harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\left( H +{\frac{1}{2}}{\frac{\dot{\Phi}}{\Phi}}\right)^{2}={\frac{1}{1 2}}\left({\frac{\dot{\Phi}}{\Phi}}\right)^{2}+{\frac{8 \pi}{3}}{\frac{\rho}{\Phi}}-{\frac{k}{a^{2}}}\, ,
L(\boldsymbol{\xi};\vartheta )\geqslant \check{c}_*(\vert \boldsymbol{\xi}\vert^2+1)\mathbf{1}_n,\boldsymbol{\xi}\in\mathbb{R}^d,0< \vartheta \leqslant 1,\quad\check{c}_*=\min \lbrace c_*;2c_4\rbrace.
d_z(f^n) =d^{n-1}d_z(f)+\sum_{k=1}^{n-1}d^{n-1-k}d_{\hat f^n(z)}(f) &\le d^{n-1}(\frac{d}{2}-1)(1+\sum_{k=1}^{n-1}d^{-k}) &<\frac{d^n}{2},
\beta_i'c_k \quad =\frac{\beta_i'(I_d-P_{w_1,\ldots,w_{k-1}})w_k}{\|(I_d-P_{w_1,\ldots,w_{k-1}})w_k\|}\quad= \frac{\beta_i'(I_d- P_{c_1,\ldots,c_{k-1}})w_k}{\|(I_d-P_{c_1,\ldots,c_{k-1}})w_k\|}&=\frac{\beta_i'w_k-\sum_{j=1}^{k-1}(\beta_i'c_j)(c_j'w_k)}{\left(w_k'w_k-\sum_{j=1}^{k-1}(c_j'w_k)^2\right)^{1/2}}.
C_{30}(p,u,v) \sim \frac 1\eta \sum_{J \le 0}2^{-J (J+1)/2 - J + J\tilde u}\sum_{L =0}^{-J}\xi_{L+1}(1/2)\, 2^{-L}\, \Gamma\left( J + L - \tilde u, 1\right).
{\frac{\kappa}{4 \pi}}\partial_{z}\Bigl ( \{\xi_{\alpha A}( x ) , \xi_{\beta R}( z ) \}\; R_{\beta \rho}( z ) \Bigr ) = R_{\rho \gamma}^{- 1}( z ) \; P_{( \alpha A ) ( \gamma R )}( x , z ) \quad .
2\tau(p) = n(n-1) + 3\sum_{i,j=1}^n g(e_i,Je_j)^2 &+ \sum_{\alpha=1}^3\sum_{i,j=1}^n \{3g(e_i,J_{\alpha}e_j)^2 +g(e_i,J_{\alpha}Je_i)\cdot g(e_j,J_{\alpha}Je_j) -g(e_i,J_{\alpha}Je_j)^2 \}&+ n^2||H||^2 - ||h||^2 \p\in M.
\frac{\partial \textbf{w}^{H}\textbf{XX}^{H}\textbf{w}}{\partial \textbf{w}^*}= \frac{\partial \textbf{w}^T \textbf{X}^* \textbf{X}^T \textbf{w}^*}{\partial \textbf{w}^*}= \textbf{X}\textbf{X}^H \textbf{w}
\begin{array}{ccc}f(x)_k = \frac{1}{\sqrt{p}}\sum_{j=1}^p\Re\left(\omega_p^{(j-1)(k-1)}\right)x_j - \Im\left(\omega_p^{(j-1)(k-1)}\right) x_{j+p}f(x)_{k+p}= \frac{1}{\sqrt{p}}\sum_{j=1}^p\Im\left(\omega_p^{(j-1)(k-1)}\right)x_j + \Re\left(\omega_p^{(j-1)(k-1)}\right) x_{j+p}\end{array}
\sum_{\substack{nm>N (n,m)=1}}\frac{r(n)^2r(m)\omega(n)\omega'(m)}{\sqrt{m}}&\qquad\leqslant N^{-\alpha}\sum_{\substack{n,m\geq 1 (n,m)=1}}\big(r(n)^2\omega(n)n^{\alpha}\big)\big(r(m)\omega'(m)m^{\alpha-1/2}\big) &\qquad=N^{-\alpha}\prod_p\big(1+r(p)^2\omega(p)p^{\alpha}+r(p)\omega'(p)p^{\alpha-1/2}\big).
S^{( 1 )}= S_{\mathrm{i n v}}^{( 1 )}+ \hat{S}^{( 1 )}, \quad \delta_{\epsilon}^{( 0 )}S_{\mathrm{i n v}}^{( 1 )}= 0 , \quad \delta_{\epsilon}^{( 0 )}\hat{S}^{( 1 )}\approx 0
I(Y,G)=Y \times_{_{Y \times Y}}(G \times Y) = Y \times_{_{Y \times Y}}(Y \times_{_{Y/G}}Y) = Y\times_{_{Y \times Y}}(Y \times Y)\times_{_{Y/G \times Y/G}}(Y/G) &= Y\times_{_{Y/G \times Y/G}}(Y/G) =Y\times_{_{Y/G}}(Y/G) \times_{_{Y/G \times Y/G}}(Y/G)= Y\times_{_{Y/G}}I(Y/G).
\frac{2m\!+\!1}{4(m\!+\!1)}\!\sum_{K_1\ni j=k}^{\nu}\!\sqrt{\Phi(x^{\ell(j+1)})-\Phi(x^{j+1})}\le\!\frac{\sqrt{2}(m\!+\!1)^2b}{\sqrt{a}}\sum_{j=k}^{\nu}\!\Xi_{j+1}+a_1(m\!+\!1)\sum_{j=k-m}^{k-1}\|x^{j+1}-x^j\| \quad\!+\!\frac{a_1(m+1)}{2}\Big[\sum_{j=k}^{\nu}\,\Xi_j\!+\!2ba^{-1}\varphi(\Phi(x^{\ell(k)})\!-\!\Phi^*)\Big].
