n-1, r_{\mathrm{M}})\in \widehat{I}_{\xi}, n+1 \ (n, r_{\mathrm{M}})\in \widehat{I}_{\xi}. \end{cases}\left(n_{\mathrm{m}}:=\begin{cases}n-1 \ (n-1, r_{\mathrm{m}})\in \widehat{I}_{\xi}, n+1 \ (n, r_{\mathrm{m}})\in \widehat{I}_{\xi}. \end{cases}\right)
\sum_{{{n\geq1},{n\neq{k}}}}\frac{1}{h_{\nu,n}^2-{h_{\nu,k}^2}}=\frac{\nu+2}{2h_{\nu,k}^2}-\frac{h_{\nu,k}^{\nu-2}}{\sqrt{\pi}2^{\nu-1}\Gamma\left(\nu+\frac{1}{2}\right)\mathbf{H}_{\nu}'(h_{\nu,k})},
cos(d_{j},q)=\frac{d_{j}\cdot q}{\|d_{j}\|\|q\|}=\frac{\sum_{i=1}^{N}d_{i,j}q_{i}}{\sqrt{\sum_{i=1}^{N}d_{i,j}^{2}}\sqrt{\sum_{i=1}^{N}q_{i}^{2}}}
A = \frac{i}{2}\{\frac{1}{2}d \ln ( \frac{d \overline{{{g}}}_{2}}{d \overline{{{z}}}_{2}}\frac{d z_{2}}{d g_{2}}) + \frac{( g_{2}d \overline{{{g}}}_{2}- \overline{{{g}}}_{2}d g_{2})}{( 1 - g_{2}\overline{{{g}}}_{2})}\},
(2\pi T_f)^{-1/2}= \frac{1}{\sqrt{2\pi T_g}}\left( 1 - \frac{\varepsilon}{2}\left( \frac{P_h}{P_g}- \frac{\rho_h}{\rho_g}\right) \right) + o(\varepsilon) =\frac{1}{\sqrt{2\pi T_g}}\left( 1- \frac{\varepsilon}{2\rho_g}\int_{-\infty}^{\infty}\left[\frac{u^2}{T_g}-1 \right] h(x,u) \mathrm{d}u \right) + o(\varepsilon).
\|\Delta f\|_{2}^2 = \|\Delta_r f \|_{2}^2 + \bigg\| \sum_{j=1}^n L_j^2f \bigg\|_{2}^2 + 2\mathbf{R}_{n}\sum_{j=1}^n \bigg\| \frac{L_jf}{|x|}\bigg\|_{2}^2 + 2\sum_{j=1}^n \bigg\| \partial_r L_jf + \mathbf{H}_n\frac{L_jf}{|x|}\bigg\|_{2}^2
\begin{aligned}b(x',y,t) ={}\sum_{x\in T^{-1}(x')}\frac{1}{\vert T'(x)\vert}(b(x,y,t)+h(x)(\frac{\partial}{\partial y}(yp_{0}(y,t)) & +\frac{1}{2}x^{2}\frac{\partial^{2}}{\partial y^{2}}p_{0}(y,t)-\frac{\partial}{\partial t}p_{0}(y,t)))\end{aligned}
C_2= K^2C_{{\rm pol}, T}^2(1+\varepsilon_0)\frac{(2(1+c_1^2))^{1/2}}{1-c_1}\mathop{}_{\mathbb T}\sum_{n\in\mathbb N}|\eta_n|^2 \ \ +\|X_*^*\|\bigl(1+\sum_{n\geq 2}\xi_n^2\bigr) \mathop{}_{\mathbb T\setminus\sigma}|h_0|.
\{\{\left|a\right|,\left|b\right|\},\left|c\right|\}= \sum_{p\in a\cap b\cap\operatorname{int}S}\{\left|\mathsf{E}_p(a,b)\right|,\left|c\right|\}-\{\left|\mathsf{e}_p(a,b)\right|,\left|c\right|\}&+\frac{1}{2}N(a,b)\{\left|a\right|\left|b\right|,\left|c\right|\}.
