harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
I_{\varphi}- I_{\alpha}= I - 4 \ln \ln{\frac{\Lambda_{0}}{\Lambda}}+ 4 \ln{\frac{\Lambda_{0}^{2}}{\Lambda^{2}}}- 4 \ln 2 +{\frac{4}{N - 1}}\, \sum_{i = 1}^{N - 1}\ln{\frac{\Lambda^{2}}{v_{i}}}\raise 2 p t \mathrm{,}
W=\left(\sum_{N<\ell\le N^2}-\sum_{N^2<\ell\le N^3}\right)\frac{\tilde\lambda(\ell)}{\ell}\sum_{m_1\mid\ell}\sum_{m_2\mid\ell}\mu(m_1)\mu(m_2)\frac{g(m_1)}{\xi(m_1)}\frac{g(m_2)}{\xi(m_2)}K\left(\frac{m_1}{m_2}\right).
Z^{\mathrm{L a r g e}N_{c}}= \int \prod_{i , j}[ d q_{+}^{\dag i}] [ d q_{+}^{i}] [ d s^{i j}] [ d \sigma_{i j}] \exp \left\{i S^{\mathrm{L a r g e}N_{c}}+ i S^{\mathrm{A u x}}\right\},
\nabla{\cal U}^{i \alpha}\equiv d{\cal U}^{i \alpha}+{\frac{1}{2}}\omega^{x}( \epsilon \sigma_{x}\epsilon^{- 1})_{j}^{i}\wedge{\cal U}^{j \alpha}+ \Delta^{\alpha \beta}\wedge{\cal U}^{i \gamma}C_{\beta \gamma}= 0 ,
\begin{aligned}G(\alpha) =\sum_{m\ge 1}(2\alpha^2 m^2-3)m^2 \exp(-\alpha^2m^2) =\frac{\sqrt{\pi^5}}{\alpha^5}\sum_{m\ge 1}\Bigl(2\Bigl(\frac{\pi}{\alpha}\Bigr)^2m^2-3\Bigr) m^2\exp\Bigl(-\Bigl(\frac{\pi}{\alpha}\Bigr)^2 m^2\Bigr). \end{aligned}
0= \int_{\mathbb{R}}\rho_3(t,u) G(t,u) du - \int_0^t \int_{\mathbb{R}}\rho_{3}(s,u) \partial_s G(s,u) du ds - \int_0^t \int_{\mathbb{R}}\rho_{5}(s,u) [ - (- \Delta)^{\gamma/2}G](s,u) du ds.
S = \int d^{D}x \, \sqrt{- g}\left[ - \sum_{k = 1}^{N}\overline{{{\psi}}}_{k}\gamma^{\mu}\nabla_{\mu}\psi_{k}+ \frac{\lambda_{0}}{2 N}( \sum_{k = 1}^{N}\overline{{{\psi}}}_{k}\psi_{k})^{2}\right] ,
\int_0^{2\pi}\rho_{(1),N}^{(\widetilde{\rm cJ})}(\theta;\beta,p,q) \, d \theta = N ={N \over 2 \pi}\delta_{k=0}\int_0^{2 \pi}e^{i k \theta}\, d \theta, k \in \mathbb Z \backslash\{0 \}.
F(n,k) =\frac{[n+1]}{2}{2n+1\brack n}\frac{(1+q^{2k-n-1})q^{-{n-2k+1\choose 2}}}{[2k]^2{n\brack k}_{q^2}^2}, begin{align*}5pt]G(n,k) =-\frac{[3n-2k+5]}{2}{2n+1\brack n}\frac{(1+q^{n+1})q^{-{n-2k+3\choose 2}}}{[2k]^2{n+1\brack k}_{q^2}^2}.
B^{rrss}\geq{}10\left( \sum_{\alpha=1}^K q_\alpha^r q_\alpha^s \right)^2- 6\sum_{\alpha=1}^K (q_\alpha^r q_\alpha^s)^2 \geq{}4\left( \sum_{\alpha=1}^K q_\alpha^r q_\alpha^s \right)^2\geq 0.
(i\Theta_{T_X}+\theta\otimes\mathrm{id}_{T_X})(\kappa_1\otimes\kappa_2,\kappa_1\otimes\kappa_2)\geq0 \forall\kappa_1,\kappa_2\in T_X~\mathrm{with}~\langle\kappa_1,\kappa_2\rangle=0
\frac{1}{2}\left (a_{ij}^2 + b_{ij}^2 + c_{ij}^2 + g_{ij}^2 \right ) = \left |\sum_{k=1}^{n}\delta_k u_{ik}v_{jk}\right |^2 + \left |\sum_{k=1}^{n}\sigma_k u_{ik}\overline{v}_{jk}\right |^2.
A_{n,k}=\frac{n!}{(n+k)!}, B_{n,k}=\frac{n!}{(n+k-1)!}\sum_{j=1}^k\frac{1}{n+j}\mbox{and}C_{n,j}=\frac{n!}{(n+j+1)!}\sum_{\ell=1}^{j+1}\frac{1}{n+\ell}
w_{1\over 2^{Q+2}}(b;S) \leq C{1\over |S|}\iint_{E\times F}|b(g)-b(g')|\, |K_j(g,g')| \, dgdg' &= C{1\over |S|}\bigg|\iint_{E\times F}(b(g)-b(g')) \,K_j(g,g')\, dgdg' \bigg| &\leq C{1\over |S|}\int_E \big| [b,R_j](\chi_F)(g) \big|\,dg.