\frac{d^l}{dx^l}\left(p^{n,k}(x)-\theta_{p}^{n,k}(x)\right)=\mu_{l}^{n,k}I_{l}^{n,k}(x)+\sum_{m=0}^{l-1}r_{m,l}^{n,k}\frac{d^m}{dx^m}\left(p^{n,k}(x)-\theta_{p}^{n,k}(x)\right).
\sum_{k=1}^{n-3}\sum_{\ell=k+1}^{n-2}(\ell-k-1)(n-\ell-1) (p-1)^2p^{n-4}&= (p-1)^2p^{n-4}\sum_{k=1}^{n-3}\sum_{\ell=k+1}^{n-2}(\ell-k-1)(n-\ell-1) &= (p-1)^2p^{n-4}\sum_{k=1}^{n-3}\binom{n-k-1}{3}&= (p-1)^2p^{n-4}\binom{n-1}{4}.
\max_{x\in Q_2(0)}e^{-a|x-y|^2}= \max_{x\in Q_2(0)}\prod_{i=1}^de^{-a|x_i-y_i|^2}=\prod_{i=1}^d\max_{-1\leq x_i\leq 1}e^{-a|x_i-y_i|^2}=\prod_{i=1}^de^{-a|x_i^*-y_i|^2},
\left\{\det \sum_{\nu}\left( 2 - u_{\nu}^{0}- u_{\nu}^{0 \dagger}\right) \right\}^{({- 5 + 1})}\left( \det \left[ - \frac 1 4 \left( u_{\lambda}^{0}- u_{\lambda}^{0 \dagger}\right)^{2}\right] \right)^{4}.
\begin{cases}U_{k_0}(-\eta_{k_0}(t)-\frac{N-1}{c_{k_0}}\log t)\geq U_{k_0}\left(H_0\right)-\frac{\log t}{t^2}-\epsilon, U_{k_0}(-\eta_{k_0}(t)-\frac{N-1}{c_{k_0}}\log t)\leq U_{k_0}\left(H^0\right)+\frac{\log t}{t^2}+\epsilon.\end{cases}
\frac{g_{1}^{2}}{4 \pi m_{0}^{2}}= \alpha \left( \sigma_{1}\right) \, \frac{\mu_{0}}{\sigma_{1}m_{0}}\sqrt{1 - \frac{1}{4}\left( \frac{\mu_{0}}{\sigma_{1}m_{0}}\right)^{2}}= \alpha_{0}+ \frac{\alpha_{1}}{\sqrt{2}}\frac{\mu_{0}}{m_{0}}+ O \left( \left( \frac{\mu_{0}}{m_{0}}\right)^{2}\right) \, ,
{\mathcal{G}}= (T_{i_1i_2}^{\epsilon})^{|x_{i_1i_2}|}\cup (T_{i_1i_3}^{\epsilon})^{|x_{i_1i_3}|}\cup (T_{i_2i_3}^{\epsilon})^{|x_{i_2i_3}|}\cup{(T_I^{\epsilon})}^{|x_I|}
L = \frac{1}{8}T_{( \mu \nu ) \sigma}T^{( \mu \nu ) \sigma}+ \frac{1}{4}T_{( \mu \nu ) \sigma}T^{( \mu \sigma ) \nu}+ \frac{1}{2}T_{( \mu \nu ) \sigma}\epsilon^{\mu \nu \alpha \beta}\partial_{\alpha}h_{\beta}^{\sigma}+ h^{\alpha \beta}\Theta_{\alpha \beta}\ ,
\left( -\frac{d^{2}}{dr^{2}}+\omega^{2}r^{2}+\frac{\mu +A_{\ell,j,m,n}-\frac{1}{4}}{r^{2}}\right) F_{k,\ell,j,m,n}(r)=E_{k,\ell,j,m,n}\F_{k,\ell,j,m,n}(r) \ .
d_k =\dim Span_{\mathbb{R}}R_{k}&\leq \dim Span_{\mathbb{R}}\left\{\prod\limits_{\substack{1\leq i\leq n 1 \leq r\leq n}}{x_i}^{a_i}y_{r}^{b_{r}}| \sum a_{i}=\sum b_{r}=k \right\}&\approx b_{22}k^{2n-2},
Z [ G_{\mu \nu}, B_{\mu \nu}, \Phi , X^{M}] = \int \frac{[ d A_{a}] [ d X^{0}] [ d \chi ]}{[ d A_{a}] [ d X^{0}] [ d \chi ]}{g a u g e \v o l u m e}e^{- S_{g a u g e d}}
\hat{Y}W : = \left[ m + e_{a}^{\mu}\gamma^{a}( p_{\mu}+ \frac{1}{2}i \nabla_{\mu}) + \frac{1}{2}i \kappa e_{a}^{\mu}\gamma^{a}R_{\mu \nu}^{b c}\left\{\frac{\partial}{\partial p_{\nu}}\cdot , \sigma_{b c}\right\}\right] W
\mathrm{T o r}_{q}^{Y}({\cal O}_{S}, \mathrm{T o r}_{0}^{X}({\cal O}_{T},{\cal O}_{Y}) ) = \mathrm{T o r}_{q}^{Y}({\cal O}_{S},{\cal O}_{W}) = \Lambda^{q}{\cal N}_{S / Y}^{\vee}|_{W},
\frac{\Gamma^3(x+\frac 23)}{\Gamma^3(x+1)}=\frac{1}{x+\frac 13 +\frac{\frac{1}{27}}{x+\frac 13+\frac{\frac{1}{54}\frac{2^3}{1}}{x+\frac 13+\frac{\frac{1}{54}\frac{4^3}{3}}{x+\frac 13+\frac{\frac{1}{54}\frac{5^3}{3}}{x+\frac 13+\ddots}}}}}.