P e^{i\oint_{\sigma_z}A}=U(z) P e^{i\oint_{\sigma_z}A^U}U^{-1}(z)= U(z)P e^{i\oint_{\sigma_z}d\sigma^\mu\int_0^1ds s\sigma^\nu U^{-1}(s\sigma)F_{\nu\mu}(s\sigma)U(s\sigma)}U^{-1}(z).
\mathsf{C}_{\Lambda}\ni S\longmapsto\vartheta_{\#}\left( S\right):=\sum_{S^{\prime}\in\mathcal{P}\left( S\right)}\left( -1\right)^{\left\vert S\right\vert -\left\vert S^{\prime}\right\vert}\log\mathfrak{Z}_{q,p}^{\#}\left( S^{\prime}\right)
\Phi^{(2)}_1 = \pi^2\;,\;\;\Phi^{(2)}_2 = \pi_\mu\psi^\mu\;,\;\;\Phi^{(2)}_{3\mu}=\frac{1}{8!}\epsilon_{\mu\nu\rho_2\ldots\rho_9}\pi^\nu\psi^{\rho_2}\ldots\psi^{\rho_9}+\frac{\alpha}{2^8}\pi_\mu\;.
W^{j}_{2}=-\varepsilon\int_{\mathbb{T}^{d}}(\partial_{x_{j}}w^{1}-\partial_{x_{j}}w_{2})\left(\Theta_{x_{j}}(t,x,\mu^{1},Dw^{1})-\Theta_{x_{j}}(t,x,\mu^{2},Dw^{2})\right)\dx,
M_{np}=\begin{cases}\delta_{p,n+1}\{\rm if}\ \zeta_{n+1}=0\ , \sqrt{\mu_{n+1}}\gamma_{n+1}\frac{\langle f_p | 1 \rangle}{(\lambda_p-\lambda_n-1) \langle f_{n+1}| 1\rangle}\{\rm if}\ \zeta_{n+1}\ne 0\ .\end{cases}
\vert\tilde{\mathcal Q}\vert\le C_\epsilon(\vert\tilde{\mathcal B}\vert+\sum_{k=\tilde N+1}^{\tilde N+N^*}\vert T^*_{\tilde B_1}(\tilde{\mathcal B}^*V_0^{(k)})\vert)\quad\mbox{in}\,\,\Omega_{1,\epsilon}.
\Vert[v]_n\Vert_{\ell_{\infty}}:=\max\limits_{0\leq k\leq n}\Big(\max\limits_{0\leq i\leq m}|v_i(k)|\Big) \leq C_{\infty}\Big(\Vert f\Vert_{L_{\infty}(D)}+\Vert p\Vert_{L_{\infty}(0,T)}+\Vert g^n\Vert_{W_2^1(0,T)}+\Vert\Phi\Vert_{L_{\infty}(0,\ell)}\Big)
\begin{aligned}t_0 :=\widetilde t(1-\beta) \in (0,1), t_{i}:=\beta \left (t_{i-1}-\frac{(1-\beta)(\widetilde t-t_{i-1})}{2 \beta}\right)+\widetilde t(1-\beta)=t_{i-1}+(\widetilde t -t_{i-1})(1-\beta)/2, i \in \mathbb{N}.\end{aligned}
\mathcal{J}_2(\varpi, f_1, f_2, f_3)= \sum_{k=1}^3\varpi_k(f_k) &-\frac{1}{2}\int_{\mathbb{T}}\sum_{k=1}^3\rho_{k-1}(u)f_k^2(u)-\left(\sum_{k=1}^3f_k(u)\rho_{k-1}(u)\right)^2du
\mathcal{K}_{S,\#}''(x,k) := \chi_0(x)T(0) \partial_x m_{+}(x,0)e^{ikx}+ \partial_x \chi_+(x) \mathcal{K}'_{+,\#}(x,k) + \partial_x \chi_-(x) \mathcal{K}'_{-,\#}(x,k).
2\mu \left(a^{22}a^{11}- a^{21}a^{12}\right) e_{1||3}^0 + \rho \left( a^{22}a^{11}- a^{21}a^{12}\right)\dot{e}_{1||3}^0 = 2\mu ae_{1||3}^0 + \rho a\dot{e}_{1||3}^0 = 0,
h_{-2}^{\mathcal{L}_{1,2,3,4}}(t)=\sum_{1\leq \beta<\gamma \leq4}h_{0}^{\mathcal{L}_{\beta,\gamma}}(t)\prod_{\alpha\neq\beta,\gamma}h_{-1}^{\mathcal{K}_\alpha}(t)-2\sum_{\beta}h_{1}^{\mathcal{K}_\beta}(t)\prod_{\alpha\neq\beta}h_{-1}^{\mathcal{K}_\alpha}(t).