G_{L}\left( z \right) = \left( \begin{array}{c c}{1}&{0}\\{\theta_{L}}&{1}\\ \end{array}\right) \left( \begin{array}{c c}{H_{L}}&{0}\\{0}&{g_{L}}\\ \end{array}\right) \left( \begin{array}{c c}{1}&{\tilde{\theta}_{L}}\\{0}&{1}\\ \end{array}\right)
\sigma^{0}= \left( \begin{array}{r r}{{1}}&{{0}}\\{{0}}&{{1}}\\ \end{array}\right) , \quad \sigma^{1}= \left( \begin{array}{r r}{{0}}&{{1}}\\{{1}}&{{0}}\\ \end{array}\right) , \quad \sigma^{2}= \left( \begin{array}{r r}{{0}}&{{- i}}\\{{i}}&{{0}}\\ \end{array}\right) , \quad \sigma^{3}= \left( \begin{array}{r r}{{1}}&{{0}}\\{{0}}&{{- 1}}\\ \end{array}\right)
\begin{cases}g_n := g_0 + h_n (n \geqslant 1){\displaystyle \varphi_0 = 1, \varphi_n = \prod_{i=0}^{n-1}(f^i)^*(\varphi)^{1/d^{i+1}}}(n \geqslant 1).\end{cases}
q_\ell = \sum_{j=0}^\ell\bigg[\binom{\ell}{j}2^{j-\ell}(k-\ell)_{\ell-j}\left(\frac{m+n+1-2k+4j}{2}\right)_{\ell-j}& \times \prod_{i=1}^{j}\left(-\frac{1}{2}\Delta - \frac{(n-m+2k-4i+3)(m+n-2k+4i-3)}{8}\right)\bigg] .
\begin{array}{ll}\frac{\mathrm{d}}{\mathrm{d}t}\int\limits_{\Omega}|\partial_t^{\ell}\mathcal{D}^{\alpha}\omega|^2\,\mathrm{d}x+ 2a\int\limits_{\Omega}|\partial_t^{\ell}\mathcal{D}^{\alpha}\omega|^2 \,\mathrm{d}x= I_1 + I_2,\end{array}
\bar{Y}^j( \alpha_j, \bar{X}^j) = (y_1^j( \alpha_j, \bar{X}^j),y_2^j( \alpha_j, \bar{X}^j),...,y_n^j( \alpha_j, \bar{X}^j))= ( I_{n}+ \alpha_j P^{'}P )^{-1}\bar{X}^j.
\hat{f}_\mathrm{L}(x)=\frac{B_\mathrm{L}f_\mathrm{L}(x)}{A_\mathrm{L}}\mathrm{e}^{-2\pi\lambda\int_{0}^{\psi_\mathrm{L}(x)}(1-p(t))t\mathrm{d}t},
\int_0^4 d\lambda\,\delta(\lambda)\,\frac{\lambda-2}{\lambda(\lambda-4)}+\frac{1}{\pi}\int_0^4 d\lambda_1\, \delta(\lambda_1)\,\mathcal{P}\!\!\,\int_0^4 d\lambda_2\, \frac{\delta'(\lambda_2)}{\lambda_2-\lambda_1}=0,
Z(1)=\int_{0}^{1}dt\frac{dZ(t)}{dt}=\sum_{n=1}^{\infty}\frac{(-)^{n-1}}{n}\int_{0}^{1}dt(e^{tad_{X}}e^{tad_{Y}}-1)^{n-1}(X+e^{tad_{X}}Y).
\Pi_{1}( m_{0}) = \frac{\lambda}{2}T \sum_{n}\int \frac{d^{3}k}{( 2 \pi )^{3}}\frac{1}{\omega_{n}^{2}+ \omega_{k}^{2}}= \frac{\lambda}{2}\int \frac{d^{3}k}{( 2 \pi )^{3}}\frac{1}{\omega_{k}( m_{0})}\left[ n_{B}( \omega ) + \frac{1}{2}\right] ,
\quad-\frac{(n+2)^{2}(2n+5)(4n+9)\sqrt{\pi}\Gamma\left(2n+\frac{7}{2}\right)}{4(2n+5)!}=\frac{3(n+2)^{2}(2n+5)\Gamma\left(2n+\frac{11}{2}\right)\Gamma\left(\frac{1}{2}\right)}{\Gamma(2n+6)\left(-3(4n+7)\right)},
| \varphi(z_1) -\varphi(z_2) |^2 =| (-2(v_1 - v_2), u_1 - u_2, -(u_1-u_2)) |^2 &=2((u_1-u_2)^2 + 2(v_1 - v_2)^2) =2|z_1 - z_2|^2,
\begin{aligned}H(\rho,\sigma) < H(\rho,k_1)=(1-\rho)\log\frac{1}{1-\rho}-(k_1-1)\rho\log\frac{1}{\rho}&\le (1-\rho)\log\frac{1}{1-\rho}-\rho\log\frac{1}{\rho}\le 0.\end{aligned}
F_1(q^2) = Z +\int\frac{{\mathrm d}^2{\vec k}_{\perp}{\mathrm d}x}{16 \pi^3}\Big[\psi^{\uparrow\ *}_{+\frac{1}{2}}(x,{\vec k'}_{\perp})\psi^{\uparrow}_{+\frac{1}{2}}(x,{\vec k}_{\perp})+\psi^{\uparrow\ *}_{-\frac{1}{2}}(x,{\vec k'}_{\perp})\psi^{\uparrow}_{-\frac{1}{2}}(x,{\vec k}_{\perp})\Big]\ ,
\begin{aligned}F(x_k)\leq a_k F(x)+ (1-a_k)F(x_{k-1})+\frac{\tilde \mu a_k^2}{2}(\|x-v_{k-1}\|^2-\|x-v_k\|^2) &- \frac{\tilde \mu-\mu}{2}\|y_k-x_k\|^2+\rho a_k\|x-x_{k-1}\|^2+\frac{ra_k^2}{2}\|x-v_{k-1}\|^2.\end{aligned}
\lambda_4 = -3a_2^2 - 4a_2 b_2 - 3b_2^2, \lambda_6 = 2(a_2+b_2)\big(a_2^2+3a_2 b_2+b_2^2\big), \lambda_8 = -a_2 b_2 \big(4a_2^2+ 7 b_2 a_2 + 4b_2^2\big), \lambda_{10}= 2 a_2^2 b_2^2 (a_2+b_2).