N^{I i}\equiv \frac{\gamma^{\tau \tau}\Pi_{\tau}^{i}+ ( \gamma^{\tau \sigma}\mp 1 / \sqrt{- \gamma}) \Pi_{\sigma}^{i}}{\gamma^{\tau \tau}\Pi_{\tau}^{+}+ ( \gamma^{\tau \sigma}\mp 1 / \sqrt{- \gamma}) \Pi_{\sigma}^{+}},
0=\langle \psi, A_n \psi\rangle =\sum_{i\neq j}\langle \psi\,,\, P_{\Omega_i}^{\perp}\psi \rangle +\langle \psi\,\, P_{vac}\psi \rangle+\langle \psi\,,\, P_{\Omega_j}^{\perp}\psi\rangle\geq \langle \psi\,,\, P_{\Omega_j}^{\perp}\psi\rangle
\Delta R = q_1^*q_2[H(p_2)-H(p_1^*)] & \stackrel{(a)}{\leq}q_1^* (1-q_1^*) [H(p_2)-H(p_1^*)] & \stackrel{(b)}{\leq}0.25[H(p_2)-H(p_1^*)] & \stackrel{(c)}{\triangleq}\overline{\Delta R},
n^{-1}_B \left(\frac{y}{|y|}\right)\cdot s =n^{-1}_B \left(\frac{y}{|y|}\right) \cdot n_{B}^{-1}\left( \frac{z-x}{|z-x|}\right)^{\perp}= \alpha n^{-1}_B\left(\frac{x}{|x|}\right) \cdot n_{B}^{-1}\left( \frac{z-x}{|z-x|}\right)^{\perp}= \alpha m(x) \cdot s,
\frac{X_j \cap M}{X_{j+1}\cap M}\cong \frac{X_j}{X_{j+1}}\frac{X_j \cap M}{X_{j+1}\cap M}= 0 \frac{X_j \cup M}{X_{j+1}\cup M}\cong 0 \frac{X_j \cup M}{X_{j+1}\cup M}= \frac{X_j}{X_{j+1}}.
X^{+}( \tau , \sigma_{i}) = X^{+}( 0 ) + \tau , \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, X^{\pm}= \sqrt{\frac{1}{2}}\left( X^{1 0}\pm X^{0}\right)
f_1( x, y ) = x^2 + y^2 + \sqrt{2 + \sqrt{2}}\, x y + \frac{\sqrt{2}- 2}{4}f_2( x, y ) = x^2 + y^2 - \sqrt{2 + \sqrt{2}}\, x y + \frac{\sqrt{2}- 2}{4}f_3( x, y ) = x^2 + y^2 - \sqrt{2 + \sqrt{2}}\, x y - \frac{\sqrt{2}- 2}{4}f_4( x, y ) = x^2 + y^2 + \sqrt{2 + \sqrt{2}}\, x y - \frac{\sqrt{2}- 2}{4}.
0=\left(\sum_{i=1}^3 \alpha_i (k_{t_i}\otimes k_{t_i}) + \sum_{i=4}^5 (\alpha_i k_{\lambda_i}\otimes C_{\theta}k_{\lambda_i}) \right) (f) ={\alpha_1}f(t_1) k_{t_1}+ \sum_{i=4}^5{\alpha_i}k_{\lambda_i}\langle f, C_{\theta}k_{\lambda_i}\rangle .
P^{-}= -{\frac{g^{2}}{2 ( 2 \pi )^{2}}}{\frac{d^{2}}{d z^{2}}}- g^{2}\int_{- L}^{L}d x^{-}T r \left( J^{+}{\frac{1}{D_{-}^{2}}}J^{+}\right) \; .
d(\mu\alpha)_i=d(\mu)_i+d(\alpha)_i=d(\alpha)_i=\max\{d(\lambda)_i,d(\lambda')_i\}-d(\lambda)_i=n-n=0, &d(\mu'\beta)_i=d(\mu')_i+d(\beta)_i=d(\beta)_i=\max\{d(\lambda)_i,d(\lambda')_i\}-d(\lambda')_i=n-n=0.
\wp ( u ) \equiv \wp ( u | 2 \omega_{1}, 2 \omega_{3}) ={\frac{1}{u^{2}}}+ \sum_{m , \, n}{}^{\prime}\left[{\frac{1}{( u - \Omega_{m , \, n})^{2}}}-{\frac{1}{\Omega_{m , \, n}^{2}}}\right] ,
<\tilde{T}_{J,J}\tilde{W}_{J_{1},m_{1}}\prod_{i=2}^{N}\tilde{T}_{J_{i},J_{i}}>= \frac{(J_{1}-m_{1})(J+J_{1}-1)}{J_{1}}<\tilde{W}_{J+J_{1}-1,J+m_{1}}\prod_{i=2}^{N}\tilde{T}_{J_{i},J_{i}}>
{\cal{Z}}_{r s}= P^{\mu}( C \Gamma_{\mu}^{( 1 1 )})_{r s}+ Z_{\bf{O}}^{\mu \nu}( C \Gamma_{\mu \nu}^{( 1 1 )})_{r s}= Z_{\bf{O}}^{[ \mu_{1}\ldots \mu_{5}]}( C \Gamma_{\mu_{1}\ldots \mu_{5}}^{( 1 1 )})_{r s}\, ,
Q\left( t\right) =i\left( -\frac{\dot{\alpha}\left( t\right)}{\sqrt{\hbar}}a^{\dag}+\frac{\overline{\dot{\alpha}\left( t\right)}}{\sqrt{\hbar}}a\right) -\frac{1}{\hbar}\operatorname{Im}\left( \alpha\left(t\right) \overline{\dot{\alpha}\left( t\right)}\right) .