\int_I F_m^{*n}(x)dx \le l_1\cdots l_d+C_d (2m+1)^{-1}l_1\cdots l_{d-1}&= |R|+ \frac{C_d}{(2m+1)l_d}|R|\le |R|+ \frac{C_d}{2\sqrt{m}}|R|.
\mathcal{L}_F({\gamma^+}|_{[0,s_1]}) =\int_0^{s_1}F(\dot{\gamma}^+)ds=\int_0^{s_1}\left[\sqrt{a_{22}(\gamma^+(s))(\dot{\gamma}^+(s))^2}+b_2(\gamma^+(s))\dot{\gamma}^+(s)\right] ds &=\int_0^{s_1}\left[\frac{m(r_0)}{1-\mu^2 m^2(r_0)}\cdot\mu-\frac{\mu m^2(r_0)}{1-\mu^2m^2(r_0)}\cdot\mu\right]ds=\frac{\mu m(r_0)}{1+\mu m(r_0)}s_1
\left(\left(1+t\right) B_\frac{t}{1+t}:t\geq 0;\,\mathbb{P}_{0\to \mu}^{0,1}\right)\stackrel{\mathcal{L}}{=}\left((1+t)B_{\left(\frac{t}{1+t}-\gamma\right)^+}:t\geq 0;\,\mathbb{P}_{0\to \mu}^{4,1-\gamma}\right).
|\breve{m}|_p\le C \|v\|_\eta^2, \max_{\,0\le k\le n}|\breve{\chi}\circ f^k|=o(n^{1/p})\;a.e., \big| \max_{1 \leq k \leq n}|\breve{\chi}\circ f^k - \breve{\chi}| \big|_p \leq C \|v\|_\eta^2\, n^{1/p}\; .
\|h\|^k\|f\| \int_{S_{k-1,0,T}}\int_{(\mathbb{R}^d)^{k-1}}\Biggl(\prod_{i=1}^{k-1}p_{\eta_n(s_i)-\eta_n(s_{i-1})}(y_{i-1},y_i)\Biggr) &\times\int_0^{T-T/n}\int_{\eta_n(s_k)}^{s_k}\int_{(\mathbb{R}^d)^2}p_{\eta_n(s_k)-\eta_n(s_{k-1})}(y_{k-1},y_k) &\times\big|\partial_{T-u}p_{T-u}(y_k,z)\big| dz dy_k du ds_k \prod_{j=1}^{k-1}dy_j \prod_{i=1}^{k-1}ds_{i}
\frac{1}{\beta W}[\frac{d^{4}W}{dr^{4}}+\frac{2}{r}\frac{d^{3}W}{dr^{3}}-\frac{1}{r^{2}}\frac{d^{2}W}{dr^{2}}+\frac{1}{r^{3}}\frac{dW}{dr}]=-\frac{1}{F}\frac{d^{2}F}{dt^{2}}=\omega^{2}
\max_{N_{\nu_q-1}\leq p<N{\nu_q}}\int_0^{2\pi}\left|\sum_{k=N_{\nu_q-1}}^p C_k^{(\nu_q)}e^{ikx}\right | \mu(x) dx<10\cdot 2^{-2q}+2\pi\cdot\left | C_{m_{q-1}}\right |.
\operatorname{d}\!J(\Omega)[\boldsymbol{V}]= \left. \frac{\operatorname{d}}{\operatorname{d}\varepsilon}J(\Omega_{\varepsilon})\right|_{\varepsilon = 0}= \int_{\Gamma}\left[ \lambda^2 - \left( \nabla u \cdot{\nu}\right)^2\right] V_{n}\operatorname{d}\!x=0,V_{n}:= \boldsymbol{V}\cdot{\nu},
\langle \Omega|T\left(\sin (gL a_{3}\left(x_{\perp}\right)/2) \sin (gL a_{3}\left(y_{\perp}\right)/2)\right)|\Omega\rangle \quad \sim \quad \langle\Omega|T\left[u\left( x_{\perp}\right) u\left( y_{\perp}\right)\right]|\Omega\rangle\ .