{}^{M\!L\!}\textrm{F}_{p}(a,b;c;z)= \frac{2}{B(b,c-b)}\int_{0}^{\frac{\pi}{2}}\sin^{2b-1}v \cos^{2c-2b-1}v(1-z\sin^{2}v))^{-a}E_{\alpha, \beta}^{\gamma}\left(-p\sec^{2}v\csc^{2}v\right)du.
\begin{aligned}I_{m,n}(k)^\perp = \prod_{i=0}^{k-1}\frac{c_{2,n}(i)}{c_{1,n}(i)}\left[\left(\prod_{i=0}^{k-1}C_{m,n}(i)\right)^{-1}\right]^t I_{m,n}(0)^\perp K_{m,n}(k)^\perp = \prod_{i=0}^{k-1}\frac{c_{2,n}(i)}{c_{1,n}(i)}\left[\left(\prod_{i=0}^{k-1}C_{m,n}(i)\right)^{-1}\right]^t K_{m,n}(0)^\perp .\end{aligned}
v=\left[{z_0^2+z_i^2+1\over 2z_0},{z_0^2+z_i^2-1\over 2z_0},{z_i\over z_0}\right],\;\;\;, l=[1,-1,\vec{0}],\;\;\;(v\cdot l)={z_0^2+z_i^2\over z_0}
2n \int_{1/n}^1 \exp(-\psi(n)/x)dx = 2n \int_{n\psi(n)}^{\psi(n)}\exp(-u) \Big(-\frac{\psi(n)}{u^2}\Big)du begin{align*}1ex] = 2n\psi(n) \int_{\psi(n)}^{n\psi(n)}\frac{1}{u^2e^u}du.
\vec{f}_F^{(1)}(x;\omega)=\frac{1+i\gamma^1}{2}\vec{f}_F(x;\omega)=\frac{1}{2}\left(\begin{array}{c}\vec{f}_F^+(x;\omega)-\vec{f}_F^-(x;\omega) -\vec{f}_F^+(x;\omega)+\vec{f}_F^-(x;\omega)\end{array}\right)
S_{\mathrm{f e r m}}= \int \! d^{2}x \, \Bigl ( i \bar{\psi}\gamma^{\mu}\partial_{\mu}\psi +{\frac{\pi F_{e}}{\mu}}\, \bar{\psi}\psi +{\frac{\pi F_{m}}{\mu}}\, \bar{\psi}^{C}\psi -{\frac{G}{2}}\, \bar{\psi}\gamma^{\mu}\psi \, \bar{\psi}\gamma_{\mu}\psi \Bigr )
P^{(0)}(z) = P_{0}^{(\infty)}(z) \left( \frac{z+x}{z}\right)^{\sigma_{3}}\begin{pmatrix}1 f(z;x) 0 1\end{pmatrix}z^{\frac{\alpha + 2}{2}\sigma_{3}}\begin{pmatrix}1 h(z) 0 1\end{pmatrix}H_{0}(z)
S_{i,0}= \sum_{\substack{n \leq x_i P(n) \leq y_0}}\frac{f(n)}{\sqrt{n}}, S_{i,j}= \sum_{\substack{y_{j-1}< m \leq x_i p | m \Rightarrow p \in (y_{j-1}, y_j]}}\frac{f(m)}{\sqrt{m}}\sum_{\substack{n \leq x_i / m P(n) \leq y_{j-1}}}\frac{f(n)}{\sqrt{n}}.
\Big |\mathbb{P}_\rho^\varepsilon(\{\mathcal{M}_{s_j \varepsilon^{-2}}= \nu_j : j=1, \dots, n\}) - \mathbb{P}_{\pi_{\rho}}(\{Y_{s_j}= \nu_j, j =1,\dots, n\}) \Big | \leq C n \varepsilon,
{\frak C}\sum_{x=0}^n |\tilde V_x(m)|^2\le \left[ |\tilde V_0(m)|\left\{\sum_{x=0}^n |M_{x,0}(m)|^2\right\}^{1/2}+ \left\{\sum_{x=0}^n |\tilde v_{x}(m)|^2\right\}^{1/2}\right] \left\{\sum_{x=0}^n |\tilde V_x(m)|^2\right\}^{1/2},
\frac{1}{\sqrt{N}\pi}=\sum_{k=0}^\infty\frac{\left(\frac{1}{2}\right)_k^3}{(k!)^3}\left(k\left(1-2\alpha_N\right)-\frac{2}{\sqrt{N}}{D_N}\!\left(e^{-\pi/\sqrt{N}}\right)\right)\left(4\alpha_N(1-\alpha_N)\right)^k.
\nabla(x_m\otimes y_n)=\sum_{\substack{(m,n) \sigma\in\Sigma_{m+n}}}(-1)^{\sharp(\sigma)}\Bigl(D_{\sigma({m+n-1})}\circ\dots\circ D_{\sigma(m)}(x_m)\Bigr)\otimes \Bigl(D_{\sigma(m-1)}\circ\dots\circ D_{\sigma(0)}(y_n)\Bigr)
\mathfrak{b}(x(v),x(v)) =\mathfrak{b}((\sigma_\mathfrak{b}(x)x)(v),v)=\mathfrak{b}((y-\sigma_\mathfrak{b}(y)-\alpha)(v),v) &=\mathfrak{b}(y(v),v)-\mathfrak{b}(\sigma_\mathfrak{b}(y)(v),v)-\alpha\mathfrak{b}(v,v) &=\mathfrak{b}(y(v),v)-\mathfrak{b}(v,y(v))-\alpha\mathfrak{b}(v,v)=\alpha\mathfrak{b}(v,v).