I = \frac{1}{2 \kappa^{2}}\int d^{D}X \sqrt{- g}\left( R - \frac{1}{2}( \nabla \vec{\phi})^{2}- \sum_{I = 1}^{k}\frac{e^{- \vec{\alpha}^{( I )}\vec{\phi}}}{2 ( d_{I}+ 1 ) !}F_{d_{I}+ 1}^{( I ) 2}\right) .
[|D_0|,[|D_0|,[\cdots[|D_0|,T]\cdots]]]\xi = 2^{-m}\sum_{k=0}^m\binom{m}{k}\Big(\frac{2}{\pi}\Big)^k\int_{\mathbb{R}^k_+}\prod_{l=1}^k\frac{\lambda_l^{1/2}D_0^2}{(\lambda_l+D_0^2)^2}\cdot \Lambda^{m+k}(T)\prod_{l=1}^k\frac{1}{\lambda_l+D_0^2}\xi d\lambda_1 d\lambda_2\cdots d\lambda_k.
N^n 2^{(H(\eta)+\eta)N^2}r^{\frac{\eta}{2}n^2}r^{\eta n^2}2^{r(H(d)+d)n^2}m(r)^{(1-\epsilon)({n\choose 2}+\frac{n}{2})}= 2^{S(n)}m(r)^{(1-\epsilon)({n\choose 2}+\frac{n}{2})}.
\left\langle u\right\rangle_{x_{i},R^{M}}^{(l_{i})}=\sup_{x\in R^{M},h>0}\frac{\left\vert D_{x_{i}}^{[l_{i}]}u(x_{1},...,x_{i}+h,...,x_{M})-D_{x_{i}}^{[l_{i}]}u(x)\right\vert}{h^{l_{i}-[l_{i}]}},
V_{m}= \left\{v_{m}(x,y) = \underset{k = 1}{\overset{m}{\sum}}\, \overline{v}_{k}(x)\, \phi_{k}(y),\enspace \mbox{with}\enspace \overline{v}_{k}(x) \in H^{1}_{0}(\Omega_{1D}),\, x \in \Omega_{1D}, \, y \in \omega\, \right \},
\lambda^k: W_7 \longrightarrow \Lambda^k W_8, \begin{array}{ll}\lambda^2(v) := e^0 \wedge v^\flat + (\imath_v \varphi) begin{align*}2mm]\lambda^4(v) := e^0 \wedge (\imath_v \ast_7 \varphi) - v^\flat \wedge \varphi begin{align*}2mm]\lambda^6(v) := \Phi \wedge \lambda^2(v) = 3 \ast \lambda^2(v)\end{array}
B_{5}^{( d - 2 )}= - \frac{4 ( d - 2 ) ( d - 3 ) ( 3 d - 8 ) ( 3 d - 1 0 )}{9 ( d - 4 ) ( d - 6 )^{3}}B_{5}^{( d )}- \frac{6 4 ( 1 5 d - 5 2 )}{9 ( d - 4 )^{4}( d - 6 )^{3}}.
M_{1}(\mathcal{C}(G))= 2n(2n-1)^{2}+ 3n(n-1)^{2}, M_{2}(\mathcal{C}(G))= \frac{2n(2n-1)^{3}+3n(n-1)^{3}}{2}, M_{1}(\mathcal{NC}(G))=66n^{3}
a_1^+ = \sigma^1_1 \mbox{and}a_\xi^+ = \sigma_{1}^3 \cdots \sigma_{\xi-1}^3 \sigma^1_\xi \mbox{for all}2 \leq \xi \leq N \, , & a_1^- = - \sigma_1^2 \mbox{and}a_\xi^- = - \sigma_{1}^3 \cdots \sigma_{\xi-1}^3 \sigma^2_\xi \mbox{for all}2 \leq \xi \leq N \, ,
\begin{aligned}\mathrm{P}_{0}\psi_{j}\left(\left|\eta\right|\right) =\sum_{l=0}^{2}\beta_{jl}\left(\left|\eta\right|\right)E_{l}^{1},\:j=0,1,2, \mathrm{P}_{0}\psi_{3}\left(\left|\eta\right|\right) =\beta_{33}\left(\left|\eta\right|\right)E_{3}^{1}, \mathrm{P}_{0}\psi_{4}\left(\left|\eta\right|\right) =\beta_{44}\left(\left|\eta\right|\right)E_{4}^{1}.\end{aligned}
N^+_{D_C}(u_i):=\{\, u_j\, |\, c_{i,j}=1\,\}N_i:=|N^+_{D_C}(u_i)|u_i N^-_{D_C}(u_i):=\{\, u_j\, |\, c_{j,i}=1\,\}I_i:=|N^-_{D_C}(u_i)|u_i
2^{\mu_*k_*t}\cdot 2^{‐\tilde m_t(s_*+1/q‐1/p_1)}n^{1/q‐1/p_1}= 2^{‐\alpha_*k_*t}\cdot 2^{‐\tilde m_t(1/q‐1/p_0)}n^{1/q‐1/p_0}\stackrel{\eqref{tilde_theta_def}}{=}n^{‐\tilde \theta};
k_{3}(x) = k_2^{l_1(x)}(h(x)) + l_2^{k_1(x)}(h(\sigma_A(x))) x \in X_A, l_{3}(x) = l_2^{l_1(x)}(h(x)) + k_2^{k_1(x)}(h(\sigma_A(x)))x \in X_A.
\P_0(|X_t|\geq r) =\int_{(0,\infty)}\P_0\left(Y\geq (2s)^{-1}r^2\right)\P(S_t\in ds) =\int_{(0,\infty)}\P(S_t\geq (2y)^{-1}r^2)\P_0(Y\in dy)\,,
\textbf{s}_m(t) = \biggl[\Re\left [s_m[t]e^{j2\pi f_0t}e^{j\psi_{1m}}\right],\ldots,\Re\left [s_m[t]e^{j2\pi f_0t}e^{j\psi_{N_sm}}\right] \biggr]^T.