(\theta' \otimes{\rm id}\otimes{\rm id})(\varphi(w^{\theta})) =(\theta' \otimes{\rm id}\otimes{\rm id})({\rm id}\otimes \theta^{2}\otimes \theta)(\varphi(w)) &=\lambda^{-1}(\theta^{3}\otimes \theta^{2}\otimes \theta)(w) =\lambda^{-1}(\theta^{2}\otimes \theta \otimes{\rm id})(\lambda w)=w^{\theta},
\frac{1}{N}\sum_{i=1}^{N}M_{t}^{i}\leq\biggl(\frac{1}{N}\sum_{i=1}^{N}{\mathbf{1}}_{\{M_{t}^{i}\geq1\}}\biggr)^{1/2}\biggl(\frac{1}{N}\sum_{i=1}^{N}\bigl(M_{t}^{i}\bigr)^{2}\biggr)^{1/2}\leq T^{1/2}\biggl(\frac{1}{N}\sum_{i=1}^{N}\bigl(M_{T}^{i}\bigr)^{2}\biggr)^{1/2}.
R_{1}=[\begin{matrix}1&10&19\\ 4&13&22\\ 7&16&25\end{matrix}]R_{2}=[\begin{matrix}2&11&20\\ 5&14&23\\ 8&17&26\end{matrix}]R_{3}=[\begin{matrix}3&12&21\\ 6&15&24\\ 9&18&27\end{matrix}]
\mathbf{L}_{\mathrm{ex}}:=\begin{array}{c}\begin{pmatrix}1 0 0 0 1 1 3 0 1 0 0 2 2 4 0 0 1 1 4 5 0\end{pmatrix}\begin{array}{ccccccc}{\scriptstyle i}{\scriptstyle j}{\scriptstyle m}{\scriptstyle \alpha}{\scriptstyle \beta}{\scriptstyle \gamma}{\scriptstyle \delta}\end{array}\end{array}
\nabla_{X_1}(JX_1)=J\nabla_{X_1}X_1=-X_1, \nabla_{X_2}(JX_2)=J\nabla_{X_2}X_2=-X_2, \nabla_{X_1}(JX_2)=J\nabla_{X_1}X_2=0,\nabla_{X_2}(JX_1)=J\nabla_{X_2}X_1=0.
s_1 \varphi_1^2(x) + s_2 \varphi_2^2(x) + s_3 \varphi_3^2(x) = \frac{d}{d t}\Big |_{t=0}u_{\frac{\pi}{3}-s_1t, \frac{\pi}{3}-s_2t, \frac{\pi}{3}-s_3t}.
\begin{cases}C_f := \underset{u \in [-2C_0,2C_0]}{\sup}f'(u), \alpha >\frac{1}{2},\ \kappa > \kappa ' >1, \frac{\alpha}{\mu}+ \frac{\kappa}{p}+\bar{\delta}\leq C_1 \leq \frac{1}{\mu}-\bar{\delta}, \end{cases}
(F,v,w)\mapsto \Phi(F,v,w):= -{(-\partial_1^2-|\partial_1|^{-1}\partial_2^2)}^{-1}{P}\Big(F+v\partial_2Rw+w\partial_2Rv+w\partial_2Rw &+\partial_2\frac{1}{2}R(w+v)^2-(w+v) \partial_1\frac{1}{2}R(w+v)^2)\Big);
X \Psi_{n_{1}n_{2}m}( \mu , \nu , \varphi ; \delta_{1}, \delta_{2}) = - \alpha \frac{n_{1}- n_{2}+ \frac{\delta_{1}- \delta_{2}}{2}}{n + \frac{\delta_{1}+ \delta_{2}}{2}}\, \Psi_{n_{1}n_{2}m}( \mu , \nu , \varphi ; \delta_{1}, \delta_{2}) ,
\mbox{\mathversion{bold}$v$}_{0}(s)=\mbox{\mathversion{bold}$v$}^{R}(s)={}^{t}(\cos(s/R),-\sin (s/R), 0), begin{align*}3mm]\mbox{\mathversion{bold}$b$}={}^{t}(\cos(L/R),-\sin (L,/R), 0).