f_{\left( k,\ell\right)}^{\left( M\right)}\left( r\right) :=\frac{2\pi}{M}{\displaystyle\sum_{s=1}^{M}}f\left( re^{i\frac{2\pi s}{M}}\right) Y_{\left( k,\ell\right)}\left(\frac{2\pi s}{M}\right)
0 = A_\rho J b^* J^{-1}- J \rho(b^*) J^{-1}A_\rho = J^2 A_\rho J^{-2}J b^* J^{-1}- J \rho(b^*) J^{-1}J^2 A_\rho J^{-2}& = J \left( J A_\rho J^{-1}b^* - \rho(b^*) J A_\rho J^{-1}\right) J^{-1}= J \left( [J A_\rho J^{-1}, b^*]_\rho \right) J^{-1}
A_1^{++--}=c_1 \left(e_2^+\cdot e_4^-\right)^{s_a}\left(K_2\cdot e_1^+\right)^{s_a}\left(K_4\cdot e_3^-\right)^{s_b}\left(K_3\cdot e_4^-\right)^{s_b-s_a}
e^{-1}{\cal L}=R+{\textstyle{15\over2}}g^2-{\textstyle{1\over12}}(S_{\hat\mu\hat\nu\hat\rho}^I)^2+e^{-1}\left[{\textstyle{1\over2}}g^{-1}S_{[3]}^I\wedge dS_{[3]}^I\right],
\Lambda(s, \operatorname{sym}^2 f)= \pi^{-3s/2}\Gamma \left( \frac{s+1}{2}\right)\Gamma \left( \frac{s+\kappa-1}{2}\right) \Gamma \left( \frac{s+\kappa}{2}\right) L(s, \operatorname{sym}^2 f).
\vert \lambda \rangle ={e^{\lambda{J_{+}}}}\left( \begin{matrix}{0}\\{1}\\ \end{matrix}\right) ,{J_{+}}= \left( \begin{matrix}{0}&{1}\\{0}&{0}\\ \end{matrix}\right) \equiv{e_{1 2}}, \langle \lambda \vert \lambda \rangle = ( 1 +{\lambda}{\bar{\lambda}})
Kf(x) = \frac{c}{\pi}a_k \log|x-x_k| + \lim_{k_0\to +\infty}\frac{c}{\pi}\sum_{k'\leq k_0 \atop{k' \neq k}}a_{k'}\log|x-x_{k'}| & =: \frac{c}{\pi}a_k \log|x-x_k| + e_k(x).
\begin{aligned}X^{(1)}_{1}(L_{1})-X^{(1)}_{2}(L_{2})+(D_{t}\varsigma_{1})L_{1}-(D_{t}\varsigma_{2})L_{2}=D_{t}A_{1}, X^{(1)}_{1}(L_{2})+X^{(1)}_{2}(L_{1})+(D_{t}\varsigma_{1})L_{2}+(D_{t}\varsigma_{2})L_{1}=D_{t}A_{2},\end{aligned}
\eta_{M,N}(q\,|\,b) = \exp\Bigl(\int\limits_0^\infty\Bigl[\bigl(\mathcal{S}_N \,\exp(-xt)\bigr)(q\,|\,b)-\bigl(\mathcal{S}_N\,\exp(-xt)\bigr)(0\,|\,b)\Bigr]f(t) dt/t^{M+1}\Bigr).
( \exp [ \langle . . \rangle_{D_{2}}+ \langle . . \rangle_{A_{2}}] )_{\log \epsilon}\approx ( \log \epsilon ) \,{\frac{1}{2 !}}( \phi \cdot \nabla_{Y})^{2}\exp [{\langle \cdots \rangle_{D_{2}}}+ \langle . . \rangle_{A_{2}}] .
||\Gamma_{k,\eta}|| \leq J_{\sigma_k+\eta}^{(1)}+ J_{\tau_k}^{(1)}+ J_{\sigma_k+\eta}^{(2)}- J_{\tau_k}^{(2)}&+ ||J_{\sigma_k+\eta}^{(3)}-J_{\tau_k}^{(3)}|| \rightarrow 0, k\rightarrow\infty,
\|\tilde \mu_{U_M \times N}(m, n)\| = \| \tilde \mu_{U_M}(m) + \mu_N(n)\| &\geq \| \tilde \mu_{U_M}(m)\| - C_N &\geq \| \mu_{U_M}(m)\| - C_N &\geq \| \mu_{U_M}(m)+ \mu_N(n)\| - 2C_N &> C_{\pi}.
\, \mathrm{o r a s}{\cal{L}}_{B I}= \beta^{2}\Biggl [ 1 - \sqrt{1 + \frac{1}{2 \beta^{2}}({\bf B}^{2}-{\bf E}^{2}) - \frac{1}{1 6 \beta^{4}}({\bf E}\cdot{\bf B})^{2}}\Biggr ] .
G^u =0\;, G^x =\tfrac{1}{4}(y^0)^2 \Phi_x\;, G^y =\tfrac{1}{4}(y^0)^2 \Phi_y\;, G^v =\tfrac{1}{4}y^0 \left(y^0 \Phi_u + 2 y^2\Phi_y + 2 y^1\Phi_x\right),
g_{i}({\bf{q}}) = [\sum_{{\bf{k}}}\frac{{\bar{n}}_{{\bf{k}}-{\bf{q}}/2}-{\bar{n}}_{{\bf{k}}+{\bf{q}}/2}}{({\tilde{\omega}}_{i}({\bf{q}}) - \frac{{\bf{k.q}}}{m})^{2}}]^{-\frac{1}{2}}
(\theta_i \theta_j, \theta_i \theta_j-\theta_j \theta_i) = (\theta_i \otimes \theta_j - \theta_j \otimes \theta_i, (\theta_i \otimes 1 + 1 \otimes \theta_i)(\theta_j \otimes 1 + 1 \otimes \theta_j)) &= (\theta_i \otimes \theta_j - \theta_j \otimes \theta_i, \theta_i \otimes \theta_j + (-1)^{p(i)p(j)}\theta_j \otimes \theta_i).
s = \left( \begin{smallmatrix}a_{1,1}0 0 a_{2,2}\end{smallmatrix}\right) = \left( \begin{smallmatrix}a_{1,1}0 0 a_{1,1}^{-1}\end{smallmatrix}\right) \, \, s = \left( \begin{smallmatrix}0 a_{1,2}-a_{2,1}0 \end{smallmatrix}\right) = \left( \begin{smallmatrix}0 a_{1,2}-a_{1,2}^{-1}0 \end{smallmatrix}\right).