\hat{S}[ \hat{B}] = \int d^{3}x [ - \frac{1}{8 \pi}\epsilon^{\mu \nu \lambda}( \hat{B}_{\mu}* \partial_{\nu}\hat{B}_{\lambda}+ \frac{2 i}{3}\hat{B}_{\mu}* \hat{B}_{\nu}* \hat{B}_{\lambda}) + \frac{1}{2 g^{2}}\hat{B}^{\mu}\hat{B}_{\mu}] + O ( \frac{1}{m}) .
x = x_L \cdot\prescript L{}{}x{}^R\cdot x_R \ell(x) = \ell(x_L) + \ell\left(\prescript L{}{}x{}^R\right) + \ell(x_R), \prescript{-L}{}{}x{}^{-R}= x_L'\cdot x \cdot x_R' \ell\left(\prescript{-L}{}{}x{}^{-R}\right) = \ell(x_L') + \ell\left(x\right) + \ell(x_R').
I_{-n}(r)\geq \frac{1}{2}\int_{1}^{r_{s'}}\rho^{-n}(\rho-1)^{2}\, d\rho\geq \frac{r_{s'}^{-n}}{2}\int_{1}^{r_{s'}}(\rho-1)^{2}\, d\rho= \frac{r_{s'}^{-n}}{6}(1-s')^{3}(r^2 -1)^{3}.
R(j;c_{1}):=\{\theta\in E_{c_{1}}(\theta_{0}): (j+1)( B / \varepsilon)^{2/(2\alpha_{0}+1)}> ||\theta-\theta_{0}||^{2}/ \varepsilon^{2}\geq j ( B / \varepsilon)^{2/(2\alpha_{0}+1)}\}.
E = - E_{n}\, , \; n \geq 1 \; \; \psi_{n , p}^{( - )}= \sqrt{\frac{E_{n}- m}{2 E_{n}}}\frac{e^{i p x_{2}}}{\sqrt{2 \pi}}e^{- \frac{1}{2}{\zeta}^{2}}\left( \begin{array}{c}{{N_{n}H_{n}( \zeta )}}\\{{- \frac{\sqrt{E_{n}^{2}- m^{2}}}{E_{n}- m}N_{n - 1}H_{n - 1}( \zeta )}}\\ \end{array}\right)
g \left( \sum_{j=1}^n \sum_{h \in H_j}h \right) g^{-1}= \sum_{j=1}^n \sum_{h \in H_j}g h g^{-1}= \sum_{j=1}^n \sum_{h \in g H_j g^{-1}}h = \sum_{j=1}^n \sum_{h \in H_j}h
\underset{\{\Delta q\}_{ij}}{}\sum_{i,j}\|\Delta q_{ij}+ \mathbf{m}_{i}(j)\|^2 & \underline{x}_{ij}\leq \Delta q_{ij}\leq \overline{x}_{ij}& \sum_{i = 1}^N \Delta q_{ij}= \mathbf{d}(j).
V_1 = \frac{1}{\sqrt{2}}\Biggl( \begin{matrix}0 1 0 -1 0 0 0 0 0 \end{matrix}\, \Biggr) , V_2 = \frac{1}{\sqrt{2}}\Biggl( \begin{matrix}0 0 1 0 0 0 -1 0 0 \end{matrix}\, \Biggr) .
\frac{\partial F}{\partial x^{1}}= \phi_{1}^{3}+ \sum_{j = 2}^{3}\phi_{j}^{3}f_{1}^{j}, \quad \quad \frac{\partial F}{\partial t}= \phi_{t}^{3}+ \sum_{j = 2}^{3}\phi_{j}^{3}f_{t}^{j}.
d x^{i}d x^{j}\wedge{}^{*}( d x^{0}d x^{l}d x^{m}) = x_{0}^{- ( d - 5 )}\delta_{l m}^{i j}d x^{1}. . . d x^{d}, \delta_{l m}^{i j}= \delta_{l}^{i}\delta_{m}^{j}- \delta_{m}^{i}\delta_{l}^{j},
\tilde{H}=\begin{pmatrix}0_{n \times n}0_n 0_{n \times n}0_n \cdots 0_{n \times n}0_n 0_{n \times n}0_n s_1^\top 0 s_2^\top 0 \cdots s_n^\top 0 0_{1 \times n}0 \end{pmatrix},
q^n-\left(3+2r\left\lceil\frac{\deg d}{2}\right\rceil\right)q^{n/2}-nq^{n/3}-\left(4r^2\left\lceil\frac{\deg d}{2}\right\rceil + 2r\right)n -\left(4r^2\left\lceil\frac{\deg d}{2}\right\rceil + 2r\left\lceil\frac{\deg d}{2}\right\rceil\right) > 0.