D_i(G_k) = \begin{cases}\langle [y,x,\overset{i-1}{\ldots},x] \rangle D_{i+1}(G_k) \langle [y,x,\overset{i-1}{\ldots},x], [y,x,\overset{i-2}{\ldots},x,y] \rangle D_{i+1}(G_k) \end{cases}
V_{I I}( \beta , \phi ) = \mu^{D}\sum_{s = 1}^{\infty}g^{s}\phi^{2 s}h ( D , s ) \biggl ( \frac{1}{2^{\frac{D}{2}- s + 2}}\Gamma ( s - \frac{D}{2}) ( \frac{m}{\mu})^{D - 2 s}+ \sum_{n = 1}^{\infty}\biggl ( \frac{m}{\mu^{2}\beta n}\biggr )^{\frac{D}{2}- s}K_{\frac{D}{2}- s}( m n \beta ) \biggr )
\sum_{i\in I}v_{i}^{2}\langle \pi_{Z_{i}}C^{\prime}f,\pi_{Z_{i}}Cf \rangle =\Vert v_{i}(C^{*}\pi_{Z_{i}}C^{\prime})^{\frac{1}{2}}f \Vert^{2}\leq(B+\epsilon^{2})\Vert f\Vert^{2}.
\sum_{S\in\mathcal{R}(G,r)}|E(G\backslash S)|+\left(q+n_1n_2^{r-1}\left(\binom{r}{2}-1\right)\right)(n_2-1) \geq \sum_{j=2}^r|E(V_1V_j)|n_1(n_2-1)n_2^{r-2}&+\sum_{i,j\neq1}|E(V_iV_j)|(n_2-1)n_1n_2^{r-2}&=|E(G)|(n_2-1)n_2^{r-2}n_1.
I_{n,q}^{r}=\frac{1}{\Gamma(1-\alpha)}\int_{0}^{1}(n+1-s)^{-\alpha}\mathrm{d}\binom{s-q+r-1}{r}=\frac{-\alpha}{\Gamma(1-\alpha)}\int_{0}^{1}(n+1-s)^{-\alpha-1}\binom{s-q+r-1}{r}\mathrm{d}s,
D^{\bar{b}_1, \ldots, \bar{b}_n, \bar{a}}_{\varepsilon}:= \sum_{\substack{\bar{v},\bar{v}' \in Q_n}}(\alpha_{\bar{v}}\cdot \alpha_{\bar{v}'}) \cdot \mu_{\bar{1}^k}\left( \left(\varphi^{\bar{v}}\land \varphi^{\bar{v}'}\right)\left(\bar{x}, \bar{b}_1, \ldots, \bar{b}_n, a_{1}, \ldots, a_{n}\right) \right).
-D_{out}= 4\left(2B_1-A_1+\frac{2f}{r^3}\left(1-\frac{3M}{r}\right)(\lambda-1)\right)\left( B_2+\frac{f}{r^3}\left(1-\frac{3M}{r}\right)(\lambda-4) \right) -\frac{f^2}{r^6}\left( 4-\frac{18M}{r}+\frac{12M^2}{r^2}\right)^2 (2\lambda-4).
h_{E}= \frac{p_{\xi}^{2}}{2 m}+ \frac{p_{\eta}^{2}}{2 m}+ \frac{p_{\phi}^{2}}{2 m}\left( \frac{1}{\xi^{2}}+ \frac{1}{\eta^{2}}\right) - E ( \xi^{2}+ \eta^{2}) - 2 e^{2}- \frac{\hbar^{2}}{8 m}\left( \frac{1}{\xi^{2}}+ \frac{1}{\eta^{2}}\right)
\left\{\frac{4n+2k-2}{m}\right\}=\begin{cases}\frac{4n+2k-2}{m}-4, ~2m\leq 2n+k-1<\frac{5m}{2}; \frac{4n+2k-2}{m}-5, ~\frac{5m}{2}\leq 2n+k-1<3m.\end{cases}
\frac{B^{''}}{AB}-\frac{B^{'}}{2AB}\left(\frac{A^{'}}{A}+\frac{B^{'}}{B}\right)+\frac{2}{rA}\left(\frac{B^{'}}{B}-\frac{A^{'}}{A}\right)+\frac{2}{Ar^2}-\frac{2}{r^2}=0 .