- \; \frac{mc}{2\pi}\frac{4 \pi^2}{\pi\!+\!g}\; \frac{1}{(2\pi)^2}\int d^2p d^2q \; \hat{C}_{m_d}(p) \hat{C}_{m_d}(q)\int_\Lambda d^2 x \; e^{ip(x-w)}e^{iq(x-z)}\; 2 \cos(\theta)
\mathfrak{D}^{n+\varsigma}_{z}f(z) := \dfrac{{\rm d}^{n}}{{\rm d}z^{n}}\left\{\mathfrak{D}^{\varsigma}_{z}f(z)\right\}(0 \leqq \varsigma < 1;\; n \in \mathbb{N}_{0}:=\mathbb{N}\cup\{0\}=\{0,1,2,3,\cdots\}),
Z_k =\left\{\prod\limits(x_ix_s)^{p_{is}}\prod\limits(x_{i}y_{s})^{l_{is}}\prod\limits({x_{i}y_{r}}-x_{r}y_{i})^{f_{ir}}|\right. \quad\left.\sum\limits_{\substack{1\leq i\leq n_1 n_1+1\leq s\leq n}}p_{is}+\sum\limits_{\substack{1\leq i\leq n_1 n_1+1\leq s\leq n}}l_{is}+ \sum\limits_{\substack{1\leq i<r\leq n_1}}f_{ir}=k \right\},
<\Psi_1|\Psi_2>=N\int\limits_{-\infty}^{\infty}\frac{d^2\vec{p}}{p^0}\int\limits_0^{2\pi}\frac{d\varphi}{(p,n)}\overline{\Psi_1(p,\varphi)}\Psi_2(p,\varphi),
(\tilde{\psi}_{t,n}^\star, \tilde{\psi}_{r,n}^\star) = \!\!\!\!\operatorname*{argmin}_{(\tilde{\psi}_{t,n}, \tilde{\psi}_{r,n}) \in \chi_{\psi}^n}\!\!\!\! \mathrm{Re}(\tilde{\vartheta}_{t,n}^* \tilde{\psi}_{t,n}) + \mathrm{Re}(\tilde{\vartheta}_{r,n}^* \tilde{\psi}_{r,n}),
\{\tilde{\Theta}_{\alpha}^{\mu}, \tilde{\Theta}_{\beta}^{\nu}\}= \{\tilde{\Theta}_{\alpha}^{\mu}, \tilde{\Psi}_{l}\}= \{\tilde{\Theta}_{\alpha}^{\mu}, \tilde{H}\}= 0 ~ ~ ; ~ ~ \{\tilde{\Psi}_{l}, \tilde{\Psi}_{n}\}\approx 0 ~ ; ~ \{\tilde{\Psi}_{l}, \tilde{H}\}\approx 0 .
\ln \frac{w_2}{w_1}(e^{\frac{2}{3}t_1+\frac{1}{3}t_2})+2\pi i = r \left[\ln \frac{w_3}{w_2}(e^{\frac{2}{3}t_1+\frac{1}{3}t_2})-2\pi i \right],~~~~~~~~~~r\in{\bf R}.
\mathrm{I m}{\cal L}_{\mathrm{s c a l a r}}^{( 1 )}= \frac{( q E ) ( q B )}{4 ( 2 \pi )^{2}}\sum_{n = 1}^{\infty}\frac{( - 1 )^{n + 1}}{n}\mathrm{c s c h}\Bigl ( \frac{n \pi B}{E}\Bigr ) \exp \Bigl ( - \frac{n \pi m^{2}}{q E}\Bigr ) .
\|\tilde{f}(l)-\mathbb{E}_{f_0,T}\tilde{f}(l)\|_\infty \leq \sum_{j=0}^{l}\max_k (|Z_{jk}|+|W_{jk}|) \sup_{x\in[0,1]}\sum_{k=0}^{2^j-1}|\psi_{jk}(x)| &\leq C\big(\sum_{j=0}^{l}2^{j/2}\max_k |Z_{jk}| +\sqrt{2^l/n}\big).
\gamma_i :=\frac{1}{f_i(r)^{2}}dr^2+\frac{r^{2}}{r_i^{2}}\sigma_i=\frac{1}{f_i(r)^{2}}dr^2+r^{2}\,\Omega, f_i(r) :=\sqrt{1-\frac{2\mu_i}{r}+\frac{q_{i}^{2}}{r^{2}}}, \varphi_{i}(r) :=\frac{q_{i}}{r},
\frac{\pi_{c,\kappa}(\mu,\sigma)}{\pi_\mathrm{P}(\mu,\sigma)}= \frac{1}{\displaystyle \frac{\frac{a^2\tilde{b}}{b^2\tilde{a}}\mu^2 + \sigma^2 + \kappa^2}{2\sigma \kappa}+c}= \frac{1}{\cosh \rho + c}.
\sup_{B_{\frac{1}{2}\lambda_t}\times Y}|\nabla^{j,\hat{g}_t}\hat{g}^\bullet_t|_{\hat{g}_t}=\lambda_t^{-j}\sup_{B_{\frac{1}{2}}\times Y}|\nabla^{j,g_t}g^\bullet_t|_{g_t}\leq C\lambda_t^{-j}\to 0.