\begin{aligned}\mathcal{D}_{\mathcal{N}_{\kappa,\gamma}}(B) \:\:\: \leq \:\:\: \sum_{t=1}^e\mathcal{D}_{\mathcal{N}_{\kappa,\gamma}}\left(B_1\times\cdots\times B_{k-1}\times E_t\times B_{k+1}\times\cdots\times B_d\right) \leq \:\:\: e\prod_{i=1}^{k-1}\left(\frac{p_i-1}{2}r_i+1\right). \end{aligned}
M ={\frac 1 2}\int d \sigma \, \sigma g^{\mu_{1}\mu_{2}}\,{\bf D}_{\mu_{1}}( \sigma ){\bf D}_{\mu_{2}}( \sigma ) , \quad N ={\frac 1 2}\int d \sigma \, \sigma g_{\mu_{1}\mu_{2}}z^{\prime \mu_{1}}z^{\prime \mu_{2}}
t_{X,Y\otimes Z}=t_{X,Y}\otimes\mathrm{Id}_{Z}+(\sigma^{-1}_{X,Y}\otimes\mathrm{Id}_{Z})\circ(\mathrm{Id}_{Y}\otimes t_{X,Z})\circ(\sigma_{X,Y}\otimes\mathrm{Id}_{Z}), t_{X\otimes Y,Z}=\mathrm{Id}_{X}\otimes t_{Y,Z}+(\mathrm{Id}_{X}\otimes\sigma^{-1}_{Y,Z})\circ(t_{X,Z}\otimes\mathrm{Id}_{Y})\circ(\mathrm{Id}_{X}\otimes\sigma_{Y,Z}),
{\cal V}_{N ; g}\, = \,{\cal C}_{g}^{o p e n}\! < \Omega | \int [{\mathrm{d}}m ]_{N}^{g}\exp \left\{\frac{1}{2}\sum_{i = 1}^{N}\sqrt{2 \alpha^{\prime}}p^{( i )}\cdot \left. \! \left[ \sqrt{2 \alpha^{\prime}}p^{( i )}+{\alpha}_{1}^{( i )}\partial_{z}\right] \log V_{i}^{\prime}( z ) \right|_{z = 0}\right\}
\psi(a_{ij}) = \begin{cases}1\otimes a_{r_ir_j}\colon q_i = q_j a_{q_iq_j}\otimes 1 \colon r_i = r_j 0 \colon (q_i-q_j)(r_i-r_j)<0 a_{q_iq_j}\otimes a_{r_ir_j}\colon (q_i-q_j)(r_i-r_j)>0 \end{cases},
\sum_{|j|<\kappa^r}|\gamma_j|^2 I_2 = \kappa^{2\beta-1}i_2(2\beta-1) \sum_{|j|<\kappa^r}|\gamma_j|^2 \Bigl[2\pi|j_h|^2 U_e^2 + \Bigl(\frac{2\pi}3|j|^2 + 2\pi j_z^2\Bigr)U_f^2\Bigr].
P^{(d)}\left(\underset{0\leq s \leq t}{\sup}|B_s|\geq kt\right)\leq P^{(d)}\left(\underset{0\leq s \leq t+\tau}{\sup}|B_s|\geq kt\right)=P^{(1)}\left(\underset{0\leq s \leq t}{\sup}R^{\rho}_s\geq kt\right).
0 = H^{n-1}(\tilde{\mathbb{P}}, \Omega_{\tilde{\mathbb{P}}}^1 \otimes \mathcal{O}(\tilde{Y}+ K_{\tilde{\mathbb{P}}})) = H^{n}(\tilde{\mathbb{P}}, \Omega_{\tilde{\mathbb{P}}}^1 \otimes \mathcal{O}(\tilde{Y}+ K_{\tilde{\mathbb{P}}}))
\begin{cases}s_{u,[k]}\;\ge\; c_{u,[k-1]}s_{e,[k]}\;\ge\; c_{e,[k-1]}s_{d,[k]}\;\ge\; c_{d,[k-1]}\end{cases},~ k = 2, 3, ... , K, ~~~ s_{d,[1]}\;\ge\;c_{u,[K]}.
W_{q}=-\frac{1}{2}\int dx\int_{0}^{\infty}\frac{d\tau}{\tau}\tau^{-\frac{2}{1+\gamma}}\langle E[\delta\left(\tilde{q}\left(1\right)\right) \exp\left(-\tau\int_{0}^{1}ds \alpha\phi_{c}^{2}\left(x+ \tau^{\sigma}\left(\tilde{q}_{s}-x\right)\right)\right)]\rangle
Z(\kappa) = \int\! [dx(t)]\, \exp\left[ -\int\! d\tau\, \Biggl({1\over 2}\Bigl({dx\over d\tau}\Bigr)^{2}- x^{2}+{\kappa x^{4}\over 2\sqrt{2}}+{1\over\sqrt{2}\kappa}\Biggr) \right]\, ,
I_1(\mu)-I_1(\mu_0)={1\over 2}(\mu^2-\mu_0^2)I_0(\mu_0) - \frac{\mu_0^3}{8 \pi}\left[{1\over 3}\left( \sqrt{\frac{\mu^2}{\mu_0^2}}-1\right)^2\left( 2 \sqrt{\frac{\mu^2}{\mu_0^2}-1}\right) \right],
\hat{h}_{0}( y ) = [ 2 y_{0}^{2}( \alpha_{1}e^{6 n}+ \alpha_{2}) / e_{l}]^{- 1 / 2}\sqrt{| y | + y_{0}}\left[ -{\frac{2}{s_{1}}}+ \ln \left({\frac{| y |}{y_{0}}}+ 1 \right) \right] ,
\begin{pmatrix}-\mu\sqrt{\lambda}\mu\sqrt{\lambda}1 1\end{pmatrix}^{-1}\begin{pmatrix}0 \mu^2\lambda 1 0\end{pmatrix}\begin{pmatrix}-\mu\sqrt{\lambda}\mu\sqrt{\lambda}1 1\end{pmatrix}=\mu\sqrt{\lambda}\begin{pmatrix}-1 0 0 1\end{pmatrix}.