P^{(i)}_{|\frac{p}{N_{0}}\rightarrow \infty}(\gamma_{})=(n+m-i)! &\times \frac{m(m-i+1)^{1-i}(\gamma_{}\omega)^{n-i+1}}{(i-1)!(m-i)!(n-i+1)!}+o\left(\left(\frac{p}{N_{0}}\right)^{-(n-i+1)}\right).
\prod_{c \in \mathrm{Art}_{\mathfrak f_0}^{-1}(\tau)}\exp(X(c)) \bmod \overline{\mathbb Q}^\times =\pi^{\zeta_{\mathfrak f_0}(0,\tau)}p_K(\tau,\sum_{\sigma \in G}\zeta_{\mathfrak f_0}(0,\sigma)\sigma).
\phi (x;\sigma )\equiv - \int_\sigma{d\Sigma_\nu (x'')d(\partial )\Delta (x-x'')\cdot}\left[{\Gamma^\nu (\partial '',\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftarrow$}}\over{-\partial''}})\phi (x'')+ (-)^{\left| \phi \right|}\eta^{-1}{S^\nu}^{\dagger}(x'')}\right].
h_{( \zeta_{\bf p}, \zeta_{\bf q})}\zeta_{\bf m}= \zeta_{\bf m}\Biggl ( 1 + i \frac{\Delta ( \zeta_{\bf p}, \zeta_{\bf q})}{R^{2}}+ O \left( \biggl ( \frac{| \mathrm{\boldmath \delta}|}{n}\biggr )^{2}\right) \Biggr ) ,
{\rm Tr}\;SP_-[\vartheta_{a'}^{(1)},P_-][\vartheta_a^{(2)},P_-]\cdots[\vartheta_a^{(2n+1)},P_-] &-{\rm Tr}\;SP_-[\vartheta_{a}^{(1)},P_-][\vartheta_a^{(2)},P_-]\cdots[\vartheta_a^{(2n+1)},P_-] &={\rm Tr}\;SP_-[(\vartheta_{a'}^{(1)}-\vartheta_{a}^{(1)}),P_-][\vartheta_a^{(2)},P_-]\cdots[\vartheta_a^{(2n+1)},P_-].
\left[ 2 \partial_{-}F_{+ i}( x^{+}, \vec{x}) , F_{+ -}( x^{+}, \vec{y}) \right] = - i \partial_{i}^{x}( \Delta_{\perp})^{2}[ \Delta_{2 - 2 \delta}]^{- 1}( x_{\perp}- y_{\perp}) \delta ({x^{-}}- y^{-})
\mbox{\mathversion{bold}$Y$}(r,t) =\mbox{\mathversion{bold}$Y$}(\xi ,t) + \mbox{\mathversion{bold}$Y$}_{\xi}(\xi ,t)(r-\xi)+O((r-\xi)^{2}), begin{align*}3mm]\mbox{\mathversion{bold}$Y$}_{\xi}(r,t) =\mbox{\mathversion{bold}$Y$}_{\xi}(\xi, t) +\mbox{\mathversion{bold}$Y$}_{\xi \xi}(\xi ,t)(r-\xi) +O((r-\xi)^{2}),
D_{- \frac{i}{2}\Lambda}( - ( 1 + i ) \tau ) , \; \; D_{- \frac{i}{2}\Lambda}( ( 1 + i ) \tau ) , \; \; D_{\frac{i}{2}\Lambda - 1}( ( 1 - i ) \tau ) , \; \; D_{\frac{i}{2}\Lambda - 1}( - ( 1 - i ) \tau ) .
(3A) \colon g(t_0:t_1:t_2:t_3) = (\epsilon_3 t_0:t_1:t_2:t_3) (3C) \colon g(t_0:t_1:t_2:t_3) = (\epsilon_3 t_0:\epsilon_3 t_1:t_2:t_3) (3D) \colon g(t_0:t_1:t_2:t_3) = (\epsilon_3 t_0:\epsilon_3^2 t_1:t_2:t_3)
\Phi \equiv \varepsilon_{a_{1}a_{2}a_{3}a_{4}}\varepsilon^{\mu \nu \lambda \sigma}( \partial_{\mu}\varphi_{a_{1}}) ( \partial_{\nu}\varphi_{a_{2}}) ( \partial_{\lambda}\varphi_{a_{3}}) ( \partial_{\sigma}\varphi_{a_{4}}) .