\lim_{t \rightarrow 0}{\cal K}[ H , \sqrt{g}, \sqrt{g}] ( \xi ~ \eta ~ \phi ~ t ; ~ \xi_{0}~ \eta_{0}~ \phi_{0}~ 0 ) = \frac{1}{\xi \eta \left( \xi^{2}+ \eta^{2}\right)}\delta ( \xi - \xi_{0}) \delta ( \eta - \eta_{0}) \delta ( \phi - \phi_{0})
\begin{cases}\dot{\bar x}_i = A\bar x_i +BR^{-1}B^T\bar y_i +G \bar m_i+\gamma \bar p_i , \dot{\bar m}_i = (A + G) \bar m_i +B\bar u +\gamma \bar p_i, \dot{\bar p}_i = -(A+G)^T \bar p_i -(I-\Gamma)^T Q [\bar x_i- (\Gamma \bar m_i+\eta)], \dot{\bar y}_i=-A^T{\bar y}_i +Q[\bar x_i-(\Gamma \bar m_i+\eta)] , \end{cases}
D_{S}^{2}(z) - \delta_{S}^{2}(z) = |z- P_{S}(z)|^{2}- | z - P_{{V+\eta}}(z)|^{2}, &=|P_{V}(z) - \frac{r_{S}}{r_1}P_{V}(z)|^{2}+ | P_{V^{\perp}}(z)-\eta|^2 -| P_{V^{\perp}}(z)-\eta|^2, =(r_{S}- r_{1})^2.
\langle P_h(\cdot, s), E_{\mathfrak{a}}^k(\cdot, \tfrac{1}{2}+ it)\rangle = \frac{\overline{\rho_\mathfrak{a}(h, \tfrac{1}{2}+ it)}}{(4\pi h)^{s-1}}\frac{\Gamma(s - \tfrac{1}{2}+ it) \Gamma(s - \tfrac{1}{2}- it)}{\Gamma(s - \frac{k}{2})}.
a_{n}^{R}= \sqrt{\frac{\alpha^{\prime}}{2 n \Omega_{n}}}\alpha_{n}^{R}, \tilde{a}_{n}^{R}= \sqrt{\frac{\alpha^{\prime}}{2 n \Omega_{n}}}\tilde{\alpha}_{n}^{R}, \mathrm{f o r a l l}n > 0
\mathbb E(f|\mathcal P_m)(s) = \sum_{n\geq 0}\mathbb E\bigl(f(s)\big| \mathcal F_{\frac{n}{2^m}}\bigr),\;\;\; s\in \Bigl(\frac{n}{2^m}, \frac{n+1}{2^m}\Bigr],n\geq 0.
\partial_t\theta+\theta\cdot \partial_v\theta=\partial_t^2v(t,y)=\partial_t\bigg(\frac{y-v(t,y)}{t}-\partial_y^{-1}\frac{\mathbb{P}_0\omega(t,y)}{t}\bigg)=-\frac{2\theta}{t}-\frac{1}{t}\partial_t\partial_y^{-1}\mathbb{P}_0\omega,
\P\left(\exists s \geq 0, u \in \mathcal{N}_s : X_u(s) \geq \sqrt{2}s+K \right) &\leq C(K+1)e^{-\sqrt{2}K}\sum_{l=1}^{\infty}\frac{1}{(l+1)^{3/2}}\leq C(K+1)e^{-\sqrt{2}K}.
\begin{pmatrix}G_{11}\left(Z,W\right) G_{12}\left(Z,W\right) G_{21}\left(Z,W\right) G_{22}\left(Z,W\right)\end{pmatrix}=\begin{pmatrix}W+\mbox{O}(2) \mbox{O}(2) \mbox{O}(2) \mbox{O}(2)\end{pmatrix},\quad\begin{pmatrix}F_{1}\left(Z,W\right) F_{2}\left(Z,W\right) \end{pmatrix}=\begin{pmatrix}\mbox{O}(1) \mbox{O}(2) \end{pmatrix}.
H=\sqrt{g_{tt}}\left[{{P^2_{\phi}}\over{h(r)r^2}}+{{P^2_r}\over{g_{rr}}}+{{P^2_{\rho}}\over{g_{\rho\rho}}}+{{(\rho P_{\phi}-N\rho^p)^2}\over{g_{\phi\phi}}}\right]^{1\over 2},
eAf\cdot eA^{d}f=eAA^{d}f=eA^{d}f\cdot eAf, eAf\cdot(eA^{d}f)^{2}=eAf\cdot e(A^{d})^2f=eA(A^{d})^2f=eA^{d}f, (eAf)^{k+1}\cdot eA^{d}f=eA^{k+1}f\cdot eA^{d}f=eA^{k+1}A^{d}f=eA^{k}f=(eAf)^{k},
\psi_0(x) = \sum_{100 \le j\le M}2^{-2j}a(2^j x), u_0(x) = \sum_{100 \le j \le M}2^{-j}(\nabla^{\perp}a)(2^j x), \omega_0(x) = \sum_{100 \le j \le M}({\Delta a})(2^j x),
\int_{\Omega}|e_n|\,dx =\int_{\Omega}|\mathcal{A}(x,\nabla u_n)\cdot\nabla u_n-\mathcal{A}(x,\nabla u)\cdot\nabla u_n-\mathcal{A}(x,\nabla u_n)\cdot\nabla u+\mathcal{A}(x,\nabla u)\cdot\nabla u|\,dx &\leq\int_{\Omega}\,w\{|\nabla u_n|^p+|\nabla u_n||\nabla u|^{p-1}+|\nabla u_n|^{p-1}|\nabla u|+|\nabla u|^p\}\,dx &\leq M,
\hat{g}_{j}(t) = \big(e_{j}^{\top}Ax_t + x_{t}^{\top}Ae_{j}+ b^{\top}e_{j}\Big)rK(r) + \left(\frac{\xi_{j}(t) - \xi^{'}_{j}(t)}{2h_{t}}\right)K(r) &= \langle \nabla f(x_t),e_{j}\rangle rK(r) + \left(\frac{\xi_{j}(t) - \xi^{'}_{j}(t)}{2h_{t}}\right)K(r),
H_{1}= \begin{pmatrix}1 0 i \frac{x_{1}}{2}\sqrt{r}1\end{pmatrix}r^{\frac{\sigma_{3}}{4}}G_{1}(-rx_{1};r\vec{x},\vec{s}) = R(-x_{1})E_{-x_{1}}(-x_{1}) \begin{pmatrix}\Psi_{1,11}\Psi_{1,12}\Psi_{1,21}\Psi_{1,22}\end{pmatrix},
A(X) =(X^2,-\Gamma^{i}_{jk}(X^1)X_2^j X_2^k), X_0(\mu)=((0,\mu),-\nu_h(\mu)) B(Y) =(Y^2,-\tilde{\Gamma}^{i}_{jk}(Y^1)Y_2^j Y_2^k), Y_0(\mu)=((0,\mu),-\nu_{h'}(\mu))
=\sum\limits_{(M_b, \mu_b) \in \mathcal{I}^{G, \mu}_{M_b, b'}}(\mathrm{Ind}^G_{P_b}\circ [\mathcal{M}_{M_b, b', \mu_b}]\circ ( \delta_{P_b}\otimes \mathrm{Jac}^G_{P^{op}_b}))\otimes [1][|\cdot|^{\langle \rho_G, \mu_b \rangle - \langle \rho_G, \mu \rangle}].