\sum_{\substack{a\in\mathcal{A}}}\mathbf{1}_{\mathcal{R}}(a)=\kappa_\mathcal{A}\frac{\#\mathcal{A}{}}{\#\mathcal{B}{}}\sum_{n< X}\mathbf{1}_{\mathcal{R}}(n)+O_{\mathcal{R},\eta}\Bigl(\frac{\#\mathcal{A}}{\log{X}\log\log{X}}\Bigr),
\int_{\Omega}|\nabla u|^{p(x)-2}\nabla u\nabla\varphi =\int_{\Omega}\biggl(\frac{f(x,Tu)|Tu|_{L^{q(x)}}^{\alpha(x)}}{\mathcal{A}(x,|Tu|_{L^{r(x)}})}+\frac{g(x,Tu)|Tu|_{L^{s(x)}}^{\gamma(x)}}{\mathcal{A}(x,|Tu|_{L^{r(x)}})}\biggl)\varphi
ds^2 = d\tilde x^{\tilde m}\tilde g_{\tilde m\tilde n}d\tilde x^{\tilde n}=e^{m}\eta_{mn}e^{n}+ e^\rho e^\rho =\rho^2 dx^2 + R^2 \,\left(\frac{d\rho}\rho\right)^2\,,
\frac{d^{m}}{d \epsilon^{m}}\vert_{\epsilon = 0}\lim_{\theta \to 1}\Bigl ( \Omega , T_{\theta ({\cal L}_{0}+ \epsilon{\cal L}_{1})}( f^{\otimes l}) \Omega \Bigr ) = i^{m}\lim_{\theta \to 1}\Bigl ( \Omega , T_{\theta{\cal L}_{0}}( ( \theta{\cal L}_{1})^{\otimes m}\otimes f^{\otimes l}) \Omega \Bigr )
I_{1,n}\leq C\sum\limits_{k=1}^n \frac{\lambda_k^2|\Psi_k|^2}{\left( T^{\alpha+\mu}\frac{1}{T^{\alpha+\beta}}\right)^2}\leq C T^{2(\beta-\mu)}\sum\limits_{k=1}^n\lambda_k^2 |\Psi_k|^2.
\sum_{j=1}^2f(I_{A_j}) + f(I_{B_j}) = \sum_{j=1}^2 s(A_j)(|c|+1+s(A_j))+s(B_j)(|c|+s(B_j)) &= |c| \cdot (s(A_1)+s(A_2)+s(B_1)+s(B_2)) + s(B_1)+s(B_2) & = |c| + s(B_1)+s(B_2),
\liminf_{n\rightarrow\infty}\nu_{n}(E,\mathrm{i}(y+2\epsilon))\geq \frac{1}{\epsilon}(L(E,\mathrm{i}(y+2\epsilon))-L(E,\mathrm{i}(y+\epsilon)))=\omega^{+}(E,\mathrm{i}y).
\alpha_{2}=\max\bigg\{\bigg(K\|\phi(0)\|_{\mathbb{X}}+MKRb +KSb\kappa_{b}\bigg), \max_{1\leq j\leq n}\bigg(KC_{\nu_{j}}&\qquad\qquad+ MKRb+KSb\kappa_{b}\bigg), \ \max_{1\leq j\leq n}C_{\nu_{j}}\bigg\}.
\vert \Delta_{n,k,m}\vert \leq (m+1)!C_1^{m+1}\left((d_n+k+1)C_3^{2n}C_2^{d_n+k}\right)^{m+1}r^{-m(m+1)}&< (C_3^3r^{-1})^{m(m+1)}
f(x)=\frac{1}{\sigma\mathrm{B}(\alpha,\beta)}\,\left[\Phi\left( \frac{x-\mu}{\sigma}\right)\right]^{\alpha-1}\,\left[1-\Phi\left(\frac{x-\mu}{\sigma}\right)\right]^{\beta-1}\,\phi\left(\frac{x-\mu}{\sigma}\right),x\in\mathbb{R},
\rho_N^{-\frac{1}{2}}\left[ 1+C_{\beta}\sqrt{2}\left(\frac{\tau}{\rho_N}\right)^{\frac{1}{4}}\right]^{-2n}= \frac{1}{D_{\beta}}\left(1+\sqrt{2}G_{\beta}\tau^{-\frac{1}{4}}\right)^{-2n}.
\mathrm{m}_q = \c_q (n) \lim_{r \to \infty}\int_{S_r}X_{(q)}^i\nu_i dS_r = \c_q(n) \int_{\mathbb{R}^n \setminus \Omega}\mathrm{div}_e X_{(q)}dV - c_q(n)\int_{\Sigma}\langle X_{(q)}, \xi \rangle d\Sigma.
\sum_{l=0}^{\infty}CC_{l}^{\left(r,s\right)}\left(x\right)\frac{\left(e^{t}-1\right)^{l}}{l!}=\sum_{l=0}^{\infty}CC_{l}^{\left(r,s\right)}\left(x\right)\sum_{n=l}^{\infty}S_{2}\left(n,l\right)\frac{t^{n}}{n!}=\sum_{n=0}^{\infty}\left(\sum_{l=0}^{n}CC_{l}^{\left(r,s\right)}\left(x\right)S_{2}\left(n,l\right)\right)\frac{t^{n}}{n!}.
\mathcal{S}= \int \frac{1}{2 l_{1 1}^{9}}\left( d^{1 1}x \sqrt{- g}R - \frac{1}{2 \cdot 4 !}d \mathcal{C}\wedge * d \mathcal{C}\right) + \int \frac{\sqrt{2}}{2^{7}\cdot 3^{3}}\mathcal{C}\wedge d \mathcal{C}\wedge d \mathcal{C}\ .
(x\ast y)\ast\alpha(z)-\alpha(x)\ast(y\ast z) =\alpha(x)\cdot (\partial(y)\cdot\partial(z))-\alpha(x)\cdot\partial(y\cdot\partial(z)) &=\alpha(x)\cdot (\partial(y)\cdot\partial(z))-\alpha(x)\cdot(\partial(y)\cdot\partial(z))=0.