\tau_{a b}= \frac{U_{0}^{7 - p}\Omega_{8 - p}}{( 2 \pi )^{7}g_{\mathrm{Y M}}^{4}}\mathrm{d i a g}\left[ \frac{9 - p}{2}, \frac{5 - p}{2}, \cdots , \frac{5 - p}{2}\right] ,
\sum_{\lambda = 0}^{n - 1}\sum_{k = 1}^{n - \lambda}( - 1 )^{k + \lambda + k \lambda + n k + k ( \vert a_{1}\vert + \dots + \vert a_{\lambda}\vert )}\mu_{n - k + 1}( a_{1}, \dots , a_{\lambda}, m_{k}( a_{\lambda + 1}, \dots , a_{\lambda + k}) , a_{\lambda + k + 1}, \dots , a_{n}) = 0 .
\vert\vert b_1^{(n)}\vert\vert \le \sum_{\ell=j(n)+1}^n \bigg\vert \frac{(\lambda_n-\lambda_{\ell-1})\dots (\lambda_n-\lambda_1)}{\alpha_{1,\ell-1}\dots \alpha_{1,1}}\bigg\vert &\le \sum_{\ell=j(n)+1}^n\frac{1}{w_{1,\ell-1}\dots w_{1,1}}\cdot\bigg\vert\frac{\lambda_n-\lambda_{j(n)}}{\lambda_\ell-\lambda_{j(\ell)}}\bigg\vert\cdot
\begin{cases}\bar u_{tt}= \bar u_{xx}, (x,t)\in(0,1)\times(0,\infty), \bar u(0,t) = 0, t\in (0,\infty) \bar u_x(1,t) = - a \bar u_t(1,t) + \frac{bd}{c}\bar u(1,t), t\in (0,\infty) \bar u(x,0) = \bar u_0(x), \bar u_t(x,0) = \bar u_1(x), x\in(0,1).\end{cases}
N_{1, \varphi; j}^{+}(x) =(\Lambda_0^+)^{-1}\big({M_{0, \varphi}^{+}(x)}-C_{0}^{(j)}\big), N_{1, \varphi; j}^{-}(x) ={M_{0, \varphi}^{-}(x)}+C_{0}^{(j)},
a_n =t^{rn}{_2F_1}\left(-n, n;\frac{1}{2};\frac{1}{2}(1-(a_1/t^r))\right)=t^{rn}T_n(a_1/t^r), b_n = b_0t^{rn}(n+1){_2F_1}\left(-n, n+2;\frac{3}{2};\frac{1}{2}(1-(a_1/t^r))\right) =b_0t^{rn}U_n(a_1/t^r),
\nu [ A ] \; = \; \lim_{R \rightarrow \infty}\; \frac{- i}{2 \pi}\; \int_{0}^{2 \pi}\; d \varphi \; \Big ( \frac{\partial}{\partial \varphi}S ( R , \varphi ) \Big ) \Big ( S ( R , \varphi ) \Big )^{- 1}\; .
\mathbb{P}\left[H(\mu_1,n_1)\le\xi_1,H(\mu_2,n_2)\le\xi_2\right] &=\lim\limits_{T\to\infty}\mathbb{P}\left[G([\mu_1T],n_1)\le \frac{q}{1-q}[\mu_1T]+\xi_1\frac{\sqrt{q}}{1-q}\sqrt{T},G([\mu_2T],n_2)\le \frac{q}{1-q}[\mu_2T]+\xi_2\frac{\sqrt{q}}{1-q}\sqrt{T}\right].
\lim_{n\rightarrow\omega}\|\widetilde{\psi}_{m,n}(x^{(i)}_nx^{(j)}_n)\widetilde{\psi}_{m,n}(1_A)-\widetilde{\psi}_{m,n}(x^{(i)}_n)\widetilde{\psi}_{m,n}(x^{(j)}_n)\|=0.