D_{i j}^{g}( q ) = \frac{n^{2}q_{i}q_{j}}{q^{2}+ i{\epsilon}}{\delta}^{\prime}( q_{-}){\pi}\mathrm{s g n}( q_{+}) - i \frac{n_{+}q_{i}q_{j}}{q^{2}+ i{\epsilon}}{\delta}( q_{-}){\int}_{- \infty}^{\infty}d k_{+}\frac{1}{( k_{+}- a )^{2}},
J_{2,+}(t_1,t_2) = \int_0^{t_2}\int_0^{\infty}\int_y^{\infty}G_{t_2-s}(x-z) \psi(s,z) \sigma_s(y) W(ds,dy) &-\int_0^{t_1}\int_0^{\infty}\int_y^{\infty}G_{t_1-s}(x-z) \psi(s,z) \sigma_s(y) W(ds,dy)
s_{i,k}\in (t_{\infty}((s_{i,k+1},\ldots,s_{p-1})),+\infty), k\in\mathbb{Z}\cap [p-N_i-1,p-2], &s_{i,k}= t_{\infty}((s_{i,k+1},\ldots,s_{k+N_i+1})), k\in\mathbb{Z}\cap [-\kappa-1,p-N_i-2]
\mathcal{S}(\tau) = q \prod_{n=1}^{\infty}\frac{(1 -q^{13n})^{2}}{(1-q^{n})^{2}}, \mathcal{T}(\tau) = \frac{\mathcal{R}(1 - 3\mathcal{R}- \mathcal{R}^{2})}{(1 + \mathcal{R}^{2})^{2}}.
L_{x}=L_{1}=(\begin{matrix}0&0&0\\ 0&0&-1\\ 0&1&0\end{matrix}),L_{y}=L_{2}=(\begin{matrix}0&0&1\\ 0&0&0\\ -1&0&0\end{matrix}),L_{z}=L_{3}=(\begin{matrix}0&-1&0\\ 1&0&0\\ 0&0&0\end{matrix})
\left(e^{\sqrt{m}\,\varrho}\, w(\varrho)\right)'=e^{\sqrt{m}\, \varrho}\, \big(w'(\varrho)+\sqrt{m}\, w(\varrho)\big)=e^{2\,\sqrt{m}\,\varrho}\, z(\varrho)\le -e^{2\,\sqrt{m}\,\varrho}\,\eta(\varrho),\varrho>r_0.
T^*_Z:=\bigoplus_{q=0}^{r-2}T^q_Z, \T^q_Z:=\Big(\mathbb{Z}[x_Z]/(x_Z^{(Z)-1}) \otimes_\mathbb{Z}A^*(\mathrm{M}_Z)\Big)^q \otimes_\mathbb{Z}\mathbb{R}.
W_{B^\perp}(y)= 1+\frac{1}{16}(- 567 + 5a + 4b + 3c + 2d + e) y &+\frac{1}{2}(1260 -10a - 10b - 9c - 7d - 4e) y^2 &+\frac{1}{16}(- 112455 + 885a + 900b + 883c + 770d + 497e)y^3+ \cdots.
\left|\begin{array}{ccccc}a_{11}a_{12}a_{13}\cdots \boxed{a_{1n}}-1 a_{22}a_{23}\cdots a_{2n}0 -1 a_{33}\cdots a_{3n}&\cdots 0 \cdots 0 -1 a_{nn}\end{array}\right|=a_{1n}+\sum_{1\leq j_1<\cdots<j_{k}<n}a_{1j_1}a_{j_1+1,2}\cdots a_{j_k+1,n}
S^{l i n}\; \equiv \; \mathrm{T r}_{1}\left\{\hat{\rho}- \hat{\rho}^{\; 2}\right\}\; = \; 1 - \; \mathrm{T r}_{1}\; \hat{\rho}^{\; 2}\; = \; 1 - \; \det \left\{\frac{1 - Y}{1 + Y}\right\}\; \; .