(\imath_{t_1,t_2}^{-1})^*\Lambda^+\imath_{t_1,t_2}^{-1}-(\imath_{t_1,t_2}^{-1})^*\Lambda^-\imath_{t_1,t_2}^{-1}=\i (\imath_{t_1,t_2}^{-1})^*G\imath_{t_1,t_2}^{-1}=\i G
\sum_{x =1}^n L(y,x)\ =\ \sum_{x'=y}^n \sum_{x=1}^n \widehat L(x,x') - \widehat L(x-1,x') =\ \sum_{x'=y}^n \widehat L(n, x')\ \leq\ \sum_{x'=1}^n \widehat L(n,x')\ =\ 0\ ,
c( b \bar{a}, \bar{a}b) = \lambda^{-1}\bigl(c(\bar{a}\bar{b}, \bar{b}a) + \sqrt{2}\,\bigr), c(\bar{a}\bar{b}, \bar{b}a) = \lambda^{-1}\bigl(c(\bar{b}a , a b) - \sqrt{2}\,\bigr), c(\bar{b}a , a b) = \lambda^{-1}\bigl(c( a b , b\bar{a}) + \sqrt{2}\,\bigr).
\langle D_1 J_1(f;v^1+sw^1,v^2),w^2 \rangle = \alpha_1 \iint_{\mathcal{O}_{1,d}\times (0,T)}(y^s-y_{1,d}) z^s dxdt &+ \mu_1 \iint_{\mathcal{O}_1 \times (0,T)}(v^1 + s w^1)w^2 dxdt
L_{B}= -{\frac{1}{2}}\left( \partial_{\mu}\phi \right)^{2}+{\frac{m}{4 !}}\epsilon^{\mu \nu \rho \sigma}\phi \partial_{[ \mu}A_{\nu \rho \sigma ]}-{\frac{1}{2 \cdot 4 !}}F^{2}{}_{\mu \nu \rho \sigma}( A )
\left\vert 1-\left\vert \frac{k_{21}k_{13}}{k_{23}k_{11}}\right\vert\right\vert =\left\vert 1-\left\vert \frac{\widehat{k_{21}}\widehat{k_{13}}}{\widehat{k_{23}}}\right\vert \right\vert \leq\delta_{12}\delta_{13},
E_{gg}(\omega_n) ={1\over 2}\sum_{n=1}^{M-1}\omega_n -{1\over 2}\lambda M \left[{a^2d\over 2}+ R_1^2\right]^{-d/2}\!\!\! + g^2 M \sum_{k=1}^{M}\left[{a^2d\over 2}+ R_k^2\right]^{-d/2}\, .
P(|W_l|_*\leqslant L) =P\left((2m-1)^{-L/2}\leqslant(2m-1)^{-\frac12|W_l|_*}\right) \leqslant (2m-1)^{-L/2}\mathbb{E}\left((2m-1)^{-\frac12|W_l|_*}\right).
=\frac{e^{4}}{(k-k^{\prime})^{4}}(\overline{v}_{k^{\prime}}\gamma^{\mu}v_{k})(\overline{v}_{k}\gamma^{\nu}v_{k^{\prime}})(\overline{u}_{p}\gamma_{\mu}u_{p^{\prime}})(\overline{u}_{p^{\prime}}\gamma_{\nu}u_{p})
\begin{multlined}I_{1,\varepsilon,t,\delta}= \dfrac{2}{n}K_n^{-\frac{n}{4}}\left(1-C_n \varepsilon-\dfrac{(n-4)^2}{8}\varepsilon \ln \delta\right. +\dfrac{(n-1)}{(n-6)(n^2-4)}(Tr_g (A_g- A_{paneitz}) \delta_\varepsilon (t)^2 1_{n\geq 8}) + o(\varepsilon)+O(\delta_\varepsilon (t)^4)\bigg),\end{multlined}
\mathbb E ({\rm Tr}\, e^{\mu D^{(N)}}) \le || \mathbb E_{\epsilon_N}( e^{\mu \epsilon_N A^{(N)}- \mu^2 (A^{(N)})^2/2}) ||_{\rm op}{\rm Tr}\, (e^{\mu D^{(N-1)}}) = || e^{-\mu^2 (A^{(N)})^2/2}\mathbb E_{\epsilon_N}( e^{\mu \epsilon_N A^{(N)}}) ||_{\rm op}{\rm Tr}\, (e^{\mu D^{(N-1)}}) ,
Z^\mathbb{I}_{|f|^2}=\sum_{i,j=1}^nP_\mathbb{I}^{ij}\frac{\partial|f|^2}{\partial x^i}\frac{\partial}{\partial x^j}+\sum_{i,j=1}^nP_\mathbb{I}^{in+j}\frac{\partial|f|^2}{\partial x^i}\frac{\partial}{\partial x^{n+j}}
\sum_{x \in E}\sum_{y\neq x}\nu(x) \frac{\mu(y)}{\mu(x)}L_{y x}= \sum_{x \in E}\frac{\nu(x)}{\mu(x)}\sum_{y \neq x}\mu(y) L_{y x}= \sum_{x \in E}\nu(x) L_{x x}= \sum_{x \in E}\nu(x) \sum_{y\neq x}L_{x y},
\frac{n^2}{m^2}\sum_{l = 2}^m\mathbb{E}_0[(D^S_{nl})^2], \ \ \frac{n^2}{m^2}\sum_{l =2}^m\mathbb{E}_0[(D^T_{nl})^2] \longrightarrow k^4 (\zeta_1^h)^2, &\frac{n}{m^2}\sum_{l = 2}^m \mathbb{E}_0\left[(D^Z_{nl})^2\right] \longrightarrow \frac{k^2 \zeta^h_1}{2},and
0=\frac{1}{\pi}\boldsymbol{B}\Big(\frac{\partial w}{\partial{x_1}}, \frac{\partial w}{\partial{x_1}}\Big)= \mathbb B_1 ( -\Phi_2, \Phi_0)={\mathbb D}( \Phi_2, \Phi_0), begin{align*}1mm] 0=\frac{1}{\pi}\boldsymbol{B}\Big(\frac{\partial w}{\partial{x_2}}, \frac{\partial w}{\partial{x_2}}\Big)= \mathbb B_1 ( i\Phi_2, i\Phi_0)={\mathbb D}( \Phi_2, \Phi_0).