\theta^1 :=du-u_xdx-u_ydy-u_zdz, \theta^2 :=du_x-u_{xx}dx-u_{xy}dy-u_{xz}dz, \theta^3 :=du_y-u_{xy}d x-u_{yy}dy-u_{yz}dz \theta^4 :=du_z-u_{xz}d x-u_{yz}dy-Fdz ,
A_{\boldsymbol{\eta}, \boldsymbol{\nu}}^{(c)}=\sum_{l=m_{1}+\cdots +m_{b-1}+1}^{m_{1}+\cdots +m_{b}}q^{c-l-\sum_{i=b+1}^{N}m_{i}/2}=q^{\sum_{i=b+1}^{N}m_{i}/2}\sum_{l=m_{1}+\cdots +m_{b-1}+1}^{m_{1}+\cdots +m_{b}}q^{\sum_{i=1}^{b}m_{i}-l}=q^{\sum_{i=b+1}^{N}m_{i}/2}\frac{1-q^{m_{b}}}{1-q}.
C_{X}= \lim_{u \uparrow R}\frac{(R-u) Z_{X}(u)}{R}= \lim_{u \uparrow R}\left( 1- \frac{u}{R}\right) \prod_{\lambda \in (W)}\frac{1}{1- \lambda u}&= \lim_{u \uparrow R}\prod_{\lambda \in (W), \atop \lambda \neq 1/R}\frac{1}{1- \lambda u}= \prod_{\lambda \in (W), \atop \lambda \neq 1/R}\frac{1}{1- \lambda R}
\tilde{\boldsymbol{\theta}}_i = \mathrm{diag}(\tilde{\boldsymbol{\beta}}_i) \tilde{\boldsymbol{\psi}}_i = \mathrm{diag}(\tilde{\boldsymbol{\psi}}_i) \tilde{\boldsymbol{\beta}}_i, \forall i \in \{t,r\}.
\begin{tabular}[c]{cc}$-\Delta u\left( s,t\right) +u\left( s,t\right) =f\left( s,t,u\left(s,t\right) \right) ,$ $\quad\left( s,t\right) \in\Omega$ $\dfrac{\partial u\left( s\right)}{\partial n_{s}}=0,$ $\quad\left(s,t\right) \in\partial\Omega$\end{tabular}
(x_1 \otimes \cdots \otimes x_b)(x_1' \otimes \cdots \otimes x_b') = q^{-\Sigma_{i<j}|x_j| \bullet |x_i'|}(-1)^{\Sigma_{i<j}p(x_j)p(x_i')}(x_1 x_1' \otimes \cdots \otimes x_b x_b').
J_{111}\lesssim \ell\sum_{q> Q}\sum_{\substack{|q-p|\leq 2 p>Q}}\sum_{Q<p'\leq q}\lambda_{p'}\|(u_1)_{p'}\|_\infty\|w_p\|_\ell\|w_q\|_\ell^{\ell-1}&\lesssim \ell\sum_{q> Q}\sum_{\substack{|q-p|\leq 2 p>Q}}\sum_{Q<p'\leq q}\lambda_{p'}^{1+\frac2r}\|(u_1)_{p'}\|_r\|w_p\|_\ell\|w_q\|_\ell^{\ell-1},
||\phi||_{k+\theta}=\max_j \{\sum_{h, |h|\leq k}\left( \sup_{z\in U_j}|D_j^h \phi_{j \alpha_1\cdots\alpha_r\beta_1\cdots\beta_s}(z)|\right)+\sup_{y,z\in U_j,|h|=k}\frac{|D_j^h \phi_{j \alpha_1\cdots\alpha_r\beta_1\cdots\beta_s}(y)-D_j^h \phi_{j \alpha_1\cdots\alpha_r\beta_1\cdots\beta_s}(z)|}{|y-z|^{\theta}}\},
\ell_1\circ h_i \circ \mu_1 =D_m(x, a^n), \ell_1\circ \hat h_i\circ \mu_2=D_n(x, a^m) \mu_1^{(-1)}\circ h_{i+1}\circ \ell_2 =D_n(x, a), \mu_2^{(-1)}\circ \hat h_{i+1}\circ \ell_2=D_m(x, a),
\frac{E_{V}}{A}= \frac{\mu}{2}\sum_{n = 1}^{\infty}\frac{1}{\Gamma ( \frac{s}{2})}\int_{0}^{\infty}d t \, t^{\frac{s}{2}- 1}e^{- ( \frac{n \pi}{a \mu})^{2}t}\int_{- \infty}^{\infty}\frac{d k_{y}d k_{z}}{( 2 \pi )^{2}}e^{- [ ( \frac{k_{y}}{\mu})^{2}+ ( \frac{k_{z}}{\mu})^{2}] t}\rfloor_{s = - 1}\, .
\begin{array}{l}{{e^{- i \chi_{s}}[ - \partial_{r}+ s ( n - \Phi^{( 0 )}+ \Upsilon ) r^{- 1}] f_{n}( r ) = ( E + m ) g_{n}( r ) ,}}\\{{e^{i \chi_{s}}[ \partial_{r}+ s ( n - \Phi^{( 0 )}+ \Upsilon + s ) r^{- 1}] g_{n}( r ) = ( E - m ) f_{n}( r ) .}}\\ \end{array}
\vec{y}= \alpha \cdot (\frac{x_1+x_2}{2},0,0,\frac{x_1+x_2}{2}, x_3, \ldots, x_m)+ (1-\alpha) \cdot (0,\frac{x_1+x_2}{2},\frac{x_1+x_2}{2},0, x_3, \ldots, x_m).
j(\gamma)j(\gamma') =([r(\gamma),e], c(\gamma))([r(\gamma'),e], c(\gamma')) =([r(\gamma),e], c(\gamma))([\gamma(s(\gamma),e)], c(\gamma')) &=([r(\gamma),e], c(\gamma))([r(\gamma),c(\gamma)], c(\gamma')) &=([r(\gamma),e], c(\gamma))([r(\gamma),e]c(\gamma), c(\gamma')) &=([r(\gamma\gamma'),e], c(\gamma\gamma'))=j(\gamma\gamma').